Subgroups — C₇×#22625 #13199841

Order 1792.

916 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#1939	C₂	1	2	yes	maximal, complemented
3	C₇×#1916	C₂	1	4	yes	maximal, complemented
4	C₇×#1797	C₂	1	2	yes	maximal
5	C₇×#1797	C₂	1	1	yes	maximal, characteristic
6	C₇×C₈×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₄×C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×C₇×C₂²⋊C₈	C₂	1	2	yes	maximal
9	C₂²×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂²×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₇	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
20	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
21	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
22	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
23	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₇×#95	C₄	1	2	yes	—
27	C₇×#336	C₂²	1	2	yes	—
28	C₇×#336	C₂²	1	4	yes	—
29	C₇×#336	C₂²	1	4	yes	—
30	C₇×#336	C₂²	1	2	yes	—
31	#22625	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	C₂³	1	4	yes	—
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	—	2	4	no	—
41	C₂×C₇×C₄⋊C₄	C₈	1	8	yes	complemented
42	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇	—	2	4	no	—
49	C₂×C₄²×C₇	—	4	4	no	—
50	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₇×C₄⋊C₈	—	2	4	no	—
59	C₇×C₄⋊C₈	—	4	4	no	—
60	C₇×C₂²⋊C₈	—	2	2	no	—
61	C₇×C₂²⋊C₈	—	2	4	no	—
62	C₇×C₂²⋊C₈	Q₈	1	4	yes	—
63	C₄×C₇×C₈	—	2	4	no	—
64	C₄×C₇×C₈	—	4	4	no	—
65	C₇×#95	C₂×C₄	1	2	yes	—
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	C₂×C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
70	#1939	C₂×C₇	1	2	yes	complemented
71	#1916	C₂×C₇	1	4	yes	complemented
72	#1797	C₂×C₇	1	1	yes	characteristic
73	#1797	C₂×C₇	1	2	yes	—
74	C₈×C₄⋊C₄	C₂×C₇	1	2	yes	complemented
75	C₄×C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic
76	C₄×C₂²⋊C₈	C₂×C₇	1	2	yes	—
77	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	—	2	2	no	—
79	C₂²×C₄×C₇	—	2	2	no	—
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	2	no	—
83	C₂²×C₄×C₇	—	2	4	no	—
84	C₂²×C₄×C₇	—	2	2	no	—
85	C₂²×C₄×C₇	—	2	4	no	—
86	C₂²×C₄×C₇	—	4	2	no	—
87	C₂²×C₄×C₇	—	4	2	no	—
88	C₂²×C₄×C₇	—	4	2	no	—
89	C₂²×C₄×C₇	—	4	1	no	—
90	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
91	C₂²×C₄×C₇	C₂×C₈	1	8	yes	—
92	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
102	C₂²×C₄×C₇	#41	1	2	yes	—
103	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₇×C₈	—	2	2	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	—	2	2	no	—
107	C₂×C₇×C₈	—	2	2	no	—
108	C₂×C₇×C₈	—	2	4	no	—
109	C₂×C₇×C₈	—	2	4	no	—
110	C₂×C₇×C₈	—	2	4	no	—
111	C₂×C₇×C₈	—	2	2	no	—
112	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
113	C₇×C₄⋊C₄	—	2	4	no	—
114	C₇×C₄⋊C₄	—	2	2	no	—
115	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
116	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
117	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
118	C₇×C₄⋊C₄	M₁₆	1	4	yes	—
119	C₄²×C₇	—	2	2	no	—
120	C₄²×C₇	—	2	2	no	—
121	C₄²×C₇	—	2	2	no	—
122	C₄²×C₇	—	2	4	no	—
123	C₄²×C₇	—	2	4	no	—
124	C₄²×C₇	—	4	2	no	—
125	C₄²×C₇	—	4	2	no	—
126	C₄²×C₇	#41	1	4	yes	—
127	C₂²×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
128	C₂²×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
129	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
130	C₂²×C₄²	C₂²×C₇	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
132	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
133	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
134	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
135	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
138	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
139	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
140	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
