Subgroups — C₇×#22721 #13199937

Order 1792.

760 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#2295	C₂	1	1	yes	maximal, characteristic
4	C₇×#1940	C₂	1	2	yes	maximal
5	C₇×#1939	C₂	1	2	yes	maximal
6	C₇×#1939	C₂	1	1	yes	maximal, characteristic
7	C₇×#1939	C₂	1	1	yes	maximal, characteristic
8	C₇×#1843	C₂	1	2	yes	maximal
9	C₇×#1842	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	C₇×#446	—	2	4	no	—
14	C₇×C₈×Q₈	—	2	4	no	—
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	1	yes	characteristic
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₇×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#391	C₄	1	1	yes	characteristic
30	C₇×#391	C₄	1	1	yes	characteristic
31	C₇×#389	C₄	1	1	yes	characteristic
32	C₇×#389	C₄	1	1	yes	characteristic
33	C₇×#387	C₄	1	2	yes	—
34	C₇×#387	C₄	1	2	yes	—
35	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
37	C₇×#376	C₂²	1	2	yes	—
38	C₄³×C₇	C₂²	1	1	yes	characteristic
39	C₇×#336	C₂²	1	1	yes	characteristic
40	C₇×#336	C₂²	1	1	yes	characteristic
41	C₇×#336	C₂²	1	2	yes	—
42	C₇×#336	C₂²	1	2	yes	—
43	#22721	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇×C₈	C₂³	1	2	yes	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₄×C₇×Q₈	D₄	1	4	yes	—
52	C₄×C₇×Q₈	Q₈	1	4	yes	—
53	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₂×C₇×C₄⋊C₄	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	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₇	C₂×C₄	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	1	yes	characteristic
66	C₂×C₄²×C₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
68	C₇×C₄⋊C₈	—	2	2	no	—
69	C₇×C₄⋊C₈	—	2	4	no	—
70	C₇×C₄⋊C₈	—	2	2	no	—
71	C₇×C₄⋊C₈	—	2	4	no	—
72	C₇×C₄⋊C₈	—	2	2	no	—
73	C₇×C₄⋊C₈	—	2	4	no	—
74	C₇×C₄⋊C₈	—	4	2	no	—
75	C₇×C₄⋊C₈	—	4	2	no	—
76	C₇×C₄⋊C₈	—	4	2	no	—
77	C₇×C₄⋊C₈	Q₈	1	4	yes	complemented
78	C₇×#97	—	2	4	no	—
79	C₄×C₇×C₈	—	2	2	no	—
80	C₄×C₇×C₈	—	2	4	no	—
81	C₄×C₇×C₈	—	2	2	no	—
82	C₇×#95	C₂×C₄	1	2	yes	—
83	C₇×#95	C₂×C₄	1	2	yes	—
84	C₇×#95	C₂×C₄	1	2	yes	—
85	C₇×#95	C₂×C₄	1	2	yes	—
86	C₇×#95	C₂×C₄	1	2	yes	—
87	C₇×#95	C₂×C₄	1	2	yes	—
88	C₇×#95	C₂×C₄	1	2	yes	—
89	C₂×#446	C₂×C₇	1	1	yes	characteristic
90	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
91	#2295	C₂×C₇	1	1	yes	characteristic
92	#1940	C₂×C₇	1	2	yes	—
93	#1939	C₂×C₇	1	2	yes	—
94	#1939	C₂×C₇	1	1	yes	characteristic
95	#1939	C₂×C₇	1	1	yes	characteristic
96	#1843	C₂×C₇	1	2	yes	—
97	#1842	C₂×C₇	1	1	yes	characteristic
98	#1768	C₂×C₇	1	2	yes	—
99	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
100	C₂×C₇×Q₈	—	2	2	no	—
101	C₂×C₇×Q₈	—	2	2	no	—
102	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
114	C₂×C₇×C₈	—	2	1	no	—
115	C₂×C₇×C₈	—	2	1	no	—
116	C₂×C₇×C₈	—	2	1	no	—
117	C₂×C₇×C₈	—	2	1	no	—
118	C₂×C₇×C₈	—	2	1	no	—
119	C₂×C₇×C₈	—	2	2	no	—
120	C₂×C₇×C₈	—	2	1	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	—	2	2	no	—
125	C₂×C₇×C₈	—	2	2	no	—
126	C₂×C₇×C₈	—	2	2	no	—
127	C₂×C₇×C₈	—	2	1	no	—
128	C₂×C₇×C₈	—	2	1	no	—
129	C₂×C₇×C₈	—	2	2	no	—
130	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
131	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
132	C₇×C₄⋊C₄	—	2	4	no	—
133	C₇×C₄⋊C₄	—	2	2	no	—
134	C₇×C₄⋊C₄	—	2	2	no	—
135	C₇×C₄⋊C₄	—	4	2	no	—
136	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
137	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
138	C₄²×C₇	—	2	2	no	—
139	C₄²×C₇	—	2	2	no	—
140	C₄²×C₇	—	2	2	no	—
141	C₄²×C₇	—	2	1	no	—
142	C₄²×C₇	—	2	2	no	—
143	C₄²×C₇	—	2	1	no	—
144	C₄²×C₇	—	2	2	no	—
145	C₄²×C₇	—	4	1	no	—
146	C₄²×C₇	C₂×D₄	1	2	yes	—
147	C₄²×C₇	C₂×D₄	1	4	yes	—
148	C₄²×C₇	C₂×Q₈	1	4	yes	—
149	C₄²×C₇	C₂×Q₈	1	2	yes	—
150	C₄²×C₇	C₂×Q₈	1	2	yes	—
151	C₄²×C₇	#41	1	2	yes	—
152	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
153	#446	—	2	4	no	—
154	C₈×Q₈	—	2	4	no	—
155	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
