Subgroups — C₇×#61342 #13238558

Order 1792.

1724 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3509	C₂	1	1	yes	maximal, characteristic
2	C₇×#3502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3377	C₂	1	4	yes	maximal, complemented
4	C₇×#3374	C₂	1	2	yes	maximal, complemented
5	C₇×#3374	C₂	1	4	yes	maximal, complemented
6	C₇×#3369	C₂	1	2	yes	maximal, complemented
7	C₇×#3364	C₂	1	2	yes	maximal, complemented
8	C₇×#3363	C₂	1	2	yes	maximal, complemented
9	C₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3137	C₂	1	2	yes	maximal
11	C₇×#3082	C₂	1	2	yes	maximal, complemented
12	C₇×#3076	C₂	1	1	yes	maximal, characteristic
13	C₇×#3044	C₂	1	2	yes	maximal
14	C₂×C₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#2962	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#2950	C₂	1	2	yes	maximal, complemented
17	C₄×C₇×#137	C₂	1	1	yes	maximal, characteristic
18	C₂×C₇×#137	C₂²	1	1	yes	characteristic
19	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
20	C₇×Q₈²	—	2	1	no	—
21	C₇×Q₈²	C₂²	1	2	yes	—
22	C₇×#557	—	2	1	no	—
23	C₇×#557	—	2	2	no	—
24	C₇×#557	C₂²	1	2	yes	—
25	C₇×#557	C₂²	1	2	yes	—
26	C₇×#557	C₂²	1	2	yes	—
27	C₇×#557	C₂²	1	4	yes	—
28	C₇×#556	C₂²	1	2	yes	—
29	C₇×#556	C₂²	1	2	yes	—
30	C₇×#552	—	2	2	no	—
31	C₇×#552	—	2	2	no	—
32	C₇×#544	—	2	2	no	—
33	C₇×#544	C₂²	1	2	yes	—
34	C₇×#541	—	2	2	no	—
35	C₇×#536	C₂²	1	2	yes	—
36	C₇×#533	—	2	1	no	—
37	C₇×#533	—	2	1	no	—
38	C₇×#533	—	2	2	no	—
39	C₇×#533	C₂²	1	1	yes	characteristic
40	C₇×#533	C₂²	1	1	yes	characteristic
41	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
42	C₇×#519	C₂²	1	2	yes	—
43	C₇×#519	C₂²	1	2	yes	—
44	C₄×C₇×#41	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
46	C₇×#491	C₂²	1	4	yes	—
47	C₇×#491	C₂²	1	4	yes	—
48	C₇×#489	C₂²	1	4	yes	—
49	C₇×#488	C₂²	1	4	yes	—
50	C₇×#487	C₂²	1	2	yes	—
51	C₇×#487	C₂²	1	4	yes	—
52	C₇×#487	C₂²	1	2	yes	—
53	C₇×#479	C₂²	1	2	yes	—
54	C₇×#479	C₂²	1	4	yes	—
55	C₇×#479	C₂²	1	4	yes	—
56	C₇×#478	C₂²	1	2	yes	—
57	C₇×#477	C₂²	1	1	yes	characteristic
58	C₇×#477	C₂²	1	4	yes	—
59	C₇×#476	C₂²	1	1	yes	characteristic
60	C₇×#475	C₂²	1	1	yes	characteristic
61	C₇×#475	C₂²	1	4	yes	—
62	C₇×#474	C₂²	1	1	yes	characteristic
63	C₇×#464	C₂²	1	4	yes	—
64	C₇×#464	C₂²	1	4	yes	—
65	C₇×#462	C₂²	1	2	yes	—
66	C₇×#462	C₂²	1	2	yes	—
67	C₇×#462	C₂²	1	4	yes	—
68	C₇×#462	C₂²	1	4	yes	—
69	C₇×#462	C₂²	1	4	yes	—
70	C₇×#446	C₂²	1	2	yes	—
71	C₇×#446	C₂²	1	4	yes	—
72	C₇×C₈×Q₈	C₂²	1	2	yes	—
73	C₇×#441	C₂²	1	2	yes	—
74	C₇×#441	C₂²	1	4	yes	—
75	C₇×#441	C₂²	1	4	yes	—
76	C₇×#440	C₂²	1	2	yes	—
77	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
78	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
79	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
80	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
81	C₇×#428	C₂²	1	1	yes	characteristic
82	C₇×#427	C₂²	1	2	yes	—
83	C₇×#424	C₂²	1	2	yes	—
84	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
85	C₇×#419	C₂²	1	1	yes	characteristic
86	C₇×#419	C₂²	1	2	yes	—
87	C₇×#419	C₂²	1	2	yes	—
88	C₇×#417	C₂²	1	1	yes	characteristic
89	C₇×#416	C₂²	1	2	yes	—
90	C₂×C₇×#103	C₂²	1	2	yes	—
91	C₇×#405	C₂²	1	2	yes	—
92	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
