Subgroups — C₇×#34392 #13211608

Order 1792.

2090 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#529	C₂	1	1	yes	maximal, characteristic
2	C₇×#2376	C₂	1	4	yes	maximal, complemented
3	C₇×#2363	C₂	1	2	yes	maximal
4	C₇×#2362	C₂	1	2	yes	maximal, complemented
5	C₇×#2357	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2356	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2351	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2349	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2347	C₂	1	2	yes	maximal, complemented
10	C₇×#2343	C₂	1	2	yes	maximal, complemented
11	C₇×#2334	C₂	1	4	yes	maximal, complemented
12	C₇×#2330	C₂	1	2	yes	maximal, complemented
13	C₇×#2326	C₂	1	2	yes	maximal, complemented
14	C₇×#2319	C₂	1	2	yes	maximal, complemented
15	C₇×#2318	C₂	1	2	yes	maximal, complemented
16	C₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
17	C₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
18	C₇×#529	C₄	1	16	yes	complemented
19	C₂×C₇×#126	C₂²	1	2	yes	—
20	C₂×C₇×#125	C₂²	1	1	yes	characteristic
21	C₂×C₇×#124	C₂²	1	1	yes	characteristic
22	C₂×C₇×#123	C₂²	1	2	yes	—
23	C₂×C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×#121	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
26	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
30	C₂×C₇×#117	C₂²	1	2	yes	—
31	C₂×C₇×#117	C₂²	1	2	yes	—
32	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
33	C₇×#391	C₂²	1	2	yes	—
34	C₇×#390	C₂²	1	4	yes	—
35	C₇×#390	C₂²	1	2	yes	—
36	C₇×#389	C₂²	1	4	yes	—
37	C₇×#389	C₂²	1	4	yes	—
38	C₇×#389	C₂²	1	2	yes	—
39	C₇×#389	C₂²	1	2	yes	—
40	C₇×#389	C₂²	1	4	yes	—
41	C₇×#388	C₂²	1	4	yes	—
42	C₇×#388	C₂²	1	4	yes	—
43	C₇×#388	C₂²	1	4	yes	—
44	C₇×#388	C₂²	1	4	yes	—
45	C₇×#388	C₂²	1	4	yes	—
46	C₇×#388	C₂²	1	4	yes	—
47	C₇×#387	C₂²	1	4	yes	—
48	C₇×#387	C₂²	1	4	yes	—
49	C₇×#387	C₂²	1	4	yes	—
50	C₇×#387	C₂²	1	4	yes	—
51	C₇×#386	C₂²	1	4	yes	—
52	C₇×#385	C₂²	1	4	yes	—
53	C₇×#385	C₂²	1	4	yes	—
54	C₇×#385	C₂²	1	4	yes	—
55	C₇×#384	C₂²	1	2	yes	—
56	C₇×#383	C₂²	1	1	yes	characteristic
57	C₇×#383	C₂²	1	1	yes	characteristic
58	C₇×#383	C₂²	1	2	yes	—
59	C₇×#382	C₂²	1	2	yes	—
60	C₇×#382	C₂²	1	1	yes	characteristic
61	C₇×#382	C₂²	1	1	yes	characteristic
62	C₇×#382	C₂²	1	2	yes	—
63	C₇×#381	C₂²	1	1	yes	characteristic
64	C₇×#381	C₂²	1	1	yes	characteristic
65	C₇×#381	C₂²	1	2	yes	—
66	C₇×#380	C₂²	1	2	yes	—
67	C₇×#380	C₂²	1	2	yes	—
68	C₇×#380	C₂²	1	2	yes	—
69	C₇×#380	C₂²	1	2	yes	—
70	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
71	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
72	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
73	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
74	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
75	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
76	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
77	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
78	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
79	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
80	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
81	C₇×#376	C₂²	1	2	yes	—
82	C₇×#376	C₂²	1	2	yes	—
83	C₂×C₇×#95	C₂²	1	2	yes	—
84	#34392	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
