Subgroups — C₇×#34396 #13211612

Order 1792.

2486 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#525	C₂	1	1	yes	maximal, characteristic
2	C₇×#2378	C₂	1	2	yes	maximal, complemented
3	C₇×#2377	C₂	1	2	yes	maximal, complemented
4	C₇×#2359	C₂	1	4	yes	maximal, complemented
5	C₇×#2357	C₂	1	2	yes	maximal, complemented
6	C₇×#2353	C₂	1	2	yes	maximal, complemented
7	C₇×#2343	C₂	1	2	yes	maximal, complemented
8	C₇×#2339	C₂	1	2	yes	maximal, complemented
9	C₇×#2333	C₂	1	2	yes	maximal, complemented
10	C₇×#2332	C₂	1	2	yes	maximal, complemented
11	C₇×#2321	C₂	1	4	yes	maximal, complemented
12	C₇×#2319	C₂	1	2	yes	maximal, complemented
13	C₇×#2318	C₂	1	2	yes	maximal, complemented
14	C₇×#2280	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₇×#381	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
18	C₇×#525	C₄	1	16	yes	complemented
19	C₂×C₇×#123	C₂²	1	2	yes	—
20	C₂×C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×#121	C₂²	1	1	yes	characteristic
23	C₂×C₇×#121	C₂²	1	1	yes	characteristic
24	C₂×C₇×#120	C₂²	1	2	yes	—
25	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
26	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
27	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₇×#117	C₂²	1	2	yes	—
29	C₂×C₇×#117	C₂²	1	1	yes	characteristic
30	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₇×#391	C₂²	1	4	yes	complemented
32	C₇×#390	C₂²	1	4	yes	—
33	C₇×#390	C₂²	1	4	yes	—
34	C₇×#389	C₂²	1	4	yes	complemented
35	C₇×#388	C₂²	1	4	yes	—
36	C₇×#388	C₂²	1	4	yes	—
37	C₇×#388	C₂²	1	4	yes	—
38	C₇×#388	C₂²	1	4	yes	—
39	C₇×#388	C₂²	1	4	yes	complemented
40	C₇×#388	C₂²	1	4	yes	complemented
41	C₇×#387	C₂²	1	4	yes	complemented
42	C₇×#387	C₂²	1	4	yes	complemented
43	C₇×#387	C₂²	1	4	yes	complemented
44	C₇×#387	C₂²	1	4	yes	complemented
45	C₇×#386	C₂²	1	4	yes	—
46	C₇×#386	C₂²	1	4	yes	—
47	C₇×#386	C₂²	1	4	yes	—
48	C₇×#386	C₂²	1	4	yes	complemented
49	C₇×#385	C₂²	1	4	yes	—
50	C₇×#385	C₂²	1	4	yes	complemented
51	C₇×#383	C₂²	1	2	yes	complemented
52	C₇×#382	C₂²	1	2	yes	complemented
53	C₇×#381	C₂²	1	1	yes	characteristic
54	C₇×#381	C₂²	1	1	yes	characteristic
55	C₇×#381	C₂²	1	2	yes	—
56	C₇×#381	C₂²	1	2	yes	—
57	C₇×#381	C₂²	1	2	yes	—
58	C₇×#381	C₂²	1	2	yes	—
59	C₇×#381	C₂²	1	2	yes	complemented
60	C₇×#380	C₂²	1	2	yes	—
61	C₇×#380	C₂²	1	2	yes	—
62	C₇×#380	C₂²	1	2	yes	complemented
63	C₇×#379	C₂²	1	2	yes	—
64	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
65	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	complemented
66	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
67	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
68	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
69	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
70	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
71	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	complemented
72	C₇×#376	C₂²	1	2	yes	complemented
73	C₂×C₇×#95	C₂²	1	2	yes	—
74	C₂×C₇×#95	C₂²	1	2	yes	—
75	C₂×C₇×#95	C₂²	1	2	yes	complemented
76	#34396	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂⁵×C₇	C₂³	1	1	yes	characteristic
78	C₂×C₇×#41	—	4	4	no	—
