Subgroups — C₇×#61377 #13238593

Order 1792.

1742 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3501	C₂	1	1	yes	maximal, characteristic
3	C₇×#3388	C₂	1	4	yes	maximal, complemented
4	C₇×#3388	C₂	1	4	yes	maximal, complemented
5	C₇×#3386	C₂	1	4	yes	maximal, complemented
6	C₇×#3385	C₂	1	4	yes	maximal, complemented
7	C₇×#3164	C₂	1	2	yes	maximal, complemented
8	C₇×#3162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3082	C₂	1	2	yes	maximal, complemented
10	C₇×#3081	C₂	1	2	yes	maximal
11	C₇×#3078	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#3077	C₂	1	1	yes	maximal, characteristic
13	C₇×#2961	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#2949	C₂	1	2	yes	maximal
15	C₂×C₇×#440	C₂	1	1	yes	maximal, characteristic
16	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
17	C₇×Q₈²	—	2	1	no	—
18	C₇×Q₈²	C₂²	1	2	yes	complemented
19	C₇×#557	—	2	1	no	—
20	C₇×#557	—	2	2	no	—
21	C₇×#557	C₂²	1	4	yes	complemented
22	C₇×#557	C₂²	1	2	yes	complemented
23	C₇×#554	—	2	1	no	—
24	C₇×#554	—	2	2	no	—
25	C₇×#554	C₂²	1	4	yes	—
26	C₇×#554	C₂²	1	2	yes	—
27	C₇×D₄×Q₈	—	2	1	no	—
28	C₇×D₄×Q₈	C₂²	1	2	yes	—
29	C₇×#544	—	2	2	no	—
30	C₇×#544	C₂²	1	2	yes	—
31	C₇×#544	C₂²	1	2	yes	—
32	C₇×#543	—	2	2	no	—
33	C₇×#533	—	2	2	no	—
34	C₇×#533	C₂²	1	1	yes	characteristic
35	C₇×#533	C₂²	1	1	yes	characteristic
36	C₇×#533	C₂²	1	1	yes	characteristic
37	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
38	C₇×#519	C₂²	1	2	yes	—
39	C₇×#518	C₂²	1	2	yes	—
40	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
42	C₇×#501	C₂²	1	2	yes	complemented
43	C₇×#501	C₂²	1	2	yes	complemented
44	C₇×#501	C₂²	1	2	yes	complemented
45	C₇×#501	C₂²	1	2	yes	complemented
46	C₇×#501	C₂²	1	4	yes	complemented
47	C₇×#501	C₂²	1	4	yes	complemented
48	C₇×#500	C₂²	1	4	yes	complemented
49	C₇×#499	C₂²	1	4	yes	complemented
50	C₇×#479	C₂²	1	4	yes	complemented
51	C₇×#479	C₂²	1	4	yes	complemented
52	C₇×#479	C₂²	1	4	yes	complemented
53	C₇×#478	C₂²	1	4	yes	—
54	C₇×#478	C₂²	1	4	yes	—
55	C₇×#478	C₂²	1	4	yes	—
56	C₇×#477	C₂²	1	2	yes	complemented
57	C₇×#477	C₂²	1	4	yes	complemented
58	C₇×#476	C₂²	1	2	yes	—
59	C₇×#476	C₂²	1	4	yes	—
60	C₇×#475	C₂²	1	2	yes	complemented
61	C₇×#475	C₂²	1	4	yes	complemented
62	C₇×#474	C₂²	1	2	yes	—
63	C₇×#474	C₂²	1	4	yes	—
64	C₇×#446	C₂²	1	4	yes	complemented
65	C₇×#446	C₂²	1	4	yes	complemented
66	C₇×#440	C₂²	1	4	yes	—
67	C₇×#440	C₂²	1	4	yes	—
68	C₇×#440	C₂²	1	4	yes	—
69	C₇×#440	C₂²	1	4	yes	—
70	C₄×C₇×SD₁₆	C₂²	1	4	yes	—
71	C₄×C₇×SD₁₆	C₂²	1	4	yes	—
72	C₇×#428	C₂²	1	2	yes	—
73	C₇×#428	C₂²	1	2	yes	—
74	C₇×#427	C₂²	1	2	yes	—
75	C₇×#427	C₂²	1	2	yes	—
76	C₂×C₇×#107	C₂²	1	1	yes	characteristic
77	C₇×#424	C₂²	1	2	yes	—
78	C₇×#423	C₂²	1	1	yes	characteristic
79	C₇×#419	C₂²	1	2	yes	—
80	C₇×#418	C₂²	1	2	yes	—
81	C₂×C₇×#103	C₂²	1	1	yes	characteristic
82	C₂×C₇×#102	C₂²	1	1	yes	characteristic
83	C₇×#405	C₂²	1	2	yes	—
84	C₂×C₇×#97	C₂²	1	1	yes	characteristic
85	#61377	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₇×Q₈	—	2	2	no	—
87	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
88	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₇	—	2	1	no	—