86	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
87	C₂²×C₇×D₄	C₂³	1	2	yes	—
88	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₇	C₂³	1	4	yes	—
90	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
92	C₂³×C₄×C₇	C₂³	1	2	yes	—
93	C₇×#126	C₂×C₄	1	16	yes	complemented
94	C₇×#125	C₂×C₄	1	8	yes	complemented
95	C₇×#124	C₂×C₄	1	8	yes	—
96	C₇×#123	C₂×C₄	1	16	yes	complemented
97	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
98	C₇×#121	C₂×C₄	1	8	yes	—
99	C₄×C₇×Q₈	—	4	4	no	—
100	C₄×C₇×Q₈	C₂×C₄	1	8	yes	complemented
101	C₄×C₇×D₄	—	4	4	no	—
102	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
103	C₄×C₇×D₄	C₂×C₄	1	16	yes	complemented
104	C₇×#117	—	4	4	no	—
105	C₇×#117	C₂×C₄	1	16	yes	complemented
106	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
107	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
108	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
109	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
110	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
112	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
113	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
114	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
115	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
116	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
117	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
118	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
119	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
120	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
121	C₂×C₇×C₂²⋊C₄	C₂³	1	2	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	4	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×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
132	C₂×C₄²×C₇	C₂×C₄	1	8	yes	complemented
133	C₂×C₄²×C₇	C₂³	1	2	yes	—
134	C₂×C₄²×C₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₇	C₂³	1	4	yes	—
136	C₂×C₄²×C₇	C₂³	1	4	yes	—
137	C₂×C₄²×C₇	C₂³	1	4	yes	—
138	C₂×C₄²×C₇	C₂³	1	4	yes	—
139	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
143	C₂×C₄²×C₇	C₂³	1	2	yes	—
144	C₇×#95	C₂³	1	1	yes	characteristic
145	C₇×#95	C₂³	1	2	yes	—
146	C₇×#95	C₂³	1	4	yes	—
147	C₇×#95	C₂³	1	1	yes	characteristic
148	C₇×#95	C₂³	1	2	yes	—
149	C₇×#95	C₂³	1	2	yes	—
150	C₇×#95	C₂³	1	2	yes	—
151	C₇×#95	C₂³	1	2	yes	—
152	C₇×#95	C₂³	1	4	yes	—
153	C₇×#95	C₂³	1	2	yes	—
154	C₇×#95	C₂³	1	1	yes	characteristic
155	C₇×#95	C₂³	1	2	yes	—
156	C₇×#95	C₂³	1	2	yes	—
157	C₇×#95	C₂³	1	4	yes	—
158	C₇×#95	C₂³	1	1	yes	characteristic
159	C₇×#95	C₂³	1	2	yes	—
160	C₇×#95	C₂³	1	2	yes	—
161	C₇×#95	C₂³	1	4	yes	—
162	C₇×#95	C₂³	1	4	yes	—
163	C₇×#95	C₂³	1	4	yes	—
164	C₂×#529	C₂×C₇	1	1	yes	characteristic
165	#2376	C₂×C₇	1	4	yes	complemented
166	#2363	C₂×C₇	1	2	yes	—
167	#2362	C₂×C₇	1	2	yes	complemented
168	#2357	C₂×C₇	1	1	yes	characteristic, complemented
169	#2356	C₂×C₇	1	1	yes	characteristic, complemented
170	#2351	C₂×C₇	1	1	yes	characteristic, complemented
171	#2349	C₂×C₇	1	1	yes	characteristic, complemented
172	#2347	C₂×C₇	1	2	yes	complemented
173	#2343	C₂×C₇	1	2	yes	complemented
174	#2334	C₂×C₇	1	4	yes	complemented
175	#2330	C₂×C₇	1	2	yes	complemented
176	#2326	C₂×C₇	1	2	yes	complemented