158	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
164	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
165	C₂×#97	C₂²×C₇	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
169	#391	C₄×C₇	1	1	yes	characteristic
170	#391	C₄×C₇	1	1	yes	characteristic
171	#389	C₄×C₇	1	1	yes	characteristic
172	#389	C₄×C₇	1	1	yes	characteristic
173	#387	C₄×C₇	1	2	yes	—
174	#387	C₄×C₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
176	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
177	#376	C₂²×C₇	1	2	yes	—
178	C₄³	C₂²×C₇	1	1	yes	characteristic
179	#336	C₂²×C₇	1	1	yes	characteristic
180	#336	C₂²×C₇	1	2	yes	—
181	#336	C₂²×C₇	1	2	yes	—
182	#336	C₂²×C₇	1	1	yes	characteristic
183	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
184	C₇×Q₈	—	2	8	no	—
185	C₂×C₄×C₇	—	2	1	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	1	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	1	no	—
195	C₂×C₄×C₇	—	2	1	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	2	no	—
200	C₂×C₄×C₇	—	2	1	no	—
201	C₂×C₄×C₇	—	2	2	no	—
202	C₂×C₄×C₇	—	2	2	no	—
203	C₂×C₄×C₇	—	2	2	no	—
204	C₂×C₄×C₇	—	2	1	no	—
205	C₂×C₄×C₇	—	2	2	no	—
206	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
207	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
208	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
209	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
210	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
211	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
212	C₂×C₄×C₇	#131	1	2	yes	—
213	C₂×C₄×C₇	#131	1	2	yes	—
214	C₂×C₄×C₇	#131	1	2	yes	—
215	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
217	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
218	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
220	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
221	C₇×C₈	—	2	2	no	—
222	C₇×C₈	—	2	2	no	—
223	C₇×C₈	—	4	1	no	—
224	C₇×C₈	—	4	2	no	—
225	C₇×C₈	—	4	1	no	—
226	C₇×C₈	—	4	2	no	—
227	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₇	1	2	yes	—
230	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
231	C₂²×C₈	C₂³×C₇	1	2	yes	—
232	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
234	C₄×Q₈	C₇×D₄	1	4	yes	—
235	C₄×Q₈	C₇×Q₈	1	4	yes	—
236	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
239	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
240	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
241	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
242	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
243	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
244	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
245	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
246	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
247	C₂×C₄²	C₂³×C₇	1	2	yes	—
248	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₇	1	2	yes	—
250	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	2	4	no	—
255	C₄⋊C₈	—	2	4	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	4	2	no	—
258	C₄⋊C₈	—	4	2	no	—
259	C₄⋊C₈	—	4	2	no	—
260	C₄⋊C₈	C₇×Q₈	1	4	yes	complemented
261	#97	—	2	4	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	2	4	no	—
264	C₄×C₈	—	2	2	no	—
265	#95	C₂×C₄×C₇	1	2	yes	—
266	#95	C₂×C₄×C₇	1	2	yes	—
267	#95	C₂×C₄×C₇	1	2	yes	—
268	#95	C₂×C₄×C₇	1	2	yes	—
269	#95	C₂×C₄×C₇	1	2	yes	—
270	#95	C₂×C₄×C₇	1	2	yes	—
271	#95	C₂×C₄×C₇	1	2	yes	—
272	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
273	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
274	C₂²×C₇	#520	1	1	yes	characteristic
275	C₂²×C₇	C₂×#131	1	1	yes	characteristic
276	C₂²×C₇	C₂×#131	1	1	yes	characteristic
277	C₂²×C₇	C₂×#131	1	1	yes	characteristic
278	C₂²×C₇	#568	1	1	yes	characteristic
279	C₄×C₇	—	2	4	no	—
280	C₄×C₇	—	2	2	no	—
281	C₄×C₇	—	2	4	no	—
282	C₄×C₇	—	2	2	no	—