93	#61342	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₇×#41	C₂³	1	1	yes	characteristic
95	C₂²×C₇×Q₈	C₂³	1	2	yes	—
96	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
97	C₇×#137	—	2	4	no	—
98	C₂×C₇×Q₁₆	—	2	8	no	—
99	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
100	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
101	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
102	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
103	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
104	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
105	C₂²×C₇×C₈	C₂³	1	2	yes	—
106	C₇×C₄⋊Q₈	—	2	2	no	—
107	C₇×C₄⋊Q₈	—	2	4	no	—
108	C₇×C₄⋊Q₈	—	2	2	no	—
109	C₇×C₄⋊Q₈	—	2	4	no	—
110	C₇×C₄⋊Q₈	—	2	2	no	—
111	C₇×C₄⋊Q₈	—	2	1	no	—
112	C₇×C₄⋊Q₈	—	2	1	no	—
113	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
114	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
115	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
116	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
117	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
118	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
119	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
120	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
121	C₇×#126	—	2	4	no	—
122	C₇×#126	—	2	4	no	—
123	C₇×#125	—	2	2	no	—
124	C₇×#125	—	2	2	no	—
125	C₇×#125	—	2	4	no	—
126	C₇×#125	—	2	4	no	—
127	C₇×#125	—	2	4	no	—
128	C₇×#125	—	2	4	no	—
129	C₇×#125	—	2	4	no	—
130	C₇×#125	—	2	2	no	—
131	C₇×#125	D₄	1	2	yes	—
132	C₇×#125	C₂³	1	2	yes	—
133	C₇×#125	C₂³	1	4	yes	—
134	C₇×#124	C₂³	1	4	yes	—
135	C₇×#123	—	2	4	no	—
136	C₇×C₂²⋊Q₈	—	2	2	no	—
137	C₇×C₂²⋊Q₈	—	2	2	no	—
138	C₇×C₂²⋊Q₈	—	2	1	no	—
139	C₇×C₂²⋊Q₈	—	2	1	no	—
140	C₇×C₂²⋊Q₈	—	2	2	no	—
141	C₇×C₂²⋊Q₈	—	2	2	no	—
142	C₇×C₂²⋊Q₈	—	2	4	no	—
143	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
144	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
145	C₄×C₇×Q₈	—	2	2	no	—
146	C₄×C₇×Q₈	—	2	2	no	—
147	C₄×C₇×Q₈	—	2	2	no	—
148	C₄×C₇×Q₈	—	2	2	no	—
149	C₄×C₇×Q₈	—	2	2	no	—
150	C₄×C₇×Q₈	—	2	2	no	—
151	C₄×C₇×Q₈	—	2	2	no	—
152	C₄×C₇×Q₈	—	2	4	no	—
153	C₄×C₇×Q₈	—	2	2	no	—
154	C₄×C₇×Q₈	—	2	1	no	—
155	C₄×C₇×Q₈	—	2	1	no	—
156	C₄×C₇×Q₈	—	2	1	no	—
157	C₄×C₇×Q₈	—	2	1	no	—
158	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
159	C₄×C₇×Q₈	C₂³	1	4	yes	—
160	C₄×C₇×Q₈	C₂³	1	4	yes	—
161	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
162	C₄×C₇×Q₈	C₂³	1	2	yes	—
163	C₄×C₇×Q₈	C₂³	1	2	yes	—
164	C₄×C₇×Q₈	C₂³	1	4	yes	—
165	C₄×C₇×Q₈	C₂³	1	4	yes	—
166	C₄×C₇×Q₈	C₂³	1	2	yes	—
167	C₄×C₇×Q₈	C₂³	1	2	yes	—
168	C₄×C₇×D₄	—	2	1	no	—
169	C₄×C₇×D₄	—	2	1	no	—
170	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
171	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
172	C₇×#117	—	2	2	no	—
173	C₇×#117	—	2	2	no	—
174	C₇×#117	—	2	4	no	—
175	C₇×#117	—	2	4	no	—
176	C₇×#117	—	2	1	no	—
177	C₇×#117	C₂³	1	1	yes	characteristic
178	C₇×#117	C₂³	1	2	yes	—
179	C₇×#117	C₂³	1	2	yes	—
180	C₂×C₇×C₄⋊C₄	—	2	1	no	—
181	C₂×C₇×C₄⋊C₄	—	2	1	no	—
182	C₂×C₇×C₄⋊C₄	—	2	1	no	—
183	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
184	C₂×C₄²×C₇	—	2	1	no	—
185	C₂×C₄²×C₇	—	2	1	no	—
186	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
187	C₇×#107	C₂³	1	2	yes	—
