Subgroups — C₇×#60799 #13238015

Order 1792.

2262 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3215	C₂	1	4	yes	maximal, complemented
4	C₇×#3213	C₂	1	4	yes	maximal, complemented
5	C₇×#3211	C₂	1	8	yes	maximal, complemented
6	C₇×#3153	C₂	1	2	yes	maximal, complemented
7	C₇×#3140	C₂	1	2	yes	maximal, complemented
8	C₇×#3040	C₂	1	4	yes	maximal, complemented
9	C₇×#3004	C₂	1	2	yes	maximal, complemented
10	C₇×#3003	C₂	1	2	yes	maximal, complemented
11	C₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×#137	C₂²	1	4	yes	complemented
15	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	4	yes	—
16	C₇×#533	C₂²	1	2	yes	complemented
17	C₇×#533	C₂²	1	1	yes	characteristic, complemented
18	C₇×#532	C₂²	1	1	yes	characteristic, complemented
19	C₇×#532	C₂²	1	2	yes	complemented
20	C₇×#529	—	2	2	no	—
21	C₇×#529	—	2	2	no	—
22	C₇×#529	C₂²	1	4	yes	complemented
23	C₂×C₇×#124	—	2	2	no	—
24	C₂×C₇×#124	—	2	2	no	—
25	C₂×C₇×#124	—	2	2	no	—
26	C₂×C₇×#124	C₂²	1	4	yes	—
27	C₂×C₇×#124	C₂²	1	4	yes	—
28	C₂×C₇×#124	C₂²	1	2	yes	—
29	C₂×C₇×#124	C₂²	1	2	yes	—
30	C₇×#525	—	2	2	no	—
31	C₇×#525	C₂²	1	2	yes	—
32	C₄×C₇×#41	—	2	2	no	—
33	C₄×C₇×#41	C₂²	1	2	yes	complemented
34	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
35	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
36	C₇×#497	C₂²	1	4	yes	complemented
37	C₇×#496	C₂²	1	4	yes	complemented
38	C₇×#495	C₂²	1	8	yes	complemented
39	C₇×#489	C₂²	1	8	yes	complemented
40	C₇×#488	C₂²	1	8	yes	complemented
41	C₇×#464	C₂²	1	16	yes	complemented
42	C₇×#463	C₂²	1	16	yes	complemented
43	C₇×#451	C₂²	1	8	yes	complemented
44	C₇×#450	C₂²	1	8	yes	—
45	C₇×#448	C₂²	1	8	yes	complemented
46	C₇×#447	C₂²	1	8	yes	—
47	C₇×#432	C₂²	1	4	yes	complemented
48	C₇×#431	C₂²	1	4	yes	complemented
49	C₇×#423	C₂²	1	2	yes	complemented
50	C₇×#419	C₂²	1	4	yes	complemented
51	C₇×#418	C₂²	1	4	yes	—
52	C₇×#407	C₂²	1	2	yes	—
53	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
54	#60799	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁵×C₇	—	2	1	no	—
56	C₂×C₇×#41	—	2	2	no	—
57	C₂×C₇×#41	C₂³	1	2	yes	—
58	C₂²×C₇×Q₈	—	2	2	no	—
59	C₂²×C₇×Q₈	—	2	1	no	—
60	C₂²×C₇×Q₈	—	4	4	no	—
61	C₂²×C₇×Q₈	C₂³	1	2	yes	—
62	C₂²×C₇×Q₈	C₂³	1	4	yes	—
63	C₂²×C₇×D₄	—	2	2	no	—
64	C₂²×C₇×D₄	—	2	1	no	—
65	C₂²×C₇×D₄	—	4	4	no	—
66	C₂²×C₇×D₄	C₂³	1	2	yes	—
67	C₂²×C₇×D₄	C₂³	1	4	yes	—
68	C₂³×C₄×C₇	—	2	1	no	—
69	C₂³×C₄×C₇	D₄	1	2	yes	—
70	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
71	C₇×#137	—	4	8	no	—
72	C₇×AGL(1,ℤ/8)	—	4	8	no	—
73	C₂×C₇×Q₁₆	C₂³	1	8	yes	complemented
74	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
75	C₂×C₇×SD₁₆	C₂³	1	8	yes	complemented
76	C₂×C₇×D₈	C₂³	1	8	yes	—
77	C₂×C₇×M₁₆	C₂³	1	4	yes	—
78	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
79	C₇×C₄⋊Q₈	C₂³	1	2	yes	complemented
80	C₇×#127	C₂³	1	1	yes	characteristic, complemented
