Subgroups — C₇×#22822 #13200038

Order 1792.

656 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₇×#1984	C₂	1	1	yes	maximal, characteristic
3	C₇×#1984	C₂	1	1	yes	maximal, characteristic
4	C₇×#1917	C₂	1	2	yes	maximal
5	C₇×#1916	C₂	1	2	yes	maximal
6	C₇×#1842	C₂	1	1	yes	maximal, characteristic
7	C₇×#1832	C₂	1	2	yes	maximal
8	C₇×#1770	C₂	1	1	yes	maximal, characteristic
9	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
10	C₇×#1768	C₂	1	2	yes	maximal
11	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
23	C₇×#391	C₄	1	1	yes	characteristic
24	C₇×#391	C₄	1	1	yes	characteristic
25	C₇×#389	C₄	1	1	yes	characteristic
26	C₇×#389	C₄	1	1	yes	characteristic
27	C₇×#387	C₄	1	2	yes	—
28	C₇×#387	C₄	1	2	yes	—
29	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
31	C₇×#376	C₂²	1	2	yes	—
32	C₄³×C₇	C₂²	1	1	yes	characteristic
33	C₇×#336	C₂²	1	2	yes	—
34	C₇×#336	C₂²	1	2	yes	—
35	C₇×#336	C₂²	1	2	yes	—
36	C₇×#336	C₂²	1	2	yes	—
37	C₇×#336	C₂²	1	1	yes	characteristic
38	C₇×#336	C₂²	1	1	yes	characteristic
39	C₇×#336	C₂²	1	2	yes	—
40	C₇×#336	C₂²	1	2	yes	—
41	#22822	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₇×C₈	C₂³	1	2	yes	—
46	C₂²×C₇×C₈	C₂³	1	2	yes	—
47	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₄×C₇×Q₈	—	2	2	no	—
50	C₄×C₇×Q₈	—	2	2	no	—
51	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
66	C₇×C₄⋊C₈	—	2	2	no	—
67	C₇×C₄⋊C₈	—	2	2	no	—
68	C₇×C₄⋊C₈	—	2	2	no	—
69	C₇×C₄⋊C₈	—	2	4	no	—
70	C₇×C₄⋊C₈	—	4	2	no	—
71	C₇×C₄⋊C₈	Q₈	1	4	yes	complemented
72	C₇×#97	—	4	2	no	—
73	C₄×C₇×C₈	—	2	2	no	—
74	C₄×C₇×C₈	—	2	2	no	—
75	C₄×C₇×C₈	—	4	2	no	—
76	C₇×#95	C₂×C₄	1	2	yes	—
77	C₇×#95	C₂×C₄	1	2	yes	—
78	C₇×#95	C₂×C₄	1	2	yes	—
79	C₇×#95	C₂×C₄	1	2	yes	—
80	C₇×#95	C₂×C₄	1	2	yes	—
81	C₇×#95	C₂×C₄	1	2	yes	—
82	C₇×#95	C₂×C₄	1	2	yes	—
83	#2295	C₂×C₇	1	1	yes	characteristic
84	#1984	C₂×C₇	1	1	yes	characteristic
85	#1984	C₂×C₇	1	1	yes	characteristic
86	#1917	C₂×C₇	1	2	yes	—
87	#1916	C₂×C₇	1	2	yes	—
88	#1842	C₂×C₇	1	1	yes	characteristic
89	#1832	C₂×C₇	1	2	yes	—
90	#1770	C₂×C₇	1	1	yes	characteristic
91	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
92	#1768	C₂×C₇	1	2	yes	—
93	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
94	C₂×C₇×Q₈	—	2	2	no	—
95	C₂×C₇×Q₈	—	2	2	no	—
96	C₂×C₇×Q₈	—	2	2	no	—
97	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
108	C₂×C₇×C₈	—	2	1	no	—
109	C₂×C₇×C₈	—	2	1	no	—
110	C₂×C₇×C₈	—	2	1	no	—
111	C₂×C₇×C₈	—	2	1	no	—
112	C₂×C₇×C₈	—	2	1	no	—
113	C₂×C₇×C₈	—	2	1	no	—
114	C₂×C₇×C₈	—	2	1	no	—
115	C₂×C₇×C₈	—	2	1	no	—
116	C₂×C₇×C₈	—	2	2	no	—
117	C₂×C₇×C₈	—	2	2	no	—
118	C₂×C₇×C₈	—	2	2	no	—
119	C₂×C₇×C₈	—	2	2	no	—
120	C₂×C₇×C₈	—	2	2	no	—
121	C₂×C₇×C₈	—	2	2	no	—
122	C₂×C₇×C₈	—	2	1	no	—
123	C₂×C₇×C₈	—	2	1	no	—
124	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
125	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
126	C₇×C₄⋊C₄	—	2	2	no	—
127	C₇×C₄⋊C₄	—	2	4	no	—
128	C₇×C₄⋊C₄	—	2	4	no	—
129	C₇×C₄⋊C₄	—	2	2	no	—
130	C₇×C₄⋊C₄	—	2	2	no	—
131	C₇×C₄⋊C₄	—	4	2	no	—
132	C₄²×C₇	—	2	2	no	—
133	C₄²×C₇	—	2	2	no	—
134	C₄²×C₇	—	2	2	no	—
135	C₄²×C₇	—	2	2	no	—
136	C₄²×C₇	—	2	2	no	—
137	C₄²×C₇	—	2	2	no	—
138	C₄²×C₇	—	2	1	no	—
139	C₄²×C₇	—	2	1	no	—
140	C₄²×C₇	—	2	2	no	—
141	C₄²×C₇	—	4	1	no	—
142	C₄²×C₇	C₂×Q₈	1	2	yes	—
143	C₄²×C₇	#41	1	2	yes	—
144	C₄²×C₇	#41	1	2	yes	—
145	C₄²×C₇	#41	1	2	yes	—
146	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