90	C₂×C₇×SD₁₆	Q₈	1	8	yes	complemented
91	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
92	C₂×C₇×M₁₆	C₂³	1	2	yes	—
93	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
94	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
95	C₇×C₄⋊Q₈	—	2	2	no	—
96	C₇×C₄⋊Q₈	—	2	4	no	—
97	C₇×C₄⋊Q₈	—	2	2	no	—
98	C₇×C₄⋊Q₈	—	2	4	no	—
99	C₇×C₄⋊Q₈	—	2	2	no	—
100	C₇×C₄⋊Q₈	—	2	2	no	—
101	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
102	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
103	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
104	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
105	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
106	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
107	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
108	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
109	C₇×#125	—	2	4	no	—
110	C₇×#125	—	2	4	no	—
111	C₇×#125	—	2	4	no	—
112	C₇×#125	—	2	2	no	—
113	C₇×#125	—	2	2	no	—
114	C₇×#125	D₄	1	2	yes	—
115	C₇×#125	C₂³	1	4	yes	—
116	C₇×#125	C₂³	1	2	yes	—
117	C₇×#125	C₂³	1	4	yes	—
118	C₇×#125	C₂³	1	2	yes	—
119	C₇×C₂²⋊Q₈	—	2	2	no	—
120	C₇×C₂²⋊Q₈	—	2	4	no	—
121	C₇×C₂²⋊Q₈	—	2	1	no	—
122	C₇×C₂²⋊Q₈	—	2	1	no	—
123	C₇×C₂²⋊Q₈	—	2	1	no	—
124	C₇×C₂²⋊Q₈	—	2	1	no	—
125	C₇×C₂²⋊Q₈	—	2	2	no	—
126	C₇×C₂²⋊Q₈	—	2	4	no	—
127	C₇×C₂²⋊Q₈	—	2	4	no	—
128	C₇×C₂²⋊Q₈	—	2	4	no	—
129	C₇×C₂²⋊Q₈	—	2	4	no	—
130	C₇×C₂²⋊Q₈	—	2	4	no	—
131	C₇×C₂²⋊Q₈	—	2	4	no	—
132	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
133	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
134	C₄×C₇×Q₈	—	2	2	no	—
135	C₄×C₇×Q₈	—	2	2	no	—
136	C₄×C₇×Q₈	—	2	2	no	—
137	C₄×C₇×Q₈	—	2	4	no	—
138	C₄×C₇×Q₈	—	2	2	no	—
139	C₄×C₇×Q₈	—	2	1	no	—
140	C₄×C₇×Q₈	—	2	1	no	—
141	C₄×C₇×Q₈	—	2	1	no	—
142	C₄×C₇×Q₈	—	2	1	no	—
143	C₄×C₇×Q₈	—	2	1	no	—
144	C₄×C₇×Q₈	—	2	1	no	—
145	C₄×C₇×Q₈	C₂³	1	2	yes	—
146	C₄×C₇×Q₈	C₂³	1	4	yes	—
147	C₄×C₇×Q₈	C₂³	1	4	yes	—
148	C₄×C₇×Q₈	C₂³	1	2	yes	—
149	C₄×C₇×Q₈	C₂³	1	2	yes	—
150	C₄×C₇×Q₈	C₂³	1	2	yes	—
151	C₄×C₇×D₄	—	2	2	no	—
152	C₄×C₇×D₄	—	2	2	no	—
153	C₄×C₇×D₄	—	2	1	no	—
154	C₄×C₇×D₄	—	2	1	no	—
155	C₄×C₇×D₄	C₂³	1	2	yes	—
156	C₄×C₇×D₄	C₂³	1	4	yes	—
157	C₄×C₇×D₄	C₂³	1	4	yes	—
158	C₄×C₇×D₄	C₂³	1	2	yes	—
159	C₇×#117	—	2	4	no	—
160	C₇×#117	C₂³	1	2	yes	—
161	C₇×#117	C₂³	1	2	yes	—
162	C₇×#117	C₂³	1	2	yes	—
163	C₂×C₇×C₄⋊C₄	—	2	1	no	—
164	C₂×C₇×C₄⋊C₄	—	2	4	no	—
165	C₂×C₇×C₄⋊C₄	—	2	2	no	—
166	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
167	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
168	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
169	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
170	C₂×C₄²×C₇	—	2	1	no	—
171	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
172	C₇×#107	C₂³	1	2	yes	—
173	C₇×#107	C₂³	1	4	yes	—
174	C₇×#107	C₂³	1	4	yes	—
175	C₇×#107	C₂³	1	2	yes	—
176	C₇×#107	C₂³	1	2	yes	—
177	C₇×#107	C₂³	1	4	yes	—
178	C₇×#107	C₂³	1	2	yes	—
179	C₇×#106	C₂³	1	2	yes	—
180	C₇×#106	C₂³	1	4	yes	—
181	C₇×#106	C₂³	1	4	yes	—
182	C₇×#106	C₂³	1	2	yes	—
183	C₇×#106	C₂³	1	2	yes	—
184	C₇×#106	C₂³	1	2	yes	—
185	C₇×C₄⋊C₈	C₂³	1	4	yes	—
186	C₇×C₄⋊C₈	C₂³	1	2	yes	—
187	C₇×C₄⋊C₈	C₂³	1	2	yes	—