81	C₇×#126	—	2	4	no	—
82	C₇×#125	—	2	2	no	—
83	C₇×#125	C₂³	1	2	yes	complemented
84	C₇×#124	—	2	2	no	—
85	C₇×#124	—	2	4	no	—
86	C₇×#124	—	2	2	no	—
87	C₇×#124	—	2	4	no	—
88	C₇×#124	—	2	4	no	—
89	C₇×#124	—	2	4	no	—
90	C₇×#124	—	2	4	no	—
91	C₇×#124	—	2	2	no	—
92	C₇×#124	C₂³	1	4	yes	—
93	C₇×#124	C₂³	1	4	yes	—
94	C₇×#124	C₂³	1	8	yes	—
95	C₇×#124	C₂³	1	2	yes	complemented
96	C₇×#123	—	2	4	no	—
97	C₇×#123	—	2	2	no	—
98	C₇×C₂²⋊Q₈	—	2	2	no	—
99	C₇×C₂²⋊Q₈	—	2	2	no	—
100	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
101	C₇×#121	—	2	2	no	—
102	C₇×#121	—	2	2	no	—
103	C₇×#121	C₂³	1	4	yes	—
104	C₇×#120	—	2	2	no	—
105	C₄×C₇×Q₈	—	2	2	no	—
106	C₄×C₇×Q₈	—	2	2	no	—
107	C₄×C₇×Q₈	C₂³	1	4	yes	complemented
108	C₄×C₇×D₄	—	2	4	no	—
109	C₄×C₇×D₄	—	2	4	no	—
110	C₄×C₇×D₄	—	2	4	no	—
111	C₄×C₇×D₄	—	2	2	no	—
112	C₄×C₇×D₄	—	2	2	no	—
113	C₄×C₇×D₄	C₂³	1	4	yes	complemented
114	C₇×#117	—	2	2	no	—
115	C₇×#117	—	2	4	no	—
116	C₇×#117	—	2	2	no	—
117	C₇×#117	C₂³	1	2	yes	—
118	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
119	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
120	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
121	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
122	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
123	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
124	C₂×C₄²×C₇	—	2	2	no	—
125	C₂×C₄²×C₇	D₄	1	2	yes	—
126	C₂×C₄²×C₇	D₄	1	2	yes	—
127	C₂×C₄²×C₇	D₄	1	2	yes	—
128	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₇	C₂³	1	2	yes	—
130	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₇	C₂³	1	2	yes	—
132	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
133	C₇×#103	C₂³	1	16	yes	complemented
134	C₇×#102	C₂³	1	16	yes	—
135	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
136	C₇×#97	C₂³	1	4	yes	complemented
137	C₄×C₇×C₈	C₂³	1	4	yes	complemented
138	#3444	C₂×C₇	1	1	yes	characteristic, complemented
139	C₂²×#124	C₂×C₇	1	1	yes	characteristic, complemented
140	#3215	C₂×C₇	1	4	yes	complemented
141	#3213	C₂×C₇	1	4	yes	complemented
142	#3211	C₂×C₇	1	8	yes	complemented
143	#3153	C₂×C₇	1	2	yes	complemented
144	#3140	C₂×C₇	1	2	yes	complemented
145	#3040	C₂×C₇	1	4	yes	complemented
146	#3004	C₂×C₇	1	2	yes	complemented
147	#3003	C₂×C₇	1	2	yes	complemented
148	#2920	C₂×C₇	1	1	yes	characteristic, complemented
149	C₂⁴×C₇	—	2	4	no	—
150	C₂⁴×C₇	—	2	2	no	—
151	C₂⁴×C₇	—	4	4	no	—
152	C₂⁴×C₇	—	4	8	no	—
153	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
154	C₇×#41	—	2	8	no	—
155	C₇×#41	—	4	8	no	—
156	C₂×C₇×Q₈	—	2	2	no	—
157	C₂×C₇×Q₈	—	2	4	no	—
158	C₂×C₇×Q₈	—	2	2	no	—
159	C₂×C₇×Q₈	—	2	8	no	—
160	C₂×C₇×Q₈	—	4	8	no	—
161	C₂×C₇×Q₈	—	4	4	no	—
162	C₂×C₇×Q₈	—	4	8	no	—
163	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
164	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
