Subgroups — C₇×#60807 #13238023

Order 1792.

2030 conjugacy classes of subgroups.

The subgroup lattice has 658 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×#536	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3225	C₂	1	8	yes	maximal, complemented
4	C₇×#3223	C₂	1	4	yes	maximal, complemented
5	C₇×#3221	C₂	1	4	yes	maximal, complemented
6	C₇×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3040	C₂	1	2	yes	maximal, complemented
11	C₇×#3039	C₂	1	2	yes	maximal, complemented
12	C₇×#3005	C₂	1	2	yes	maximal, complemented
13	C₇×#3004	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×Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×#137	C₂²	1	2	yes	—
18	C₂×C₇×#137	C₂²	1	2	yes	—
19	C₂×C₇×#137	C₂²	1	2	yes	—
20	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
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×#536	—	2	4	no	—
27	C₇×#536	C₂²	1	4	yes	—
28	C₇×#536	C₂²	1	4	yes	—
29	C₇×#533	C₂²	1	1	yes	characteristic
30	C₇×#532	C₂²	1	1	yes	characteristic
31	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂×C₇×C₄⋊Q₈	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×#123	—	2	2	no	—
38	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
39	C₄×C₇×#41	C₂²	1	2	yes	—
40	C₂×C₇×#117	C₂²	1	1	yes	characteristic
41	C₇×#497	C₂²	1	2	yes	complemented
42	C₇×#497	C₂²	1	4	yes	complemented
43	C₇×#496	C₂²	1	2	yes	complemented
44	C₇×#495	C₂²	1	4	yes	complemented
45	C₇×#494	C₂²	1	2	yes	complemented
46	C₇×#492	C₂²	1	2	yes	complemented
47	C₇×#491	C₂²	1	4	yes	complemented
48	C₇×#489	C₂²	1	4	yes	complemented
49	C₇×#488	C₂²	1	4	yes	complemented
50	C₇×#487	C₂²	1	4	yes	complemented
51	C₇×#464	C₂²	1	8	yes	complemented
52	C₇×#463	C₂²	1	8	yes	complemented
53	C₇×#462	C₂²	1	8	yes	complemented
54	C₇×#461	C₂²	1	8	yes	complemented
55	C₇×#452	C₂²	1	8	yes	—
56	C₇×#451	C₂²	1	8	yes	complemented
57	C₇×#449	C₂²	1	8	yes	—
58	C₇×#448	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	—
62	C₇×#419	C₂²	1	2	yes	—
63	C₇×#419	C₂²	1	2	yes	—
64	C₇×#419	C₂²	1	2	yes	—
65	C₇×#418	C₂²	1	2	yes	—
66	C₇×#407	C₂²	1	2	yes	—
67	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
68	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
69	#60807	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₇×#41	—	2	2	no	—
71	C₂×C₇×#41	—	4	4	no	—
72	C₂×C₇×#41	C₂³	1	2	yes	—
73	C₂×C₇×#41	C₂³	1	4	yes	—
74	C₂²×C₇×Q₈	—	2	1	no	—
75	C₂²×C₇×Q₈	—	2	2	no	—
76	C₂²×C₇×Q₈	—	4	4	no	—
77	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
78	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
79	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
80	C₂²×C₇×Q₈	C₂³	1	4	yes	—
81	C₂²×C₇×D₄	—	2	1	no	—
82	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₇	—	2	1	no	—
84	C₂³×C₄×C₇	—	2	1	no	—
85	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
86	C₇×#137	—	4	4	no	—
87	C₇×#137	—	4	4	no	—
88	C₇×#137	—	4	4	no	—
89	C₇×AGL(1,ℤ/8)	—	4	4	no	—
90	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
91	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
92	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
