Subgroups — C₇×#34405 #13211621

Order 1792.

1990 conjugacy classes of subgroups.

The subgroup lattice has 608 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×#2382	C₂	1	2	yes	maximal, complemented
3	C₇×#2379	C₂	1	2	yes	maximal, complemented
4	C₇×#2373	C₂	1	4	yes	maximal, complemented
5	C₇×#2356	C₂	1	2	yes	maximal, complemented
6	C₇×#2353	C₂	1	2	yes	maximal, complemented
7	C₇×#2347	C₂	1	2	yes	maximal, complemented
8	C₇×#2343	C₂	1	2	yes	maximal, complemented
9	C₇×#2335	C₂	1	4	yes	maximal, complemented
10	C₇×#2333	C₂	1	2	yes	maximal
11	C₇×#2332	C₂	1	2	yes	maximal, complemented
12	C₇×#2319	C₂	1	2	yes	maximal, complemented
13	C₇×#2318	C₂	1	2	yes	maximal, complemented
14	C₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#529	C₄	1	16	yes	complemented
17	C₂×C₇×#126	C₂²	1	2	yes	—
18	C₂×C₇×#125	C₂²	1	1	yes	characteristic
19	C₂×C₇×#124	C₂²	1	1	yes	characteristic
20	C₂×C₇×#123	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×#121	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
26	C₂×C₇×#117	C₂²	1	2	yes	—
27	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
28	C₇×#391	C₂²	1	4	yes	—
29	C₇×#390	C₂²	1	4	yes	—
30	C₇×#389	C₂²	1	4	yes	—
31	C₇×#389	C₂²	1	4	yes	—
32	C₇×#388	C₂²	1	4	yes	—
33	C₇×#388	C₂²	1	4	yes	—
34	C₇×#388	C₂²	1	4	yes	—
35	C₇×#388	C₂²	1	4	yes	—
36	C₇×#388	C₂²	1	4	yes	—
37	C₇×#388	C₂²	1	4	yes	—
38	C₇×#387	C₂²	1	4	yes	—
39	C₇×#387	C₂²	1	4	yes	—
40	C₇×#387	C₂²	1	4	yes	—
41	C₇×#387	C₂²	1	4	yes	—
42	C₇×#387	C₂²	1	4	yes	—
43	C₇×#387	C₂²	1	4	yes	—
44	C₇×#386	C₂²	1	4	yes	—
45	C₇×#386	C₂²	1	4	yes	—
46	C₇×#385	C₂²	1	4	yes	—
47	C₇×#385	C₂²	1	4	yes	—
48	C₇×#383	C₂²	1	2	yes	—
49	C₇×#383	C₂²	1	2	yes	—
50	C₇×#383	C₂²	1	2	yes	—
51	C₇×#382	C₂²	1	2	yes	—
52	C₇×#382	C₂²	1	2	yes	—
53	C₇×#382	C₂²	1	2	yes	—
54	C₇×#381	C₂²	1	2	yes	—
55	C₇×#381	C₂²	1	2	yes	—
56	C₇×#381	C₂²	1	2	yes	—
57	C₇×#380	C₂²	1	1	yes	characteristic
58	C₇×#380	C₂²	1	1	yes	characteristic
59	C₇×#380	C₂²	1	2	yes	—
60	C₇×#380	C₂²	1	2	yes	—
61	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
62	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
63	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
64	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
65	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
66	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
67	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
68	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
69	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
70	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
71	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
72	C₇×#376	C₂²	1	2	yes	—
73	C₇×#376	C₂²	1	2	yes	—
74	C₇×#376	C₂²	1	2	yes	—
75	C₂×C₇×#95	C₂²	1	2	yes	—
76	#34405	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
78	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
79	C₂²×C₇×D₄	C₂³	1	2	yes	—
80	C₂³×C₄×C₇	C₂³	1	4	yes	—
81	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₇	C₂³	1	2	yes	—
85	C₇×#126	C₂×C₄	1	16	yes	complemented
86	C₇×#125	C₂×C₄	1	8	yes	complemented
87	C₇×#124	C₂×C₄	1	8	yes	—
88	C₇×#123	C₂×C₄	1	16	yes	complemented
89	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
90	C₇×#121	C₂×C₄	1	8	yes	—
91	C₄×C₇×Q₈	C₂×C₄	1	8	yes	complemented
92	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