79	C₂×C₇×#41	C₂×C₄	1	8	yes	—
80	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
81	C₂²×C₇×D₄	C₂³	1	2	yes	—
82	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
83	C₂²×C₇×D₄	C₂³	1	2	yes	—
84	C₂²×C₇×D₄	C₂³	1	2	yes	—
85	C₂³×C₄×C₇	—	4	4	no	—
86	C₂³×C₄×C₇	C₂×C₄	1	8	yes	complemented
87	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₇	C₂³	1	4	yes	—
90	C₂³×C₄×C₇	C₂³	1	4	yes	—
91	C₂³×C₄×C₇	C₂³	1	2	yes	—
92	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
93	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
94	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
95	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
96	C₇×#123	C₂×C₄	1	16	yes	complemented
97	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
98	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
99	C₇×#121	C₂×C₄	1	8	yes	complemented
100	C₇×#121	C₂×C₄	1	8	yes	complemented
101	C₇×#120	C₂×C₄	1	16	yes	complemented
102	C₄×C₇×D₄	—	4	8	no	—
103	C₄×C₇×D₄	C₂×C₄	1	16	yes	complemented
104	C₄×C₇×D₄	C₂×C₄	1	16	yes	complemented
105	C₇×#117	—	4	4	no	—
106	C₇×#117	C₂×C₄	1	8	yes	complemented
107	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
108	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
109	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
110	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
112	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
114	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
115	C₂×C₇×C₂²⋊C₄	—	4	4	no	—
116	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
117	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
118	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
119	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
120	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
121	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
122	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
123	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
124	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
125	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
126	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
128	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
130	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
131	C₂×C₄²×C₇	C₂³	1	4	yes	—
132	C₂×C₄²×C₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₇	C₂³	1	2	yes	—
134	C₂×C₄²×C₇	C₂³	1	4	yes	—
135	C₂×C₄²×C₇	C₂³	1	4	yes	—
136	C₂×C₄²×C₇	C₂³	1	2	yes	—
137	C₇×#95	C₂³	1	2	yes	—
138	C₇×#95	C₂³	1	2	yes	—
139	C₇×#95	C₂³	1	4	yes	—
140	C₇×#95	C₂³	1	2	yes	—
141	C₇×#95	C₂³	1	4	yes	—
142	C₇×#95	C₂³	1	2	yes	—
143	C₇×#95	C₂³	1	2	yes	—
144	C₇×#95	C₂³	1	4	yes	—
145	C₇×#95	C₂³	1	4	yes	—
146	C₇×#95	C₂³	1	2	yes	—
147	C₇×#95	C₂³	1	4	yes	—
148	C₇×#95	C₂³	1	4	yes	—
149	C₇×#95	C₂³	1	4	yes	—
150	C₇×#95	C₂³	1	4	yes	—
151	C₇×#95	C₂³	1	4	yes	—
152	C₂×#525	C₂×C₇	1	1	yes	characteristic
153	#2378	C₂×C₇	1	2	yes	complemented
154	#2377	C₂×C₇	1	2	yes	complemented
155	#2359	C₂×C₇	1	4	yes	complemented
156	#2357	C₂×C₇	1	2	yes	complemented
157	#2353	C₂×C₇	1	2	yes	complemented
158	#2343	C₂×C₇	1	2	yes	complemented