93	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
94	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
95	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
96	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
97	C₂×C₇×D₈	C₂³	1	4	yes	—
98	C₂×C₇×M₁₆	C₂³	1	2	yes	—
99	C₂×C₇×M₁₆	C₂³	1	2	yes	—
100	C₇×C₄⋊Q₈	—	2	2	no	—
101	C₇×C₄⋊Q₈	—	2	2	no	—
102	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
103	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
104	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
105	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
106	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
107	C₇×#127	C₂³	1	1	yes	characteristic
108	C₇×#126	—	2	4	no	—
109	C₇×#126	—	2	4	no	—
110	C₇×#125	—	2	4	no	—
111	C₇×#125	C₂³	1	2	yes	—
112	C₇×#124	—	2	4	no	—
113	C₇×#124	—	2	2	no	—
114	C₇×#124	—	2	2	no	—
115	C₇×#124	C₂³	1	2	yes	—
116	C₇×#124	C₂³	1	2	yes	—
117	C₇×#124	C₂³	1	2	yes	—
118	C₇×#124	C₂³	1	4	yes	—
119	C₇×#123	—	2	2	no	—
120	C₇×#123	—	2	4	no	—
121	C₇×#123	—	2	8	no	—
122	C₇×#123	—	2	8	no	—
123	C₇×#123	—	2	4	no	—
124	C₇×C₂²⋊Q₈	—	2	2	no	—
125	C₇×C₂²⋊Q₈	—	2	8	no	—
126	C₇×C₂²⋊Q₈	—	2	8	no	—
127	C₇×C₂²⋊Q₈	—	2	4	no	—
128	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
129	C₇×#121	—	2	2	no	—
130	C₇×#121	—	2	4	no	—
131	C₇×#121	C₂³	1	4	yes	—
132	C₇×#120	—	2	2	no	—
133	C₄×C₇×Q₈	—	2	2	no	—
134	C₄×C₇×Q₈	C₂³	1	4	yes	—
135	C₄×C₇×D₄	—	2	4	no	—
136	C₄×C₇×D₄	—	2	4	no	—
137	C₄×C₇×D₄	—	2	2	no	—
138	C₄×C₇×D₄	C₂³	1	4	yes	—
139	C₇×#117	—	2	2	no	—
140	C₇×#117	D₄	1	4	yes	—
141	C₇×#117	C₂³	1	2	yes	—
142	C₇×#117	C₂³	1	2	yes	—
143	C₇×#117	C₂³	1	2	yes	—
144	C₂×C₇×C₄⋊C₄	—	2	2	no	—
145	C₂×C₇×C₄⋊C₄	—	2	2	no	—
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	—
154	C₇×#103	C₂³	1	8	yes	—
155	C₇×#103	C₂³	1	8	yes	—
156	C₇×#103	C₂³	1	8	yes	—
157	C₇×#102	C₂³	1	8	yes	—
158	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
159	C₇×#97	C₂³	1	2	yes	—
160	C₇×#97	C₂³	1	2	yes	—
161	C₄×C₇×C₈	C₂³	1	2	yes	—
162	C₄×C₇×C₈	C₂³	1	2	yes	—
163	#3455	C₂×C₇	1	1	yes	characteristic, complemented
164	C₂×#536	C₂×C₇	1	1	yes	characteristic, complemented
165	#3225	C₂×C₇	1	8	yes	complemented
166	#3223	C₂×C₇	1	4	yes	complemented
167	#3221	C₂×C₇	1	4	yes	complemented
168	#3153	C₂×C₇	1	1	yes	characteristic, complemented
169	#3152	C₂×C₇	1	1	yes	characteristic, complemented
170	#3140	C₂×C₇	1	1	yes	characteristic, complemented
171	#3139	C₂×C₇	1	1	yes	characteristic, complemented
172	#3040	C₂×C₇	1	2	yes	complemented
173	#3039	C₂×C₇	1	2	yes	complemented
174	#3005	C₂×C₇	1	2	yes	complemented
175	#3004	C₂×C₇	1	2	yes	complemented
176	#2921	C₂×C₇	1	1	yes	characteristic, complemented
177	C₂⁴×C₇	—	2	1	no	—
178	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
179	C₇×#41	—	2	4	no	—
180	C₇×#41	—	2	4	no	—
181	C₇×#41	—	4	8	no	—
182	C₇×#41	—	4	8	no	—
183	C₂×C₇×Q₈	—	2	4	no	—
184	C₂×C₇×Q₈	—	2	1	no	—
185	C₂×C₇×Q₈	—	2	1	no	—
186	C₂×C₇×Q₈	—	2	1	no	—
187	C₂×C₇×Q₈	—	2	2	no	—
188	C₂×C₇×Q₈	—	2	4	no	—
189	C₂×C₇×Q₈	—	2	4	no	—
190	C₂×C₇×Q₈	—	2	4	no	—
191	C₂×C₇×Q₈	—	4	8	no	—
192	C₂×C₇×Q₈	—	4	4	no	—