188	C₇×C₄⋊C₈	C₂³	1	4	yes	—
189	C₇×#103	C₂³	1	4	yes	—
190	C₇×#103	C₂³	1	4	yes	—
191	C₇×#103	C₂³	1	4	yes	—
192	C₇×#102	C₂³	1	4	yes	—
193	C₇×#102	C₂³	1	4	yes	—
194	C₇×#102	C₂³	1	4	yes	—
195	C₇×#97	C₂³	1	2	yes	—
196	C₇×#97	C₂³	1	2	yes	—
197	C₇×#97	C₂³	1	4	yes	—
198	C₄×C₇×C₈	C₂³	1	4	yes	—
199	#3502	C₂×C₇	1	1	yes	characteristic, complemented
200	#3501	C₂×C₇	1	1	yes	characteristic
201	#3388	C₂×C₇	1	4	yes	complemented
202	#3388	C₂×C₇	1	4	yes	complemented
203	#3386	C₂×C₇	1	4	yes	complemented
204	#3385	C₂×C₇	1	4	yes	complemented
205	#3164	C₂×C₇	1	2	yes	complemented
206	#3162	C₂×C₇	1	1	yes	characteristic, complemented
207	#3082	C₂×C₇	1	2	yes	complemented
208	#3081	C₂×C₇	1	2	yes	—
209	#3078	C₂×C₇	1	1	yes	characteristic, complemented
210	#3077	C₂×C₇	1	1	yes	characteristic
211	#2961	C₂×C₇	1	1	yes	characteristic, complemented
212	#2949	C₂×C₇	1	2	yes	—
213	C₂×#440	C₂×C₇	1	1	yes	characteristic
214	C₂⁴×C₇	—	2	1	no	—
215	C₂×C₇×Q₈	—	2	1	no	—
216	C₂×C₇×Q₈	—	2	1	no	—
217	C₂×C₇×Q₈	—	2	1	no	—
218	C₂×C₇×Q₈	—	2	1	no	—
219	C₂×C₇×Q₈	—	2	1	no	—
220	C₂×C₇×Q₈	—	2	4	no	—
221	C₂×C₇×Q₈	—	2	4	no	—
222	C₂×C₇×Q₈	—	2	2	no	—
223	C₂×C₇×Q₈	—	8	4	no	—
224	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
225	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
226	C₂×C₇×Q₈	C₂×Q₈	1	4	yes	complemented
227	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
228	C₂×C₇×D₄	—	2	1	no	—
229	C₂×C₇×D₄	—	2	2	no	—
230	C₂×C₇×D₄	C₂×Q₈	1	4	yes	—
231	C₂×C₇×D₄	C₂⁴	1	2	yes	—
232	C₂²×C₄×C₇	—	2	1	no	—
233	C₂²×C₄×C₇	—	2	1	no	—
234	C₂²×C₄×C₇	—	2	1	no	—
235	C₂²×C₄×C₇	—	2	2	no	—
236	C₂²×C₄×C₇	—	2	4	no	—
237	C₂²×C₄×C₇	—	2	4	no	—
238	C₂²×C₄×C₇	—	2	2	no	—
239	C₂²×C₄×C₇	—	8	2	no	—
240	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
241	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
242	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
243	C₇×SD₁₆	C₂×Q₈	1	16	yes	complemented
244	C₇×M₁₆	—	2	4	no	—
245	C₂×C₇×C₈	—	2	2	no	—
246	C₂×C₇×C₈	C₂×Q₈	1	4	yes	complemented
247	C₂×C₇×C₈	C₂⁴	1	4	yes	—
248	C₂×C₇×C₈	C₂⁴	1	2	yes	—
249	C₂×C₇×C₈	C₂⁴	1	2	yes	—
250	C₇×C₄⋊C₄	—	2	2	no	—
251	C₇×C₄⋊C₄	—	2	1	no	—
252	C₇×C₄⋊C₄	—	2	2	no	—
253	C₇×C₄⋊C₄	—	2	2	no	—
254	C₇×C₄⋊C₄	—	2	4	no	—
255	C₇×C₄⋊C₄	—	2	4	no	—
256	C₇×C₄⋊C₄	—	2	4	no	—
257	C₇×C₄⋊C₄	—	2	1	no	—
258	C₇×C₄⋊C₄	—	2	1	no	—
259	C₇×C₄⋊C₄	—	2	4	no	—
260	C₇×C₄⋊C₄	—	2	4	no	—
261	C₇×C₄⋊C₄	—	2	2	no	—
262	C₇×C₄⋊C₄	—	2	1	no	—
263	C₇×C₄⋊C₄	—	2	2	no	—
264	C₇×C₄⋊C₄	—	2	1	no	—
265	C₇×C₄⋊C₄	—	2	1	no	—
266	C₇×C₄⋊C₄	—	2	4	no	—
267	C₇×C₄⋊C₄	—	2	4	no	—
268	C₇×C₄⋊C₄	—	2	2	no	—
269	C₇×C₄⋊C₄	—	2	4	no	—
270	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
271	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
272	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
273	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
274	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
275	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
276	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
277	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
278	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