159	#2339	C₂×C₇	1	2	yes	complemented
160	#2333	C₂×C₇	1	2	yes	complemented
161	#2332	C₂×C₇	1	2	yes	complemented
162	#2321	C₂×C₇	1	4	yes	complemented
163	#2319	C₂×C₇	1	2	yes	complemented
164	#2318	C₂×C₇	1	2	yes	complemented
165	#2280	C₂×C₇	1	1	yes	characteristic, complemented
166	C₂×#381	C₂×C₇	1	1	yes	characteristic, complemented
167	C₂⁴×C₇	—	2	4	no	—
168	C₂⁴×C₇	—	4	2	no	—
169	C₂⁴×C₇	—	4	2	no	—
170	C₂⁴×C₇	C₂²×C₄	1	4	yes	—
171	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
172	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
173	C₂⁴×C₇	C₂⁴	1	2	yes	—
174	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
175	C₇×#41	—	4	16	no	—
176	C₂×C₇×Q₈	—	4	2	no	—
177	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
178	C₂×C₇×D₄	—	2	8	no	—
179	C₂×C₇×D₄	—	4	2	no	—
180	C₂×C₇×D₄	—	4	4	no	—
181	C₂×C₇×D₄	—	4	2	no	—
182	C₂×C₇×D₄	—	4	2	no	—
183	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
184	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
185	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
186	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
187	C₂²×C₄×C₇	—	2	4	no	—
188	C₂²×C₄×C₇	—	2	4	no	—
189	C₂²×C₄×C₇	—	2	8	no	—
190	C₂²×C₄×C₇	—	4	16	no	—
191	C₂²×C₄×C₇	—	4	2	no	—
192	C₂²×C₄×C₇	—	4	2	no	—
193	C₂²×C₄×C₇	—	4	2	no	—
194	C₂²×C₄×C₇	—	4	2	no	—
195	C₂²×C₄×C₇	—	4	4	no	—
196	C₂²×C₄×C₇	—	4	4	no	—
197	C₂²×C₄×C₇	—	4	4	no	—
198	C₂²×C₄×C₇	—	4	4	no	—
199	C₂²×C₄×C₇	—	4	4	no	—
200	C₂²×C₄×C₇	—	4	4	no	—
201	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
202	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
203	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
204	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
205	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
206	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
207	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
208	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
209	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
210	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
211	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
212	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
214	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
215	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
216	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
218	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
219	C₇×C₄⋊C₄	—	4	4	no	—
220	C₇×C₄⋊C₄	—	4	4	no	—
221	C₇×C₄⋊C₄	—	4	4	no	—
222	C₇×C₄⋊C₄	—	4	4	no	—
223	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
224	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
225	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
226	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
227	C₇×C₂²⋊C₄	—	2	8	no	—
228	C₇×C₂²⋊C₄	—	4	2	no	—
229	C₇×C₂²⋊C₄	—	4	4	no	—
230	C₇×C₂²⋊C₄	—	4	2	no	—
231	C₇×C₂²⋊C₄	—	4	4	no	—
232	C₇×C₂²⋊C₄	—	4	2	no	—
233	C₇×C₂²⋊C₄	—	4	4	no	—
234	C₇×C₂²⋊C₄	—	4	2	no	—
235	C₇×C₂²⋊C₄	—	4	4	no	—
236	C₇×C₂²⋊C₄	—	4	4	no	—
237	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
238	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