193	C₂×C₇×Q₈	—	4	8	no	—
194	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
195	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
196	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
197	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
198	C₂×C₇×D₄	—	2	2	no	—
199	C₂×C₇×D₄	—	2	4	no	—
200	C₂×C₇×D₄	—	2	4	no	—
201	C₂×C₇×D₄	—	2	1	no	—
202	C₂×C₇×D₄	—	2	2	no	—
203	C₂×C₇×D₄	—	2	4	no	—
204	C₂×C₇×D₄	—	4	4	no	—
205	C₂×C₇×D₄	C₂⁴	1	2	yes	—
206	C₂×C₇×D₄	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₇	—	2	4	no	—
208	C₂²×C₄×C₇	—	2	1	no	—
209	C₂²×C₄×C₇	—	2	1	no	—
210	C₂²×C₄×C₇	—	2	1	no	—
211	C₂²×C₄×C₇	—	2	4	no	—
212	C₂²×C₄×C₇	—	2	2	no	—
213	C₂²×C₄×C₇	—	2	4	no	—
214	C₂²×C₄×C₇	—	2	4	no	—
215	C₂²×C₄×C₇	—	4	4	no	—
216	C₂²×C₄×C₇	—	4	4	no	—
217	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
218	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
219	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
220	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
221	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
222	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
223	C₇×Q₁₆	—	4	4	no	—
224	C₇×Q₁₆	—	4	4	no	—
225	C₇×Q₁₆	—	4	4	no	—
226	C₇×SD₁₆	—	4	4	no	—
227	C₇×SD₁₆	—	4	4	no	—
228	C₇×SD₁₆	—	4	4	no	—
229	C₇×SD₁₆	—	4	4	no	—
230	C₇×D₈	—	4	4	no	—
231	C₇×M₁₆	—	2	4	no	—
232	C₇×M₁₆	—	2	4	no	—
233	C₂×C₇×C₈	C₂⁴	1	4	yes	—
234	C₂×C₇×C₈	C₂⁴	1	4	yes	—
235	C₇×C₄⋊C₄	—	2	8	no	—
236	C₇×C₄⋊C₄	—	2	4	no	—
237	C₇×C₄⋊C₄	—	2	4	no	—
238	C₇×C₄⋊C₄	—	2	4	no	—
239	C₇×C₄⋊C₄	—	2	2	no	—
240	C₇×C₄⋊C₄	—	2	4	no	—
241	C₇×C₄⋊C₄	—	2	4	no	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
244	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
245	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
246	C₇×C₂²⋊C₄	—	2	2	no	—
247	C₇×C₂²⋊C₄	—	2	4	no	—
248	C₇×C₂²⋊C₄	—	2	8	no	—
249	C₇×C₂²⋊C₄	—	2	4	no	—
250	C₇×C₂²⋊C₄	—	2	4	no	—
251	C₇×C₂²⋊C₄	—	2	4	no	—
252	C₇×C₂²⋊C₄	—	2	4	no	—
253	C₇×C₂²⋊C₄	—	2	4	no	—
254	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
255	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
256	C₄²×C₇	—	2	2	no	—
257	C₄²×C₇	—	2	4	no	—
258	C₄²×C₇	C₂×D₄	1	2	yes	—
259	C₄²×C₇	C₂×D₄	1	2	yes	—
260	C₄²×C₇	C₂⁴	1	1	yes	characteristic
261	C₄²×C₇	C₂⁴	1	1	yes	characteristic
262	C₄²×C₇	C₂⁴	1	2	yes	—
263	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
264	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
265	C₂×#137	C₂²×C₇	1	2	yes	—
266	C₂×#137	C₂²×C₇	1	2	yes	—
267	C₂×#137	C₂²×C₇	1	2	yes	—
268	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
269	#542	C₂²×C₇	1	4	yes	complemented
270	#541	C₂²×C₇	1	2	yes	complemented
271	#540	C₂²×C₇	1	2	yes	complemented
272	#539	—	2	2	no	—
273	#538	—	2	2	no	—
274	#536	—	2	4	no	—
275	#536	C₂²×C₇	1	4	yes	—
276	#536	C₂²×C₇	1	4	yes	—
277	#533	C₂²×C₇	1	1	yes	characteristic
278	#532	C₂²×C₇	1	1	yes	characteristic
279	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
280	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
281	#529	—	2	4	no	—
282	C₂×#124	C₂²×C₇	1	1	yes	characteristic
283	C₂×#124	C₂²×C₇	1	1	yes	characteristic
