Subgroups — C₇×#23242 #13200458

Order 1792.

764 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1984	C₂	1	2	yes	maximal, complemented
3	C₇×#1917	C₂	1	2	yes	maximal, complemented
4	C₇×#1917	C₂	1	2	yes	maximal, complemented
5	C₇×#1851	C₂	1	2	yes	maximal
6	C₇×#1771	C₂	1	2	yes	maximal, complemented
7	C₇×#1753	C₂	1	2	yes	maximal
8	C₇×#1749	C₂	1	1	yes	maximal, characteristic
9	C₇×#1728	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
11	C₂×C₇×C₂²⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₇	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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×#97	C₂²	1	2	yes	—
19	C₂×C₇×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
21	C₇×#391	C₄	1	1	yes	characteristic
22	C₇×#389	C₄	1	1	yes	characteristic
23	C₇×#387	C₄	1	2	yes	—
24	C₇×#385	C₄	1	2	yes	—
25	C₇×#380	C₄	1	1	yes	characteristic
26	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	C₇×#336	C₂²	1	1	yes	characteristic
31	C₇×#336	C₂²	1	1	yes	characteristic
32	C₇×#336	C₂²	1	2	yes	—
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	#23242	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₇×M₁₆	—	2	4	no	—
43	C₂×C₇×M₁₆	—	2	4	no	—
44	C₂×C₇×M₁₆	—	4	2	no	—
45	C₂²×C₇×C₈	C₂³	1	2	yes	—
46	C₂²×C₇×C₈	C₂³	1	2	yes	—
47	C₂²×C₇×C₈	C₂³	1	2	yes	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₇×C₂²⋊Q₈	—	2	4	no	—
50	C₄×C₇×Q₈	—	4	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×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₇	—	4	2	no	—
60	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₇	C₂³	1	2	yes	—
66	C₇×C₄⋊C₈	—	2	4	no	—
67	C₇×C₄⋊C₈	—	4	2	no	—
68	C₇×C₂²⋊C₈	—	2	2	no	—
69	C₇×C₂²⋊C₈	Q₈	1	4	yes	complemented
70	C₇×#97	—	2	4	no	—
71	C₇×#97	—	4	2	no	—
72	C₄×C₇×C₈	—	4	2	no	—
73	C₇×#95	C₂×C₄	1	2	yes	—
74	C₇×#95	C₂×C₄	1	2	yes	—
75	C₇×#95	C₂×C₄	1	2	yes	—
76	C₇×#95	C₂×C₄	1	2	yes	—
77	C₇×#95	C₂×C₄	1	2	yes	—
78	C₄×C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic
79	#1984	C₂×C₇	1	2	yes	complemented
80	#1917	C₂×C₇	1	2	yes	complemented
81	#1917	C₂×C₇	1	2	yes	complemented
82	#1851	C₂×C₇	1	2	yes	—
83	#1771	C₂×C₇	1	2	yes	complemented
84	#1753	C₂×C₇	1	2	yes	—
85	#1749	C₂×C₇	1	1	yes	characteristic
86	#1728	C₂×C₇	1	1	yes	characteristic
87	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×Q₈	—	4	1	no	—
89	C₂×C₇×Q₈	—	4	1	no	—
90	C₂×C₇×Q₈	M₁₆	1	4	yes	complemented
91	C₂²×C₄×C₇	—	2	2	no	—
92	C₂²×C₄×C₇	—	2	2	no	—
93	C₂²×C₄×C₇	—	2	2	no	—
94	C₂²×C₄×C₇	—	4	1	no	—
95	C₂²×C₄×C₇	—	4	1	no	—
96	C₂²×C₄×C₇	—	4	1	no	—
97	C₂²×C₄×C₇	—	4	1	no	—
98	C₂²×C₄×C₇	—	4	1	no	—
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	1	yes	characteristic
102	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
109	C₂²×C₄×C₇	#41	1	2	yes	—
110	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
111	C₇×M₁₆	—	4	8	no	—
112	C₂×C₇×C₈	—	2	2	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	2	no	—
115	C₂×C₇×C₈	—	2	2	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	C₂×Q₈	1	4	yes	—
123	C₂×C₇×C₈	#41	1	4	yes	—
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	—	2	4	no	—
126	C₇×C₄⋊C₄	—	4	1	no	—
127	C₇×C₄⋊C₄	—	4	2	no	—
128	C₇×C₄⋊C₄	—	4	1	no	—
129	C₇×C₄⋊C₄	—	4	2	no	—
130	C₇×C₂²⋊C₄	—	2	4	no	—
131	C₇×C₂²⋊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	—	4	1	no	—
139	C₄²×C₇	—	4	1	no	—
140	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
141	C₂×C₂²⋊Q₈	C₄×C₇	1	1	yes	characteristic
142	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
143	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
144	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
146	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
147	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
148	C₂×#97	C₂²×C₇	1	2	yes	—