279	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
280	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
281	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
282	C₇×C₂²⋊C₄	—	2	2	no	—
283	C₇×C₂²⋊C₄	—	2	2	no	—
284	C₇×C₂²⋊C₄	—	2	4	no	—
285	C₇×C₂²⋊C₄	—	2	4	no	—
286	C₇×C₂²⋊C₄	—	2	2	no	—
287	C₇×C₂²⋊C₄	—	2	2	no	—
288	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
289	C₄²×C₇	—	2	2	no	—
290	C₄²×C₇	—	2	2	no	—
291	C₄²×C₇	—	2	4	no	—
292	C₄²×C₇	—	2	4	no	—
293	C₄²×C₇	—	2	2	no	—
294	C₄²×C₇	—	2	2	no	—
295	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
296	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
297	C₄²×C₇	C₂⁴	1	4	yes	—
298	C₄²×C₇	C₂⁴	1	1	yes	characteristic
299	C₄²×C₇	C₂⁴	1	1	yes	characteristic
300	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
301	Q₈²	—	2	1	no	—
302	Q₈²	C₂²×C₇	1	2	yes	complemented
303	#557	—	2	1	no	—
304	#557	—	2	2	no	—
305	#557	C₂²×C₇	1	4	yes	complemented
306	#557	C₂²×C₇	1	2	yes	complemented
307	#554	—	2	1	no	—
308	#554	—	2	2	no	—
309	#554	C₂²×C₇	1	4	yes	—
310	#554	C₂²×C₇	1	2	yes	—
311	D₄×Q₈	—	2	1	no	—
312	D₄×Q₈	C₂²×C₇	1	2	yes	—
313	#544	—	2	2	no	—
314	#544	C₂²×C₇	1	2	yes	—
315	#544	C₂²×C₇	1	2	yes	—
316	#543	—	2	2	no	—
317	#533	—	2	2	no	—
318	#533	C₂²×C₇	1	1	yes	characteristic
319	#533	C₂²×C₇	1	1	yes	characteristic
320	#533	C₂²×C₇	1	1	yes	characteristic
321	C₂×C₂²⋊Q₈	—	2	2	no	—
322	#519	C₂²×C₇	1	2	yes	—
323	#518	C₂²×C₇	1	2	yes	—
324	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
325	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
326	#501	C₂²×C₇	1	2	yes	complemented
327	#501	C₂²×C₇	1	4	yes	complemented
328	#501	C₂²×C₇	1	2	yes	complemented
329	#501	C₂²×C₇	1	4	yes	complemented
330	#501	C₂²×C₇	1	2	yes	complemented
331	#501	C₂²×C₇	1	2	yes	complemented
332	#500	C₂²×C₇	1	4	yes	complemented
333	#499	C₂²×C₇	1	4	yes	complemented
334	#479	C₂²×C₇	1	4	yes	complemented
335	#479	C₂²×C₇	1	4	yes	complemented
336	#479	C₂²×C₇	1	4	yes	complemented
337	#478	C₂²×C₇	1	4	yes	—
338	#478	C₂²×C₇	1	4	yes	—
339	#478	C₂²×C₇	1	4	yes	—
340	#477	C₂²×C₇	1	2	yes	complemented
341	#477	C₂²×C₇	1	4	yes	complemented
342	#476	C₂²×C₇	1	2	yes	—
343	#476	C₂²×C₇	1	4	yes	—
344	#475	C₂²×C₇	1	2	yes	complemented
345	#475	C₂²×C₇	1	4	yes	complemented
346	#474	C₂²×C₇	1	2	yes	—
347	#474	C₂²×C₇	1	4	yes	—
348	#446	C₂²×C₇	1	4	yes	complemented
349	#446	C₂²×C₇	1	4	yes	complemented
350	#440	C₂²×C₇	1	4	yes	—
351	#440	C₂²×C₇	1	4	yes	—
352	#440	C₂²×C₇	1	4	yes	—
353	#440	C₂²×C₇	1	4	yes	—
354	C₄×SD₁₆	C₂²×C₇	1	4	yes	—
355	C₄×SD₁₆	C₂²×C₇	1	4	yes	—
356	#428	C₂²×C₇	1	2	yes	—
357	#428	C₂²×C₇	1	2	yes	—
358	#427	C₂²×C₇	1	2	yes	—
359	#427	C₂²×C₇	1	2	yes	—
360	C₂×#107	C₂²×C₇	1	1	yes	characteristic
361	#424	C₂²×C₇	1	2	yes	—
362	#423	C₂²×C₇	1	1	yes	characteristic
363	#419	C₂²×C₇	1	2	yes	—
364	#418	C₂²×C₇	1	2	yes	—
365	C₂×#103	C₂²×C₇	1	1	yes	characteristic
366	C₂×#102	C₂²×C₇	1	1	yes	characteristic
367	#405	C₂²×C₇	1	2	yes	—
368	C₂×#97	C₂²×C₇	1	1	yes	characteristic
369	C₂³×C₇	—	2	2	no	—
370	C₂³×C₇	—	2	4	no	—
371	C₂³×C₇	—	8	1	no	—
372	C₂³×C₇	—	8	1	no	—
373	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
374	C₇×Q₈	—	2	4	no	—