153	C₂×#97	C₂²×C₇	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
157	#391	C₄×C₇	1	1	yes	characteristic
158	#391	C₄×C₇	1	1	yes	characteristic
159	#389	C₄×C₇	1	1	yes	characteristic
160	#389	C₄×C₇	1	1	yes	characteristic
161	#387	C₄×C₇	1	2	yes	—
162	#387	C₄×C₇	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
164	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
165	#376	C₂²×C₇	1	2	yes	—
166	C₄³	C₂²×C₇	1	1	yes	characteristic
167	#336	C₂²×C₇	1	2	yes	—
168	#336	C₂²×C₇	1	2	yes	—
169	#336	C₂²×C₇	1	2	yes	—
170	#336	C₂²×C₇	1	2	yes	—
171	#336	C₂²×C₇	1	1	yes	characteristic
172	#336	C₂²×C₇	1	2	yes	—
173	#336	C₂²×C₇	1	1	yes	characteristic
174	#336	C₂²×C₇	1	2	yes	—
175	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
176	C₇×Q₈	—	4	4	no	—
177	C₂×C₄×C₇	—	2	1	no	—
178	C₂×C₄×C₇	—	2	1	no	—
179	C₂×C₄×C₇	—	2	1	no	—
180	C₂×C₄×C₇	—	2	1	no	—
181	C₂×C₄×C₇	—	2	1	no	—
182	C₂×C₄×C₇	—	2	1	no	—
183	C₂×C₄×C₇	—	2	1	no	—
184	C₂×C₄×C₇	—	2	1	no	—
185	C₂×C₄×C₇	—	2	2	no	—
186	C₂×C₄×C₇	—	2	2	no	—
187	C₂×C₄×C₇	—	2	2	no	—
188	C₂×C₄×C₇	—	2	1	no	—
189	C₂×C₄×C₇	—	2	2	no	—
190	C₂×C₄×C₇	—	2	1	no	—
191	C₂×C₄×C₇	—	2	2	no	—
192	C₂×C₄×C₇	—	2	2	no	—
193	C₂×C₄×C₇	—	2	2	no	—
194	C₂×C₄×C₇	—	2	2	no	—
195	C₂×C₄×C₇	—	2	2	no	—
196	C₂×C₄×C₇	—	2	2	no	—
197	C₂×C₄×C₇	—	2	1	no	—
198	C₂×C₄×C₇	—	2	2	no	—
199	C₂×C₄×C₇	—	2	2	no	—
200	C₂×C₄×C₇	#117	1	2	yes	—
201	C₂×C₄×C₇	#117	1	2	yes	—
202	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
203	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
204	C₂×C₄×C₇	#131	1	2	yes	—
205	C₂×C₄×C₇	#131	1	2	yes	—
206	C₂×C₄×C₇	#131	1	2	yes	—
207	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
208	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
213	C₇×C₈	—	2	2	no	—
214	C₇×C₈	—	2	2	no	—
215	C₇×C₈	—	4	1	no	—
216	C₇×C₈	—	4	1	no	—
217	C₇×C₈	—	4	2	no	—
218	C₇×C₈	—	4	2	no	—
219	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₇	1	2	yes	—
223	C₂²×C₈	C₂³×C₇	1	2	yes	—
224	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
230	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
234	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
235	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
236	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
237	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₇	1	2	yes	—
239	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₇	1	2	yes	—
241	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
242	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	4	no	—
247	C₄⋊C₈	—	4	2	no	—
248	C₄⋊C₈	C₇×Q₈	1	4	yes	complemented
249	#97	—	4	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	—	4	2	no	—
253	#95	C₂×C₄×C₇	1	2	yes	—
254	#95	C₂×C₄×C₇	1	2	yes	—
255	#95	C₂×C₄×C₇	1	2	yes	—
256	#95	C₂×C₄×C₇	1	2	yes	—
257	#95	C₂×C₄×C₇	1	2	yes	—
258	#95	C₂×C₄×C₇	1	2	yes	—
259	#95	C₂×C₄×C₇	1	2	yes	—
260	C₂²×C₇	C₂×#117	1	1	yes	characteristic
261	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
262	C₂²×C₇	#520	1	1	yes	characteristic
263	C₂²×C₇	C₂×#131	1	1	yes	characteristic
264	C₂²×C₇	C₂×#131	1	1	yes	characteristic
265	C₂²×C₇	C₂×#131	1	1	yes	characteristic
266	C₂²×C₇	#568	1	1	yes	characteristic
267	C₄×C₇	—	2	2	no	—
268	C₄×C₇	—	2	2	no	—
269	C₄×C₇	—	2	2	no	—
270	C₄×C₇	—	2	2	no	—
271	C₄×C₇	—	4	2	no	—
272	C₄×C₇	—	4	2	no	—
273	C₄×C₇	—	4	1	no	—
274	C₄×C₇	—	4	2	no	—
275	C₄×C₇	—	4	2	no	—
276	C₄×C₇	#433	1	4	yes	—