141	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
142	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
143	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
144	C₂×#95	C₄×C₇	1	2	yes	—
145	#336	C₂²×C₇	1	2	yes	—
146	#336	C₂²×C₇	1	4	yes	—
147	#336	C₂²×C₇	1	2	yes	—
148	#336	C₂²×C₇	1	4	yes	—
149	C₂³×C₇	—	2	1	no	—
150	C₂³×C₇	—	2	1	no	—
151	C₂³×C₇	—	2	1	no	—
152	C₂³×C₇	—	2	1	no	—
153	C₂³×C₇	C₄×Q₈	1	2	yes	—
154	C₂³×C₇	C₂²×C₈	1	2	yes	—
155	C₂³×C₇	#131	1	2	yes	—
156	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
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	2	no	—
162	C₂×C₄×C₇	—	2	1	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	1	no	—
165	C₂×C₄×C₇	—	2	2	no	—
166	C₂×C₄×C₇	—	2	1	no	—
167	C₂×C₄×C₇	—	2	2	no	—
168	C₂×C₄×C₇	—	2	2	no	—
169	C₂×C₄×C₇	—	2	4	no	—
170	C₂×C₄×C₇	—	2	4	no	—
171	C₂×C₄×C₇	—	2	8	no	—
172	C₂×C₄×C₇	—	2	1	no	—
173	C₂×C₄×C₇	—	2	2	no	—
174	C₂×C₄×C₇	—	2	2	no	—
175	C₂×C₄×C₇	—	2	1	no	—
176	C₂×C₄×C₇	—	2	2	no	—
177	C₂×C₄×C₇	—	2	1	no	—
178	C₂×C₄×C₇	—	4	8	no	—
179	C₂×C₄×C₇	—	4	4	no	—
180	C₂×C₄×C₇	—	4	8	no	—
181	C₂×C₄×C₇	#117	1	2	yes	—
182	C₂×C₄×C₇	#117	1	2	yes	—
183	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
184	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
185	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
186	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
187	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
188	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
189	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
190	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
191	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₇	C₂×#41	1	2	yes	—
193	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
194	C₇×C₈	—	2	4	no	—
195	C₇×C₈	—	4	2	no	—
196	C₇×C₈	—	4	4	no	—
197	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
198	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
199	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
200	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
201	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₇	1	2	yes	—
203	C₂²×C₈	C₂³×C₇	1	4	yes	—
204	C₂²×C₈	C₂³×C₇	1	2	yes	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	C₇×C₈	1	8	yes	complemented
207	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	—	4	4	no	—
215	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
216	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
217	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
218	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
219	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₇	1	2	yes	—
221	C₂×C₄²	C₂³×C₇	1	2	yes	—
222	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	4	4	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	4	no	—
227	C₂²⋊C₈	C₇×Q₈	1	4	yes	—
228	C₄×C₈	—	2	4	no	—
229	C₄×C₈	—	4	4	no	—
230	#95	C₂×C₄×C₇	1	2	yes	—
231	#95	C₂×C₄×C₇	1	2	yes	—
232	#95	C₂×C₄×C₇	1	2	yes	—
233	#95	C₂×C₄×C₇	1	2	yes	—
234	C₂²×C₇	—	2	2	no	—
235	C₂²×C₇	—	2	2	no	—
236	C₂²×C₇	—	2	2	no	—
237	C₂²×C₇	—	2	2	no	—
238	C₂²×C₇	—	2	2	no	—
239	C₂²×C₇	—	2	2	no	—
240	C₂²×C₇	C₈×Q₈	1	4	yes	—
241	C₂²×C₇	C₂×#117	1	1	yes	characteristic
242	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
243	C₂²×C₇	#518	1	1	yes	characteristic
244	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
245	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
246	C₂²×C₇	C₂×#131	1	1	yes	characteristic
247	C₂²×C₇	#568	1	1	yes	characteristic
248	C₄×C₇	—	2	2	no	—
249	C₄×C₇	—	2	2	no	—
250	C₄×C₇	—	2	2	no	—
251	C₄×C₇	—	2	4	no	—
252	C₄×C₇	—	2	4	no	—
253	C₄×C₇	—	2	4	no	—
254	C₄×C₇	—	4	1	no	—
255	C₄×C₇	—	4	2	no	—
256	C₄×C₇	#431	1	4	yes	—
257	C₄×C₇	#553	1	2	yes	—
258	C₄×C₇	#554	1	2	yes	—
259	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
273	C₂²×C₄	C₂×C₇×C₈	1	8	yes	—
274	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