165	C₂×C₇×D₄	—	2	2	no	—
166	C₂×C₇×D₄	—	2	4	no	—
167	C₂×C₇×D₄	—	2	2	no	—
168	C₂×C₇×D₄	—	2	4	no	—
169	C₂×C₇×D₄	—	2	2	no	—
170	C₂×C₇×D₄	—	2	8	no	—
171	C₂×C₇×D₄	—	4	8	no	—
172	C₂×C₇×D₄	—	4	4	no	—
173	C₂×C₇×D₄	—	4	8	no	—
174	C₂×C₇×D₄	C₂⁴	1	4	yes	—
175	C₂×C₇×D₄	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₇	—	2	4	no	—
177	C₂²×C₄×C₇	—	2	4	no	—
178	C₂²×C₄×C₇	—	2	2	no	—
179	C₂²×C₄×C₇	—	2	2	no	—
180	C₂²×C₄×C₇	—	2	4	no	—
181	C₂²×C₄×C₇	—	4	4	no	—
182	C₂²×C₄×C₇	—	4	4	no	—
183	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
189	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
190	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
193	C₇×Q₁₆	—	4	8	no	—
194	C₇×SD₁₆	—	4	8	no	—
195	C₇×SD₁₆	—	4	8	no	—
196	C₇×D₈	—	4	8	no	—
197	C₇×M₁₆	—	2	8	no	—
198	C₂×C₇×C₈	C₂⁴	1	8	yes	—
199	C₇×C₄⋊C₄	—	2	2	no	—
200	C₇×C₄⋊C₄	—	2	2	no	—
201	C₇×C₄⋊C₄	—	2	4	no	—
202	C₇×C₄⋊C₄	—	2	4	no	—
203	C₇×C₄⋊C₄	—	2	4	no	—
204	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
205	C₇×C₂²⋊C₄	—	2	4	no	—
206	C₇×C₂²⋊C₄	—	2	2	no	—
207	C₇×C₂²⋊C₄	—	2	8	no	—
208	C₇×C₂²⋊C₄	—	2	8	no	—
209	C₇×C₂²⋊C₄	—	2	8	no	—
210	C₇×C₂²⋊C₄	—	2	8	no	—
211	C₇×C₂²⋊C₄	—	2	4	no	—
212	C₇×C₂²⋊C₄	—	2	4	no	—
213	C₇×C₂²⋊C₄	—	2	4	no	—
214	C₄²×C₇	—	2	4	no	—
215	C₄²×C₇	—	2	2	no	—
216	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₇	C₂×D₄	1	2	yes	—
218	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₇	C₂×D₄	1	2	yes	—
220	C₄²×C₇	C₂×D₄	1	2	yes	—
221	C₄²×C₇	C₂×D₄	1	2	yes	—
222	C₄²×C₇	C₂×D₄	1	2	yes	—
223	C₄²×C₇	C₂⁴	1	1	yes	characteristic
224	C₄²×C₇	C₂⁴	1	1	yes	characteristic
225	C₄²×C₇	C₂⁴	1	2	yes	—
226	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
227	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
228	C₂×#137	C₂²×C₇	1	4	yes	complemented
229	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	4	yes	—
230	#533	C₂²×C₇	1	2	yes	complemented
231	#533	C₂²×C₇	1	1	yes	characteristic, complemented
232	#532	C₂²×C₇	1	1	yes	characteristic, complemented
233	#532	C₂²×C₇	1	2	yes	complemented
234	#529	—	2	2	no	—
235	#529	—	2	2	no	—
236	#529	C₂²×C₇	1	4	yes	complemented
237	C₂×#124	—	2	2	no	—
238	C₂×#124	—	2	2	no	—
239	C₂×#124	—	2	2	no	—
240	C₂×#124	C₂²×C₇	1	2	yes	—
241	C₂×#124	C₂²×C₇	1	4	yes	—
242	C₂×#124	C₂²×C₇	1	4	yes	—
243	C₂×#124	C₂²×C₇	1	2	yes	—
244	#525	—	2	2	no	—
245	#525	C₂²×C₇	1	2	yes	—
246	C₄×#41	—	2	2	no	—
247	C₄×#41	C₂²×C₇	1	2	yes	complemented
248	C₂²×C₂²⋊C₄	—	2	2	no	—
249	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
250	#497	C₂²×C₇	1	4	yes	complemented
251	#496	C₂²×C₇	1	4	yes	complemented
252	#495	C₂²×C₇	1	8	yes	complemented
253	#489	C₂²×C₇	1	8	yes	complemented
254	#488	C₂²×C₇	1	8	yes	complemented
255	#464	C₂²×C₇	1	16	yes	complemented
256	#463	C₂²×C₇	1	16	yes	complemented
257	#451	C₂²×C₇	1	8	yes	complemented
258	#450	C₂²×C₇	1	8	yes	—
