Subgroups — C₇×#23343 #13200559

Order 1792.

700 conjugacy classes of subgroups.

The subgroup lattice has 418 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×#1940	C₂	1	2	yes	maximal, complemented
4	C₇×#1924	C₂	1	2	yes	maximal
5	C₇×#1923	C₂	1	2	yes	maximal
6	C₇×#1917	C₂	1	2	yes	maximal, complemented
7	C₇×#1916	C₂	1	2	yes	maximal, complemented
8	C₇×#1817	C₂	1	1	yes	maximal, characteristic
9	C₇×#1797	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×C₂²⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
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×C₂²⋊C₈	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	2	yes	—
31	C₇×#336	C₂²	1	2	yes	—
32	C₇×#336	C₂²	1	2	yes	—
33	C₇×#336	C₂²	1	1	yes	characteristic
34	C₇×#336	C₂²	1	1	yes	characteristic
35	C₇×#336	C₂²	1	2	yes	—
36	C₇×#336	C₂²	1	2	yes	—
37	#23343	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	C₂²×C₇×C₈	C₂³	1	2	yes	—
43	C₂²×C₇×C₈	C₂³	1	2	yes	—
44	C₂²×C₇×C₈	C₂³	1	2	yes	—
45	C₂²×C₇×C₈	C₂³	1	2	yes	—
46	C₇×C₂²⋊Q₈	—	2	4	no	—
47	C₄×C₇×Q₈	—	4	2	no	—
48	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₇	—	4	2	no	—
57	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₇	C₂³	1	2	yes	—
63	C₇×C₄⋊C₈	—	4	2	no	—
64	C₇×C₄⋊C₈	—	4	2	no	—
65	C₇×C₄⋊C₈	—	4	2	no	—
66	C₇×C₂²⋊C₈	—	4	2	no	—
67	C₇×C₂²⋊C₈	—	4	1	no	—
68	C₇×C₂²⋊C₈	—	4	1	no	—
69	C₇×#97	—	4	2	no	—
70	C₄×C₇×C₈	—	4	2	no	—
71	C₇×#95	C₂×C₄	1	2	yes	—
72	C₇×#95	C₂×C₄	1	2	yes	—
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	C₄×C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic
77	#1984	C₂×C₇	1	2	yes	complemented
78	#1940	C₂×C₇	1	2	yes	complemented
79	#1924	C₂×C₇	1	2	yes	—
80	#1923	C₂×C₇	1	2	yes	—
81	#1917	C₂×C₇	1	2	yes	complemented
82	#1916	C₂×C₇	1	2	yes	complemented
83	#1817	C₂×C₇	1	1	yes	characteristic
84	#1797	C₂×C₇	1	1	yes	characteristic
85	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₇×Q₈	—	4	1	no	—
87	C₂×C₇×Q₈	—	4	1	no	—
88	C₂×C₇×Q₈	M₁₆	1	4	yes	—
89	C₂²×C₄×C₇	—	2	2	no	—
90	C₂²×C₄×C₇	—	2	2	no	—
91	C₂²×C₄×C₇	—	2	2	no	—
92	C₂²×C₄×C₇	—	2	1	no	—
93	C₂²×C₄×C₇	—	2	2	no	—
94	C₂²×C₄×C₇	—	2	2	no	—
95	C₂²×C₄×C₇	—	2	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	1	yes	characteristic
100	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
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	2	yes	—
106	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
109	C₂×C₇×C₈	—	2	2	no	—
110	C₂×C₇×C₈	—	2	2	no	—
111	C₂×C₇×C₈	—	2	2	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	C₇×C₄⋊C₄	—	2	2	no	—
122	C₇×C₄⋊C₄	—	2	4	no	—
123	C₇×C₄⋊C₄	—	2	4	no	—
124	C₇×C₄⋊C₄	—	4	1	no	—
125	C₇×C₄⋊C₄	—	4	1	no	—
126	C₇×C₄⋊C₄	—	4	2	no	—
127	C₇×C₂²⋊C₄	—	2	4	no	—
128	C₇×C₂²⋊C₄	—	2	4	no	—
129	C₄²×C₇	—	2	2	no	—
130	C₄²×C₇	—	2	2	no	—
131	C₄²×C₇	—	2	2	no	—
132	C₄²×C₇	—	2	2	no	—
133	C₄²×C₇	—	2	2	no	—
134	C₄²×C₇	—	2	2	no	—
135	C₄²×C₇	—	4	1	no	—
136	C₄²×C₇	—	4	1	no	—
137	C₂×C₂²⋊Q₈	C₄×C₇	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
139	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
140	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
141	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
142	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
143	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
144	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
145	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
146	C₂×#97	C₂²×C₇	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