177	#2319	C₂×C₇	1	2	yes	complemented
178	#2318	C₂×C₇	1	2	yes	complemented
179	#2285	C₂×C₇	1	1	yes	characteristic, complemented
180	#2283	C₂×C₇	1	1	yes	characteristic, complemented
181	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
182	C₂⁴×C₇	C₂⁴	1	2	yes	—
183	C₂×C₇×Q₈	—	2	2	no	—
184	C₂×C₇×Q₈	—	2	2	no	—
185	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
186	C₂×C₇×D₄	—	2	2	no	—
187	C₂×C₇×D₄	—	2	2	no	—
188	C₂×C₇×D₄	—	4	4	no	—
189	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
190	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
191	C₂²×C₄×C₇	—	2	2	no	—
192	C₂²×C₄×C₇	—	2	2	no	—
193	C₂²×C₄×C₇	—	4	4	no	—
194	C₂²×C₄×C₇	—	4	2	no	—
195	C₂²×C₄×C₇	—	4	2	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	C₂²×C₄	1	4	yes	—
200	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
201	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
202	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
203	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
207	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
208	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
209	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
215	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
216	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
217	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
218	C₇×C₄⋊C₄	—	2	4	no	—
219	C₇×C₄⋊C₄	—	2	4	no	—
220	C₇×C₄⋊C₄	—	4	4	no	—
221	C₇×C₄⋊C₄	—	4	4	no	—
222	C₇×C₄⋊C₄	—	4	2	no	—
223	C₇×C₄⋊C₄	—	4	4	no	—
224	C₇×C₄⋊C₄	—	4	4	no	—
225	C₇×C₄⋊C₄	—	4	2	no	—
226	C₇×C₄⋊C₄	—	4	4	no	—
227	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
228	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
229	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
230	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
231	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
232	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
233	C₇×C₂²⋊C₄	—	4	2	no	—
234	C₇×C₂²⋊C₄	—	4	4	no	—
235	C₇×C₂²⋊C₄	—	4	2	no	—
236	C₇×C₂²⋊C₄	—	4	4	no	—
237	C₇×C₂²⋊C₄	—	4	4	no	—
238	C₇×C₂²⋊C₄	—	4	4	no	—
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	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
243	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
244	C₄²×C₇	—	2	4	no	—
245	C₄²×C₇	—	2	4	no	—
246	C₄²×C₇	—	4	4	no	—
247	C₄²×C₇	—	4	4	no	—
248	C₄²×C₇	—	4	4	no	—
249	C₄²×C₇	—	4	4	no	—
250	C₄²×C₇	C₂²×C₄	1	4	yes	—
251	C₄²×C₇	C₂²×C₄	1	4	yes	—
252	C₄²×C₇	C₂²×C₄	1	4	yes	—
253	C₄²×C₇	C₂²×C₄	1	4	yes	—
254	C₄²×C₇	C₂²×C₄	1	8	yes	—
255	#529	C₄×C₇	1	16	yes	complemented
256	C₂×#126	C₂²×C₇	1	2	yes	—
257	C₂×#125	C₂²×C₇	1	1	yes	characteristic
258	C₂×#124	C₂²×C₇	1	1	yes	characteristic
259	C₂×#123	C₂²×C₇	1	2	yes	—
260	C₂×C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
261	C₂×#121	C₂²×C₇	1	1	yes	characteristic
262	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
263	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
266	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
267	C₂×#117	C₂²×C₇	1	2	yes	—
268	C₂×#117	C₂²×C₇	1	2	yes	—
269	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
270	#391	C₂²×C₇	1	2	yes	—
271	#390	C₂²×C₇	1	4	yes	—
272	#390	C₂²×C₇	1	2	yes	—