239	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
240	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
241	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
242	C₄²×C₇	—	4	4	no	—
243	C₄²×C₇	—	4	4	no	—
244	C₄²×C₇	—	4	2	no	—
245	C₄²×C₇	—	4	2	no	—
246	C₄²×C₇	—	4	4	no	—
247	C₄²×C₇	C₂²×C₄	1	8	yes	—
248	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
249	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
250	#525	C₄×C₇	1	16	yes	complemented
251	C₂×#123	C₂²×C₇	1	2	yes	—
252	C₂×C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
253	C₂×C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
254	C₂×#121	C₂²×C₇	1	1	yes	characteristic
255	C₂×#121	C₂²×C₇	1	1	yes	characteristic
256	C₂×#120	C₂²×C₇	1	2	yes	—
257	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
258	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
259	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
260	C₂×#117	C₂²×C₇	1	2	yes	—
261	C₂×#117	C₂²×C₇	1	1	yes	characteristic
262	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
263	#391	C₂²×C₇	1	4	yes	complemented
264	#390	C₂²×C₇	1	4	yes	—
265	#390	C₂²×C₇	1	4	yes	—
266	#389	C₂²×C₇	1	4	yes	complemented
267	#388	C₂²×C₇	1	4	yes	—
268	#388	C₂²×C₇	1	4	yes	—
269	#388	C₂²×C₇	1	4	yes	—
270	#388	C₂²×C₇	1	4	yes	—
271	#388	C₂²×C₇	1	4	yes	complemented
272	#388	C₂²×C₇	1	4	yes	complemented
273	#387	C₂²×C₇	1	4	yes	complemented
274	#387	C₂²×C₇	1	4	yes	complemented
275	#387	C₂²×C₇	1	4	yes	complemented
276	#387	C₂²×C₇	1	4	yes	complemented
277	#386	C₂²×C₇	1	4	yes	—
278	#386	C₂²×C₇	1	4	yes	—
279	#386	C₂²×C₇	1	4	yes	—
280	#386	C₂²×C₇	1	4	yes	complemented
281	#385	C₂²×C₇	1	4	yes	—
282	#385	C₂²×C₇	1	4	yes	complemented
283	#383	C₂²×C₇	1	2	yes	complemented
284	#382	C₂²×C₇	1	2	yes	complemented
285	#381	C₂²×C₇	1	1	yes	characteristic
286	#381	C₂²×C₇	1	1	yes	characteristic
287	#381	C₂²×C₇	1	2	yes	—
288	#381	C₂²×C₇	1	2	yes	—
289	#381	C₂²×C₇	1	2	yes	—
290	#381	C₂²×C₇	1	2	yes	—
291	#381	C₂²×C₇	1	2	yes	complemented
292	#380	C₂²×C₇	1	2	yes	—
293	#380	C₂²×C₇	1	2	yes	—
294	#380	C₂²×C₇	1	2	yes	complemented
295	#379	C₂²×C₇	1	2	yes	—
296	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
297	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	complemented
298	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
299	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
300	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
301	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
302	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	complemented
303	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	complemented
304	#376	C₂²×C₇	1	2	yes	complemented
305	C₂×#95	C₂²×C₇	1	2	yes	—
306	C₂×#95	C₂²×C₇	1	2	yes	—
307	C₂×#95	C₂²×C₇	1	2	yes	complemented
308	C₂³×C₇	—	2	2	no	—
309	C₂³×C₇	—	2	2	no	—
310	C₂³×C₇	—	2	8	no	—
311	C₂³×C₇	—	2	2	no	—
312	C₂³×C₇	—	2	2	no	—
313	C₂³×C₇	—	2	4	no	—
314	C₂³×C₇	—	2	2	no	—
315	C₂³×C₇	—	2	2	no	—
316	C₂³×C₇	—	2	2	no	—
317	C₂³×C₇	—	2	4	no	—
318	C₂³×C₇	—	2	2	no	—
319	C₂³×C₇	—	2	2	no	—
320	C₂³×C₇	—	2	2	no	—
321	C₂³×C₇	—	4	4	no	—
322	C₂³×C₇	—	4	4	no	—