149	C₂×#97	C₂²×C₇	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
151	#391	C₄×C₇	1	1	yes	characteristic
152	#389	C₄×C₇	1	1	yes	characteristic
153	#387	C₄×C₇	1	2	yes	—
154	#385	C₄×C₇	1	2	yes	—
155	#380	C₄×C₇	1	1	yes	characteristic
156	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
157	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
158	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
159	#336	C₂²×C₇	1	2	yes	—
160	#336	C₂²×C₇	1	2	yes	—
161	#336	C₂²×C₇	1	1	yes	characteristic
162	#336	C₂²×C₇	1	2	yes	—
163	#336	C₂²×C₇	1	2	yes	—
164	#336	C₂²×C₇	1	2	yes	—
165	#336	C₂²×C₇	1	2	yes	—
166	#336	C₂²×C₇	1	1	yes	characteristic
167	C₂³×C₇	—	2	1	no	—
168	C₂³×C₇	—	2	1	no	—
169	C₂³×C₇	—	2	1	no	—
170	C₂³×C₇	—	2	1	no	—
171	C₂³×C₇	—	2	1	no	—
172	C₂³×C₇	—	2	1	no	—
173	C₂³×C₇	C₄×Q₈	1	2	yes	—
174	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
175	C₇×Q₈	—	4	4	no	—
176	C₂×C₄×C₇	—	2	1	no	—
177	C₂×C₄×C₇	—	2	1	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	1	no	—
180	C₂×C₄×C₇	—	2	1	no	—
181	C₂×C₄×C₇	—	2	2	no	—
182	C₂×C₄×C₇	—	2	4	no	—
183	C₂×C₄×C₇	—	2	4	no	—
184	C₂×C₄×C₇	—	2	2	no	—
185	C₂×C₄×C₇	—	2	2	no	—
186	C₂×C₄×C₇	—	2	1	no	—
187	C₂×C₄×C₇	—	2	1	no	—
188	C₂×C₄×C₇	—	2	1	no	—
189	C₂×C₄×C₇	—	2	2	no	—
190	C₂×C₄×C₇	—	2	2	no	—
191	C₂×C₄×C₇	—	2	1	no	—
192	C₂×C₄×C₇	—	2	2	no	—
193	C₂×C₄×C₇	—	2	2	no	—
194	C₂×C₄×C₇	—	2	1	no	—
195	C₂×C₄×C₇	—	2	2	no	—
196	C₂×C₄×C₇	—	2	2	no	—
197	C₂×C₄×C₇	—	2	2	no	—
198	C₂×C₄×C₇	—	2	1	no	—
199	C₂×C₄×C₇	—	2	1	no	—
200	C₂×C₄×C₇	—	2	1	no	—
201	C₂×C₄×C₇	—	2	1	no	—
202	C₂×C₄×C₇	—	4	4	no	—
203	C₂×C₄×C₇	—	4	4	no	—
204	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
205	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
206	C₂×C₄×C₇	C₂×M₁₆	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	2	yes	—
210	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
211	C₇×C₈	—	2	4	no	—
212	C₇×C₈	—	2	4	no	—
213	C₇×C₈	—	4	2	no	—
214	C₇×C₈	—	4	2	no	—
215	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
216	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
217	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
218	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	—	2	4	no	—
221	C₂×M₁₆	—	4	2	no	—
222	C₂²×C₈	C₂³×C₇	1	2	yes	—
223	C₂²×C₈	C₂³×C₇	1	2	yes	—
224	C₂²×C₈	C₂³×C₇	1	2	yes	—
225	C₂²×C₈	C₂³×C₇	1	2	yes	—
226	C₂²⋊Q₈	—	2	4	no	—
227	C₄×Q₈	—	4	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	1	yes	characteristic
231	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
234	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
235	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
236	C₂×C₄²	—	4	2	no	—
237	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
238	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₇	1	2	yes	—
241	C₂×C₄²	C₂³×C₇	1	2	yes	—
242	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
243	C₄⋊C₈	—	2	4	no	—
244	C₄⋊C₈	—	4	2	no	—
245	C₂²⋊C₈	—	2	2	no	—
246	C₂²⋊C₈	C₇×Q₈	1	4	yes	complemented
247	#97	—	2	4	no	—
248	#97	—	4	2	no	—
249	C₄×C₈	—	4	2	no	—
250	#95	C₂×C₄×C₇	1	2	yes	—
251	#95	C₂×C₄×C₇	1	2	yes	—
252	#95	C₂×C₄×C₇	1	2	yes	—
253	#95	C₂×C₄×C₇	1	2	yes	—
254	#95	C₂×C₄×C₇	1	2	yes	—
255	C₂²×C₇	—	2	2	no	—
256	C₂²×C₇	—	2	2	no	—
257	C₂²×C₇	—	2	2	no	—
258	C₂²×C₇	—	4	1	no	—
259	C₂²×C₇	—	4	1	no	—
260	C₂²×C₇	—	4	1	no	—
261	C₂²×C₇	—	4	1	no	—
262	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
263	C₂²×C₇	C₄×#41	1	1	yes	characteristic
264	C₂²×C₇	#518	1	1	yes	characteristic
265	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
266	C₂²×C₇	#568	1	1	yes	characteristic
267	C₂²×C₇	#568	1	1	yes	characteristic
268	C₂²×C₇	#568	1	1	yes	characteristic
269	C₄×C₇	—	2	2	no	—
270	C₄×C₇	—	2	2	no	—
271	C₄×C₇	—	2	4	no	—
272	C₄×C₇	—	4	2	no	—
273	C₄×C₇	—	4	2	no	—
274	C₄×C₇	—	4	1	no	—