188	C₇×#107	C₂³	1	2	yes	—
189	C₇×#107	C₂³	1	2	yes	—
190	C₇×#106	C₂³	1	2	yes	—
191	C₇×#106	C₂³	1	2	yes	—
192	C₇×#106	C₂³	1	2	yes	—
193	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
194	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
195	C₇×C₄⋊C₈	C₂³	1	4	yes	—
196	C₇×C₄⋊C₈	C₂³	1	4	yes	—
197	C₇×C₄⋊C₈	C₂³	1	2	yes	—
198	C₇×#103	C₂³	1	4	yes	—
199	C₇×#103	C₂³	1	4	yes	—
200	C₇×#103	C₂³	1	2	yes	—
201	C₇×#103	C₂³	1	2	yes	—
202	C₇×#103	C₂³	1	4	yes	—
203	C₇×#103	C₂³	1	4	yes	—
204	C₇×#103	C₂³	1	4	yes	—
205	C₇×#103	C₂³	1	2	yes	—
206	C₇×#103	C₂³	1	4	yes	—
207	C₇×#103	C₂³	1	2	yes	—
208	C₇×#103	C₂³	1	2	yes	—
209	C₇×#102	C₂³	1	2	yes	—
210	C₇×#102	C₂³	1	2	yes	—
211	C₇×#102	C₂³	1	2	yes	—
212	C₇×#97	C₂³	1	4	yes	—
213	C₇×#97	C₂³	1	2	yes	—
214	C₄×C₇×C₈	C₂³	1	2	yes	—
215	C₄×C₇×C₈	C₂³	1	4	yes	—
216	#3509	C₂×C₇	1	1	yes	characteristic
217	#3502	C₂×C₇	1	1	yes	characteristic, complemented
218	#3377	C₂×C₇	1	4	yes	complemented
219	#3374	C₂×C₇	1	2	yes	complemented
220	#3374	C₂×C₇	1	4	yes	complemented
221	#3369	C₂×C₇	1	2	yes	complemented
222	#3364	C₂×C₇	1	2	yes	complemented
223	#3363	C₂×C₇	1	2	yes	complemented
224	#3139	C₂×C₇	1	1	yes	characteristic, complemented
225	#3137	C₂×C₇	1	2	yes	—
226	#3082	C₂×C₇	1	2	yes	complemented
227	#3076	C₂×C₇	1	1	yes	characteristic
228	#3044	C₂×C₇	1	2	yes	—
229	C₂×#462	C₂×C₇	1	1	yes	characteristic, complemented
230	#2962	C₂×C₇	1	1	yes	characteristic, complemented
231	#2950	C₂×C₇	1	2	yes	complemented
232	C₄×#137	C₂×C₇	1	1	yes	characteristic
233	C₇×#41	—	2	2	no	—
234	C₇×#41	—	2	2	no	—
235	C₂×C₇×Q₈	—	2	1	no	—
236	C₂×C₇×Q₈	—	2	1	no	—
237	C₂×C₇×Q₈	—	2	2	no	—
238	C₂×C₇×Q₈	—	2	1	no	—
239	C₂×C₇×Q₈	—	2	4	no	—
240	C₂×C₇×Q₈	—	2	2	no	—
241	C₂×C₇×Q₈	—	2	4	no	—
242	C₂×C₇×Q₈	—	2	4	no	—
243	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
244	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
245	C₂×C₇×Q₈	#41	1	4	yes	—
246	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
247	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
248	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
249	C₂×C₇×D₄	—	2	1	no	—
250	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
251	C₂²×C₄×C₇	—	2	1	no	—
252	C₂²×C₄×C₇	—	2	2	no	—
253	C₂²×C₄×C₇	—	2	1	no	—
254	C₂²×C₄×C₇	—	2	1	no	—
255	C₂²×C₄×C₇	—	2	1	no	—
256	C₂²×C₄×C₇	—	2	1	no	—
257	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
258	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
259	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
260	C₇×Q₁₆	—	2	8	no	—
261	C₇×Q₁₆	—	2	8	no	—
262	C₇×Q₁₆	—	2	4	no	—
263	C₇×SD₁₆	—	2	4	no	—
264	C₇×M₁₆	—	2	2	no	—
265	C₇×M₁₆	—	2	2	no	—
266	C₂×C₇×C₈	—	2	4	no	—
267	C₂×C₇×C₈	C₂⁴	1	2	yes	—
268	C₂×C₇×C₈	C₂⁴	1	2	yes	—
269	C₂×C₇×C₈	C₂⁴	1	2	yes	—
270	C₂×C₇×C₈	C₂⁴	1	2	yes	—
271	C₇×C₄⋊C₄	—	2	2	no	—
272	C₇×C₄⋊C₄	—	2	2	no	—
273	C₇×C₄⋊C₄	—	2	2	no	—
274	C₇×C₄⋊C₄	—	2	2	no	—
275	C₇×C₄⋊C₄	—	2	2	no	—
276	C₇×C₄⋊C₄	—	2	2	no	—
277	C₇×C₄⋊C₄	—	2	1	no	—
278	C₇×C₄⋊C₄	—	2	1	no	—
279	C₇×C₄⋊C₄	—	2	1	no	—
280	C₇×C₄⋊C₄	—	2	4	no	—
281	C₇×C₄⋊C₄	—	2	4	no	—
282	C₇×C₄⋊C₄	—	2	4	no	—
283	C₇×C₄⋊C₄	—	2	4	no	—
284	C₇×C₄⋊C₄	—	2	2	no	—
285	C₇×C₄⋊C₄	—	2	1	no	—
286	C₇×C₄⋊C₄	—	2	4	no	—
287	C₇×C₄⋊C₄	—	2	4	no	—
288	C₇×C₄⋊C₄	—	2	1	no	—
289	C₇×C₄⋊C₄	—	2	4	no	—