323	C₂³×C₇	—	4	16	no	—
324	C₂³×C₇	C₂³×C₄	1	2	yes	—
325	C₂³×C₇	C₂³×C₄	1	2	yes	—
326	C₂³×C₇	C₂³×C₄	1	4	yes	—
327	C₂³×C₇	C₂³×C₄	1	2	yes	—
328	C₂³×C₇	H(5,𝔽₂)	1	4	yes	—
329	C₂³×C₇	C₂⁵	1	1	yes	characteristic
330	C₇×Q₈	—	4	4	no	—
331	C₇×D₄	—	2	16	no	—
332	C₇×D₄	—	4	8	no	—
333	C₇×D₄	—	4	8	no	—
334	C₇×D₄	—	4	4	no	—
335	C₂×C₄×C₇	—	2	8	no	—
336	C₂×C₄×C₇	—	2	8	no	—
337	C₂×C₄×C₇	—	2	2	no	—
338	C₂×C₄×C₇	—	2	4	no	—
339	C₂×C₄×C₇	—	2	2	no	—
340	C₂×C₄×C₇	—	2	2	no	—
341	C₂×C₄×C₇	—	2	2	no	—
342	C₂×C₄×C₇	—	2	2	no	—
343	C₂×C₄×C₇	—	2	4	no	—
344	C₂×C₄×C₇	—	2	2	no	—
345	C₂×C₄×C₇	—	2	4	no	—
346	C₂×C₄×C₇	—	2	4	no	—
347	C₂×C₄×C₇	—	2	4	no	—
348	C₂×C₄×C₇	—	2	4	no	—
349	C₂×C₄×C₇	—	2	4	no	—
350	C₂×C₄×C₇	—	2	4	no	—
351	C₂×C₄×C₇	—	2	4	no	—
352	C₂×C₄×C₇	—	2	4	no	—
353	C₂×C₄×C₇	—	2	4	no	—
354	C₂×C₄×C₇	—	2	4	no	—
355	C₂×C₄×C₇	—	2	4	no	—
356	C₂×C₄×C₇	—	2	4	no	—
357	C₂×C₄×C₇	—	4	4	no	—
358	C₂×C₄×C₇	—	4	4	no	—
359	C₂×C₄×C₇	—	4	4	no	—
360	C₂×C₄×C₇	—	4	4	no	—
361	C₂×C₄×C₇	—	4	8	no	—
362	C₂×C₄×C₇	—	8	8	no	—
363	C₂×C₄×C₇	—	8	8	no	—
364	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
365	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
366	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
367	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
368	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
369	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
370	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
371	C₂⁵	C₂³×C₇	1	1	yes	characteristic
372	C₂×#41	—	4	4	no	—
373	C₂×#41	C₂×C₄×C₇	1	8	yes	—
374	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
375	C₂²×D₄	C₂³×C₇	1	2	yes	—
376	C₂²×D₄	C₂³×C₇	1	2	yes	—
377	C₂²×D₄	C₂³×C₇	1	2	yes	—
378	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
379	C₂³×C₄	—	4	4	no	—
380	C₂³×C₄	C₂×C₄×C₇	1	8	yes	complemented
381	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₇	1	4	yes	—
384	C₂³×C₄	C₂³×C₇	1	4	yes	—
385	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₇	1	2	yes	—
388	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
390	#123	C₂×C₄×C₇	1	16	yes	complemented
391	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	complemented
392	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	complemented
393	#121	C₂×C₄×C₇	1	8	yes	complemented
394	#121	C₂×C₄×C₇	1	8	yes	complemented
395	#120	C₂×C₄×C₇	1	16	yes	complemented
396	C₄×D₄	—	4	8	no	—
397	C₄×D₄	C₂×C₄×C₇	1	16	yes	complemented
398	C₄×D₄	C₂×C₄×C₇	1	16	yes	complemented
399	#117	—	4	4	no	—
400	#117	C₂×C₄×C₇	1	8	yes	complemented
401	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
409	C₂×C₂²⋊C₄	—	4	4	no	—
410	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
425	C₂×C₄²	C₂³×C₇	1	2	yes	—
426	C₂×C₄²	C₂³×C₇	1	2	yes	—
427	C₂×C₄²	C₂³×C₇	1	4	yes	—
428	C₂×C₄²	C₂³×C₇	1	4	yes	—
429	C₂×C₄²	C₂³×C₇	1	4	yes	—
430	C₂×C₄²	C₂³×C₇	1	2	yes	—
431	#95	C₂³×C₇	1	2	yes	—
432	#95	C₂³×C₇	1	2	yes	—
433	#95	C₂³×C₇	1	4	yes	—
434	#95	C₂³×C₇	1	2	yes	—
435	#95	C₂³×C₇	1	4	yes	—
436	#95	C₂³×C₇	1	2	yes	—
437	#95	C₂³×C₇	1	2	yes	—
438	#95	C₂³×C₇	1	4	yes	—
439	#95	C₂³×C₇	1	4	yes	—
440	#95	C₂³×C₇	1	2	yes	—
441	#95	C₂³×C₇	1	4	yes	—