259	#448	C₂²×C₇	1	8	yes	complemented
260	#447	C₂²×C₇	1	8	yes	—
261	#432	C₂²×C₇	1	4	yes	complemented
262	#431	C₂²×C₇	1	4	yes	complemented
263	#423	C₂²×C₇	1	2	yes	complemented
264	#419	C₂²×C₇	1	4	yes	complemented
265	#418	C₂²×C₇	1	4	yes	—
266	#407	C₂²×C₇	1	2	yes	—
267	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
268	C₂³×C₇	—	2	2	no	—
269	C₂³×C₇	—	2	4	no	—
270	C₂³×C₇	—	2	4	no	—
271	C₂³×C₇	—	2	2	no	—
272	C₂³×C₇	—	4	8	no	—
273	C₂³×C₇	—	4	8	no	—
274	C₂³×C₇	—	4	8	no	—
275	C₂³×C₇	—	4	4	no	—
276	C₂³×C₇	—	4	8	no	—
277	C₂³×C₇	—	8	16	no	—
278	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
279	C₂³×C₇	C₂²×D₄	1	2	yes	—
280	C₇×Q₈	—	2	4	no	—
281	C₇×Q₈	—	2	8	no	—
282	C₇×Q₈	—	4	8	no	—
283	C₇×Q₈	—	4	4	no	—
284	C₇×Q₈	—	4	4	no	—
285	C₇×D₄	—	2	4	no	—
286	C₇×D₄	—	2	8	no	—
287	C₇×D₄	—	4	8	no	—
288	C₇×D₄	—	4	4	no	—
289	C₇×D₄	—	4	4	no	—
290	C₇×D₄	—	4	8	no	—
291	C₇×D₄	—	4	4	no	—
292	C₂×C₄×C₇	—	2	2	no	—
293	C₂×C₄×C₇	—	2	4	no	—
294	C₂×C₄×C₇	—	2	4	no	—
295	C₂×C₄×C₇	—	2	4	no	—
296	C₂×C₄×C₇	—	2	4	no	—
297	C₂×C₄×C₇	—	2	2	no	—
298	C₂×C₄×C₇	—	2	4	no	—
299	C₂×C₄×C₇	—	2	4	no	—
300	C₂×C₄×C₇	—	4	2	no	—
301	C₂×C₄×C₇	—	4	2	no	—
302	C₂×C₄×C₇	—	4	2	no	—
303	C₂×C₄×C₇	—	4	2	no	—
304	C₂×C₄×C₇	—	4	8	no	—
305	C₂×C₄×C₇	—	4	8	no	—
306	C₂×C₄×C₇	—	4	4	no	—
307	C₂×C₄×C₇	—	4	4	no	—
308	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
309	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
310	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
311	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
312	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
313	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
315	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
316	C₇×C₈	—	2	8	no	—
317	C₂⁵	—	2	1	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	C₂³×C₇	1	2	yes	—
320	C₂²×Q₈	—	2	2	no	—
321	C₂²×Q₈	—	2	1	no	—
322	C₂²×Q₈	—	4	4	no	—
323	C₂²×Q₈	C₂³×C₇	1	2	yes	—
324	C₂²×Q₈	C₂³×C₇	1	4	yes	—
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	—	4	4	no	—
328	C₂²×D₄	C₂³×C₇	1	2	yes	—
329	C₂²×D₄	C₂³×C₇	1	4	yes	—
330	C₂³×C₄	—	2	1	no	—
331	C₂³×C₄	C₇×D₄	1	2	yes	—
332	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
333	#137	—	4	8	no	—
334	AGL(1,ℤ/8)	—	4	8	no	—
335	C₂×Q₁₆	C₂³×C₇	1	8	yes	complemented
336	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
337	C₂×SD₁₆	C₂³×C₇	1	8	yes	complemented
338	C₂×D₈	C₂³×C₇	1	8	yes	—
339	C₂×M₁₆	C₂³×C₇	1	4	yes	—
340	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic, complemented
341	C₄⋊Q₈	C₂³×C₇	1	2	yes	complemented
342	#127	C₂³×C₇	1	1	yes	characteristic, complemented
343	#126	—	2	4	no	—
344	#125	—	2	2	no	—
345	#125	C₂³×C₇	1	2	yes	complemented
346	#124	—	2	2	no	—
347	#124	—	2	4	no	—
348	#124	—	2	2	no	—
349	#124	—	2	4	no	—
350	#124	—	2	4	no	—
351	#124	—	2	4	no	—
352	#124	—	2	4	no	—
