Subgroups — C₇×#60806 #13238022

Order 1792.

2222 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#535	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3224	C₂	1	8	yes	maximal, complemented
4	C₇×#3222	C₂	1	4	yes	maximal, complemented
5	C₇×#3220	C₂	1	4	yes	maximal, complemented
6	C₇×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3040	C₂	1	2	yes	maximal, complemented
11	C₇×#3038	C₂	1	2	yes	maximal, complemented
12	C₇×#3005	C₂	1	2	yes	maximal, complemented
13	C₇×#3003	C₂	1	2	yes	maximal, complemented
14	C₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
16	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×#137	C₂²	1	2	yes	complemented
18	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
19	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
20	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
21	C₇×#542	C₂²	1	4	yes	complemented
22	C₇×#541	C₂²	1	2	yes	complemented
23	C₇×#540	C₂²	1	2	yes	complemented
24	C₇×#539	—	2	2	no	—
25	C₇×#538	—	2	2	no	—
26	C₇×#535	—	2	4	no	—
27	C₇×#535	C₂²	1	4	yes	—
28	C₇×#535	C₂²	1	4	yes	—
29	C₇×#533	C₂²	1	1	yes	characteristic, complemented
30	C₇×#532	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₇×#127	C₂²	1	1	yes	characteristic
32	C₂×C₇×#127	C₂²	1	1	yes	characteristic
33	C₇×#529	—	2	4	no	—
34	C₂×C₇×#124	C₂²	1	1	yes	characteristic
35	C₂×C₇×#124	C₂²	1	1	yes	characteristic
36	C₇×#525	C₂²	1	2	yes	—
37	C₂×C₇×#121	—	2	2	no	—
38	C₂×C₇×#120	—	2	2	no	—
39	C₄×C₇×#41	C₂²	1	2	yes	complemented
40	C₂×C₇×#117	C₂²	1	1	yes	characteristic
41	C₇×#497	C₂²	1	2	yes	complemented
42	C₇×#496	C₂²	1	2	yes	complemented
43	C₇×#496	C₂²	1	4	yes	complemented
44	C₇×#495	C₂²	1	4	yes	complemented
45	C₇×#493	C₂²	1	2	yes	complemented
46	C₇×#492	C₂²	1	2	yes	complemented
47	C₇×#490	C₂²	1	4	yes	complemented
48	C₇×#489	C₂²	1	4	yes	complemented
49	C₇×#488	C₂²	1	4	yes	complemented
50	C₇×#486	C₂²	1	4	yes	complemented
51	C₇×#464	C₂²	1	8	yes	complemented
52	C₇×#463	C₂²	1	8	yes	complemented
53	C₇×#460	C₂²	1	8	yes	complemented
54	C₇×#459	C₂²	1	8	yes	complemented
55	C₇×#452	C₂²	1	8	yes	complemented
56	C₇×#450	C₂²	1	8	yes	complemented
57	C₇×#449	C₂²	1	8	yes	complemented
58	C₇×#447	C₂²	1	8	yes	complemented
59	C₇×#433	C₂²	1	4	yes	complemented
60	C₇×#433	C₂²	1	4	yes	complemented
61	C₇×#423	C₂²	1	2	yes	complemented
62	C₇×#419	C₂²	1	2	yes	complemented
63	C₇×#418	C₂²	1	2	yes	—
64	C₇×#418	C₂²	1	2	yes	—
65	C₇×#418	C₂²	1	2	yes	complemented
66	C₇×#407	C₂²	1	2	yes	—
67	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
68	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
69	#60806	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂⁵×C₇	—	2	1	no	—
71	C₂×C₇×#41	—	2	2	no	—
72	C₂×C₇×#41	—	4	4	no	—
73	C₂×C₇×#41	C₂³	1	2	yes	—
74	C₂×C₇×#41	C₂³	1	4	yes	—
75	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
76	C₂²×C₇×D₄	—	2	1	no	—
77	C₂²×C₇×D₄	—	2	1	no	—
78	C₂²×C₇×D₄	—	2	2	no	—
79	C₂²×C₇×D₄	—	2	2	no	—
80	C₂²×C₇×D₄	—	2	2	no	—
81	C₂²×C₇×D₄	—	4	4	no	—
82	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
83	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
84	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
85	C₂²×C₇×D₄	C₂³	1	4	yes	—