273	#389	C₂²×C₇	1	4	yes	—
274	#389	C₂²×C₇	1	4	yes	—
275	#389	C₂²×C₇	1	2	yes	—
276	#389	C₂²×C₇	1	2	yes	—
277	#389	C₂²×C₇	1	4	yes	—
278	#388	C₂²×C₇	1	4	yes	—
279	#388	C₂²×C₇	1	4	yes	—
280	#388	C₂²×C₇	1	4	yes	—
281	#388	C₂²×C₇	1	4	yes	—
282	#388	C₂²×C₇	1	4	yes	—
283	#388	C₂²×C₇	1	4	yes	—
284	#387	C₂²×C₇	1	4	yes	—
285	#387	C₂²×C₇	1	4	yes	—
286	#387	C₂²×C₇	1	4	yes	—
287	#387	C₂²×C₇	1	4	yes	—
288	#386	C₂²×C₇	1	4	yes	—
289	#385	C₂²×C₇	1	4	yes	—
290	#385	C₂²×C₇	1	4	yes	—
291	#385	C₂²×C₇	1	4	yes	—
292	#384	C₂²×C₇	1	2	yes	—
293	#383	C₂²×C₇	1	2	yes	—
294	#383	C₂²×C₇	1	1	yes	characteristic
295	#383	C₂²×C₇	1	1	yes	characteristic
296	#382	C₂²×C₇	1	2	yes	—
297	#382	C₂²×C₇	1	2	yes	—
298	#382	C₂²×C₇	1	1	yes	characteristic
299	#382	C₂²×C₇	1	1	yes	characteristic
300	#381	C₂²×C₇	1	1	yes	characteristic
301	#381	C₂²×C₇	1	1	yes	characteristic
302	#381	C₂²×C₇	1	2	yes	—
303	#380	C₂²×C₇	1	2	yes	—
304	#380	C₂²×C₇	1	2	yes	—
305	#380	C₂²×C₇	1	2	yes	—
306	#380	C₂²×C₇	1	2	yes	—
307	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
308	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
309	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
310	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
311	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
312	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
313	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
314	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
315	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
316	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
317	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
318	#376	C₂²×C₇	1	2	yes	—
319	#376	C₂²×C₇	1	2	yes	—
320	C₂×#95	C₂²×C₇	1	2	yes	—
321	C₂³×C₇	—	2	1	no	—
322	C₂³×C₇	—	2	2	no	—
323	C₂³×C₇	—	2	1	no	—
324	C₂³×C₇	—	2	2	no	—
325	C₂³×C₇	—	2	2	no	—
326	C₂³×C₇	—	2	4	no	—
327	C₂³×C₇	—	2	4	no	—
328	C₂³×C₇	—	2	2	no	—
329	C₂³×C₇	C₂³×C₄	1	4	yes	—
330	C₂³×C₇	C₂³×C₄	1	2	yes	—
331	C₂³×C₇	C₂⁵	1	1	yes	characteristic
332	C₇×Q₈	—	2	8	no	—
333	C₇×D₄	—	2	8	no	—
334	C₇×D₄	—	4	8	no	—
335	C₂×C₄×C₇	—	2	1	no	—
336	C₂×C₄×C₇	—	2	2	no	—
337	C₂×C₄×C₇	—	2	1	no	—
338	C₂×C₄×C₇	—	2	2	no	—
339	C₂×C₄×C₇	—	2	4	no	—
340	C₂×C₄×C₇	—	2	4	no	—
341	C₂×C₄×C₇	—	2	2	no	—
342	C₂×C₄×C₇	—	2	2	no	—
343	C₂×C₄×C₇	—	2	2	no	—
344	C₂×C₄×C₇	—	2	2	no	—
345	C₂×C₄×C₇	—	2	8	no	—
346	C₂×C₄×C₇	—	2	2	no	—
347	C₂×C₄×C₇	—	2	2	no	—
348	C₂×C₄×C₇	—	2	4	no	—
349	C₂×C₄×C₇	—	2	4	no	—
350	C₂×C₄×C₇	—	2	2	no	—
351	C₂×C₄×C₇	—	2	2	no	—
352	C₂×C₄×C₇	—	2	2	no	—
353	C₂×C₄×C₇	—	2	2	no	—
354	C₂×C₄×C₇	—	2	4	no	—
355	C₂×C₄×C₇	—	2	4	no	—
356	C₂×C₄×C₇	—	2	2	no	—
357	C₂×C₄×C₇	—	2	2	no	—
358	C₂×C₄×C₇	—	2	2	no	—
359	C₂×C₄×C₇	—	2	4	no	—
360	C₂×C₄×C₇	—	2	2	no	—
361	C₂×C₄×C₇	—	2	2	no	—
362	C₂×C₄×C₇	—	2	4	no	—
363	C₂×C₄×C₇	—	2	2	no	—
364	C₂×C₄×C₇	—	4	4	no	—
365	C₂×C₄×C₇	—	4	8	no	—
366	C₂×C₄×C₇	—	4	4	no	—
367	C₂×C₄×C₇	—	8	8	no	—
368	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
369	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
370	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
371	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
372	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
373	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