290	C₇×C₄⋊C₄	—	2	4	no	—
291	C₇×C₄⋊C₄	—	2	2	no	—
292	C₇×C₄⋊C₄	—	2	4	no	—
293	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
294	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
295	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
296	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
297	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
298	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
299	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
300	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
301	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
302	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
303	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
304	C₇×C₂²⋊C₄	—	2	1	no	—
305	C₇×C₂²⋊C₄	—	2	2	no	—
306	C₇×C₂²⋊C₄	—	2	1	no	—
307	C₇×C₂²⋊C₄	—	2	1	no	—
308	C₇×C₂²⋊C₄	—	2	4	no	—
309	C₇×C₂²⋊C₄	—	2	4	no	—
310	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
311	C₄²×C₇	—	2	1	no	—
312	C₄²×C₇	—	2	1	no	—
313	C₄²×C₇	—	2	1	no	—
314	C₄²×C₇	—	2	2	no	—
315	C₄²×C₇	—	2	2	no	—
316	C₄²×C₇	—	2	4	no	—
317	C₄²×C₇	—	2	4	no	—
318	C₄²×C₇	—	2	4	no	—
319	C₄²×C₇	—	2	4	no	—
320	C₄²×C₇	—	2	2	no	—
321	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
322	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
323	C₄²×C₇	C₂⁴	1	1	yes	characteristic
324	C₄²×C₇	C₂⁴	1	4	yes	—
325	C₄²×C₇	C₂⁴	1	1	yes	characteristic
326	C₂×#137	C₂²×C₇	1	1	yes	characteristic
327	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
328	Q₈²	—	2	1	no	—
329	Q₈²	C₂²×C₇	1	2	yes	—
330	#557	—	2	1	no	—
331	#557	—	2	2	no	—
332	#557	C₂²×C₇	1	2	yes	—
333	#557	C₂²×C₇	1	2	yes	—
334	#557	C₂²×C₇	1	2	yes	—
335	#557	C₂²×C₇	1	4	yes	—
336	#556	C₂²×C₇	1	2	yes	—
337	#556	C₂²×C₇	1	2	yes	—
338	#552	—	2	2	no	—
339	#552	—	2	2	no	—
340	#544	—	2	2	no	—
341	#544	C₂²×C₇	1	2	yes	—
342	#541	—	2	2	no	—
343	#536	C₂²×C₇	1	2	yes	—
344	#533	—	2	1	no	—
345	#533	—	2	1	no	—
346	#533	—	2	2	no	—
347	#533	C₂²×C₇	1	1	yes	characteristic
348	#533	C₂²×C₇	1	1	yes	characteristic
349	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
350	#519	C₂²×C₇	1	2	yes	—
351	#519	C₂²×C₇	1	2	yes	—
352	C₄×#41	C₂²×C₇	1	1	yes	characteristic
353	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
354	#491	C₂²×C₇	1	4	yes	—
355	#491	C₂²×C₇	1	4	yes	—
356	#489	C₂²×C₇	1	4	yes	—
357	#488	C₂²×C₇	1	4	yes	—
358	#487	C₂²×C₇	1	2	yes	—
359	#487	C₂²×C₇	1	4	yes	—
360	#487	C₂²×C₇	1	2	yes	—
361	#479	C₂²×C₇	1	2	yes	—
362	#479	C₂²×C₇	1	4	yes	—
363	#479	C₂²×C₇	1	4	yes	—
364	#478	C₂²×C₇	1	2	yes	—
365	#477	C₂²×C₇	1	1	yes	characteristic
366	#477	C₂²×C₇	1	4	yes	—
367	#476	C₂²×C₇	1	1	yes	characteristic
368	#475	C₂²×C₇	1	1	yes	characteristic
369	#475	C₂²×C₇	1	4	yes	—
370	#474	C₂²×C₇	1	1	yes	characteristic
371	#464	C₂²×C₇	1	4	yes	—
372	#464	C₂²×C₇	1	4	yes	—
373	#462	C₂²×C₇	1	2	yes	—
374	#462	C₂²×C₇	1	4	yes	—
375	#462	C₂²×C₇	1	2	yes	—
376	#462	C₂²×C₇	1	4	yes	—
377	#462	C₂²×C₇	1	4	yes	—
378	#446	C₂²×C₇	1	2	yes	—
379	#446	C₂²×C₇	1	4	yes	—
380	C₈×Q₈	C₂²×C₇	1	2	yes	—
381	#441	C₂²×C₇	1	4	yes	—
382	#441	C₂²×C₇	1	2	yes	—
383	#441	C₂²×C₇	1	4	yes	—
384	#440	C₂²×C₇	1	2	yes	—
385	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
386	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