284	#525	C₂²×C₇	1	2	yes	—
285	C₂×#123	—	2	2	no	—
286	C₂×C₂²⋊Q₈	—	2	2	no	—
287	C₄×#41	C₂²×C₇	1	2	yes	—
288	C₂×#117	C₂²×C₇	1	1	yes	characteristic
289	#497	C₂²×C₇	1	4	yes	complemented
290	#497	C₂²×C₇	1	2	yes	complemented
291	#496	C₂²×C₇	1	2	yes	complemented
292	#495	C₂²×C₇	1	4	yes	complemented
293	#494	C₂²×C₇	1	2	yes	complemented
294	#492	C₂²×C₇	1	2	yes	complemented
295	#491	C₂²×C₇	1	4	yes	complemented
296	#489	C₂²×C₇	1	4	yes	complemented
297	#488	C₂²×C₇	1	4	yes	complemented
298	#487	C₂²×C₇	1	4	yes	complemented
299	#464	C₂²×C₇	1	8	yes	complemented
300	#463	C₂²×C₇	1	8	yes	complemented
301	#462	C₂²×C₇	1	8	yes	complemented
302	#461	C₂²×C₇	1	8	yes	complemented
303	#452	C₂²×C₇	1	8	yes	—
304	#451	C₂²×C₇	1	8	yes	complemented
305	#449	C₂²×C₇	1	8	yes	—
306	#448	C₂²×C₇	1	8	yes	complemented
307	#433	C₂²×C₇	1	4	yes	complemented
308	#433	C₂²×C₇	1	4	yes	complemented
309	#423	C₂²×C₇	1	2	yes	—
310	#419	C₂²×C₇	1	2	yes	—
311	#419	C₂²×C₇	1	2	yes	—
312	#419	C₂²×C₇	1	2	yes	—
313	#418	C₂²×C₇	1	2	yes	—
314	#407	C₂²×C₇	1	2	yes	—
315	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
316	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
317	C₂³×C₇	—	2	2	no	—
318	C₂³×C₇	—	2	2	no	—
319	C₂³×C₇	—	2	2	no	—
320	C₂³×C₇	—	4	2	no	—
321	C₂³×C₇	—	4	4	no	—
322	C₂³×C₇	—	4	2	no	—
323	C₂³×C₇	C₂²×D₄	1	2	yes	—
324	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
325	C₇×Q₈	—	2	2	no	—
326	C₇×Q₈	—	2	4	no	—
327	C₇×Q₈	—	2	4	no	—
328	C₇×Q₈	—	2	4	no	—
329	C₇×Q₈	—	2	2	no	—
330	C₇×Q₈	—	4	8	no	—
331	C₇×Q₈	—	4	2	no	—
332	C₇×Q₈	—	4	2	no	—
333	C₇×Q₈	—	4	2	no	—
334	C₇×Q₈	—	8	2	no	—
335	C₇×D₄	—	2	2	no	—
336	C₇×D₄	—	2	4	no	—
337	C₇×D₄	—	2	2	no	—
338	C₇×D₄	—	4	8	no	—
339	C₇×D₄	—	4	4	no	—
340	C₇×D₄	—	4	2	no	—
341	C₇×D₄	—	8	2	no	—
342	C₇×D₄	—	8	4	no	—
343	C₂×C₄×C₇	—	2	2	no	—
344	C₂×C₄×C₇	—	2	4	no	—
345	C₂×C₄×C₇	—	2	4	no	—
346	C₂×C₄×C₇	—	2	2	no	—
347	C₂×C₄×C₇	—	2	4	no	—
348	C₂×C₄×C₇	—	2	2	no	—
349	C₂×C₄×C₇	—	2	2	no	—
350	C₂×C₄×C₇	—	2	2	no	—
351	C₂×C₄×C₇	—	4	2	no	—
352	C₂×C₄×C₇	—	4	2	no	—
353	C₂×C₄×C₇	—	4	8	no	—
354	C₂×C₄×C₇	—	4	8	no	—
355	C₂×C₄×C₇	—	4	2	no	—
356	C₂×C₄×C₇	—	4	2	no	—
357	C₂×C₄×C₇	—	4	2	no	—
358	C₂×C₄×C₇	—	4	4	no	—
359	C₂×C₄×C₇	—	8	4	no	—
360	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
361	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
362	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
363	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
364	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
365	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
366	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
367	C₇×C₈	—	2	4	no	—
368	C₇×C₈	—	2	4	no	—
369	C₂×#41	—	2	2	no	—
370	C₂×#41	—	4	4	no	—
371	C₂×#41	C₂³×C₇	1	2	yes	—
372	C₂×#41	C₂³×C₇	1	4	yes	—
373	C₂²×Q₈	—	2	1	no	—
374	C₂²×Q₈	—	2	2	no	—
375	C₂²×Q₈	—	4	4	no	—
376	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
377	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
378	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
379	C₂²×Q₈	C₂³×C₇	1	4	yes	—
380	C₂²×D₄	—	2	1	no	—
381	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