353	#124	—	2	2	no	—
354	#124	C₂³×C₇	1	8	yes	—
355	#124	C₂³×C₇	1	4	yes	—
356	#124	C₂³×C₇	1	4	yes	—
357	#124	C₂³×C₇	1	2	yes	complemented
358	#123	—	2	4	no	—
359	#123	—	2	2	no	—
360	C₂²⋊Q₈	—	2	2	no	—
361	C₂²⋊Q₈	—	2	2	no	—
362	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
363	#121	—	2	2	no	—
364	#121	—	2	2	no	—
365	#121	C₂³×C₇	1	4	yes	—
366	#120	—	2	2	no	—
367	C₄×Q₈	—	2	2	no	—
368	C₄×Q₈	—	2	2	no	—
369	C₄×Q₈	C₂³×C₇	1	4	yes	complemented
370	C₄×D₄	—	2	4	no	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	4	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	C₂³×C₇	1	4	yes	complemented
376	#117	—	2	2	no	—
377	#117	—	2	4	no	—
378	#117	—	2	2	no	—
379	#117	C₂³×C₇	1	2	yes	—
380	C₂×C₂²⋊C₄	—	2	4	no	—
381	C₂×C₂²⋊C₄	—	2	4	no	—
382	C₂×C₂²⋊C₄	—	2	4	no	—
383	C₂×C₂²⋊C₄	—	2	4	no	—
384	C₂×C₂²⋊C₄	—	2	4	no	—
385	C₂×C₂²⋊C₄	—	2	4	no	—
386	C₂×C₄²	—	2	2	no	—
387	C₂×C₄²	C₇×D₄	1	2	yes	—
388	C₂×C₄²	C₇×D₄	1	2	yes	—
389	C₂×C₄²	C₇×D₄	1	2	yes	—
390	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₇	1	2	yes	—
392	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₇	1	2	yes	—
394	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
395	#103	C₂³×C₇	1	16	yes	complemented
396	#102	C₂³×C₇	1	16	yes	—
397	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
398	#97	C₂³×C₇	1	4	yes	complemented
399	C₄×C₈	C₂³×C₇	1	4	yes	complemented
400	C₂²×C₇	—	2	8	no	—
401	C₂²×C₇	—	2	2	no	—
402	C₂²×C₇	—	2	2	no	—
403	C₂²×C₇	—	2	2	no	—
404	C₂²×C₇	—	4	2	no	—
405	C₂²×C₇	—	4	4	no	—
406	C₂²×C₇	—	4	4	no	—
407	C₂²×C₇	—	4	8	no	—
408	C₂²×C₇	—	8	8	no	—
409	C₂²×C₇	—	8	16	no	—
410	C₂²×C₇	#535	1	2	yes	—
411	C₂²×C₇	#575	1	4	yes	—
412	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
413	C₄×C₇	—	2	2	no	—
414	C₄×C₇	—	2	2	no	—
415	C₄×C₇	—	2	2	no	—
416	C₄×C₇	—	2	2	no	—
417	C₄×C₇	—	2	2	no	—
418	C₄×C₇	—	4	4	no	—
419	C₄×C₇	—	4	2	no	—
420	C₄×C₇	—	4	4	no	—
421	C₄×C₇	#535	1	2	yes	—
422	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
423	C₂⁴	—	2	4	no	—
424	C₂⁴	—	2	2	no	—
425	C₂⁴	—	4	4	no	—
426	C₂⁴	—	4	8	no	—
427	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
428	#41	—	2	8	no	—
429	#41	—	4	8	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	—	2	8	no	—
434	C₂×Q₈	—	4	8	no	—
435	C₂×Q₈	—	4	4	no	—
436	C₂×Q₈	—	4	8	no	—
437	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
438	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	4	8	no	—
446	C₂×D₄	—	4	4	no	—
447	C₂×D₄	—	4	8	no	—
448	C₂×D₄	C₂⁴×C₇	1	4	yes	—
449	C₂×D₄	C₂⁴×C₇	1	2	yes	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
458	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
459	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
460	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
461	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
462	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