375	C₇×Q₈	—	2	2	no	—
376	C₇×Q₈	—	4	2	no	—
377	C₇×Q₈	—	4	2	no	—
378	C₇×Q₈	—	4	2	no	—
379	C₇×Q₈	—	4	2	no	—
380	C₇×Q₈	—	8	1	no	—
381	C₇×Q₈	—	8	4	no	—
382	C₇×Q₈	—	8	1	no	—
383	C₇×Q₈	—	8	4	no	—
384	C₇×Q₈	C₂²×Q₈	1	4	yes	—
385	C₇×D₄	—	2	4	no	—
386	C₇×D₄	—	2	2	no	—
387	C₇×D₄	C₂²×Q₈	1	4	yes	—
388	C₂×C₄×C₇	—	2	4	no	—
389	C₂×C₄×C₇	—	2	2	no	—
390	C₂×C₄×C₇	—	2	2	no	—
391	C₂×C₄×C₇	—	2	2	no	—
392	C₂×C₄×C₇	—	2	2	no	—
393	C₂×C₄×C₇	—	2	2	no	—
394	C₂×C₄×C₇	—	2	2	no	—
395	C₂×C₄×C₇	—	2	2	no	—
396	C₂×C₄×C₇	—	4	1	no	—
397	C₂×C₄×C₇	—	4	1	no	—
398	C₂×C₄×C₇	—	4	4	no	—
399	C₂×C₄×C₇	—	8	4	no	—
400	C₂×C₄×C₇	—	8	2	no	—
401	C₂×C₄×C₇	—	8	4	no	—
402	C₂×C₄×C₇	—	8	1	no	—
403	C₂×C₄×C₇	—	8	1	no	—
404	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
405	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
406	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
407	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
408	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
412	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
413	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
414	C₇×C₈	—	2	2	no	—
415	C₇×C₈	—	2	4	no	—
416	C₇×C₈	C₂²×Q₈	1	4	yes	—
417	C₂²×Q₈	—	2	2	no	—
418	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
419	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
420	C₂³×C₄	—	2	1	no	—
421	C₂×SD₁₆	C₇×Q₈	1	8	yes	complemented
422	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
423	C₂×M₁₆	C₂³×C₇	1	2	yes	—
424	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
425	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
426	C₄⋊Q₈	—	2	2	no	—
427	C₄⋊Q₈	—	2	4	no	—
428	C₄⋊Q₈	—	2	2	no	—
429	C₄⋊Q₈	—	2	4	no	—
430	C₄⋊Q₈	—	2	2	no	—
431	C₄⋊Q₈	—	2	2	no	—
432	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
433	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
434	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
435	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
436	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
437	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
438	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
439	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
440	#125	—	2	4	no	—
441	#125	—	2	4	no	—
442	#125	—	2	4	no	—
443	#125	—	2	2	no	—
444	#125	—	2	2	no	—
445	#125	C₇×D₄	1	2	yes	—
446	#125	C₂³×C₇	1	2	yes	—
447	#125	C₂³×C₇	1	2	yes	—
448	#125	C₂³×C₇	1	4	yes	—
449	#125	C₂³×C₇	1	4	yes	—
450	C₂²⋊Q₈	—	2	2	no	—
451	C₂²⋊Q₈	—	2	4	no	—
452	C₂²⋊Q₈	—	2	1	no	—
453	C₂²⋊Q₈	—	2	1	no	—
454	C₂²⋊Q₈	—	2	1	no	—
455	C₂²⋊Q₈	—	2	1	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	4	no	—
458	C₂²⋊Q₈	—	2	4	no	—
459	C₂²⋊Q₈	—	2	4	no	—
460	C₂²⋊Q₈	—	2	4	no	—
461	C₂²⋊Q₈	—	2	4	no	—
462	C₂²⋊Q₈	—	2	4	no	—
463	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
464	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
465	C₄×Q₈	—	2	2	no	—
466	C₄×Q₈	—	2	2	no	—
467	C₄×Q₈	—	2	2	no	—
468	C₄×Q₈	—	2	4	no	—
469	C₄×Q₈	—	2	2	no	—
470	C₄×Q₈	—	2	1	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	1	no	—
473	C₄×Q₈	—	2	1	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×Q₈	C₂³×C₇	1	2	yes	—