382	C₂³×C₄	—	2	1	no	—
383	C₂³×C₄	—	2	1	no	—
384	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
385	#137	—	4	4	no	—
386	#137	—	4	4	no	—
387	#137	—	4	4	no	—
388	AGL(1,ℤ/8)	—	4	4	no	—
389	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
390	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
391	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
392	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
393	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
394	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
395	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
396	C₂×D₈	C₂³×C₇	1	4	yes	—
397	C₂×M₁₆	C₂³×C₇	1	2	yes	—
398	C₂×M₁₆	C₂³×C₇	1	2	yes	—
399	C₄⋊Q₈	—	2	2	no	—
400	C₄⋊Q₈	—	2	2	no	—
401	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
402	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
403	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
404	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
405	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
406	#127	C₂³×C₇	1	1	yes	characteristic
407	#126	—	2	4	no	—
408	#126	—	2	4	no	—
409	#125	—	2	4	no	—
410	#125	C₂³×C₇	1	2	yes	—
411	#124	—	2	4	no	—
412	#124	—	2	2	no	—
413	#124	—	2	2	no	—
414	#124	C₂³×C₇	1	4	yes	—
415	#124	C₂³×C₇	1	2	yes	—
416	#124	C₂³×C₇	1	2	yes	—
417	#124	C₂³×C₇	1	2	yes	—
418	#123	—	2	2	no	—
419	#123	—	2	4	no	—
420	#123	—	2	8	no	—
421	#123	—	2	8	no	—
422	#123	—	2	4	no	—
423	C₂²⋊Q₈	—	2	2	no	—
424	C₂²⋊Q₈	—	2	8	no	—
425	C₂²⋊Q₈	—	2	8	no	—
426	C₂²⋊Q₈	—	2	4	no	—
427	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
428	#121	—	2	2	no	—
429	#121	—	2	4	no	—
430	#121	C₂³×C₇	1	4	yes	—
431	#120	—	2	2	no	—
432	C₄×Q₈	—	2	2	no	—
433	C₄×Q₈	C₂³×C₇	1	4	yes	—
434	C₄×D₄	—	2	4	no	—
435	C₄×D₄	—	2	4	no	—
436	C₄×D₄	—	2	2	no	—
437	C₄×D₄	C₂³×C₇	1	4	yes	—
438	#117	—	2	2	no	—
439	#117	C₇×D₄	1	4	yes	—
440	#117	C₂³×C₇	1	2	yes	—
441	#117	C₂³×C₇	1	2	yes	—
442	#117	C₂³×C₇	1	2	yes	—
443	C₂×C₄⋊C₄	—	2	2	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
446	C₂×C₂²⋊C₄	—	2	2	no	—
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
449	C₂×C₄²	—	2	2	no	—
450	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
452	C₄⋊C₈	C₂³×C₇	1	8	yes	—
453	#103	C₂³×C₇	1	8	yes	—
454	#103	C₂³×C₇	1	8	yes	—
455	#103	C₂³×C₇	1	8	yes	—
456	#102	C₂³×C₇	1	8	yes	—
457	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
458	#97	C₂³×C₇	1	2	yes	—
459	#97	C₂³×C₇	1	2	yes	—
460	C₄×C₈	C₂³×C₇	1	2	yes	—
461	C₄×C₈	C₂³×C₇	1	2	yes	—
462	C₂²×C₇	—	2	2	no	—
463	C₂²×C₇	—	2	4	no	—
464	C₂²×C₇	—	2	2	no	—
465	C₂²×C₇	—	4	4	no	—
466	C₂²×C₇	—	4	4	no	—
467	C₂²×C₇	—	4	4	no	—
468	C₂²×C₇	—	4	2	no	—
469	C₂²×C₇	—	4	2	no	—
470	C₂²×C₇	—	8	4	no	—
471	C₂²×C₇	#535	1	2	yes	—
472	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
473	C₄×C₇	—	2	4	no	—
474	C₄×C₇	—	2	2	no	—
475	C₄×C₇	—	2	2	no	—
476	C₄×C₇	—	2	2	no	—
477	C₄×C₇	—	4	4	no	—
478	C₄×C₇	—	4	2	no	—
479	C₄×C₇	—	4	2	no	—
480	C₄×C₇	—	4	2	no	—
481	C₄×C₇	—	4	2	no	—
482	C₄×C₇	#535	1	2	yes	—
483	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
484	C₂⁴	—	2	1	no	—