93	C₄×C₇×D₄	C₂×C₄	1	16	yes	complemented
94	C₇×#117	C₂×C₄	1	16	yes	complemented
95	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
96	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
97	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
98	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
99	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
100	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
101	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
102	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
103	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
104	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
105	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
106	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
107	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
108	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
109	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
110	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
111	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
112	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
113	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
114	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
115	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
116	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
117	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
118	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
119	C₂×C₄²×C₇	C₂×C₄	1	8	yes	complemented
120	C₂×C₄²×C₇	C₂³	1	2	yes	—
121	C₂×C₄²×C₇	C₂³	1	4	yes	—
122	C₂×C₄²×C₇	C₂³	1	2	yes	—
123	C₂×C₄²×C₇	C₂³	1	4	yes	—
124	C₂×C₄²×C₇	C₂³	1	4	yes	—
125	C₂×C₄²×C₇	C₂³	1	4	yes	—
126	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₇	C₂³	1	2	yes	—
131	C₇×#95	C₂³	1	2	yes	—
132	C₇×#95	C₂³	1	2	yes	—
133	C₇×#95	C₂³	1	4	yes	—
134	C₇×#95	C₂³	1	2	yes	—
135	C₇×#95	C₂³	1	4	yes	—
136	C₇×#95	C₂³	1	4	yes	—
137	C₇×#95	C₂³	1	4	yes	—
138	C₇×#95	C₂³	1	4	yes	—
139	C₇×#95	C₂³	1	2	yes	—
140	C₇×#95	C₂³	1	4	yes	—
141	C₇×#95	C₂³	1	4	yes	—
142	C₇×#95	C₂³	1	4	yes	—
143	C₇×#95	C₂³	1	4	yes	—
144	C₇×#95	C₂³	1	4	yes	—
145	C₇×#95	C₂³	1	4	yes	—
146	C₇×#95	C₂³	1	4	yes	—
147	C₂×#529	C₂×C₇	1	1	yes	characteristic
148	#2382	C₂×C₇	1	2	yes	complemented
149	#2379	C₂×C₇	1	2	yes	complemented
150	#2373	C₂×C₇	1	4	yes	complemented
151	#2356	C₂×C₇	1	2	yes	complemented
152	#2353	C₂×C₇	1	2	yes	complemented
153	#2347	C₂×C₇	1	2	yes	complemented
154	#2343	C₂×C₇	1	2	yes	complemented
155	#2335	C₂×C₇	1	4	yes	complemented
156	#2333	C₂×C₇	1	2	yes	—
157	#2332	C₂×C₇	1	2	yes	complemented
158	#2319	C₂×C₇	1	2	yes	complemented
159	#2318	C₂×C₇	1	2	yes	complemented
160	#2283	C₂×C₇	1	1	yes	characteristic, complemented
161	#2283	C₂×C₇	1	1	yes	characteristic, complemented
162	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
163	C₂⁴×C₇	C₂⁴	1	2	yes	—
164	C₂×C₇×Q₈	—	4	1	no	—
165	C₂×C₇×Q₈	—	4	1	no	—
166	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
167	C₂×C₇×D₄	—	4	4	no	—
168	C₂×C₇×D₄	—	4	1	no	—
169	C₂×C₇×D₄	—	4	1	no	—
170	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
171	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
172	C₂²×C₄×C₇	—	4	2	no	—
173	C₂²×C₄×C₇	—	4	1	no	—
174	C₂²×C₄×C₇	—	4	1	no	—
175	C₂²×C₄×C₇	—	4	4	no	—