374	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
375	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
376	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
377	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
378	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
379	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
380	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
381	C₂²×D₄	C₂³×C₇	1	2	yes	—
382	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₇	1	4	yes	—
384	C₂³×C₄	C₂³×C₇	1	2	yes	—
385	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
387	#126	C₂×C₄×C₇	1	16	yes	complemented
388	#125	C₂×C₄×C₇	1	8	yes	complemented
389	#124	C₂×C₄×C₇	1	8	yes	—
390	#123	C₂×C₄×C₇	1	16	yes	complemented
391	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	complemented
392	#121	C₂×C₄×C₇	1	8	yes	—
393	C₄×Q₈	—	4	4	no	—
394	C₄×Q₈	C₂×C₄×C₇	1	8	yes	complemented
395	C₄×D₄	—	4	4	no	—
396	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
397	C₄×D₄	C₂×C₄×C₇	1	16	yes	complemented
398	#117	—	4	4	no	—
399	#117	C₂×C₄×C₇	1	16	yes	complemented
400	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
407	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
408	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₇	1	4	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	2	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₇	1	4	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×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
426	C₂×C₄²	C₂×C₄×C₇	1	8	yes	complemented
427	C₂×C₄²	C₂³×C₇	1	2	yes	—
428	C₂×C₄²	C₂³×C₇	1	2	yes	—
429	C₂×C₄²	C₂³×C₇	1	4	yes	—
430	C₂×C₄²	C₂³×C₇	1	4	yes	—
431	C₂×C₄²	C₂³×C₇	1	4	yes	—
432	C₂×C₄²	C₂³×C₇	1	4	yes	—
433	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₇	1	2	yes	—
435	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
438	#95	C₂³×C₇	1	1	yes	characteristic
439	#95	C₂³×C₇	1	2	yes	—
440	#95	C₂³×C₇	1	4	yes	—
441	#95	C₂³×C₇	1	1	yes	characteristic
442	#95	C₂³×C₇	1	2	yes	—
443	#95	C₂³×C₇	1	2	yes	—
444	#95	C₂³×C₇	1	2	yes	—
445	#95	C₂³×C₇	1	2	yes	—
446	#95	C₂³×C₇	1	4	yes	—
447	#95	C₂³×C₇	1	2	yes	—
448	#95	C₂³×C₇	1	1	yes	characteristic
449	#95	C₂³×C₇	1	2	yes	—
450	#95	C₂³×C₇	1	2	yes	—
451	#95	C₂³×C₇	1	4	yes	—
452	#95	C₂³×C₇	1	1	yes	characteristic
453	#95	C₂³×C₇	1	2	yes	—
454	#95	C₂³×C₇	1	2	yes	—
455	#95	C₂³×C₇	1	4	yes	—
456	#95	C₂³×C₇	1	4	yes	—
457	#95	C₂³×C₇	1	4	yes	—
458	C₂²×C₇	—	2	2	no	—
459	C₂²×C₇	—	2	4	no	—
460	C₂²×C₇	—	2	2	no	—
461	C₂²×C₇	—	2	2	no	—
462	C₂²×C₇	—	2	4	no	—
463	C₂²×C₇	—	2	4	no	—
464	C₂²×C₇	—	4	4	no	—
465	C₂²×C₇	—	4	2	no	—
466	C₂²×C₇	—	4	2	no	—
467	C₂²×C₇	—	4	4	no	—
468	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
469	C₂²×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
470	C₂²×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
471	C₂²×C₇	#585	1	2	yes	—
472	C₂²×C₇	#585	1	2	yes	—
473	C₄×C₇	—	2	2	no	—
474	C₄×C₇	—	2	4	no	—
475	C₄×C₇	—	2	2	no	—
476	C₄×C₇	—	2	4	no	—
477	C₄×C₇	—	2	2	no	—
478	C₄×C₇	—	2	2	no	—
479	C₄×C₇	—	2	2	no	—
480	C₄×C₇	—	2	4	no	—
481	C₄×C₇	—	4	2	no	—
482	C₄×C₇	—	4	4	no	—
483	C₄×C₇	—	4	4	no	—
484	C₄×C₇	—	4	2	no	—
485	C₄×C₇	—	4	2	no	—
486	C₄×C₇	—	4	2	no	—
487	C₄×C₇	#520	1	4	yes	—