86	C₂³×C₄×C₇	—	2	1	no	—
87	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
88	C₇×#137	—	4	4	no	—
89	C₇×AGL(1,ℤ/8)	—	4	4	no	—
90	C₇×AGL(1,ℤ/8)	—	4	4	no	—
91	C₇×AGL(1,ℤ/8)	—	4	4	no	—
92	C₂×C₇×Q₁₆	C₂³	1	4	yes	complemented
93	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
94	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
95	C₂×C₇×SD₁₆	C₂³	1	4	yes	complemented
96	C₂×C₇×SD₁₆	C₂³	1	4	yes	complemented
97	C₂×C₇×D₈	C₂³	1	4	yes	—
98	C₂×C₇×D₈	C₂³	1	4	yes	—
99	C₂×C₇×D₈	C₂³	1	4	yes	complemented
100	C₂×C₇×M₁₆	C₂³	1	2	yes	—
101	C₂×C₇×M₁₆	C₂³	1	2	yes	—
102	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
103	C₇×C₄⋊Q₈	C₂³	1	2	yes	complemented
104	C₇×#127	—	2	2	no	—
105	C₇×#127	—	2	2	no	—
106	C₇×#127	C₂³	1	2	yes	—
107	C₇×#127	C₂³	1	2	yes	—
108	C₇×#127	C₂³	1	4	yes	—
109	C₇×#127	C₂³	1	1	yes	characteristic, complemented
110	C₇×#126	—	2	4	no	—
111	C₇×#126	—	2	4	no	—
112	C₇×#125	—	2	4	no	—
113	C₇×#125	C₂³	1	2	yes	complemented
114	C₇×#124	—	2	4	no	—
115	C₇×#124	—	2	2	no	—
116	C₇×#124	—	2	2	no	—
117	C₇×#124	C₂³	1	2	yes	—
118	C₇×#124	C₂³	1	2	yes	—
119	C₇×#124	C₂³	1	4	yes	—
120	C₇×#124	C₂³	1	2	yes	complemented
121	C₇×#123	—	2	2	no	—
122	C₇×#123	—	2	4	no	—
123	C₇×#123	—	2	4	no	—
124	C₇×C₂²⋊Q₈	—	2	2	no	—
125	C₇×C₂²⋊Q₈	—	2	4	no	—
126	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
127	C₇×#121	—	2	2	no	—
128	C₇×#121	—	2	8	no	—
129	C₇×#121	—	2	8	no	—
130	C₇×#121	—	2	4	no	—
131	C₇×#121	C₂³	1	4	yes	—
132	C₇×#120	—	2	2	no	—
133	C₇×#120	—	2	8	no	—
134	C₇×#120	—	2	8	no	—
135	C₄×C₇×Q₈	—	2	2	no	—
136	C₄×C₇×Q₈	C₂³	1	4	yes	complemented
137	C₄×C₇×D₄	—	2	4	no	—
138	C₄×C₇×D₄	—	2	4	no	—
139	C₄×C₇×D₄	—	2	2	no	—
140	C₄×C₇×D₄	C₂³	1	4	yes	complemented
141	C₇×#117	—	2	2	no	—
142	C₇×#117	D₄	1	4	yes	—
143	C₇×#117	C₂³	1	2	yes	—
144	C₇×#117	C₂³	1	2	yes	—
145	C₇×#117	C₂³	1	2	yes	—
146	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
147	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
149	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
150	C₂×C₄²×C₇	—	2	2	no	—
151	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
153	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
154	C₇×#103	C₂³	1	8	yes	complemented
155	C₇×#102	C₂³	1	8	yes	—
156	C₇×#102	C₂³	1	8	yes	—
157	C₇×#102	C₂³	1	8	yes	complemented
158	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
159	C₇×#97	C₂³	1	2	yes	complemented
160	C₇×#97	C₂³	1	2	yes	complemented
161	C₄×C₇×C₈	C₂³	1	2	yes	complemented
162	C₄×C₇×C₈	C₂³	1	2	yes	complemented
163	#3455	C₂×C₇	1	1	yes	characteristic, complemented
164	C₂×#535	C₂×C₇	1	1	yes	characteristic, complemented
165	#3224	C₂×C₇	1	8	yes	complemented
166	#3222	C₂×C₇	1	4	yes	complemented
167	#3220	C₂×C₇	1	4	yes	complemented
168	#3153	C₂×C₇	1	1	yes	characteristic, complemented
169	#3151	C₂×C₇	1	1	yes	characteristic, complemented
170	#3140	C₂×C₇	1	1	yes	characteristic, complemented
171	#3138	C₂×C₇	1	1	yes	characteristic, complemented
172	#3040	C₂×C₇	1	2	yes	complemented
173	#3038	C₂×C₇	1	2	yes	complemented
174	#3005	C₂×C₇	1	2	yes	complemented
175	#3003	C₂×C₇	1	2	yes	complemented
176	#2921	C₂×C₇	1	1	yes	characteristic, complemented
177	C₂⁴×C₇	—	2	1	no	—