176	C₂²×C₄×C₇	—	4	2	no	—
177	C₂²×C₄×C₇	—	4	4	no	—
178	C₂²×C₄×C₇	—	4	4	no	—
179	C₂²×C₄×C₇	—	4	4	no	—
180	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
181	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
182	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
183	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
184	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
197	C₇×C₄⋊C₄	—	4	4	no	—
198	C₇×C₄⋊C₄	—	4	4	no	—
199	C₇×C₄⋊C₄	—	4	2	no	—
200	C₇×C₄⋊C₄	—	4	2	no	—
201	C₇×C₄⋊C₄	—	4	2	no	—
202	C₇×C₄⋊C₄	—	4	2	no	—
203	C₇×C₄⋊C₄	—	4	4	no	—
204	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
205	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
206	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
207	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
208	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
209	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
210	C₇×C₂²⋊C₄	—	4	4	no	—
211	C₇×C₂²⋊C₄	—	4	2	no	—
212	C₇×C₂²⋊C₄	—	4	4	no	—
213	C₇×C₂²⋊C₄	—	4	4	no	—
214	C₇×C₂²⋊C₄	—	4	4	no	—
215	C₇×C₂²⋊C₄	—	4	2	no	—
216	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
217	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
218	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
219	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
220	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
221	C₄²×C₇	—	4	2	no	—
222	C₄²×C₇	—	4	4	no	—
223	C₄²×C₇	—	4	2	no	—
224	C₄²×C₇	—	4	4	no	—
225	C₄²×C₇	—	4	4	no	—
226	C₄²×C₇	—	4	4	no	—
227	C₄²×C₇	C₂²×C₄	1	4	yes	—
228	C₄²×C₇	C₂²×C₄	1	4	yes	—
229	C₄²×C₇	C₂²×C₄	1	4	yes	—
230	C₄²×C₇	C₂²×C₄	1	4	yes	—
231	C₄²×C₇	C₂²×C₄	1	8	yes	—
232	#529	C₄×C₇	1	16	yes	complemented
233	C₂×#126	C₂²×C₇	1	2	yes	—
234	C₂×#125	C₂²×C₇	1	1	yes	characteristic
235	C₂×#124	C₂²×C₇	1	1	yes	characteristic
236	C₂×#123	C₂²×C₇	1	2	yes	—
237	C₂×C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
238	C₂×#121	C₂²×C₇	1	1	yes	characteristic
239	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
240	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
241	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
242	C₂×#117	C₂²×C₇	1	2	yes	—
243	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
244	#391	C₂²×C₇	1	4	yes	—
245	#390	C₂²×C₇	1	4	yes	—
246	#389	C₂²×C₇	1	4	yes	—
247	#389	C₂²×C₇	1	4	yes	—
248	#388	C₂²×C₇	1	4	yes	—
249	#388	C₂²×C₇	1	4	yes	—
250	#388	C₂²×C₇	1	4	yes	—
251	#388	C₂²×C₇	1	4	yes	—
252	#388	C₂²×C₇	1	4	yes	—
253	#388	C₂²×C₇	1	4	yes	—
254	#387	C₂²×C₇	1	4	yes	—
255	#387	C₂²×C₇	1	4	yes	—
256	#387	C₂²×C₇	1	4	yes	—
257	#387	C₂²×C₇	1	4	yes	—
258	#387	C₂²×C₇	1	4	yes	—
259	#387	C₂²×C₇	1	4	yes	—
260	#386	C₂²×C₇	1	4	yes	—
261	#386	C₂²×C₇	1	4	yes	—
262	#385	C₂²×C₇	1	4	yes	—
263	#385	C₂²×C₇	1	4	yes	—
264	#383	C₂²×C₇	1	2	yes	—
265	#383	C₂²×C₇	1	2	yes	—
266	#383	C₂²×C₇	1	2	yes	—
267	#382	C₂²×C₇	1	2	yes	—
268	#382	C₂²×C₇	1	2	yes	—
269	#382	C₂²×C₇	1	2	yes	—
270	#381	C₂²×C₇	1	2	yes	—
271	#381	C₂²×C₇	1	2	yes	—
272	#381	C₂²×C₇	1	2	yes	—
273	#380	C₂²×C₇	1	1	yes	characteristic
274	#380	C₂²×C₇	1	1	yes	characteristic
275	#380	C₂²×C₇	1	2	yes	—
276	#380	C₂²×C₇	1	2	yes	—
277	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
278	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
279	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