387	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
388	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
389	#428	C₂²×C₇	1	1	yes	characteristic
390	#427	C₂²×C₇	1	2	yes	—
391	#424	C₂²×C₇	1	2	yes	—
392	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
393	#419	C₂²×C₇	1	2	yes	—
394	#419	C₂²×C₇	1	2	yes	—
395	#419	C₂²×C₇	1	1	yes	characteristic
396	#417	C₂²×C₇	1	1	yes	characteristic
397	#416	C₂²×C₇	1	2	yes	—
398	C₂×#103	C₂²×C₇	1	2	yes	—
399	#405	C₂²×C₇	1	2	yes	—
400	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
401	C₂³×C₇	—	2	1	no	—
402	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
403	C₇×Q₈	—	2	4	no	—
404	C₇×Q₈	—	2	4	no	—
405	C₇×Q₈	—	2	2	no	—
406	C₇×Q₈	—	2	4	no	—
407	C₇×Q₈	—	2	1	no	—
408	C₇×Q₈	—	2	4	no	—
409	C₇×Q₈	—	2	1	no	—
410	C₇×Q₈	—	4	2	no	—
411	C₇×Q₈	—	4	4	no	—
412	C₇×Q₈	—	4	2	no	—
413	C₇×Q₈	—	8	1	no	—
414	C₇×Q₈	—	8	4	no	—
415	C₇×Q₈	C₂×#41	1	4	yes	—
416	C₇×D₄	—	2	1	no	—
417	C₇×D₄	—	2	1	no	—
418	C₇×D₄	—	4	2	no	—
419	C₂×C₄×C₇	—	2	4	no	—
420	C₂×C₄×C₇	—	2	2	no	—
421	C₂×C₄×C₇	—	2	2	no	—
422	C₂×C₄×C₇	—	2	1	no	—
423	C₂×C₄×C₇	—	2	1	no	—
424	C₂×C₄×C₇	—	2	2	no	—
425	C₂×C₄×C₇	—	2	2	no	—
426	C₂×C₄×C₇	—	2	4	no	—
427	C₂×C₄×C₇	—	2	1	no	—
428	C₂×C₄×C₇	—	2	2	no	—
429	C₂×C₄×C₇	—	4	1	no	—
430	C₂×C₄×C₇	—	4	2	no	—
431	C₂×C₄×C₇	—	4	4	no	—
432	C₂×C₄×C₇	—	4	2	no	—
433	C₂×C₄×C₇	—	4	2	no	—
434	C₂×C₄×C₇	—	8	1	no	—
435	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
436	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
437	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
438	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
439	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
440	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
441	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
442	C₂×C₄×C₇	C₂×#41	1	2	yes	—
443	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
444	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
445	C₇×C₈	—	2	2	no	—
446	C₇×C₈	—	2	2	no	—
447	C₇×C₈	—	2	2	no	—
448	C₇×C₈	—	2	2	no	—
449	C₂×#41	C₂³×C₇	1	1	yes	characteristic
450	C₂²×Q₈	C₂³×C₇	1	2	yes	—
451	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
452	#137	—	2	4	no	—
453	C₂×Q₁₆	—	2	8	no	—
454	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
455	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
456	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
457	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
458	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
459	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
460	C₂²×C₈	C₂³×C₇	1	2	yes	—
461	C₄⋊Q₈	—	2	2	no	—
462	C₄⋊Q₈	—	2	4	no	—
463	C₄⋊Q₈	—	2	2	no	—
464	C₄⋊Q₈	—	2	4	no	—
465	C₄⋊Q₈	—	2	2	no	—
466	C₄⋊Q₈	—	2	1	no	—
467	C₄⋊Q₈	—	2	1	no	—
468	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
469	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
470	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
471	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
472	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
473	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
474	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
475	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
476	#126	—	2	4	no	—