178	C₂⁴×C₇	—	2	1	no	—
179	C₂⁴×C₇	—	2	1	no	—
180	C₂⁴×C₇	—	2	4	no	—
181	C₂⁴×C₇	—	4	4	no	—
182	C₂⁴×C₇	—	8	4	no	—
183	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
184	C₇×#41	—	2	4	no	—
185	C₇×#41	—	2	4	no	—
186	C₇×#41	—	4	8	no	—
187	C₇×#41	—	4	8	no	—
188	C₂×C₇×Q₈	—	2	1	no	—
189	C₂×C₇×Q₈	—	2	2	no	—
190	C₂×C₇×Q₈	—	2	4	no	—
191	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
192	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
193	C₂×C₇×D₄	—	2	2	no	—
194	C₂×C₇×D₄	—	2	4	no	—
195	C₂×C₇×D₄	—	2	4	no	—
196	C₂×C₇×D₄	—	2	4	no	—
197	C₂×C₇×D₄	—	2	1	no	—
198	C₂×C₇×D₄	—	2	1	no	—
199	C₂×C₇×D₄	—	2	1	no	—
200	C₂×C₇×D₄	—	2	2	no	—
201	C₂×C₇×D₄	—	2	8	no	—
202	C₂×C₇×D₄	—	2	4	no	—
203	C₂×C₇×D₄	—	2	4	no	—
204	C₂×C₇×D₄	—	2	4	no	—
205	C₂×C₇×D₄	—	2	4	no	—
206	C₂×C₇×D₄	—	2	4	no	—
207	C₂×C₇×D₄	—	4	8	no	—
208	C₂×C₇×D₄	—	4	4	no	—
209	C₂×C₇×D₄	—	4	8	no	—
210	C₂×C₇×D₄	—	4	4	no	—
211	C₂×C₇×D₄	—	8	4	no	—
212	C₂×C₇×D₄	—	8	4	no	—
213	C₂×C₇×D₄	C₂⁴	1	2	yes	—
214	C₂×C₇×D₄	C₂⁴	1	2	yes	—
215	C₂×C₇×D₄	C₂⁴	1	2	yes	—
216	C₂×C₇×D₄	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₇	—	2	4	no	—
218	C₂²×C₄×C₇	—	2	1	no	—
219	C₂²×C₄×C₇	—	2	2	no	—
220	C₂²×C₄×C₇	—	2	4	no	—
221	C₂²×C₄×C₇	—	2	4	no	—
222	C₂²×C₄×C₇	—	4	4	no	—
223	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
224	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
225	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
226	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
227	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
228	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
229	C₇×Q₁₆	—	4	4	no	—
230	C₇×SD₁₆	—	4	4	no	—
231	C₇×SD₁₆	—	4	4	no	—
232	C₇×SD₁₆	—	4	4	no	—
233	C₇×SD₁₆	—	4	4	no	—
234	C₇×D₈	—	4	4	no	—
235	C₇×D₈	—	4	4	no	—
236	C₇×D₈	—	4	4	no	—
237	C₇×M₁₆	—	2	4	no	—
238	C₇×M₁₆	—	2	4	no	—
239	C₂×C₇×C₈	C₂⁴	1	4	yes	—
240	C₂×C₇×C₈	C₂⁴	1	4	yes	—
241	C₇×C₄⋊C₄	—	2	4	no	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	—	2	4	no	—
244	C₇×C₄⋊C₄	—	2	4	no	—
245	C₇×C₄⋊C₄	—	2	2	no	—
246	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
247	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
248	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
249	C₇×C₂²⋊C₄	—	2	2	no	—
250	C₇×C₂²⋊C₄	—	2	4	no	—
251	C₇×C₂²⋊C₄	—	2	8	no	—
252	C₇×C₂²⋊C₄	—	2	4	no	—
253	C₇×C₂²⋊C₄	—	2	4	no	—
254	C₇×C₂²⋊C₄	—	2	4	no	—
255	C₇×C₂²⋊C₄	—	2	4	no	—
256	C₇×C₂²⋊C₄	—	2	4	no	—
257	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
258	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
259	C₄²×C₇	—	2	2	no	—
260	C₄²×C₇	—	2	4	no	—
261	C₄²×C₇	C₂×D₄	1	2	yes	—
262	C₄²×C₇	C₂×D₄	1	2	yes	—
263	C₄²×C₇	C₂⁴	1	1	yes	characteristic
264	C₄²×C₇	C₂⁴	1	1	yes	characteristic
265	C₄²×C₇	C₂⁴	1	2	yes	—
266	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
267	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
268	C₂×#137	C₂²×C₇	1	2	yes	complemented
269	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
270	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
271	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	complemented
272	#542	C₂²×C₇	1	4	yes	complemented
273	#541	C₂²×C₇	1	2	yes	complemented
274	#540	C₂²×C₇	1	2	yes	complemented