277	C₄×C₇	#554	1	1	yes	characteristic
278	C₄×C₇	#556	1	1	yes	characteristic
279	C₄×C₇	#557	1	1	yes	characteristic
280	C₄×C₇	#559	1	1	yes	characteristic
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
287	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
288	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
289	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
292	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
293	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
294	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
312	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	1	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	4	1	no	—
329	C₄²	C₂×C₇×Q₈	1	2	yes	—
330	C₄²	C₇×#41	1	2	yes	—
331	C₄²	C₇×#41	1	2	yes	—
332	C₄²	C₇×#41	1	2	yes	—
333	C₂×C₇	#2355	1	1	yes	characteristic
334	C₂×C₇	C₂×#433	1	1	yes	characteristic
335	C₂×C₇	#2925	1	1	yes	characteristic
336	C₂×C₇	#2960	1	1	yes	characteristic
337	C₂×C₇	#2962	1	1	yes	characteristic
338	C₂×C₇	#2993	1	1	yes	characteristic
339	C₂×C₇	#2994	1	1	yes	characteristic
340	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
341	Q₈	—	4	4	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	C₇×#117	1	2	yes	—
366	C₂×C₄	C₇×#117	1	2	yes	—
367	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
368	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
369	C₂×C₄	C₇×#131	1	2	yes	—
370	C₂×C₄	C₇×#131	1	2	yes	—
371	C₂×C₄	C₇×#131	1	2	yes	—
372	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
373	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
374	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
375	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
376	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
377	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	—	2	2	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	1	no	—
382	C₈	—	4	2	no	—
383	C₈	—	4	2	no	—
384	C₇	#22822	1	1	yes	characteristic, direct product factor, Sylow
385	C₂²	C₂×C₇×#117	1	1	yes	characteristic
386	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
387	C₂²	C₇×#520	1	1	yes	characteristic
388	C₂²	C₂×C₇×#131	1	1	yes	characteristic
389	C₂²	C₂×C₇×#131	1	1	yes	characteristic
390	C₂²	C₂×C₇×#131	1	1	yes	characteristic
391	C₂²	C₇×#568	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	2	2	no	—
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	2	no	—
398	C₄	—	4	1	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	2	no	—
401	C₄	C₇×#433	1	4	yes	—
402	C₄	C₇×#554	1	1	yes	characteristic
403	C₄	C₇×#556	1	1	yes	characteristic
404	C₄	C₇×#557	1	1	yes	characteristic
405	C₄	C₇×#559	1	1	yes	characteristic
406	C₂	C₇×#2355	1	1	yes	characteristic
407	C₂	C₂×C₇×#433	1	1	yes	characteristic
408	C₂	C₇×#2925	1	1	yes	characteristic
409	C₂	C₇×#2960	1	1	yes	characteristic
410	C₂	C₇×#2962	1	1	yes	characteristic
411	C₂	C₇×#2993	1	1	yes	characteristic
412	C₂	C₇×#2994	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
41. #22822	C₇	direct (N × H)
71. C₇×C₄⋊C₈	Q₈	semidirect
248. C₄⋊C₈	C₇×Q₈	semidirect
384. C₇	#22822	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2295	2
C₇×#1984	2
C₇×#1917	2
C₇×#1916	2
C₇×#1842	2
C₇×#1832	2
C₇×#1770	2
C₇×C₈×C₄⋊C₄	2
C₇×#1768	2
C₄×C₇×C₄⋊C₈	2
#22822	7

Subgroups — C7×#22822 #13200038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22822 #13200038