148	#391	C₄×C₇	1	1	yes	characteristic
149	#389	C₄×C₇	1	1	yes	characteristic
150	#387	C₄×C₇	1	2	yes	—
151	#385	C₄×C₇	1	2	yes	—
152	#380	C₄×C₇	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
154	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
155	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
156	#336	C₂²×C₇	1	2	yes	—
157	#336	C₂²×C₇	1	1	yes	characteristic
158	#336	C₂²×C₇	1	2	yes	—
159	#336	C₂²×C₇	1	1	yes	characteristic
160	#336	C₂²×C₇	1	2	yes	—
161	#336	C₂²×C₇	1	2	yes	—
162	#336	C₂²×C₇	1	2	yes	—
163	#336	C₂²×C₇	1	2	yes	—
164	C₂³×C₇	—	2	1	no	—
165	C₂³×C₇	—	2	1	no	—
166	C₂³×C₇	—	2	1	no	—
167	C₂³×C₇	—	2	1	no	—
168	C₂³×C₇	—	2	1	no	—
169	C₂³×C₇	—	2	1	no	—
170	C₂³×C₇	#131	1	2	yes	—
171	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
172	C₇×Q₈	—	4	4	no	—
173	C₂×C₄×C₇	—	2	1	no	—
174	C₂×C₄×C₇	—	2	1	no	—
175	C₂×C₄×C₇	—	2	2	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	1	no	—
183	C₂×C₄×C₇	—	2	1	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	2	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	2	no	—
192	C₂×C₄×C₇	—	2	2	no	—
193	C₂×C₄×C₇	—	2	1	no	—
194	C₂×C₄×C₇	—	2	1	no	—
195	C₂×C₄×C₇	—	2	2	no	—
196	C₂×C₄×C₇	—	2	1	no	—
197	C₂×C₄×C₇	—	4	4	no	—
198	C₂×C₄×C₇	—	4	4	no	—
199	C₂×C₄×C₇	—	4	2	no	—
200	C₂×C₄×C₇	—	4	2	no	—
201	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
202	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₇	#131	1	2	yes	—
204	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
205	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
206	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
207	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
208	C₇×C₈	—	4	2	no	—
209	C₇×C₈	—	4	2	no	—
210	C₇×C₈	—	4	2	no	—
211	C₇×C₈	—	4	2	no	—
212	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
213	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
214	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
215	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₇	1	2	yes	—
217	C₂²×C₈	C₂³×C₇	1	2	yes	—
218	C₂²×C₈	C₂³×C₇	1	2	yes	—
219	C₂²×C₈	C₂³×C₇	1	2	yes	—
220	C₂²⋊Q₈	—	2	4	no	—
221	C₄×Q₈	—	4	2	no	—
222	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
224	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
229	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
230	C₂×C₄²	—	4	2	no	—
231	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
232	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₇	1	2	yes	—
235	C₂×C₄²	C₂³×C₇	1	2	yes	—
236	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
237	C₄⋊C₈	—	4	2	no	—
238	C₄⋊C₈	—	4	2	no	—
239	C₄⋊C₈	—	4	2	no	—
240	C₂²⋊C₈	—	4	2	no	—
241	C₂²⋊C₈	—	4	1	no	—
242	C₂²⋊C₈	—	4	1	no	—
243	#97	—	4	2	no	—
244	C₄×C₈	—	4	2	no	—
245	#95	C₂×C₄×C₇	1	2	yes	—
246	#95	C₂×C₄×C₇	1	2	yes	—
247	#95	C₂×C₄×C₇	1	2	yes	—
248	#95	C₂×C₄×C₇	1	2	yes	—
249	#95	C₂×C₄×C₇	1	2	yes	—
250	C₂²×C₇	—	2	2	no	—
251	C₂²×C₇	—	2	2	no	—
252	C₂²×C₇	—	2	2	no	—
253	C₂²×C₇	—	4	1	no	—
254	C₂²×C₇	—	4	1	no	—
255	C₂²×C₇	—	4	1	no	—
256	C₂²×C₇	—	4	1	no	—
257	C₂²×C₇	#518	1	1	yes	characteristic
258	C₂²×C₇	#519	1	1	yes	characteristic
259	C₂²×C₇	#520	1	1	yes	characteristic
260	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
261	C₂²×C₇	C₂×#131	1	1	yes	characteristic
262	C₂²×C₇	C₂×#131	1	1	yes	characteristic
263	C₂²×C₇	#568	1	1	yes	characteristic
264	C₄×C₇	—	2	2	no	—
265	C₄×C₇	—	2	4	no	—
266	C₄×C₇	—	2	2	no	—