280	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
282	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
283	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
284	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
285	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
286	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
287	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
288	#376	C₂²×C₇	1	2	yes	—
289	#376	C₂²×C₇	1	2	yes	—
290	#376	C₂²×C₇	1	2	yes	—
291	C₂×#95	C₂²×C₇	1	2	yes	—
292	C₂³×C₇	—	2	1	no	—
293	C₂³×C₇	—	2	2	no	—
294	C₂³×C₇	—	2	1	no	—
295	C₂³×C₇	—	2	2	no	—
296	C₂³×C₇	—	2	4	no	—
297	C₂³×C₇	—	2	4	no	—
298	C₂³×C₇	—	2	2	no	—
299	C₂³×C₇	—	2	2	no	—
300	C₂³×C₇	C₂³×C₄	1	4	yes	—
301	C₂³×C₇	C₂³×C₄	1	2	yes	—
302	C₂³×C₇	C₂⁵	1	1	yes	characteristic
303	C₇×Q₈	—	4	4	no	—
304	C₇×D₄	—	4	4	no	—
305	C₇×D₄	—	4	8	no	—
306	C₂×C₄×C₇	—	2	1	no	—
307	C₂×C₄×C₇	—	2	2	no	—
308	C₂×C₄×C₇	—	2	1	no	—
309	C₂×C₄×C₇	—	2	2	no	—
310	C₂×C₄×C₇	—	2	4	no	—
311	C₂×C₄×C₇	—	2	4	no	—
312	C₂×C₄×C₇	—	2	2	no	—
313	C₂×C₄×C₇	—	2	2	no	—
314	C₂×C₄×C₇	—	2	2	no	—
315	C₂×C₄×C₇	—	2	2	no	—
316	C₂×C₄×C₇	—	2	1	no	—
317	C₂×C₄×C₇	—	2	1	no	—
318	C₂×C₄×C₇	—	2	4	no	—
319	C₂×C₄×C₇	—	2	4	no	—
320	C₂×C₄×C₇	—	2	4	no	—
321	C₂×C₄×C₇	—	2	4	no	—
322	C₂×C₄×C₇	—	2	4	no	—
323	C₂×C₄×C₇	—	2	4	no	—
324	C₂×C₄×C₇	—	2	4	no	—
325	C₂×C₄×C₇	—	2	4	no	—
326	C₂×C₄×C₇	—	2	4	no	—
327	C₂×C₄×C₇	—	2	4	no	—
328	C₂×C₄×C₇	—	2	4	no	—
329	C₂×C₄×C₇	—	2	4	no	—
330	C₂×C₄×C₇	—	4	4	no	—
331	C₂×C₄×C₇	—	4	8	no	—
332	C₂×C₄×C₇	—	4	4	no	—
333	C₂×C₄×C₇	—	4	4	no	—
334	C₂×C₄×C₇	—	8	8	no	—
335	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
336	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
337	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
338	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
339	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
340	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
341	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
342	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
343	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
344	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
345	C₂²×D₄	C₂³×C₇	1	2	yes	—
346	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₇	1	4	yes	—
348	C₂³×C₄	C₂³×C₇	1	2	yes	—
349	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
352	#126	C₂×C₄×C₇	1	16	yes	complemented
353	#125	C₂×C₄×C₇	1	8	yes	complemented
354	#124	C₂×C₄×C₇	1	8	yes	—
355	#123	C₂×C₄×C₇	1	16	yes	complemented
356	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	complemented
357	#121	C₂×C₄×C₇	1	8	yes	—
358	C₄×Q₈	C₂×C₄×C₇	1	8	yes	complemented
359	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
360	C₄×D₄	C₂×C₄×C₇	1	16	yes	complemented
361	#117	C₂×C₄×C₇	1	16	yes	complemented
362	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
363	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
366	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
386	C₂×C₄²	C₂×C₄×C₇	1	8	yes	complemented
387	C₂×C₄²	C₂³×C₇	1	2	yes	—
388	C₂×C₄²	C₂³×C₇	1	4	yes	—
389	C₂×C₄²	C₂³×C₇	1	2	yes	—
390	C₂×C₄²	C₂³×C₇	1	4	yes	—
391	C₂×C₄²	C₂³×C₇	1	4	yes	—
392	C₂×C₄²	C₂³×C₇	1	4	yes	—
393	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₇	1	2	yes	—
395	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