477	C₄×Q₈	C₂³×C₇	1	2	yes	—
478	C₄×Q₈	C₂³×C₇	1	4	yes	—
479	C₄×Q₈	C₂³×C₇	1	4	yes	—
480	C₄×Q₈	C₂³×C₇	1	2	yes	—
481	C₄×Q₈	C₂³×C₇	1	2	yes	—
482	C₄×D₄	—	2	2	no	—
483	C₄×D₄	—	2	2	no	—
484	C₄×D₄	—	2	1	no	—
485	C₄×D₄	—	2	1	no	—
486	C₄×D₄	C₂³×C₇	1	2	yes	—
487	C₄×D₄	C₂³×C₇	1	2	yes	—
488	C₄×D₄	C₂³×C₇	1	4	yes	—
489	C₄×D₄	C₂³×C₇	1	4	yes	—
490	#117	—	2	4	no	—
491	#117	C₂³×C₇	1	2	yes	—
492	#117	C₂³×C₇	1	2	yes	—
493	#117	C₂³×C₇	1	2	yes	—
494	C₂×C₄⋊C₄	—	2	1	no	—
495	C₂×C₄⋊C₄	—	2	4	no	—
496	C₂×C₄⋊C₄	—	2	2	no	—
497	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
498	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
499	C₂×C₂²⋊C₄	—	2	1	no	—
500	C₂×C₂²⋊C₄	—	2	4	no	—
501	C₂×C₄²	—	2	1	no	—
502	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
503	#107	C₂³×C₇	1	4	yes	—
504	#107	C₂³×C₇	1	2	yes	—
505	#107	C₂³×C₇	1	2	yes	—
506	#107	C₂³×C₇	1	2	yes	—
507	#107	C₂³×C₇	1	2	yes	—
508	#107	C₂³×C₇	1	4	yes	—
509	#107	C₂³×C₇	1	4	yes	—
510	#106	C₂³×C₇	1	2	yes	—
511	#106	C₂³×C₇	1	2	yes	—
512	#106	C₂³×C₇	1	2	yes	—
513	#106	C₂³×C₇	1	2	yes	—
514	#106	C₂³×C₇	1	4	yes	—
515	#106	C₂³×C₇	1	4	yes	—
516	C₄⋊C₈	C₂³×C₇	1	2	yes	—
517	C₄⋊C₈	C₂³×C₇	1	2	yes	—
518	C₄⋊C₈	C₂³×C₇	1	4	yes	—
519	C₄⋊C₈	C₂³×C₇	1	4	yes	—
520	#103	C₂³×C₇	1	4	yes	—
521	#103	C₂³×C₇	1	4	yes	—
522	#103	C₂³×C₇	1	4	yes	—
523	#102	C₂³×C₇	1	4	yes	—
524	#102	C₂³×C₇	1	4	yes	—
525	#102	C₂³×C₇	1	4	yes	—
526	#97	C₂³×C₇	1	2	yes	—
527	#97	C₂³×C₇	1	2	yes	—
528	#97	C₂³×C₇	1	4	yes	—
529	C₄×C₈	C₂³×C₇	1	4	yes	—
530	C₂²×C₇	—	2	1	no	—
531	C₂²×C₇	—	2	1	no	—
532	C₂²×C₇	—	2	4	no	—
533	C₂²×C₇	—	4	2	no	—
534	C₂²×C₇	—	4	2	no	—
535	C₂²×C₇	—	8	4	no	—
536	C₂²×C₇	D₄×Q₈	1	2	yes	—
537	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
538	C₄×C₇	—	2	1	no	—
539	C₄×C₇	—	2	1	no	—
540	C₄×C₇	—	2	4	no	—
541	C₄×C₇	—	2	1	no	—
542	C₄×C₇	—	2	1	no	—
543	C₄×C₇	—	2	4	no	—
544	C₄×C₇	—	2	4	no	—
545	C₄×C₇	—	4	2	no	—
546	C₄×C₇	—	4	2	no	—
547	C₄×C₇	—	4	2	no	—
548	C₄×C₇	—	4	2	no	—
549	C₄×C₇	—	4	2	no	—
550	C₄×C₇	—	4	2	no	—
551	C₄×C₇	D₄×Q₈	1	2	yes	—
552	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
553	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
554	C₂⁴	—	2	1	no	—
555	C₂×Q₈	—	2	1	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	1	no	—
559	C₂×Q₈	—	2	1	no	—
560	C₂×Q₈	—	2	4	no	—
561	C₂×Q₈	—	2	4	no	—
562	C₂×Q₈	—	2	2	no	—
563	C₂×Q₈	—	8	4	no	—
564	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
565	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
566	C₂×Q₈	C₂×C₇×Q₈	1	4	yes	complemented
567	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
568	C₂×D₄	—	2	1	no	—
569	C₂×D₄	—	2	2	no	—
570	C₂×D₄	C₂×C₇×Q₈	1	4	yes	—
571	C₂×D₄	C₂⁴×C₇	1	2	yes	—
572	C₂²×C₄	—	2	1	no	—
573	C₂²×C₄	—	2	1	no	—
574	C₂²×C₄	—	2	1	no	—
575	C₂²×C₄	—	2	2	no	—
576	C₂²×C₄	—	2	4	no	—
577	C₂²×C₄	—	2	4	no	—
578	C₂²×C₄	—	2	2	no	—
579	C₂²×C₄	—	8	2	no	—
580	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
581	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
582	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
583	SD₁₆	C₂×C₇×Q₈	1	16	yes	complemented
584	M₁₆	—	2	4	no	—
585	C₂×C₈	—	2	2	no	—
586	C₂×C₈	C₂×C₇×Q₈	1	4	yes	complemented