477	#126	—	2	4	no	—
478	#125	—	2	2	no	—
479	#125	—	2	2	no	—
480	#125	—	2	4	no	—
481	#125	—	2	4	no	—
482	#125	—	2	4	no	—
483	#125	—	2	4	no	—
484	#125	—	2	4	no	—
485	#125	—	2	2	no	—
486	#125	C₇×D₄	1	2	yes	—
487	#125	C₂³×C₇	1	2	yes	—
488	#125	C₂³×C₇	1	4	yes	—
489	#124	C₂³×C₇	1	4	yes	—
490	#123	—	2	4	no	—
491	C₂²⋊Q₈	—	2	2	no	—
492	C₂²⋊Q₈	—	2	2	no	—
493	C₂²⋊Q₈	—	2	1	no	—
494	C₂²⋊Q₈	—	2	1	no	—
495	C₂²⋊Q₈	—	2	2	no	—
496	C₂²⋊Q₈	—	2	2	no	—
497	C₂²⋊Q₈	—	2	4	no	—
498	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
499	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
500	C₄×Q₈	—	2	2	no	—
501	C₄×Q₈	—	2	2	no	—
502	C₄×Q₈	—	2	2	no	—
503	C₄×Q₈	—	2	2	no	—
504	C₄×Q₈	—	2	2	no	—
505	C₄×Q₈	—	2	2	no	—
506	C₄×Q₈	—	2	2	no	—
507	C₄×Q₈	—	2	4	no	—
508	C₄×Q₈	—	2	2	no	—
509	C₄×Q₈	—	2	1	no	—
510	C₄×Q₈	—	2	1	no	—
511	C₄×Q₈	—	2	1	no	—
512	C₄×Q₈	—	2	1	no	—
513	C₄×Q₈	C₂³×C₇	1	2	yes	—
514	C₄×Q₈	C₂³×C₇	1	2	yes	—
515	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
516	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
517	C₄×Q₈	C₂³×C₇	1	4	yes	—
518	C₄×Q₈	C₂³×C₇	1	4	yes	—
519	C₄×Q₈	C₂³×C₇	1	4	yes	—
520	C₄×Q₈	C₂³×C₇	1	4	yes	—
521	C₄×Q₈	C₂³×C₇	1	2	yes	—
522	C₄×Q₈	C₂³×C₇	1	2	yes	—
523	C₄×D₄	—	2	1	no	—
524	C₄×D₄	—	2	1	no	—
525	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
526	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
527	#117	—	2	2	no	—
528	#117	—	2	2	no	—
529	#117	—	2	4	no	—
530	#117	—	2	4	no	—
531	#117	—	2	1	no	—
532	#117	C₂³×C₇	1	1	yes	characteristic
533	#117	C₂³×C₇	1	2	yes	—
534	#117	C₂³×C₇	1	2	yes	—
535	C₂×C₄⋊C₄	—	2	1	no	—
536	C₂×C₄⋊C₄	—	2	1	no	—
537	C₂×C₄⋊C₄	—	2	1	no	—
538	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
539	C₂×C₄²	—	2	1	no	—
540	C₂×C₄²	—	2	1	no	—
541	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
542	#107	C₂³×C₇	1	2	yes	—
543	#107	C₂³×C₇	1	2	yes	—
544	#107	C₂³×C₇	1	2	yes	—
545	#106	C₂³×C₇	1	2	yes	—
546	#106	C₂³×C₇	1	2	yes	—
547	#106	C₂³×C₇	1	2	yes	—
548	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
549	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
550	C₄⋊C₈	C₂³×C₇	1	2	yes	—
551	C₄⋊C₈	C₂³×C₇	1	4	yes	—
552	C₄⋊C₈	C₂³×C₇	1	4	yes	—
553	#103	C₂³×C₇	1	4	yes	—
554	#103	C₂³×C₇	1	4	yes	—
555	#103	C₂³×C₇	1	2	yes	—
556	#103	C₂³×C₇	1	2	yes	—
557	#103	C₂³×C₇	1	2	yes	—
558	#103	C₂³×C₇	1	2	yes	—
559	#103	C₂³×C₇	1	2	yes	—
560	#103	C₂³×C₇	1	4	yes	—
561	#103	C₂³×C₇	1	4	yes	—
562	#103	C₂³×C₇	1	4	yes	—
563	#103	C₂³×C₇	1	4	yes	—
564	#102	C₂³×C₇	1	2	yes	—
565	#102	C₂³×C₇	1	2	yes	—
566	#102	C₂³×C₇	1	2	yes	—
567	#97	C₂³×C₇	1	2	yes	—
568	#97	C₂³×C₇	1	4	yes	—
569	C₄×C₈	C₂³×C₇	1	2	yes	—
570	C₄×C₈	C₂³×C₇	1	4	yes	—
571	C₂²×C₇	—	2	1	no	—
572	C₂²×C₇	—	2	1	no	—
573	C₂²×C₇	—	4	1	no	—
574	C₂²×C₇	—	4	1	no	—
575	C₂²×C₇	—	4	1	no	—
576	C₂²×C₇	#548	1	2	yes	—
577	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
578	C₄×C₇	—	2	1	no	—
579	C₄×C₇	—	2	1	no	—
580	C₄×C₇	—	2	4	no	—
581	C₄×C₇	—	2	4	no	—
582	C₄×C₇	—	2	1	no	—
583	C₄×C₇	—	2	4	no	—
584	C₄×C₇	—	4	1	no	—
585	C₄×C₇	—	4	2	no	—
586	C₄×C₇	—	4	2	no	—
587	C₄×C₇	—	4	1	no	—
588	C₄×C₇	—	4	4	no	—
589	C₄×C₇	—	4	1	no	—
590	C₄×C₇	—	4	2	no	—
591	C₄×C₇	#548	1	2	yes	—
592	C₄×C₇	#578	1	2	yes	—
593	C₄×C₇	C₂²×#41	1	1	yes	characteristic