275	#539	—	2	2	no	—
276	#538	—	2	2	no	—
277	#535	—	2	4	no	—
278	#535	C₂²×C₇	1	4	yes	—
279	#535	C₂²×C₇	1	4	yes	—
280	#533	C₂²×C₇	1	1	yes	characteristic, complemented
281	#532	C₂²×C₇	1	1	yes	characteristic, complemented
282	C₂×#127	C₂²×C₇	1	1	yes	characteristic
283	C₂×#127	C₂²×C₇	1	1	yes	characteristic
284	#529	—	2	4	no	—
285	C₂×#124	C₂²×C₇	1	1	yes	characteristic
286	C₂×#124	C₂²×C₇	1	1	yes	characteristic
287	#525	C₂²×C₇	1	2	yes	—
288	C₂×#121	—	2	2	no	—
289	C₂×#120	—	2	2	no	—
290	C₄×#41	C₂²×C₇	1	2	yes	complemented
291	C₂×#117	C₂²×C₇	1	1	yes	characteristic
292	#497	C₂²×C₇	1	2	yes	complemented
293	#496	C₂²×C₇	1	2	yes	complemented
294	#496	C₂²×C₇	1	4	yes	complemented
295	#495	C₂²×C₇	1	4	yes	complemented
296	#493	C₂²×C₇	1	2	yes	complemented
297	#492	C₂²×C₇	1	2	yes	complemented
298	#490	C₂²×C₇	1	4	yes	complemented
299	#489	C₂²×C₇	1	4	yes	complemented
300	#488	C₂²×C₇	1	4	yes	complemented
301	#486	C₂²×C₇	1	4	yes	complemented
302	#464	C₂²×C₇	1	8	yes	complemented
303	#463	C₂²×C₇	1	8	yes	complemented
304	#460	C₂²×C₇	1	8	yes	complemented
305	#459	C₂²×C₇	1	8	yes	complemented
306	#452	C₂²×C₇	1	8	yes	complemented
307	#450	C₂²×C₇	1	8	yes	complemented
308	#449	C₂²×C₇	1	8	yes	complemented
309	#447	C₂²×C₇	1	8	yes	complemented
310	#433	C₂²×C₇	1	4	yes	complemented
311	#433	C₂²×C₇	1	4	yes	complemented
312	#423	C₂²×C₇	1	2	yes	complemented
313	#419	C₂²×C₇	1	2	yes	complemented
314	#418	C₂²×C₇	1	2	yes	—
315	#418	C₂²×C₇	1	2	yes	—
316	#418	C₂²×C₇	1	2	yes	complemented
317	#407	C₂²×C₇	1	2	yes	—
318	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
319	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
320	C₂³×C₇	—	2	2	no	—
321	C₂³×C₇	—	2	2	no	—
322	C₂³×C₇	—	2	2	no	—
323	C₂³×C₇	—	2	2	no	—
324	C₂³×C₇	—	2	2	no	—
325	C₂³×C₇	—	4	2	no	—
326	C₂³×C₇	—	4	8	no	—
327	C₂³×C₇	—	4	4	no	—
328	C₂³×C₇	—	4	4	no	—
329	C₂³×C₇	—	4	4	no	—
330	C₂³×C₇	—	4	2	no	—
331	C₂³×C₇	—	4	2	no	—
332	C₂³×C₇	—	4	2	no	—
333	C₂³×C₇	—	8	8	no	—
334	C₂³×C₇	—	8	16	no	—
335	C₂³×C₇	C₂²×D₄	1	2	yes	—
336	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
337	C₇×Q₈	—	2	2	no	—
338	C₇×Q₈	—	2	2	no	—
339	C₇×Q₈	—	2	4	no	—
340	C₇×Q₈	—	4	2	no	—
341	C₇×Q₈	—	8	2	no	—
342	C₇×D₄	—	2	2	no	—
343	C₇×D₄	—	2	2	no	—
344	C₇×D₄	—	2	4	no	—
345	C₇×D₄	—	2	4	no	—
346	C₇×D₄	—	2	4	no	—
347	C₇×D₄	—	4	8	no	—
348	C₇×D₄	—	4	8	no	—
349	C₇×D₄	—	4	4	no	—
350	C₇×D₄	—	4	2	no	—
351	C₇×D₄	—	4	2	no	—
352	C₇×D₄	—	4	2	no	—
353	C₇×D₄	—	8	2	no	—
354	C₇×D₄	—	8	8	no	—
355	C₇×D₄	—	8	4	no	—
356	C₇×D₄	—	8	4	no	—
357	C₇×D₄	—	8	4	no	—
358	C₂×C₄×C₇	—	2	2	no	—
359	C₂×C₄×C₇	—	2	4	no	—
360	C₂×C₄×C₇	—	2	4	no	—
361	C₂×C₄×C₇	—	2	2	no	—
362	C₂×C₄×C₇	—	2	4	no	—
363	C₂×C₄×C₇	—	2	2	no	—
364	C₂×C₄×C₇	—	4	2	no	—
365	C₂×C₄×C₇	—	4	2	no	—
366	C₂×C₄×C₇	—	4	8	no	—
367	C₂×C₄×C₇	—	4	2	no	—
368	C₂×C₄×C₇	—	4	4	no	—
369	C₂×C₄×C₇	—	8	4	no	—
370	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
371	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
372	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
373	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
374	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
375	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