463	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
464	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
467	Q₁₆	—	4	8	no	—
468	SD₁₆	—	4	8	no	—
469	SD₁₆	—	4	8	no	—
470	D₈	—	4	8	no	—
471	M₁₆	—	2	8	no	—
472	C₂×C₈	C₂⁴×C₇	1	8	yes	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	2	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₄²	—	2	4	no	—
489	C₄²	—	2	2	no	—
490	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
491	C₄²	C₂×C₇×D₄	1	2	yes	—
492	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
493	C₄²	C₂×C₇×D₄	1	2	yes	—
494	C₄²	C₂×C₇×D₄	1	2	yes	—
495	C₄²	C₂×C₇×D₄	1	2	yes	—
496	C₄²	C₂×C₇×D₄	1	2	yes	—
497	C₄²	C₂⁴×C₇	1	1	yes	characteristic
498	C₄²	C₂⁴×C₇	1	1	yes	characteristic
499	C₄²	C₂⁴×C₇	1	2	yes	—
500	C₂×C₇	—	2	2	no	—
501	C₂×C₇	—	2	4	no	—
502	C₂×C₇	—	8	2	no	—
503	C₂×C₇	—	8	4	no	—
504	C₂×C₇	C₂×#535	1	1	yes	characteristic
505	C₂×C₇	C₂×#575	1	2	yes	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	4	8	no	—
511	C₂³	—	4	8	no	—
512	C₂³	—	4	8	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	8	no	—
515	C₂³	—	8	16	no	—
516	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
517	C₂³	C₂²×C₇×D₄	1	2	yes	—
518	Q₈	—	2	4	no	—
519	Q₈	—	2	8	no	—
520	Q₈	—	4	8	no	—
521	Q₈	—	4	4	no	—
522	Q₈	—	4	4	no	—
523	D₄	—	2	4	no	—
524	D₄	—	2	8	no	—
525	D₄	—	4	8	no	—
526	D₄	—	4	4	no	—
527	D₄	—	4	4	no	—
528	D₄	—	4	8	no	—
529	D₄	—	4	4	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	8	no	—
543	C₂×C₄	—	4	8	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
547	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
548	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
549	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
550	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
551	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
552	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
553	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
554	C₈	—	2	8	no	—
555	C₇	#60799	1	1	yes	characteristic, direct product factor, Sylow
556	C₂²	—	2	8	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	8	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	8	8	no	—
565	C₂²	—	8	16	no	—
566	C₂²	C₇×#535	1	2	yes	—
567	C₂²	C₇×#575	1	4	yes	—
568	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
569	C₄	—	2	2	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	2	no	—
577	C₄	C₇×#535	1	2	yes	—
578	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
579	C₂	—	2	2	no	—
580	C₂	—	2	4	no	—
581	C₂	—	8	4	no	—
582	C₂	—	8	2	no	—
583	C₂	C₂×C₇×#535	1	1	yes	characteristic
584	C₂	C₂×C₇×#575	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 ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3444	C₂	semidirect
2. C₂²×C₇×#124	C₂	semidirect
3. C₇×#3215	C₂	semidirect