594	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
595	#41	—	2	2	no	—
596	#41	—	2	2	no	—
597	C₂×Q₈	—	2	1	no	—
598	C₂×Q₈	—	2	1	no	—
599	C₂×Q₈	—	2	2	no	—
600	C₂×Q₈	—	2	1	no	—
601	C₂×Q₈	—	2	4	no	—
602	C₂×Q₈	—	2	2	no	—
603	C₂×Q₈	—	2	4	no	—
604	C₂×Q₈	—	2	4	no	—
605	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
606	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
607	C₂×Q₈	C₇×#41	1	4	yes	—
608	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
609	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
610	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
611	C₂×D₄	—	2	1	no	—
612	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
613	C₂²×C₄	—	2	1	no	—
614	C₂²×C₄	—	2	2	no	—
615	C₂²×C₄	—	2	1	no	—
616	C₂²×C₄	—	2	1	no	—
617	C₂²×C₄	—	2	1	no	—
618	C₂²×C₄	—	2	1	no	—
619	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
620	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
621	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
622	Q₁₆	—	2	8	no	—
623	Q₁₆	—	2	8	no	—
624	Q₁₆	—	2	4	no	—
625	SD₁₆	—	2	4	no	—
626	M₁₆	—	2	2	no	—
627	M₁₆	—	2	2	no	—
628	C₂×C₈	—	2	4	no	—
629	C₂×C₈	C₂⁴×C₇	1	2	yes	—
630	C₂×C₈	C₂⁴×C₇	1	2	yes	—
631	C₂×C₈	C₂⁴×C₇	1	2	yes	—
632	C₂×C₈	C₂⁴×C₇	1	2	yes	—
633	C₄⋊C₄	—	2	2	no	—
634	C₄⋊C₄	—	2	2	no	—
635	C₄⋊C₄	—	2	2	no	—
636	C₄⋊C₄	—	2	2	no	—
637	C₄⋊C₄	—	2	2	no	—
638	C₄⋊C₄	—	2	2	no	—
639	C₄⋊C₄	—	2	1	no	—
640	C₄⋊C₄	—	2	1	no	—
641	C₄⋊C₄	—	2	1	no	—
642	C₄⋊C₄	—	2	4	no	—
643	C₄⋊C₄	—	2	4	no	—
644	C₄⋊C₄	—	2	4	no	—
645	C₄⋊C₄	—	2	4	no	—
646	C₄⋊C₄	—	2	2	no	—
647	C₄⋊C₄	—	2	1	no	—
648	C₄⋊C₄	—	2	4	no	—
649	C₄⋊C₄	—	2	4	no	—
650	C₄⋊C₄	—	2	1	no	—
651	C₄⋊C₄	—	2	4	no	—
652	C₄⋊C₄	—	2	4	no	—
653	C₄⋊C₄	—	2	2	no	—
654	C₄⋊C₄	—	2	4	no	—
655	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
656	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
657	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
658	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
659	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
660	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
661	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
662	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
663	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
664	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
665	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
666	C₂²⋊C₄	—	2	1	no	—
667	C₂²⋊C₄	—	2	2	no	—
668	C₂²⋊C₄	—	2	1	no	—
669	C₂²⋊C₄	—	2	1	no	—
670	C₂²⋊C₄	—	2	4	no	—
671	C₂²⋊C₄	—	2	4	no	—
672	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
673	C₄²	—	2	1	no	—
674	C₄²	—	2	1	no	—
675	C₄²	—	2	1	no	—
676	C₄²	—	2	2	no	—
677	C₄²	—	2	2	no	—
678	C₄²	—	2	4	no	—
679	C₄²	—	2	4	no	—
680	C₄²	—	2	4	no	—
681	C₄²	—	2	4	no	—
682	C₄²	—	2	2	no	—
683	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
684	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
685	C₄²	C₂⁴×C₇	1	1	yes	characteristic
686	C₄²	C₂⁴×C₇	1	4	yes	—
687	C₄²	C₂⁴×C₇	1	1	yes	characteristic
688	C₂×C₇	—	2	2	no	—
689	C₂×C₇	—	8	1	no	—
690	C₂×C₇	C₂×#548	1	1	yes	characteristic
691	C₂×C₇	C₂×#578	1	1	yes	characteristic
692	C₂×C₇	#3586	1	1	yes	characteristic
693	C₂³	—	2	1	no	—
694	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
695	Q₈	—	2	2	no	—