376	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
377	C₇×C₈	—	2	4	no	—
378	C₇×C₈	—	2	4	no	—
379	C₂⁵	—	2	1	no	—
380	C₂×#41	—	2	2	no	—
381	C₂×#41	—	4	4	no	—
382	C₂×#41	C₂³×C₇	1	2	yes	—
383	C₂×#41	C₂³×C₇	1	4	yes	—
384	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
385	C₂²×D₄	—	2	1	no	—
386	C₂²×D₄	—	2	1	no	—
387	C₂²×D₄	—	2	2	no	—
388	C₂²×D₄	—	2	2	no	—
389	C₂²×D₄	—	2	2	no	—
390	C₂²×D₄	—	4	4	no	—
391	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
392	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
394	C₂²×D₄	C₂³×C₇	1	4	yes	—
395	C₂³×C₄	—	2	1	no	—
396	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
397	#137	—	4	4	no	—
398	AGL(1,ℤ/8)	—	4	4	no	—
399	AGL(1,ℤ/8)	—	4	4	no	—
400	AGL(1,ℤ/8)	—	4	4	no	—
401	C₂×Q₁₆	C₂³×C₇	1	4	yes	complemented
402	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
403	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
404	C₂×SD₁₆	C₂³×C₇	1	4	yes	complemented
405	C₂×SD₁₆	C₂³×C₇	1	4	yes	complemented
406	C₂×D₈	C₂³×C₇	1	4	yes	—
407	C₂×D₈	C₂³×C₇	1	4	yes	—
408	C₂×D₈	C₂³×C₇	1	4	yes	complemented
409	C₂×M₁₆	C₂³×C₇	1	2	yes	—
410	C₂×M₁₆	C₂³×C₇	1	2	yes	—
411	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic, complemented
412	C₄⋊Q₈	C₂³×C₇	1	2	yes	complemented
413	#127	—	2	2	no	—
414	#127	—	2	2	no	—
415	#127	C₂³×C₇	1	4	yes	—
416	#127	C₂³×C₇	1	2	yes	—
417	#127	C₂³×C₇	1	2	yes	—
418	#127	C₂³×C₇	1	1	yes	characteristic, complemented
419	#126	—	2	4	no	—
420	#126	—	2	4	no	—
421	#125	—	2	4	no	—
422	#125	C₂³×C₇	1	2	yes	complemented
423	#124	—	2	4	no	—
424	#124	—	2	2	no	—
425	#124	—	2	2	no	—
426	#124	C₂³×C₇	1	4	yes	—
427	#124	C₂³×C₇	1	2	yes	—
428	#124	C₂³×C₇	1	2	yes	—
429	#124	C₂³×C₇	1	2	yes	complemented
430	#123	—	2	2	no	—
431	#123	—	2	4	no	—
432	#123	—	2	4	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	C₂²⋊Q₈	—	2	4	no	—
435	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
436	#121	—	2	2	no	—
437	#121	—	2	8	no	—
438	#121	—	2	8	no	—
439	#121	—	2	4	no	—
440	#121	C₂³×C₇	1	4	yes	—
441	#120	—	2	2	no	—
442	#120	—	2	8	no	—
443	#120	—	2	8	no	—
444	C₄×Q₈	—	2	2	no	—
445	C₄×Q₈	C₂³×C₇	1	4	yes	complemented
446	C₄×D₄	—	2	4	no	—
447	C₄×D₄	—	2	4	no	—
448	C₄×D₄	—	2	2	no	—
449	C₄×D₄	C₂³×C₇	1	4	yes	complemented
450	#117	—	2	2	no	—
451	#117	C₇×D₄	1	4	yes	—
452	#117	C₂³×C₇	1	2	yes	—
453	#117	C₂³×C₇	1	2	yes	—
454	#117	C₂³×C₇	1	2	yes	—
455	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
456	C₂×C₂²⋊C₄	—	2	2	no	—
457	C₂×C₂²⋊C₄	—	2	2	no	—
458	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
459	C₂×C₄²	—	2	2	no	—
460	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
461	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
462	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
463	#103	C₂³×C₇	1	8	yes	complemented
464	#102	C₂³×C₇	1	8	yes	—
465	#102	C₂³×C₇	1	8	yes	—
466	#102	C₂³×C₇	1	8	yes	complemented
467	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
468	#97	C₂³×C₇	1	2	yes	complemented
469	#97	C₂³×C₇	1	2	yes	complemented
470	C₄×C₈	C₂³×C₇	1	2	yes	complemented
471	C₄×C₈	C₂³×C₇	1	2	yes	complemented
472	C₂²×C₇	—	2	2	no	—
473	C₂²×C₇	—	2	4	no	—
474	C₂²×C₇	—	2	2	no	—
475	C₂²×C₇	—	4	4	no	—