442	#95	C₂³×C₇	1	4	yes	—
443	#95	C₂³×C₇	1	4	yes	—
444	#95	C₂³×C₇	1	4	yes	—
445	#95	C₂³×C₇	1	4	yes	—
446	C₂²×C₇	—	2	2	no	—
447	C₂²×C₇	—	2	2	no	—
448	C₂²×C₇	—	2	8	no	—
449	C₂²×C₇	—	2	2	no	—
450	C₂²×C₇	—	2	2	no	—
451	C₂²×C₇	—	2	4	no	—
452	C₂²×C₇	—	2	2	no	—
453	C₂²×C₇	—	2	2	no	—
454	C₂²×C₇	—	2	4	no	—
455	C₂²×C₇	—	2	4	no	—
456	C₂²×C₇	—	2	2	no	—
457	C₂²×C₇	—	2	2	no	—
458	C₂²×C₇	—	4	16	no	—
459	C₂²×C₇	—	4	4	no	—
460	C₂²×C₇	—	4	4	no	—
461	C₂²×C₇	—	4	4	no	—
462	C₂²×C₇	—	4	4	no	—
463	C₂²×C₇	#518	1	4	yes	—
464	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
465	C₂²×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
466	C₂²×C₇	#585	1	2	yes	—
467	C₂²×C₇	#585	1	2	yes	—
468	C₄×C₇	—	2	2	no	—
469	C₄×C₇	—	2	2	no	—
470	C₄×C₇	—	2	2	no	—
471	C₄×C₇	—	2	2	no	—
472	C₄×C₇	—	2	4	no	—
473	C₄×C₇	—	2	4	no	—
474	C₄×C₇	—	2	4	no	—
475	C₄×C₇	—	4	4	no	—
476	C₄×C₇	—	4	4	no	—
477	C₄×C₇	—	4	4	no	—
478	C₄×C₇	—	4	4	no	—
479	C₂⁴	—	2	4	no	—
480	C₂⁴	—	4	2	no	—
481	C₂⁴	—	4	2	no	—
482	C₂⁴	C₂²×C₄×C₇	1	4	yes	—
483	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
484	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
485	C₂⁴	C₂⁴×C₇	1	2	yes	—
486	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
487	#41	—	4	16	no	—
488	C₂×Q₈	—	4	2	no	—
489	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
490	C₂×D₄	—	2	8	no	—
491	C₂×D₄	—	4	2	no	—
492	C₂×D₄	—	4	2	no	—
493	C₂×D₄	—	4	4	no	—
494	C₂×D₄	—	4	2	no	—
495	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
496	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
497	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
498	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	8	no	—
502	C₂²×C₄	—	4	16	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	—	4	4	no	—
508	C₂²×C₄	—	4	4	no	—
509	C₂²×C₄	—	4	4	no	—
510	C₂²×C₄	—	4	4	no	—
511	C₂²×C₄	—	4	4	no	—
512	C₂²×C₄	—	4	4	no	—
513	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
514	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
515	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
516	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
517	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
518	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
519	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
520	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
521	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
522	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
523	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
524	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
525	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
527	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
530	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
531	C₄⋊C₄	—	4	4	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₄⋊C₄	—	4	4	no	—
534	C₄⋊C₄	—	4	4	no	—
535	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
536	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
537	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
538	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
539	C₂²⋊C₄	—	2	8	no	—