696	Q₈	—	2	4	no	—
697	Q₈	—	2	4	no	—
698	Q₈	—	2	4	no	—
699	Q₈	—	2	1	no	—
700	Q₈	—	2	4	no	—
701	Q₈	—	2	1	no	—
702	Q₈	—	4	2	no	—
703	Q₈	—	4	4	no	—
704	Q₈	—	4	2	no	—
705	Q₈	—	8	1	no	—
706	Q₈	—	8	4	no	—
707	Q₈	C₂×C₇×#41	1	4	yes	—
708	D₄	—	2	1	no	—
709	D₄	—	2	1	no	—
710	D₄	—	4	2	no	—
711	C₂×C₄	—	2	2	no	—
712	C₂×C₄	—	2	4	no	—
713	C₂×C₄	—	2	2	no	—
714	C₂×C₄	—	2	1	no	—
715	C₂×C₄	—	2	1	no	—
716	C₂×C₄	—	2	2	no	—
717	C₂×C₄	—	2	2	no	—
718	C₂×C₄	—	2	4	no	—
719	C₂×C₄	—	2	1	no	—
720	C₂×C₄	—	2	2	no	—
721	C₂×C₄	—	4	1	no	—
722	C₂×C₄	—	4	2	no	—
723	C₂×C₄	—	4	4	no	—
724	C₂×C₄	—	4	2	no	—
725	C₂×C₄	—	4	2	no	—
726	C₂×C₄	—	8	1	no	—
727	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
728	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
729	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
730	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
731	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
732	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
733	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
734	C₂×C₄	C₂×C₇×#41	1	2	yes	—
735	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
736	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
737	C₈	—	2	2	no	—
738	C₈	—	2	2	no	—
739	C₈	—	2	2	no	—
740	C₈	—	2	2	no	—
741	C₇	#61342	1	1	yes	characteristic, direct product factor, Sylow
742	C₂²	—	2	1	no	—
743	C₂²	—	2	1	no	—
744	C₂²	—	4	1	no	—
745	C₂²	—	4	1	no	—
746	C₂²	—	4	1	no	—
747	C₂²	C₇×#548	1	2	yes	—
748	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
749	C₄	—	2	1	no	—
750	C₄	—	2	4	no	—
751	C₄	—	2	1	no	—
752	C₄	—	2	1	no	—
753	C₄	—	2	4	no	—
754	C₄	—	2	4	no	—
755	C₄	—	4	1	no	—
756	C₄	—	4	2	no	—
757	C₄	—	4	1	no	—
758	C₄	—	4	2	no	—
759	C₄	—	4	2	no	—
760	C₄	—	4	1	no	—
761	C₄	—	4	4	no	—
762	C₄	C₇×#548	1	2	yes	—
763	C₄	C₇×#578	1	2	yes	—
764	C₄	C₂²×C₇×#41	1	1	yes	characteristic
765	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
766	C₂	—	2	2	no	—
767	C₂	—	8	1	no	—
768	C₂	C₂×C₇×#548	1	1	yes	characteristic
769	C₂	C₂×C₇×#578	1	1	yes	characteristic
770	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	C₂×C₇×#41	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
2. C₇×#3502	C₂	semidirect
3. C₇×#3377	C₂	semidirect
4. C₇×#3374	C₂	semidirect
5. C₇×#3374	C₂	semidirect
6. C₇×#3369	C₂	semidirect
7. C₇×#3364	C₂	semidirect
8. C₇×#3363	C₂	semidirect
9. C₇×#3139	C₂	semidirect
11. C₇×#3082	C₂	semidirect
14. C₂×C₇×#462	C₂	semidirect
15. C₇×#2962	C₂	semidirect
16. C₇×#2950	C₂	semidirect
93. #61342	C₇	direct (N × H)
217. #3502	C₂×C₇	semidirect
218. #3377	C₂×C₇	semidirect
219. #3374	C₂×C₇	semidirect
220. #3374	C₂×C₇	semidirect
221. #3369	C₂×C₇	semidirect
222. #3364	C₂×C₇	semidirect
223. #3363	C₂×C₇	semidirect
224. #3139	C₂×C₇	semidirect
226. #3082	C₂×C₇	semidirect
229. C₂×#462	C₂×C₇	semidirect
230. #2962	C₂×C₇	semidirect
231. #2950	C₂×C₇	semidirect
741. C₇	#61342	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×#3509	2
C₇×#3502	2
C₇×#3377	2
C₇×#3374	2
C₇×#3369	2
C₇×#3364	2
C₇×#3363	2
C₇×#3139	2
C₇×#3137	2
C₇×#3082	2
C₇×#3076	2
C₇×#3044	2
C₂×C₇×#462	2
C₇×#2962	2
C₇×#2950	2
C₄×C₇×#137	2
#61342	7

Subgroups — C7×#61342 #13238558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61342 #13238558