Subgroups — C₇×#22821 #13200037

Order 1792.

784 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1984	C₂	1	2	yes	maximal
3	C₇×#1916	C₂	1	4	yes	maximal
4	C₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₇×#1831	C₂	1	2	yes	maximal
6	C₇×C₈×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₄×C₇×C₄⋊C₈	C₂	1	2	yes	maximal
8	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
19	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
21	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
22	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄³×C₇	C₂²	1	2	yes	—
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#336	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	4	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	C₇×#336	C₂²	1	4	yes	—
31	#22821	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	2	yes	—
34	C₂²×C₇×C₈	C₂³	1	4	yes	—
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₄×C₇×Q₈	—	2	4	no	—
37	C₄×C₇×Q₈	—	2	2	no	—
38	C₄×C₇×Q₈	—	2	2	no	—
39	C₄×C₇×Q₈	C₈	1	8	yes	complemented
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
59	C₇×C₄⋊C₈	—	2	4	no	—
60	C₇×C₄⋊C₈	—	2	2	no	—
61	C₇×C₄⋊C₈	—	2	4	no	—
62	C₇×C₄⋊C₈	—	4	2	no	—
63	C₇×C₄⋊C₈	Q₈	1	4	yes	complemented
64	C₄×C₇×C₈	—	2	4	no	—
65	C₄×C₇×C₈	—	4	4	no	—
66	C₇×#95	C₂×C₄	1	2	yes	—
67	C₇×#95	C₂×C₄	1	2	yes	—
68	C₇×#95	C₂×C₄	1	2	yes	—
69	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic
70	#1984	C₂×C₇	1	2	yes	—
71	#1916	C₂×C₇	1	4	yes	—
72	#1842	C₂×C₇	1	1	yes	characteristic
73	#1831	C₂×C₇	1	2	yes	—
74	C₈×C₄⋊C₄	C₂×C₇	1	2	yes	—
75	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
76	C₄×C₄⋊C₈	C₂×C₇	1	2	yes	—
77	C₂×C₇×Q₈	—	2	2	no	—
78	C₂×C₇×Q₈	—	2	2	no	—
79	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
80	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	4	no	—
96	C₂×C₇×C₈	—	2	4	no	—
97	C₂×C₇×C₈	—	2	4	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
104	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
105	C₇×C₄⋊C₄	M₁₆	1	4	yes	—
106	C₄²×C₇	—	2	4	no	—
107	C₄²×C₇	—	2	2	no	—
108	C₄²×C₇	—	2	2	no	—
109	C₄²×C₇	—	2	4	no	—
110	C₄²×C₇	—	2	2	no	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	—	2	1	no	—
114	C₄²×C₇	—	2	2	no	—
115	C₄²×C₇	—	2	1	no	—
116	C₄²×C₇	—	4	1	no	—
117	C₄²×C₇	—	4	2	no	—
118	C₄²×C₇	C₂×C₈	1	8	yes	—
119	C₄²×C₇	C₂×C₈	1	4	yes	—
120	C₄²×C₇	C₂×Q₈	1	2	yes	—
121	C₄²×C₇	#41	1	2	yes	—
122	C₄²×C₇	#41	1	2	yes	—
123	C₄²×C₇	#41	1	2	yes	—
124	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
127	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
128	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
134	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
135	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
136	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
137	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
138	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
139	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
140	C₄³	C₂²×C₇	1	2	yes	—
141	C₄³	C₂²×C₇	1	1	yes	characteristic
142	#336	C₂²×C₇	1	4	yes	—
143	#336	C₂²×C₇	1	4	yes	—
144	#336	C₂²×C₇	1	2	yes	—
145	#336	C₂²×C₇	1	4	yes	—
146	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
147	C₇×Q₈	—	2	8	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	1	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	1	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	1	no	—
157	C₂×C₄×C₇	—	2	1	no	—
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	1	no	—
162	C₂×C₄×C₇	—	2	1	no	—
163	C₂×C₄×C₇	—	2	2	no	—
164	C₂×C₄×C₇	—	2	1	no	—
165	C₂×C₄×C₇	—	2	2	no	—
166	C₂×C₄×C₇	#117	1	2	yes	—
167	C₂×C₄×C₇	#117	1	2	yes	—
168	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
170	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
171	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
172	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
173	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
174	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
175	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
176	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
177	C₂×C₄×C₇	#131	1	2	yes	—
178	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
184	C₇×C₈	—	2	4	no	—
185	C₇×C₈	—	4	2	no	—
186	C₇×C₈	—	4	4	no	—
187	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₇	1	2	yes	—
189	C₂²×C₈	C₂³×C₇	1	4	yes	—
190	C₂²×C₈	C₂³×C₇	1	2	yes	—
191	C₄×Q₈	—	2	4	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	C₇×C₈	1	8	yes	complemented
195	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
203	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