488	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
489	C₂⁴	C₂⁴×C₇	1	2	yes	—
490	C₂×Q₈	—	2	2	no	—
491	C₂×Q₈	—	2	2	no	—
492	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	4	4	no	—
496	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
497	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	4	4	no	—
501	C₂²×C₄	—	4	2	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	4	4	no	—
504	C₂²×C₄	—	4	4	no	—
505	C₂²×C₄	—	4	4	no	—
506	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
507	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
508	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
509	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
510	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
511	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
516	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
518	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
522	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
523	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
524	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	4	4	no	—
528	C₄⋊C₄	—	4	4	no	—
529	C₄⋊C₄	—	4	4	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	4	4	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
535	C₄⋊C₄	C₂²×C₄×C₇	1	8	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	4	yes	—
539	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
540	C₂²⋊C₄	—	4	4	no	—
541	C₂²⋊C₄	—	4	4	no	—
542	C₂²⋊C₄	—	4	2	no	—
543	C₂²⋊C₄	—	4	4	no	—
544	C₂²⋊C₄	—	4	2	no	—
545	C₂²⋊C₄	—	4	4	no	—
546	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
547	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
548	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
549	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
550	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
551	C₄²	—	2	4	no	—
552	C₄²	—	2	4	no	—
553	C₄²	—	4	4	no	—
554	C₄²	—	4	4	no	—
555	C₄²	—	4	4	no	—
556	C₄²	—	4	4	no	—
557	C₄²	C₂²×C₄×C₇	1	4	yes	—
558	C₄²	C₂²×C₄×C₇	1	4	yes	—
559	C₄²	C₂²×C₄×C₇	1	4	yes	—
560	C₄²	C₂²×C₄×C₇	1	4	yes	—
561	C₄²	C₂²×C₄×C₇	1	8	yes	—
562	C₂×C₇	—	4	2	no	—
563	C₂×C₇	—	4	4	no	—
564	C₂×C₇	C₂×#520	1	1	yes	characteristic
565	C₂×C₇	#3428	1	1	yes	characteristic
566	C₂×C₇	#3428	1	1	yes	characteristic
567	C₂×C₇	#3562	1	2	yes	—
568	C₂×C₇	#3565	1	2	yes	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	2	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	2	2	no	—
574	C₂³	—	2	4	no	—
575	C₂³	—	2	4	no	—
576	C₂³	—	2	2	no	—
577	C₂³	C₂³×C₄×C₇	1	4	yes	—
578	C₂³	C₂³×C₄×C₇	1	2	yes	—
579	C₂³	C₂⁵×C₇	1	1	yes	characteristic
580	Q₈	—	2	8	no	—
581	D₄	—	2	8	no	—
582	D₄	—	4	8	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	8	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	4	4	no	—
613	C₂×C₄	—	4	8	no	—
614	C₂×C₄	—	4	4	no	—
615	C₂×C₄	—	8	8	no	—
616	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
617	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
618	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
619	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
620	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
621	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