587	C₂×C₈	C₂⁴×C₇	1	4	yes	—
588	C₂×C₈	C₂⁴×C₇	1	2	yes	—
589	C₂×C₈	C₂⁴×C₇	1	2	yes	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	2	no	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	4	no	—
595	C₄⋊C₄	—	2	4	no	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	1	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	4	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	—	2	4	no	—
608	C₄⋊C₄	—	2	2	no	—
609	C₄⋊C₄	—	2	4	no	—
610	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
611	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
612	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
613	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
614	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
615	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
616	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
617	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
618	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
619	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
620	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
621	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
622	C₂²⋊C₄	—	2	2	no	—
623	C₂²⋊C₄	—	2	2	no	—
624	C₂²⋊C₄	—	2	4	no	—
625	C₂²⋊C₄	—	2	4	no	—
626	C₂²⋊C₄	—	2	2	no	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
629	C₄²	—	2	2	no	—
630	C₄²	—	2	2	no	—
631	C₄²	—	2	4	no	—
632	C₄²	—	2	4	no	—
633	C₄²	—	2	2	no	—
634	C₄²	—	2	2	no	—
635	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
636	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
637	C₄²	C₂⁴×C₇	1	4	yes	—
638	C₄²	C₂⁴×C₇	1	1	yes	characteristic
639	C₄²	C₂⁴×C₇	1	1	yes	characteristic
640	C₂×C₇	—	2	2	no	—
641	C₂×C₇	—	4	4	no	—
642	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
643	C₂×C₇	#3585	1	1	yes	characteristic
644	C₂×C₇	#3586	1	1	yes	characteristic
645	C₂³	—	2	2	no	—
646	C₂³	—	2	4	no	—
647	C₂³	—	8	1	no	—
648	C₂³	—	8	1	no	—
649	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
650	Q₈	—	2	2	no	—
651	Q₈	—	2	4	no	—
652	Q₈	—	4	2	no	—
653	Q₈	—	4	2	no	—
654	Q₈	—	4	2	no	—
655	Q₈	—	4	2	no	—
656	Q₈	—	8	1	no	—
657	Q₈	—	8	4	no	—
658	Q₈	—	8	1	no	—
659	Q₈	—	8	4	no	—
660	Q₈	C₂²×C₇×Q₈	1	4	yes	—
661	D₄	—	2	2	no	—
662	D₄	—	2	4	no	—
663	D₄	C₂²×C₇×Q₈	1	4	yes	—
664	C₂×C₄	—	2	2	no	—
665	C₂×C₄	—	2	4	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	4	no	—
675	C₂×C₄	—	8	4	no	—
676	C₂×C₄	—	8	2	no	—
677	C₂×C₄	—	8	4	no	—
678	C₂×C₄	—	8	1	no	—
679	C₂×C₄	—	8	1	no	—
680	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
681	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
682	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
683	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
684	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
685	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
686	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
687	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
688	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
689	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
690	C₈	—	2	2	no	—
691	C₈	—	2	4	no	—
692	C₈	C₂²×C₇×Q₈	1	4	yes	—