476	C₂²×C₇	—	4	4	no	—
477	C₂²×C₇	—	4	4	no	—
478	C₂²×C₇	—	4	4	no	—
479	C₂²×C₇	—	4	4	no	—
480	C₂²×C₇	—	4	2	no	—
481	C₂²×C₇	—	4	2	no	—
482	C₂²×C₇	—	4	2	no	—
483	C₂²×C₇	—	4	2	no	—
484	C₂²×C₇	—	8	4	no	—
485	C₂²×C₇	—	8	4	no	—
486	C₂²×C₇	—	8	4	no	—
487	C₂²×C₇	—	8	16	no	—
488	C₂²×C₇	#535	1	2	yes	—
489	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
490	C₄×C₇	—	2	4	no	—
491	C₄×C₇	—	2	2	no	—
492	C₄×C₇	—	2	2	no	—
493	C₄×C₇	—	2	2	no	—
494	C₄×C₇	—	4	4	no	—
495	C₄×C₇	—	4	2	no	—
496	C₄×C₇	—	4	2	no	—
497	C₄×C₇	#535	1	2	yes	—
498	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
499	C₂⁴	—	2	1	no	—
500	C₂⁴	—	2	1	no	—
501	C₂⁴	—	2	1	no	—
502	C₂⁴	—	2	4	no	—
503	C₂⁴	—	4	4	no	—
504	C₂⁴	—	8	4	no	—
505	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
506	#41	—	2	4	no	—
507	#41	—	2	4	no	—
508	#41	—	4	8	no	—
509	#41	—	4	8	no	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	2	no	—
512	C₂×Q₈	—	2	4	no	—
513	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
514	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	2	4	no	—
517	C₂×D₄	—	2	4	no	—
518	C₂×D₄	—	2	4	no	—
519	C₂×D₄	—	2	1	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	2	8	no	—
524	C₂×D₄	—	2	4	no	—
525	C₂×D₄	—	2	4	no	—
526	C₂×D₄	—	2	4	no	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	—	2	4	no	—
529	C₂×D₄	—	4	8	no	—
530	C₂×D₄	—	4	4	no	—
531	C₂×D₄	—	4	8	no	—
532	C₂×D₄	—	4	4	no	—
533	C₂×D₄	—	8	4	no	—
534	C₂×D₄	—	8	4	no	—
535	C₂×D₄	C₂⁴×C₇	1	2	yes	—
536	C₂×D₄	C₂⁴×C₇	1	2	yes	—
537	C₂×D₄	C₂⁴×C₇	1	2	yes	—
538	C₂×D₄	C₂⁴×C₇	1	2	yes	—
539	C₂²×C₄	—	2	4	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	4	no	—
543	C₂²×C₄	—	2	4	no	—
544	C₂²×C₄	—	4	4	no	—
545	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
546	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
547	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
548	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
549	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
550	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
551	Q₁₆	—	4	4	no	—
552	SD₁₆	—	4	4	no	—
553	SD₁₆	—	4	4	no	—
554	SD₁₆	—	4	4	no	—
555	SD₁₆	—	4	4	no	—
556	D₈	—	4	4	no	—
557	D₈	—	4	4	no	—
558	D₈	—	4	4	no	—
559	M₁₆	—	2	4	no	—
560	M₁₆	—	2	4	no	—
561	C₂×C₈	C₂⁴×C₇	1	4	yes	—
562	C₂×C₈	C₂⁴×C₇	1	4	yes	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
569	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
570	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	2	4	no	—
573	C₂²⋊C₄	—	2	8	no	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
580	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
581	C₄²	—	2	2	no	—
582	C₄²	—	2	4	no	—
583	C₄²	C₂×C₇×D₄	1	2	yes	—
584	C₄²	C₂×C₇×D₄	1	2	yes	—
585	C₄²	C₂⁴×C₇	1	1	yes	characteristic
586	C₄²	C₂⁴×C₇	1	1	yes	characteristic
587	C₄²	C₂⁴×C₇	1	2	yes	—
588	C₂×C₇	—	2	2	no	—
589	C₂×C₇	—	4	2	no	—
590	C₂×C₇	—	8	2	no	—
591	C₂×C₇	—	8	2	no	—
592	C₂×C₇	—	8	2	no	—
593	C₂×C₇	—	8	2	no	—
594	C₂×C₇	C₂×#535	1	1	yes	characteristic
595	C₂×C₇	#3585	1	2	yes	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	2	2	no	—
599	C₂³	—	2	2	no	—
600	C₂³	—	2	2	no	—
601	C₂³	—	4	2	no	—
602	C₂³	—	4	8	no	—
603	C₂³	—	4	4	no	—
604	C₂³	—	4	4	no	—