398	#95	C₂³×C₇	1	2	yes	—
399	#95	C₂³×C₇	1	2	yes	—
400	#95	C₂³×C₇	1	4	yes	—
401	#95	C₂³×C₇	1	2	yes	—
402	#95	C₂³×C₇	1	4	yes	—
403	#95	C₂³×C₇	1	4	yes	—
404	#95	C₂³×C₇	1	4	yes	—
405	#95	C₂³×C₇	1	4	yes	—
406	#95	C₂³×C₇	1	2	yes	—
407	#95	C₂³×C₇	1	4	yes	—
408	#95	C₂³×C₇	1	4	yes	—
409	#95	C₂³×C₇	1	4	yes	—
410	#95	C₂³×C₇	1	4	yes	—
411	#95	C₂³×C₇	1	4	yes	—
412	#95	C₂³×C₇	1	4	yes	—
413	#95	C₂³×C₇	1	4	yes	—
414	C₂²×C₇	—	2	2	no	—
415	C₂²×C₇	—	2	2	no	—
416	C₂²×C₇	—	2	2	no	—
417	C₂²×C₇	—	2	4	no	—
418	C₂²×C₇	—	2	4	no	—
419	C₂²×C₇	—	2	4	no	—
420	C₂²×C₇	—	4	2	no	—
421	C₂²×C₇	—	4	4	no	—
422	C₂²×C₇	—	4	4	no	—
423	C₂²×C₇	—	4	2	no	—
424	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
425	C₂²×C₇	#585	1	1	yes	characteristic
426	C₂²×C₇	#585	1	2	yes	—
427	C₂²×C₇	#585	1	2	yes	—
428	C₂²×C₇	#585	1	1	yes	characteristic
429	C₄×C₇	—	2	2	no	—
430	C₄×C₇	—	2	2	no	—
431	C₄×C₇	—	2	2	no	—
432	C₄×C₇	—	2	4	no	—
433	C₄×C₇	—	2	2	no	—
434	C₄×C₇	—	2	2	no	—
435	C₄×C₇	—	2	2	no	—
436	C₄×C₇	—	2	4	no	—
437	C₄×C₇	—	2	4	no	—
438	C₄×C₇	—	4	4	no	—
439	C₄×C₇	—	4	4	no	—
440	C₄×C₇	—	4	4	no	—
441	C₄×C₇	—	4	4	no	—
442	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
443	C₂⁴	C₂⁴×C₇	1	2	yes	—
444	C₂×Q₈	—	4	1	no	—
445	C₂×Q₈	—	4	1	no	—
446	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
447	C₂×D₄	—	4	4	no	—
448	C₂×D₄	—	4	1	no	—
449	C₂×D₄	—	4	1	no	—
450	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
451	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	1	no	—
454	C₂²×C₄	—	4	1	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	—	4	4	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
461	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
462	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
463	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
464	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
465	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
466	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
468	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
477	C₄⋊C₄	—	4	4	no	—
478	C₄⋊C₄	—	4	4	no	—
479	C₄⋊C₄	—	4	2	no	—
480	C₄⋊C₄	—	4	2	no	—
481	C₄⋊C₄	—	4	2	no	—
482	C₄⋊C₄	—	4	2	no	—
483	C₄⋊C₄	—	4	4	no	—
484	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
485	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
486	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
487	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
488	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
489	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
490	C₂²⋊C₄	—	4	4	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	4	no	—
493	C₂²⋊C₄	—	4	4	no	—
494	C₂²⋊C₄	—	4	4	no	—
495	C₂²⋊C₄	—	4	2	no	—
496	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
497	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
498	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
499	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
500	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
501	C₄²	—	4	2	no	—
502	C₄²	—	4	4	no	—
503	C₄²	—	4	2	no	—
504	C₄²	—	4	4	no	—
505	C₄²	—	4	4	no	—
506	C₄²	—	4	4	no	—
507	C₄²	C₂²×C₄×C₇	1	4	yes	—
508	C₄²	C₂²×C₄×C₇	1	4	yes	—
509	C₄²	C₂²×C₄×C₇	1	4	yes	—
510	C₄²	C₂²×C₄×C₇	1	4	yes	—
511	C₄²	C₂²×C₄×C₇	1	8	yes	—
512	C₂×C₇	—	4	2	no	—