540	C₂²⋊C₄	—	4	2	no	—
541	C₂²⋊C₄	—	4	4	no	—
542	C₂²⋊C₄	—	4	2	no	—
543	C₂²⋊C₄	—	4	4	no	—
544	C₂²⋊C₄	—	4	4	no	—
545	C₂²⋊C₄	—	4	2	no	—
546	C₂²⋊C₄	—	4	4	no	—
547	C₂²⋊C₄	—	4	2	no	—
548	C₂²⋊C₄	—	4	4	no	—
549	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
550	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
551	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
552	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
553	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
554	C₄²	—	4	2	no	—
555	C₄²	—	4	2	no	—
556	C₄²	—	4	4	no	—
557	C₄²	—	4	4	no	—
558	C₄²	—	4	4	no	—
559	C₄²	C₂²×C₄×C₇	1	8	yes	—
560	C₂×C₇	—	2	4	no	—
561	C₂×C₇	—	4	4	no	—
562	C₂×C₇	—	4	2	no	—
563	C₂×C₇	—	4	2	no	—
564	C₂×C₇	C₂×#518	1	1	yes	characteristic
565	C₂×C₇	#3428	1	1	yes	characteristic
566	C₂×C₇	#3428	1	1	yes	characteristic
567	C₂×C₇	#3533	1	4	yes	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	2	no	—
570	C₂³	—	2	2	no	—
571	C₂³	—	2	2	no	—
572	C₂³	—	2	4	no	—
573	C₂³	—	2	2	no	—
574	C₂³	—	2	2	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	4	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	8	no	—
581	C₂³	—	4	4	no	—
582	C₂³	—	4	4	no	—
583	C₂³	—	4	16	no	—
584	C₂³	C₂³×C₄×C₇	1	2	yes	—
585	C₂³	C₂³×C₄×C₇	1	2	yes	—
586	C₂³	C₂³×C₄×C₇	1	4	yes	—
587	C₂³	C₂³×C₄×C₇	1	2	yes	—
588	C₂³	C₇×H(5,𝔽₂)	1	4	yes	—
589	C₂³	C₂⁵×C₇	1	1	yes	characteristic
590	Q₈	—	4	4	no	—
591	D₄	—	2	16	no	—
592	D₄	—	4	8	no	—
593	D₄	—	4	8	no	—
594	D₄	—	4	4	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	8	no	—
604	C₂×C₄	—	2	8	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	4	4	no	—
620	C₂×C₄	—	4	4	no	—
621	C₂×C₄	—	4	8	no	—
622	C₂×C₄	—	8	8	no	—
623	C₂×C₄	—	8	8	no	—
624	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
625	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
626	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
627	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
628	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
629	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
630	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
631	C₇	#34396	1	1	yes	characteristic, direct product factor, Sylow
632	C₂²	—	2	2	no	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	8	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	4	no	—
641	C₂²	—	2	4	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	4	16	no	—
645	C₂²	—	4	4	no	—
646	C₂²	—	4	4	no	—
647	C₂²	—	4	4	no	—
648	C₂²	—	4	4	no	—
649	C₂²	C₇×#518	1	4	yes	—
650	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
651	C₂²	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
652	C₂²	C₇×#585	1	2	yes	—
653	C₂²	C₇×#585	1	2	yes	—
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	2	2	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	4	no	—
659	C₄	—	2	4	no	—
660	C₄	—	2	4	no	—
661	C₄	—	4	4	no	—
662	C₄	—	4	4	no	—
663	C₄	—	4	4	no	—
664	C₄	—	4	4	no	—
665	C₂	—	2	4	no	—
666	C₂	—	4	2	no	—
667	C₂	—	4	2	no	—
668	C₂	—	4	4	no	—
669	C₂	C₂×C₇×#518	1	1	yes	characteristic
670	C₂	C₇×#3428	1	1	yes	characteristic
671	C₂	C₇×#3428	1	1	yes	characteristic