693	C₇	#61377	1	1	yes	characteristic, direct product factor, Sylow
694	C₂²	—	2	1	no	—
695	C₂²	—	2	1	no	—
696	C₂²	—	2	4	no	—
697	C₂²	—	4	2	no	—
698	C₂²	—	4	2	no	—
699	C₂²	—	8	4	no	—
700	C₂²	C₇×D₄×Q₈	1	2	yes	—
701	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
702	C₄	—	2	1	no	—
703	C₄	—	2	1	no	—
704	C₄	—	2	4	no	—
705	C₄	—	2	1	no	—
706	C₄	—	2	1	no	—
707	C₄	—	2	4	no	—
708	C₄	—	2	4	no	—
709	C₄	—	4	2	no	—
710	C₄	—	4	2	no	—
711	C₄	—	4	2	no	—
712	C₄	—	4	2	no	—
713	C₄	—	4	2	no	—
714	C₄	—	4	2	no	—
715	C₄	C₇×D₄×Q₈	1	2	yes	—
716	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
717	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
718	C₂	—	2	2	no	—
719	C₂	—	4	4	no	—
720	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
721	C₂	C₇×#3585	1	1	yes	characteristic
722	C₂	C₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3502	C₂	semidirect
3. C₇×#3388	C₂	semidirect
4. C₇×#3388	C₂	semidirect
5. C₇×#3386	C₂	semidirect
6. C₇×#3385	C₂	semidirect
7. C₇×#3164	C₂	semidirect
8. C₇×#3162	C₂	semidirect
9. C₇×#3082	C₂	semidirect
11. C₇×#3078	C₂	semidirect
13. C₇×#2961	C₂	semidirect
18. C₇×Q₈²	C₂²	semidirect
21. C₇×#557	C₂²	semidirect
22. C₇×#557	C₂²	semidirect
42. C₇×#501	C₂²	semidirect
43. C₇×#501	C₂²	semidirect
44. C₇×#501	C₂²	semidirect
45. C₇×#501	C₂²	semidirect
46. C₇×#501	C₂²	semidirect
47. C₇×#501	C₂²	semidirect
48. C₇×#500	C₂²	semidirect
49. C₇×#499	C₂²	semidirect
50. C₇×#479	C₂²	semidirect
51. C₇×#479	C₂²	semidirect
52. C₇×#479	C₂²	semidirect
56. C₇×#477	C₂²	semidirect
57. C₇×#477	C₂²	semidirect
60. C₇×#475	C₂²	semidirect
61. C₇×#475	C₂²	semidirect
64. C₇×#446	C₂²	semidirect
65. C₇×#446	C₂²	semidirect
85. #61377	C₇	direct (N × H)
90. C₂×C₇×SD₁₆	Q₈	semidirect
199. #3502	C₂×C₇	semidirect
201. #3388	C₂×C₇	semidirect
202. #3388	C₂×C₇	semidirect
203. #3386	C₂×C₇	semidirect
204. #3385	C₂×C₇	semidirect
205. #3164	C₂×C₇	semidirect
206. #3162	C₂×C₇	semidirect
207. #3082	C₂×C₇	semidirect
209. #3078	C₂×C₇	semidirect
211. #2961	C₂×C₇	semidirect
226. C₂×C₇×Q₈	C₂×Q₈	semidirect
243. C₇×SD₁₆	C₂×Q₈	semidirect
246. C₂×C₇×C₈	C₂×Q₈	semidirect
302. Q₈²	C₂²×C₇	semidirect
305. #557	C₂²×C₇	semidirect
306. #557	C₂²×C₇	semidirect
326. #501	C₂²×C₇	semidirect
327. #501	C₂²×C₇	semidirect
328. #501	C₂²×C₇	semidirect
329. #501	C₂²×C₇	semidirect
330. #501	C₂²×C₇	semidirect
331. #501	C₂²×C₇	semidirect
332. #500	C₂²×C₇	semidirect
333. #499	C₂²×C₇	semidirect
334. #479	C₂²×C₇	semidirect
335. #479	C₂²×C₇	semidirect
336. #479	C₂²×C₇	semidirect
340. #477	C₂²×C₇	semidirect
341. #477	C₂²×C₇	semidirect
344. #475	C₂²×C₇	semidirect
345. #475	C₂²×C₇	semidirect
348. #446	C₂²×C₇	semidirect
349. #446	C₂²×C₇	semidirect
421. C₂×SD₁₆	C₇×Q₈	semidirect
566. C₂×Q₈	C₂×C₇×Q₈	semidirect
583. SD₁₆	C₂×C₇×Q₈	semidirect
586. C₂×C₈	C₂×C₇×Q₈	semidirect
693. C₇	#61377	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×#3502	2
C₇×#3501	2
C₇×#3388	2
C₇×#3386	2
C₇×#3385	2
C₇×#3164	2
C₇×#3162	2
C₇×#3082	2
C₇×#3081	2
C₇×#3078	2
C₇×#3077	2
C₇×#2961	2
C₇×#2949	2
C₂×C₇×#440	2
#61377	7

Subgroups — C7×#61377 #13238593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61377 #13238593