605	C₂³	—	4	4	no	—
606	C₂³	—	4	2	no	—
607	C₂³	—	4	2	no	—
608	C₂³	—	4	2	no	—
609	C₂³	—	8	8	no	—
610	C₂³	—	8	16	no	—
611	C₂³	C₂²×C₇×D₄	1	2	yes	—
612	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
613	Q₈	—	2	2	no	—
614	Q₈	—	2	2	no	—
615	Q₈	—	2	4	no	—
616	Q₈	—	4	2	no	—
617	Q₈	—	8	2	no	—
618	D₄	—	2	2	no	—
619	D₄	—	2	2	no	—
620	D₄	—	2	4	no	—
621	D₄	—	2	4	no	—
622	D₄	—	2	4	no	—
623	D₄	—	4	8	no	—
624	D₄	—	4	8	no	—
625	D₄	—	4	4	no	—
626	D₄	—	4	2	no	—
627	D₄	—	4	2	no	—
628	D₄	—	4	2	no	—
629	D₄	—	8	2	no	—
630	D₄	—	8	8	no	—
631	D₄	—	8	4	no	—
632	D₄	—	8	4	no	—
633	D₄	—	8	4	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	4	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	4	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	4	2	no	—
641	C₂×C₄	—	4	2	no	—
642	C₂×C₄	—	4	8	no	—
643	C₂×C₄	—	4	2	no	—
644	C₂×C₄	—	4	4	no	—
645	C₂×C₄	—	8	4	no	—
646	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
647	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
648	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
649	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
650	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
651	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
652	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
653	C₈	—	2	4	no	—
654	C₈	—	2	4	no	—
655	C₇	#60806	1	1	yes	characteristic, direct product factor, Sylow
656	C₂²	—	2	2	no	—
657	C₂²	—	2	4	no	—
658	C₂²	—	2	2	no	—
659	C₂²	—	4	4	no	—
660	C₂²	—	4	4	no	—
661	C₂²	—	4	4	no	—
662	C₂²	—	4	4	no	—
663	C₂²	—	4	4	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	2	no	—
668	C₂²	—	8	4	no	—
669	C₂²	—	8	4	no	—
670	C₂²	—	8	4	no	—
671	C₂²	—	8	16	no	—
672	C₂²	C₇×#535	1	2	yes	—
673	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
674	C₄	—	2	4	no	—
675	C₄	—	2	2	no	—
676	C₄	—	2	2	no	—
677	C₄	—	2	2	no	—
678	C₄	—	4	4	no	—
679	C₄	—	4	2	no	—
680	C₄	—	4	2	no	—
681	C₄	C₇×#535	1	2	yes	—
682	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
683	C₂	—	2	2	no	—
684	C₂	—	4	2	no	—
685	C₂	—	8	2	no	—
686	C₂	—	8	2	no	—
687	C₂	—	8	2	no	—
688	C₂	—	8	2	no	—
689	C₂	C₂×C₇×#535	1	1	yes	characteristic
690	C₂	C₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3455	C₂	semidirect
2. C₂×C₇×#535	C₂	semidirect
3. C₇×#3224	C₂	semidirect
4. C₇×#3222	C₂	semidirect
5. C₇×#3220	C₂	semidirect
6. C₇×#3153	C₂	semidirect
7. C₇×#3151	C₂	semidirect
8. C₇×#3140	C₂	semidirect
9. C₇×#3138	C₂	semidirect
10. C₇×#3040	C₂	semidirect
11. C₇×#3038	C₂	semidirect
12. C₇×#3005	C₂	semidirect
13. C₇×#3003	C₂	semidirect
14. C₇×#2921	C₂	semidirect
17. C₂×C₇×#137	C₂²	semidirect
20. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
21. C₇×#542	C₂²	semidirect
22. C₇×#541	C₂²	semidirect
23. C₇×#540	C₂²	semidirect
29. C₇×#533	C₂²	semidirect
30. C₇×#532	C₂²	semidirect
39. C₄×C₇×#41	C₂²	semidirect
41. C₇×#497	C₂²	semidirect
42. C₇×#496	C₂²	semidirect
43. C₇×#496	C₂²	semidirect
44. C₇×#495	C₂²	semidirect
45. C₇×#493	C₂²	semidirect
46. C₇×#492	C₂²	semidirect
47. C₇×#490	C₂²	semidirect
48. C₇×#489	C₂²	semidirect
49. C₇×#488	C₂²	semidirect
50. C₇×#486	C₂²	semidirect
51. C₇×#464	C₂²	semidirect
52. C₇×#463	C₂²	semidirect