513	C₂×C₇	—	4	4	no	—
514	C₂×C₇	#3428	1	1	yes	characteristic
515	C₂×C₇	#3428	1	1	yes	characteristic
516	C₂×C₇	#3428	1	1	yes	characteristic
517	C₂×C₇	#3567	1	2	yes	—
518	C₂×C₇	#3568	1	2	yes	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	1	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	C₂³×C₄×C₇	1	4	yes	—
528	C₂³	C₂³×C₄×C₇	1	2	yes	—
529	C₂³	C₂⁵×C₇	1	1	yes	characteristic
530	Q₈	—	4	4	no	—
531	D₄	—	4	4	no	—
532	D₄	—	4	8	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	8	8	no	—
562	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
563	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
566	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
567	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
568	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
569	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
570	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
571	C₇	#34405	1	1	yes	characteristic, direct product factor, Sylow
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	2	no	—
582	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
583	C₂²	C₇×#585	1	1	yes	characteristic
584	C₂²	C₇×#585	1	2	yes	—
585	C₂²	C₇×#585	1	2	yes	—
586	C₂²	C₇×#585	1	1	yes	characteristic
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	4	no	—
595	C₄	—	2	4	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	4	no	—
598	C₄	—	4	4	no	—
599	C₄	—	4	4	no	—
600	C₂	—	4	2	no	—
601	C₂	—	4	4	no	—
602	C₂	C₇×#3428	1	1	yes	characteristic
603	C₂	C₇×#3428	1	1	yes	characteristic
604	C₂	C₇×#3428	1	1	yes	characteristic
605	C₂	C₇×#3567	1	2	yes	—
606	C₂	C₇×#3568	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×#2382	C₂	semidirect
3. C₇×#2379	C₂	semidirect
4. C₇×#2373	C₂	semidirect
5. C₇×#2356	C₂	semidirect
6. C₇×#2353	C₂	semidirect
7. C₇×#2347	C₂	semidirect
8. C₇×#2343	C₂	semidirect
9. C₇×#2335	C₂	semidirect
11. C₇×#2332	C₂	semidirect
12. C₇×#2319	C₂	semidirect
13. C₇×#2318	C₂	semidirect
14. C₇×#2283	C₂	semidirect
15. C₇×#2283	C₂	semidirect
16. C₇×#529	C₄	semidirect
76. #34405	C₇	direct (N × H)
85. C₇×#126	C₂×C₄	semidirect
86. C₇×#125	C₂×C₄	semidirect
88. C₇×#123	C₂×C₄	semidirect
89. C₇×C₂²⋊Q₈	C₂×C₄	semidirect
91. C₄×C₇×Q₈	C₂×C₄	semidirect
93. C₄×C₇×D₄	C₂×C₄	semidirect
94. C₇×#117	C₂×C₄	semidirect
119. C₂×C₄²×C₇	C₂×C₄	semidirect
148. #2382	C₂×C₇	semidirect
149. #2379	C₂×C₇	semidirect
150. #2373	C₂×C₇	semidirect
151. #2356	C₂×C₇	semidirect
152. #2353	C₂×C₇	semidirect
153. #2347	C₂×C₇	semidirect
154. #2343	C₂×C₇	semidirect
155. #2335	C₂×C₇	semidirect
157. #2332	C₂×C₇	semidirect
158. #2319	C₂×C₇	semidirect
159. #2318	C₂×C₇	semidirect
160. #2283	C₂×C₇	semidirect
161. #2283	C₂×C₇	semidirect
232. #529	C₄×C₇	semidirect
352. #126	C₂×C₄×C₇	semidirect
353. #125	C₂×C₄×C₇	semidirect
355. #123	C₂×C₄×C₇	semidirect
356. C₂²⋊Q₈	C₂×C₄×C₇	semidirect
358. C₄×Q₈	C₂×C₄×C₇	semidirect
360. C₄×D₄	C₂×C₄×C₇	semidirect
361. #117	C₂×C₄×C₇	semidirect
386. C₂×C₄²	C₂×C₄×C₇	semidirect
571. C₇	#34405	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×#2382	2
C₇×#2379	2
C₇×#2373	2
C₇×#2356	2
C₇×#2353	2
C₇×#2347	2
C₇×#2343	2
C₇×#2335	2
C₇×#2333	2
C₇×#2332	2
C₇×#2319	2
C₇×#2318	2
C₇×#2283	2
#34405	7

Subgroups — C7×#34405 #13211621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34405 #13211621