4. C₇×#3213	C₂	semidirect
5. C₇×#3211	C₂	semidirect
6. C₇×#3153	C₂	semidirect
7. C₇×#3140	C₂	semidirect
8. C₇×#3040	C₂	semidirect
9. C₇×#3004	C₂	semidirect
10. C₇×#3003	C₂	semidirect
11. C₇×#2920	C₂	semidirect
14. C₂×C₇×#137	C₂²	semidirect
16. C₇×#533	C₂²	semidirect
17. C₇×#533	C₂²	semidirect
18. C₇×#532	C₂²	semidirect
19. C₇×#532	C₂²	semidirect
22. C₇×#529	C₂²	semidirect
33. C₄×C₇×#41	C₂²	semidirect
36. C₇×#497	C₂²	semidirect
37. C₇×#496	C₂²	semidirect
38. C₇×#495	C₂²	semidirect
39. C₇×#489	C₂²	semidirect
40. C₇×#488	C₂²	semidirect
41. C₇×#464	C₂²	semidirect
42. C₇×#463	C₂²	semidirect
43. C₇×#451	C₂²	semidirect
45. C₇×#448	C₂²	semidirect
47. C₇×#432	C₂²	semidirect
48. C₇×#431	C₂²	semidirect
49. C₇×#423	C₂²	semidirect
50. C₇×#419	C₂²	semidirect
53. C₄×C₇×M₁₆	C₂²	semidirect
54. #60799	C₇	direct (N × H)
73. C₂×C₇×Q₁₆	C₂³	semidirect
75. C₂×C₇×SD₁₆	C₂³	semidirect
78. C₇×C₄⋊Q₈	C₂³	semidirect
79. C₇×C₄⋊Q₈	C₂³	semidirect
80. C₇×#127	C₂³	semidirect
83. C₇×#125	C₂³	semidirect
95. C₇×#124	C₂³	semidirect
107. C₄×C₇×Q₈	C₂³	semidirect
113. C₄×C₇×D₄	C₂³	semidirect
132. C₇×C₄⋊C₈	C₂³	semidirect
133. C₇×#103	C₂³	semidirect
136. C₇×#97	C₂³	semidirect
137. C₄×C₇×C₈	C₂³	semidirect
138. #3444	C₂×C₇	semidirect
139. C₂²×#124	C₂×C₇	semidirect
140. #3215	C₂×C₇	semidirect
141. #3213	C₂×C₇	semidirect
142. #3211	C₂×C₇	semidirect
143. #3153	C₂×C₇	semidirect
144. #3140	C₂×C₇	semidirect
145. #3040	C₂×C₇	semidirect
146. #3004	C₂×C₇	semidirect
147. #3003	C₂×C₇	semidirect
148. #2920	C₂×C₇	semidirect
228. C₂×#137	C₂²×C₇	semidirect
230. #533	C₂²×C₇	semidirect
231. #533	C₂²×C₇	semidirect
232. #532	C₂²×C₇	semidirect
233. #532	C₂²×C₇	semidirect
236. #529	C₂²×C₇	semidirect
247. C₄×#41	C₂²×C₇	semidirect
250. #497	C₂²×C₇	semidirect
251. #496	C₂²×C₇	semidirect
252. #495	C₂²×C₇	semidirect
253. #489	C₂²×C₇	semidirect
254. #488	C₂²×C₇	semidirect
255. #464	C₂²×C₇	semidirect
256. #463	C₂²×C₇	semidirect
257. #451	C₂²×C₇	semidirect
259. #448	C₂²×C₇	semidirect
261. #432	C₂²×C₇	semidirect
262. #431	C₂²×C₇	semidirect
263. #423	C₂²×C₇	semidirect
264. #419	C₂²×C₇	semidirect
267. C₄×M₁₆	C₂²×C₇	semidirect
335. C₂×Q₁₆	C₂³×C₇	semidirect
337. C₂×SD₁₆	C₂³×C₇	semidirect
340. C₄⋊Q₈	C₂³×C₇	semidirect
341. C₄⋊Q₈	C₂³×C₇	semidirect
342. #127	C₂³×C₇	semidirect
345. #125	C₂³×C₇	semidirect
357. #124	C₂³×C₇	semidirect
369. C₄×Q₈	C₂³×C₇	semidirect
375. C₄×D₄	C₂³×C₇	semidirect
394. C₄⋊C₈	C₂³×C₇	semidirect
395. #103	C₂³×C₇	semidirect
398. #97	C₂³×C₇	semidirect
399. C₄×C₈	C₂³×C₇	semidirect
555. C₇	#60799	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×#3444	2
C₂²×C₇×#124	2
C₇×#3215	2
C₇×#3213	2
C₇×#3211	2
C₇×#3153	2
C₇×#3140	2
C₇×#3040	2
C₇×#3004	2
C₇×#3003	2
C₇×#2920	2
#60799	7

Subgroups — C7×#60799 #13238015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60799 #13238015