276	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
277	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
280	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
282	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
283	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
284	C₂²×C₄	C₇×#41	1	2	yes	—
285	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	2	4	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	C₂×C₇×C₈	1	8	yes	—
298	C₄⋊C₄	C₂×C₇×C₈	1	4	yes	—
299	C₄⋊C₄	C₂×C₇×C₈	1	4	yes	—
300	C₄⋊C₄	C₇×M₁₆	1	4	yes	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	4	no	—
305	C₄²	—	2	4	no	—
306	C₄²	—	4	2	no	—
307	C₄²	—	4	2	no	—
308	C₄²	C₇×#41	1	4	yes	—
309	C₂×C₇	—	2	4	no	—
310	C₂×C₇	#2345	1	1	yes	characteristic
311	C₂×C₇	C₂×#431	1	1	yes	characteristic
312	C₂×C₇	#2924	1	1	yes	characteristic
313	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
314	C₂×C₇	Q₈×M₁₆	1	1	yes	characteristic
315	C₂×C₇	#2972	1	1	yes	characteristic
316	C₂×C₇	#2974	1	1	yes	characteristic
317	C₂³	—	2	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	2	1	no	—
320	C₂³	—	2	1	no	—
321	C₂³	C₄×C₇×Q₈	1	2	yes	—
322	C₂³	C₂²×C₇×C₈	1	2	yes	—
323	C₂³	C₇×#131	1	2	yes	—
324	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	8	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	8	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	8	no	—
349	C₂×C₄	C₇×#117	1	2	yes	—
350	C₂×C₄	C₇×#117	1	2	yes	—
351	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
352	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
353	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
354	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
355	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
356	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
357	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
358	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
359	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
360	C₂×C₄	C₂×C₇×#41	1	2	yes	—
361	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
362	C₈	—	2	4	no	—
363	C₈	—	4	2	no	—
364	C₈	—	4	4	no	—
365	C₇	#22625	1	1	yes	characteristic, direct product factor, Sylow
366	C₂²	—	2	2	no	—
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	2	no	—
372	C₂²	C₇×C₈×Q₈	1	4	yes	—
373	C₂²	C₂×C₇×#117	1	1	yes	characteristic
374	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
375	C₂²	C₇×#518	1	1	yes	characteristic
376	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
377	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
378	C₂²	C₂×C₇×#131	1	1	yes	characteristic
379	C₂²	C₇×#568	1	1	yes	characteristic
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	4	no	—
383	C₄	—	2	4	no	—
384	C₄	—	2	4	no	—
385	C₄	—	2	2	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	2	no	—
388	C₄	C₇×#431	1	4	yes	—
389	C₄	C₇×#553	1	2	yes	—
390	C₄	C₇×#554	1	2	yes	—
391	C₂	—	2	4	no	—
392	C₂	C₇×#2345	1	1	yes	characteristic
393	C₂	C₂×C₇×#431	1	1	yes	characteristic
394	C₂	C₇×#2924	1	1	yes	characteristic
395	C₂	C₂×C₇×C₈×Q₈	1	1	yes	characteristic
396	C₂	C₇×Q₈×M₁₆	1	1	yes	characteristic
397	C₂	C₇×#2972	1	1	yes	characteristic
398	C₂	C₇×#2974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×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
2. C₇×#1939	C₂	semidirect
3. C₇×#1916	C₂	semidirect
6. C₇×C₈×C₄⋊C₄	C₂	semidirect
31. #22625	C₇	direct (N × H)
41. C₂×C₇×C₄⋊C₄	C₈	semidirect
70. #1939	C₂×C₇	semidirect
71. #1916	C₂×C₇	semidirect
74. C₈×C₄⋊C₄	C₂×C₇	semidirect
206. C₂×C₄⋊C₄	C₇×C₈	semidirect
365. C₇	#22625	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×C₄×C₇×C₄⋊C₄	2
C₇×#1939	2
C₇×#1916	2
C₇×#1797	2
C₇×C₈×C₄⋊C₄	2
C₄×C₇×C₂²⋊C₈	2
#22625	7

Subgroups — C7×#22625 #13199841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22625 #13199841