275	C₄×C₇	—	4	2	no	—
276	C₄×C₇	—	4	1	no	—
277	C₄×C₇	—	4	1	no	—
278	C₄×C₇	#551	1	2	yes	—
279	C₄×C₇	#554	1	2	yes	—
280	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
281	C₂×Q₈	—	4	1	no	—
282	C₂×Q₈	—	4	1	no	—
283	C₂×Q₈	C₇×M₁₆	1	4	yes	complemented
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
293	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
295	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
296	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
297	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
298	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
300	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
302	C₂²×C₄	C₇×#41	1	2	yes	—
303	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
304	M₁₆	—	4	8	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	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
316	C₂×C₈	C₇×#41	1	4	yes	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	4	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	2	no	—
323	C₂²⋊C₄	—	2	4	no	—
324	C₂²⋊C₄	—	4	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₂×C₇	—	4	2	no	—
334	C₂×C₇	#2368	1	1	yes	characteristic
335	C₂×C₇	#2961	1	1	yes	characteristic
336	C₂×C₇	Q₈×M₁₆	1	1	yes	characteristic
337	C₂×C₇	#2967	1	1	yes	characteristic
338	C₂×C₇	#2971	1	1	yes	characteristic
339	C₂×C₇	#2973	1	1	yes	characteristic
340	C₂×C₇	#2978	1	1	yes	characteristic
341	C₂³	—	2	1	no	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	1	no	—
347	C₂³	C₄×C₇×Q₈	1	2	yes	—
348	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
349	Q₈	—	4	4	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	—	4	4	no	—
378	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
379	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
380	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
381	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
382	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
383	C₂×C₄	C₂×C₇×#41	1	2	yes	—
384	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
385	C₈	—	2	4	no	—
386	C₈	—	2	4	no	—
387	C₈	—	4	2	no	—
388	C₈	—	4	2	no	—
389	C₇	#23242	1	1	yes	characteristic, direct product factor, Sylow
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
398	C₂²	C₄×C₇×#41	1	1	yes	characteristic
399	C₂²	C₇×#518	1	1	yes	characteristic
400	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
401	C₂²	C₇×#568	1	1	yes	characteristic
402	C₂²	C₇×#568	1	1	yes	characteristic
403	C₂²	C₇×#568	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	4	no	—
406	C₄	—	2	2	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	1	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	4	2	no	—
413	C₄	C₇×#551	1	2	yes	—
414	C₄	C₇×#554	1	2	yes	—
415	C₂	—	4	2	no	—
416	C₂	C₇×#2368	1	1	yes	characteristic
417	C₂	C₇×#2961	1	1	yes	characteristic
418	C₂	C₇×Q₈×M₁₆	1	1	yes	characteristic
419	C₂	C₇×#2967	1	1	yes	characteristic
420	C₂	C₇×#2971	1	1	yes	characteristic
421	C₂	C₇×#2973	1	1	yes	characteristic
422	C₂	C₇×#2978	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×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
2. C₇×#1984	C₂	semidirect
3. C₇×#1917	C₂	semidirect
4. C₇×#1917	C₂	semidirect
6. C₇×#1771	C₂	semidirect
37. #23242	C₇	direct (N × H)
69. C₇×C₂²⋊C₈	Q₈	semidirect
79. #1984	C₂×C₇	semidirect
80. #1917	C₂×C₇	semidirect
81. #1917	C₂×C₇	semidirect
83. #1771	C₂×C₇	semidirect
90. C₂×C₇×Q₈	M₁₆	semidirect
246. C₂²⋊C₈	C₇×Q₈	semidirect
283. C₂×Q₈	C₇×M₁₆	semidirect
389. C₇	#23242	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×C₂²⋊Q₈	2
C₇×#1984	2
C₇×#1917	2
C₇×#1851	2
C₇×#1771	2
C₇×#1753	2
C₇×#1749	2
C₇×#1728	2
#23242	7

Subgroups — C7×#23242 #13200458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23242 #13200458