267	C₄×C₇	—	4	1	no	—
268	C₄×C₇	—	4	1	no	—
269	C₄×C₇	—	4	2	no	—
270	C₄×C₇	—	4	1	no	—
271	C₄×C₇	—	4	2	no	—
272	C₄×C₇	—	4	2	no	—
273	C₄×C₇	#562	1	2	yes	—
274	C₄×C₇	#563	1	2	yes	—
275	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	C₇×M₁₆	1	4	yes	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
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	2	yes	—
294	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
295	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
296	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
297	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
298	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	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	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	4	no	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	2	4	no	—
318	C₂²⋊C₄	—	2	4	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	—	4	1	no	—
326	C₄²	—	4	1	no	—
327	C₂×C₇	—	4	2	no	—
328	C₂×C₇	#2378	1	1	yes	characteristic
329	C₂×C₇	#2973	1	1	yes	characteristic
330	C₂×C₇	#2977	1	1	yes	characteristic
331	C₂×C₇	#2984	1	1	yes	characteristic
332	C₂×C₇	#2985	1	1	yes	characteristic
333	C₂×C₇	#2991	1	1	yes	characteristic
334	C₂×C₇	#2992	1	1	yes	characteristic
335	C₂³	—	2	1	no	—
336	C₂³	—	2	1	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	C₂³	—	2	1	no	—
340	C₂³	—	2	1	no	—
341	C₂³	C₇×#131	1	2	yes	—
342	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
343	Q₈	—	4	4	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	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	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
373	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
374	C₂×C₄	C₇×#131	1	2	yes	—
375	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
376	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
377	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
378	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
379	C₈	—	4	2	no	—
380	C₈	—	4	2	no	—
381	C₈	—	4	2	no	—
382	C₈	—	4	2	no	—
383	C₇	#23343	1	1	yes	characteristic, direct product factor, Sylow
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	C₇×#518	1	1	yes	characteristic
392	C₂²	C₇×#519	1	1	yes	characteristic
393	C₂²	C₇×#520	1	1	yes	characteristic
394	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
395	C₂²	C₂×C₇×#131	1	1	yes	characteristic
396	C₂²	C₂×C₇×#131	1	1	yes	characteristic
397	C₂²	C₇×#568	1	1	yes	characteristic
398	C₄	—	2	2	no	—
399	C₄	—	2	4	no	—
400	C₄	—	2	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	4	1	no	—
407	C₄	C₇×#562	1	2	yes	—
408	C₄	C₇×#563	1	2	yes	—
409	C₂	—	4	2	no	—
410	C₂	C₇×#2378	1	1	yes	characteristic
411	C₂	C₇×#2973	1	1	yes	characteristic
412	C₂	C₇×#2977	1	1	yes	characteristic
413	C₂	C₇×#2984	1	1	yes	characteristic
414	C₂	C₇×#2985	1	1	yes	characteristic
415	C₂	C₇×#2991	1	1	yes	characteristic
416	C₂	C₇×#2992	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×#1940	C₂	semidirect
6. C₇×#1917	C₂	semidirect
7. C₇×#1916	C₂	semidirect
37. #23343	C₇	direct (N × H)
77. #1984	C₂×C₇	semidirect
78. #1940	C₂×C₇	semidirect
81. #1917	C₂×C₇	semidirect
82. #1916	C₂×C₇	semidirect
383. C₇	#23343	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×#1940	2
C₇×#1924	2
C₇×#1923	2
C₇×#1917	2
C₇×#1916	2
C₇×#1817	2
C₇×#1797	2
#23343	7

Subgroups — C7×#23343 #13200559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23343 #13200559