283	C₄×C₇	—	2	2	no	—
284	C₄×C₇	—	2	2	no	—
285	C₄×C₇	—	4	1	no	—
286	C₄×C₇	—	4	2	no	—
287	C₄×C₇	—	4	2	no	—
288	C₄×C₇	#424	1	4	yes	—
289	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
290	C₄×C₇	#550	1	1	yes	characteristic
291	C₄×C₇	#557	1	1	yes	characteristic
292	C₄×C₇	Q₈²	1	1	yes	characteristic
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
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	C₂²×C₄×C₇	1	2	yes	—
302	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
303	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
304	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
305	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
306	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
324	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
325	C₄⋊C₄	—	2	4	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
330	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	4	1	no	—
339	C₄²	C₂×C₇×D₄	1	2	yes	—
340	C₄²	C₂×C₇×D₄	1	4	yes	—
341	C₄²	C₂×C₇×Q₈	1	4	yes	—
342	C₄²	C₂×C₇×Q₈	1	2	yes	—
343	C₄²	C₂×C₇×Q₈	1	2	yes	—
344	C₄²	C₇×#41	1	2	yes	—
345	C₂×C₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
346	C₂×C₇	C₂×#424	1	1	yes	characteristic
347	C₂×C₇	#2906	1	1	yes	characteristic
348	C₂×C₇	#2960	1	1	yes	characteristic
349	C₂×C₇	#2962	1	1	yes	characteristic
350	C₂×C₇	#2993	1	1	yes	characteristic
351	C₂×C₇	#2994	1	1	yes	characteristic
352	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
353	Q₈	—	2	8	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	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	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
376	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
377	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
378	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
379	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
380	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
381	C₂×C₄	C₇×#131	1	2	yes	—
382	C₂×C₄	C₇×#131	1	2	yes	—
383	C₂×C₄	C₇×#131	1	2	yes	—
384	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
385	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
386	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
387	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
388	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
389	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	2	2	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	2	no	—
394	C₈	—	4	1	no	—
395	C₈	—	4	2	no	—
396	C₇	#22721	1	1	yes	characteristic, direct product factor, Sylow
397	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
398	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
399	C₂²	C₇×#520	1	1	yes	characteristic
400	C₂²	C₂×C₇×#131	1	1	yes	characteristic
401	C₂²	C₂×C₇×#131	1	1	yes	characteristic
402	C₂²	C₂×C₇×#131	1	1	yes	characteristic
403	C₂²	C₇×#568	1	1	yes	characteristic
404	C₄	—	2	4	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	C₇×#424	1	4	yes	—
414	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
415	C₄	C₇×#550	1	1	yes	characteristic
416	C₄	C₇×#557	1	1	yes	characteristic
417	C₄	C₇×Q₈²	1	1	yes	characteristic
418	C₂	C₇×Q₈×C₄⋊C₄	1	1	yes	characteristic
419	C₂	C₂×C₇×#424	1	1	yes	characteristic
420	C₂	C₇×#2906	1	1	yes	characteristic
421	C₂	C₇×#2960	1	1	yes	characteristic
422	C₂	C₇×#2962	1	1	yes	characteristic
423	C₂	C₇×#2993	1	1	yes	characteristic
424	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
43. #22721	C₇	direct (N × H)
77. C₇×C₄⋊C₈	Q₈	semidirect
260. C₄⋊C₈	C₇×Q₈	semidirect
396. C₇	#22721	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×C₇×#446	2
C₂×C₇×C₈×Q₈	2
C₇×#2295	2
C₇×#1940	2
C₇×#1939	2
C₇×#1843	2
C₇×#1842	2
C₇×#1768	2
C₄×C₇×C₄⋊C₈	2
#22721	7

Subgroups — C7×#22721 #13199937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22721 #13199937