622	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
623	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
624	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
625	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
626	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
627	C₇	#34392	1	1	yes	characteristic, direct product factor, Sylow
628	C₂²	—	2	2	no	—
629	C₂²	—	2	4	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	4	no	—
633	C₂²	—	2	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	4	no	—
638	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
639	C₂²	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
640	C₂²	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
641	C₂²	C₇×#585	1	2	yes	—
642	C₂²	C₇×#585	1	2	yes	—
643	C₄	—	2	2	no	—
644	C₄	—	2	4	no	—
645	C₄	—	2	2	no	—
646	C₄	—	2	4	no	—
647	C₄	—	2	2	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	4	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	4	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	4	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	2	no	—
657	C₄	C₇×#520	1	4	yes	—
658	C₂	—	4	2	no	—
659	C₂	—	4	4	no	—
660	C₂	C₂×C₇×#520	1	1	yes	characteristic
661	C₂	C₇×#3428	1	1	yes	characteristic
662	C₂	C₇×#3428	1	1	yes	characteristic
663	C₂	C₇×#3562	1	2	yes	—
664	C₂	C₇×#3565	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂×C₇×#121	C₂²
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×#2376	C₂	semidirect
4. C₇×#2362	C₂	semidirect
5. C₇×#2357	C₂	semidirect
6. C₇×#2356	C₂	semidirect
7. C₇×#2351	C₂	semidirect
8. C₇×#2349	C₂	semidirect
9. C₇×#2347	C₂	semidirect
10. C₇×#2343	C₂	semidirect
11. C₇×#2334	C₂	semidirect
12. C₇×#2330	C₂	semidirect
13. C₇×#2326	C₂	semidirect
14. C₇×#2319	C₂	semidirect
15. C₇×#2318	C₂	semidirect
16. C₇×#2285	C₂	semidirect
17. C₇×#2283	C₂	semidirect
18. C₇×#529	C₄	semidirect
84. #34392	C₇	direct (N × H)
93. C₇×#126	C₂×C₄	semidirect
94. C₇×#125	C₂×C₄	semidirect
96. C₇×#123	C₂×C₄	semidirect
97. C₇×C₂²⋊Q₈	C₂×C₄	semidirect
100. C₄×C₇×Q₈	C₂×C₄	semidirect
103. C₄×C₇×D₄	C₂×C₄	semidirect
105. C₇×#117	C₂×C₄	semidirect
132. C₂×C₄²×C₇	C₂×C₄	semidirect
165. #2376	C₂×C₇	semidirect
167. #2362	C₂×C₇	semidirect
168. #2357	C₂×C₇	semidirect
169. #2356	C₂×C₇	semidirect
170. #2351	C₂×C₇	semidirect
171. #2349	C₂×C₇	semidirect
172. #2347	C₂×C₇	semidirect
173. #2343	C₂×C₇	semidirect
174. #2334	C₂×C₇	semidirect
175. #2330	C₂×C₇	semidirect
176. #2326	C₂×C₇	semidirect
177. #2319	C₂×C₇	semidirect
178. #2318	C₂×C₇	semidirect
179. #2285	C₂×C₇	semidirect
180. #2283	C₂×C₇	semidirect
255. #529	C₄×C₇	semidirect
387. #126	C₂×C₄×C₇	semidirect
388. #125	C₂×C₄×C₇	semidirect
390. #123	C₂×C₄×C₇	semidirect
391. C₂²⋊Q₈	C₂×C₄×C₇	semidirect
394. C₄×Q₈	C₂×C₄×C₇	semidirect
397. C₄×D₄	C₂×C₄×C₇	semidirect
399. #117	C₂×C₄×C₇	semidirect
426. C₂×C₄²	C₂×C₄×C₇	semidirect
627. C₇	#34392	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×#529	2
C₇×#2376	2
C₇×#2363	2
C₇×#2362	2
C₇×#2357	2
C₇×#2356	2
C₇×#2351	2
C₇×#2349	2
C₇×#2347	2
C₇×#2343	2
C₇×#2334	2
C₇×#2330	2
C₇×#2326	2
C₇×#2319	2
C₇×#2318	2
C₇×#2285	2
C₇×#2283	2
#34392	7

Subgroups — C7×#34392 #13211608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34392 #13211608