485	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
486	#41	—	2	4	no	—
487	#41	—	2	4	no	—
488	#41	—	4	8	no	—
489	#41	—	4	8	no	—
490	C₂×Q₈	—	2	4	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	—	2	2	no	—
495	C₂×Q₈	—	2	4	no	—
496	C₂×Q₈	—	2	4	no	—
497	C₂×Q₈	—	2	4	no	—
498	C₂×Q₈	—	4	8	no	—
499	C₂×Q₈	—	4	4	no	—
500	C₂×Q₈	—	4	8	no	—
501	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
502	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
503	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
504	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	—	2	4	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	4	4	no	—
512	C₂×D₄	C₂⁴×C₇	1	2	yes	—
513	C₂×D₄	C₂⁴×C₇	1	2	yes	—
514	C₂²×C₄	—	2	4	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	4	no	—
522	C₂²×C₄	—	4	4	no	—
523	C₂²×C₄	—	4	4	no	—
524	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
525	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
526	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
527	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
530	Q₁₆	—	4	4	no	—
531	Q₁₆	—	4	4	no	—
532	Q₁₆	—	4	4	no	—
533	SD₁₆	—	4	4	no	—
534	SD₁₆	—	4	4	no	—
535	SD₁₆	—	4	4	no	—
536	SD₁₆	—	4	4	no	—
537	D₈	—	4	4	no	—
538	M₁₆	—	2	4	no	—
539	M₁₆	—	2	4	no	—
540	C₂×C₈	C₂⁴×C₇	1	4	yes	—
541	C₂×C₈	C₂⁴×C₇	1	4	yes	—
542	C₄⋊C₄	—	2	8	no	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
551	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
552	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
553	C₂²⋊C₄	—	2	2	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	8	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
562	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
563	C₄²	—	2	2	no	—
564	C₄²	—	2	4	no	—
565	C₄²	C₂×C₇×D₄	1	2	yes	—
566	C₄²	C₂×C₇×D₄	1	2	yes	—
567	C₄²	C₂⁴×C₇	1	1	yes	characteristic
568	C₄²	C₂⁴×C₇	1	1	yes	characteristic
569	C₄²	C₂⁴×C₇	1	2	yes	—
570	C₂×C₇	—	2	2	no	—
571	C₂×C₇	—	4	2	no	—
572	C₂×C₇	—	8	2	no	—
573	C₂×C₇	—	8	2	no	—
574	C₂×C₇	C₂×#535	1	1	yes	characteristic
575	C₂×C₇	#3586	1	2	yes	—
576	C₂³	—	2	2	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	4	2	no	—
580	C₂³	—	4	4	no	—
581	C₂³	—	4	2	no	—
582	C₂³	C₂²×C₇×D₄	1	2	yes	—
583	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
584	Q₈	—	2	2	no	—
585	Q₈	—	2	4	no	—
586	Q₈	—	2	4	no	—
587	Q₈	—	2	4	no	—
588	Q₈	—	2	2	no	—
589	Q₈	—	4	8	no	—
590	Q₈	—	4	2	no	—
591	Q₈	—	4	2	no	—
592	Q₈	—	4	2	no	—
593	Q₈	—	8	2	no	—
594	D₄	—	2	2	no	—
595	D₄	—	2	4	no	—
596	D₄	—	2	2	no	—
597	D₄	—	4	8	no	—
598	D₄	—	4	4	no	—
599	D₄	—	4	2	no	—
600	D₄	—	8	2	no	—
601	D₄	—	8	4	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	8	no	—
613	C₂×C₄	—	4	8	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	8	4	no	—
619	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
620	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
621	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
622	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
623	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
624	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