672	C₂	C₇×#3533	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂×C₇×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2378	C₂	semidirect
3. C₇×#2377	C₂	semidirect
4. C₇×#2359	C₂	semidirect
5. C₇×#2357	C₂	semidirect
6. C₇×#2353	C₂	semidirect
7. C₇×#2343	C₂	semidirect
8. C₇×#2339	C₂	semidirect
9. C₇×#2333	C₂	semidirect
10. C₇×#2332	C₂	semidirect
11. C₇×#2321	C₂	semidirect
12. C₇×#2319	C₂	semidirect
13. C₇×#2318	C₂	semidirect
14. C₇×#2280	C₂	semidirect
15. C₂×C₇×#381	C₂	semidirect
18. C₇×#525	C₄	semidirect
31. C₇×#391	C₂²	semidirect
34. C₇×#389	C₂²	semidirect
39. C₇×#388	C₂²	semidirect
40. C₇×#388	C₂²	semidirect
41. C₇×#387	C₂²	semidirect
42. C₇×#387	C₂²	semidirect
43. C₇×#387	C₂²	semidirect
44. C₇×#387	C₂²	semidirect
48. C₇×#386	C₂²	semidirect
50. C₇×#385	C₂²	semidirect
51. C₇×#383	C₂²	semidirect
52. C₇×#382	C₂²	semidirect
59. C₇×#381	C₂²	semidirect
62. C₇×#380	C₂²	semidirect
64. C₄×C₇×C₄⋊C₄	C₂²	semidirect
65. C₄×C₇×C₄⋊C₄	C₂²	semidirect
70. C₄×C₇×C₂²⋊C₄	C₂²	semidirect
71. C₄×C₇×C₂²⋊C₄	C₂²	semidirect
72. C₇×#376	C₂²	semidirect
75. C₂×C₇×#95	C₂²	semidirect
76. #34396	C₇	direct (N × H)
86. C₂³×C₄×C₇	C₂×C₄	semidirect
96. C₇×#123	C₂×C₄	semidirect
97. C₇×C₂²⋊Q₈	C₂×C₄	semidirect
98. C₇×C₂²⋊Q₈	C₂×C₄	semidirect
99. C₇×#121	C₂×C₄	semidirect
100. C₇×#121	C₂×C₄	semidirect
101. C₇×#120	C₂×C₄	semidirect
103. C₄×C₇×D₄	C₂×C₄	semidirect
104. C₄×C₇×D₄	C₂×C₄	semidirect
106. C₇×#117	C₂×C₄	semidirect
153. #2378	C₂×C₇	semidirect
154. #2377	C₂×C₇	semidirect
155. #2359	C₂×C₇	semidirect
156. #2357	C₂×C₇	semidirect
157. #2353	C₂×C₇	semidirect
158. #2343	C₂×C₇	semidirect
159. #2339	C₂×C₇	semidirect
160. #2333	C₂×C₇	semidirect
161. #2332	C₂×C₇	semidirect
162. #2321	C₂×C₇	semidirect
163. #2319	C₂×C₇	semidirect
164. #2318	C₂×C₇	semidirect
165. #2280	C₂×C₇	semidirect
166. C₂×#381	C₂×C₇	semidirect
250. #525	C₄×C₇	semidirect
263. #391	C₂²×C₇	semidirect
266. #389	C₂²×C₇	semidirect
271. #388	C₂²×C₇	semidirect
272. #388	C₂²×C₇	semidirect
273. #387	C₂²×C₇	semidirect
274. #387	C₂²×C₇	semidirect
275. #387	C₂²×C₇	semidirect
276. #387	C₂²×C₇	semidirect
280. #386	C₂²×C₇	semidirect
282. #385	C₂²×C₇	semidirect
283. #383	C₂²×C₇	semidirect
284. #382	C₂²×C₇	semidirect
291. #381	C₂²×C₇	semidirect
294. #380	C₂²×C₇	semidirect
296. C₄×C₄⋊C₄	C₂²×C₇	semidirect
297. C₄×C₄⋊C₄	C₂²×C₇	semidirect
302. C₄×C₂²⋊C₄	C₂²×C₇	semidirect
303. C₄×C₂²⋊C₄	C₂²×C₇	semidirect
304. #376	C₂²×C₇	semidirect
307. C₂×#95	C₂²×C₇	semidirect
380. C₂³×C₄	C₂×C₄×C₇	semidirect
390. #123	C₂×C₄×C₇	semidirect
391. C₂²⋊Q₈	C₂×C₄×C₇	semidirect
392. C₂²⋊Q₈	C₂×C₄×C₇	semidirect
393. #121	C₂×C₄×C₇	semidirect
394. #121	C₂×C₄×C₇	semidirect
395. #120	C₂×C₄×C₇	semidirect
397. C₄×D₄	C₂×C₄×C₇	semidirect
398. C₄×D₄	C₂×C₄×C₇	semidirect
400. #117	C₂×C₄×C₇	semidirect
631. C₇	#34396	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#525	2
C₇×#2378	2
C₇×#2377	2
C₇×#2359	2
C₇×#2357	2
C₇×#2353	2
C₇×#2343	2
C₇×#2339	2
C₇×#2333	2
C₇×#2332	2
C₇×#2321	2
C₇×#2319	2
C₇×#2318	2
C₇×#2280	2
C₂×C₇×#381	2
#34396	7

Subgroups — C7×#34396 #13211612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34396 #13211612