205	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
206	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
207	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
208	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₇	1	2	yes	—
210	C₂×C₄²	C₂³×C₇	1	2	yes	—
211	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	4	2	no	—
218	C₄⋊C₈	C₇×Q₈	1	4	yes	complemented
219	C₄×C₈	—	2	4	no	—
220	C₄×C₈	—	4	4	no	—
221	#95	C₂×C₄×C₇	1	2	yes	—
222	#95	C₂×C₄×C₇	1	2	yes	—
223	#95	C₂×C₄×C₇	1	2	yes	—
224	C₂²×C₇	C₂×#117	1	1	yes	characteristic
225	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
226	C₂²×C₇	#520	1	1	yes	characteristic
227	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
228	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
229	C₂²×C₇	C₂×#131	1	1	yes	characteristic
230	C₂²×C₇	#568	1	1	yes	characteristic
231	C₄×C₇	—	2	4	no	—
232	C₄×C₇	—	2	2	no	—
233	C₄×C₇	—	2	2	no	—
234	C₄×C₇	—	2	4	no	—
235	C₄×C₇	—	2	2	no	—
236	C₄×C₇	—	2	4	no	—
237	C₄×C₇	—	4	2	no	—
238	C₄×C₇	—	4	1	no	—
239	C₄×C₇	#431	1	4	yes	—
240	C₄×C₇	C₈×Q₈	1	4	yes	—
241	C₄×C₇	#554	1	1	yes	characteristic
242	C₄×C₇	#556	1	1	yes	characteristic
243	C₄×C₇	#557	1	1	yes	characteristic
244	C₄×C₇	#559	1	1	yes	characteristic
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	C₂×C₇×C₈	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
253	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
256	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
258	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	C₂×C₇×C₈	1	8	yes	—
272	C₄⋊C₄	C₂×C₇×C₈	1	4	yes	—
273	C₄⋊C₄	C₇×M₁₆	1	4	yes	—
274	C₄²	—	2	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	2	no	—
286	C₄²	C₂×C₇×C₈	1	8	yes	—
287	C₄²	C₂×C₇×C₈	1	4	yes	—
288	C₄²	C₂×C₇×Q₈	1	2	yes	—
289	C₄²	C₇×#41	1	2	yes	—
290	C₄²	C₇×#41	1	2	yes	—
291	C₄²	C₇×#41	1	2	yes	—
292	C₂×C₇	#2355	1	1	yes	characteristic
293	C₂×C₇	C₂×#431	1	1	yes	characteristic
294	C₂×C₇	#2925	1	1	yes	characteristic
295	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
296	C₂×C₇	Q₈×M₁₆	1	1	yes	characteristic
297	C₂×C₇	#2988	1	1	yes	characteristic
298	C₂×C₇	#2990	1	1	yes	characteristic
299	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	C₇×#117	1	2	yes	—
320	C₂×C₄	C₇×#117	1	2	yes	—
321	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
322	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
323	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
324	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
325	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
326	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
327	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
328	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
329	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
330	C₂×C₄	C₇×#131	1	2	yes	—
331	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
332	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
333	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
334	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
335	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
336	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	4	no	—
340	C₇	#22821	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²	C₂×C₇×#117	1	1	yes	characteristic
342	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
343	C₂²	C₇×#520	1	1	yes	characteristic
344	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
345	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
346	C₂²	C₂×C₇×#131	1	1	yes	characteristic
347	C₂²	C₇×#568	1	1	yes	characteristic
348	C₄	—	2	4	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	4	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	1	no	—
356	C₄	C₇×#431	1	4	yes	—
357	C₄	C₇×C₈×Q₈	1	4	yes	—
358	C₄	C₇×#554	1	1	yes	characteristic
359	C₄	C₇×#556	1	1	yes	characteristic
360	C₄	C₇×#557	1	1	yes	characteristic
361	C₄	C₇×#559	1	1	yes	characteristic
362	C₂	C₇×#2355	1	1	yes	characteristic
363	C₂	C₂×C₇×#431	1	1	yes	characteristic
364	C₂	C₇×#2925	1	1	yes	characteristic
365	C₂	C₂×C₇×C₈×Q₈	1	1	yes	characteristic
366	C₂	C₇×Q₈×M₁₆	1	1	yes	characteristic
367	C₂	C₇×#2988	1	1	yes	characteristic
368	C₂	C₇×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
31. #22821	C₇	direct (N × H)
39. C₄×C₇×Q₈	C₈	semidirect
63. C₇×C₄⋊C₈	Q₈	semidirect
194. C₄×Q₈	C₇×C₈	semidirect
218. C₄⋊C₈	C₇×Q₈	semidirect
340. C₇	#22821	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₇×Q₈	2
C₇×#1984	2
C₇×#1916	2
C₇×#1842	2
C₇×#1831	2
C₇×C₈×C₄⋊C₄	2
C₄×C₇×C₄⋊C₈	2
#22821	7

Subgroups — C7×#22821 #13200037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22821 #13200037