53. C₇×#460	C₂²	semidirect
54. C₇×#459	C₂²	semidirect
55. C₇×#452	C₂²	semidirect
56. C₇×#450	C₂²	semidirect
57. C₇×#449	C₂²	semidirect
58. C₇×#447	C₂²	semidirect
59. C₇×#433	C₂²	semidirect
60. C₇×#433	C₂²	semidirect
61. C₇×#423	C₂²	semidirect
62. C₇×#419	C₂²	semidirect
65. C₇×#418	C₂²	semidirect
67. C₄×C₇×M₁₆	C₂²	semidirect
68. C₄×C₇×M₁₆	C₂²	semidirect
69. #60806	C₇	direct (N × H)
92. C₂×C₇×Q₁₆	C₂³	semidirect
95. C₂×C₇×SD₁₆	C₂³	semidirect
96. C₂×C₇×SD₁₆	C₂³	semidirect
99. C₂×C₇×D₈	C₂³	semidirect
102. C₇×C₄⋊Q₈	C₂³	semidirect
103. C₇×C₄⋊Q₈	C₂³	semidirect
109. C₇×#127	C₂³	semidirect
113. C₇×#125	C₂³	semidirect
120. C₇×#124	C₂³	semidirect
136. C₄×C₇×Q₈	C₂³	semidirect
140. C₄×C₇×D₄	C₂³	semidirect
153. C₇×C₄⋊C₈	C₂³	semidirect
154. C₇×#103	C₂³	semidirect
157. C₇×#102	C₂³	semidirect
159. C₇×#97	C₂³	semidirect
160. C₇×#97	C₂³	semidirect
161. C₄×C₇×C₈	C₂³	semidirect
162. C₄×C₇×C₈	C₂³	semidirect
163. #3455	C₂×C₇	semidirect
164. C₂×#535	C₂×C₇	semidirect
165. #3224	C₂×C₇	semidirect
166. #3222	C₂×C₇	semidirect
167. #3220	C₂×C₇	semidirect
168. #3153	C₂×C₇	semidirect
169. #3151	C₂×C₇	semidirect
170. #3140	C₂×C₇	semidirect
171. #3138	C₂×C₇	semidirect
172. #3040	C₂×C₇	semidirect
173. #3038	C₂×C₇	semidirect
174. #3005	C₂×C₇	semidirect
175. #3003	C₂×C₇	semidirect
176. #2921	C₂×C₇	semidirect
268. C₂×#137	C₂²×C₇	semidirect
271. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
272. #542	C₂²×C₇	semidirect
273. #541	C₂²×C₇	semidirect
274. #540	C₂²×C₇	semidirect
280. #533	C₂²×C₇	semidirect
281. #532	C₂²×C₇	semidirect
290. C₄×#41	C₂²×C₇	semidirect
292. #497	C₂²×C₇	semidirect
293. #496	C₂²×C₇	semidirect
294. #496	C₂²×C₇	semidirect
295. #495	C₂²×C₇	semidirect
296. #493	C₂²×C₇	semidirect
297. #492	C₂²×C₇	semidirect
298. #490	C₂²×C₇	semidirect
299. #489	C₂²×C₇	semidirect
300. #488	C₂²×C₇	semidirect
301. #486	C₂²×C₇	semidirect
302. #464	C₂²×C₇	semidirect
303. #463	C₂²×C₇	semidirect
304. #460	C₂²×C₇	semidirect
305. #459	C₂²×C₇	semidirect
306. #452	C₂²×C₇	semidirect
307. #450	C₂²×C₇	semidirect
308. #449	C₂²×C₇	semidirect
309. #447	C₂²×C₇	semidirect
310. #433	C₂²×C₇	semidirect
311. #433	C₂²×C₇	semidirect
312. #423	C₂²×C₇	semidirect
313. #419	C₂²×C₇	semidirect
316. #418	C₂²×C₇	semidirect
318. C₄×M₁₆	C₂²×C₇	semidirect
319. C₄×M₁₆	C₂²×C₇	semidirect
401. C₂×Q₁₆	C₂³×C₇	semidirect
404. C₂×SD₁₆	C₂³×C₇	semidirect
405. C₂×SD₁₆	C₂³×C₇	semidirect
408. C₂×D₈	C₂³×C₇	semidirect
411. C₄⋊Q₈	C₂³×C₇	semidirect
412. C₄⋊Q₈	C₂³×C₇	semidirect
418. #127	C₂³×C₇	semidirect
422. #125	C₂³×C₇	semidirect
429. #124	C₂³×C₇	semidirect
445. C₄×Q₈	C₂³×C₇	semidirect
449. C₄×D₄	C₂³×C₇	semidirect
462. C₄⋊C₈	C₂³×C₇	semidirect
463. #103	C₂³×C₇	semidirect
466. #102	C₂³×C₇	semidirect
468. #97	C₂³×C₇	semidirect
469. #97	C₂³×C₇	semidirect
470. C₄×C₈	C₂³×C₇	semidirect
471. C₄×C₈	C₂³×C₇	semidirect
655. C₇	#60806	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3455	2
C₂×C₇×#535	2
C₇×#3224	2
C₇×#3222	2
C₇×#3220	2
C₇×#3153	2
C₇×#3151	2
C₇×#3140	2
C₇×#3138	2
C₇×#3040	2
C₇×#3038	2
C₇×#3005	2
C₇×#3003	2
C₇×#2921	2
#60806	7

Subgroups — C7×#60806 #13238022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60806 #13238022