625	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
626	C₈	—	2	4	no	—
627	C₈	—	2	4	no	—
628	C₇	#60807	1	1	yes	characteristic, direct product factor, Sylow
629	C₂²	—	2	2	no	—
630	C₂²	—	2	4	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	4	4	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	8	4	no	—
638	C₂²	C₇×#535	1	2	yes	—
639	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
640	C₄	—	2	4	no	—
641	C₄	—	2	2	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	4	4	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	C₇×#535	1	2	yes	—
650	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
651	C₂	—	2	2	no	—
652	C₂	—	4	2	no	—
653	C₂	—	8	2	no	—
654	C₂	—	8	2	no	—
655	C₂	C₂×C₇×#535	1	1	yes	characteristic
656	C₂	C₇×#3586	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×#536	C₂	semidirect
3. C₇×#3225	C₂	semidirect
4. C₇×#3223	C₂	semidirect
5. C₇×#3221	C₂	semidirect
6. C₇×#3153	C₂	semidirect
7. C₇×#3152	C₂	semidirect
8. C₇×#3140	C₂	semidirect
9. C₇×#3139	C₂	semidirect
10. C₇×#3040	C₂	semidirect
11. C₇×#3039	C₂	semidirect
12. C₇×#3005	C₂	semidirect
13. C₇×#3004	C₂	semidirect
14. C₇×#2921	C₂	semidirect
21. C₇×#542	C₂²	semidirect
22. C₇×#541	C₂²	semidirect
23. C₇×#540	C₂²	semidirect
41. C₇×#497	C₂²	semidirect
42. C₇×#497	C₂²	semidirect
43. C₇×#496	C₂²	semidirect
44. C₇×#495	C₂²	semidirect
45. C₇×#494	C₂²	semidirect
46. C₇×#492	C₂²	semidirect
47. C₇×#491	C₂²	semidirect
48. C₇×#489	C₂²	semidirect
49. C₇×#488	C₂²	semidirect
50. C₇×#487	C₂²	semidirect
51. C₇×#464	C₂²	semidirect
52. C₇×#463	C₂²	semidirect
53. C₇×#462	C₂²	semidirect
54. C₇×#461	C₂²	semidirect
56. C₇×#451	C₂²	semidirect
58. C₇×#448	C₂²	semidirect
59. C₇×#433	C₂²	semidirect
60. C₇×#433	C₂²	semidirect
69. #60807	C₇	direct (N × H)
163. #3455	C₂×C₇	semidirect
164. C₂×#536	C₂×C₇	semidirect
165. #3225	C₂×C₇	semidirect
166. #3223	C₂×C₇	semidirect
167. #3221	C₂×C₇	semidirect
168. #3153	C₂×C₇	semidirect
169. #3152	C₂×C₇	semidirect
170. #3140	C₂×C₇	semidirect
171. #3139	C₂×C₇	semidirect
172. #3040	C₂×C₇	semidirect
173. #3039	C₂×C₇	semidirect
174. #3005	C₂×C₇	semidirect
175. #3004	C₂×C₇	semidirect
176. #2921	C₂×C₇	semidirect
269. #542	C₂²×C₇	semidirect
270. #541	C₂²×C₇	semidirect
271. #540	C₂²×C₇	semidirect
289. #497	C₂²×C₇	semidirect
290. #497	C₂²×C₇	semidirect
291. #496	C₂²×C₇	semidirect
292. #495	C₂²×C₇	semidirect
293. #494	C₂²×C₇	semidirect
294. #492	C₂²×C₇	semidirect
295. #491	C₂²×C₇	semidirect
296. #489	C₂²×C₇	semidirect
297. #488	C₂²×C₇	semidirect
298. #487	C₂²×C₇	semidirect
299. #464	C₂²×C₇	semidirect
300. #463	C₂²×C₇	semidirect
301. #462	C₂²×C₇	semidirect
302. #461	C₂²×C₇	semidirect
304. #451	C₂²×C₇	semidirect
306. #448	C₂²×C₇	semidirect
307. #433	C₂²×C₇	semidirect
308. #433	C₂²×C₇	semidirect
628. C₇	#60807	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×#536	2
C₇×#3225	2
C₇×#3223	2
C₇×#3221	2
C₇×#3153	2
C₇×#3152	2
C₇×#3140	2
C₇×#3139	2
C₇×#3040	2
C₇×#3039	2
C₇×#3005	2
C₇×#3004	2
C₇×#2921	2
#60807	7

Subgroups — C7×#60807 #13238023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60807 #13238023