Subgroups — C₇×#22701 #13199917

Order 1792.

916 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#2293	C₂	1	1	yes	maximal, characteristic
4	C₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1851	C₂	1	2	yes	maximal, complemented
7	C₇×#1850	C₂	1	2	yes	maximal, complemented
8	C₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1798	C₂	1	2	yes	maximal, complemented
10	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1729	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
13	C₂³×C₇×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×#117	C₂²	1	2	yes	—
16	C₇×#436	—	2	4	no	—
17	C₇×C₈×D₄	—	2	4	no	—
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#390	C₄	1	1	yes	characteristic
29	C₇×#390	C₄	1	1	yes	characteristic
30	C₇×#389	C₄	1	1	yes	characteristic
31	C₇×#389	C₄	1	1	yes	characteristic
32	C₇×#388	C₄	1	2	yes	—
33	C₇×#388	C₄	1	2	yes	—
34	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
36	C₄³×C₇	C₂²	1	1	yes	characteristic
37	C₇×#336	C₂²	1	1	yes	characteristic
38	C₇×#336	C₂²	1	2	yes	—
39	C₇×#336	C₂²	1	2	yes	—
40	C₇×#336	C₂²	1	2	yes	—
41	C₇×#336	C₂²	1	1	yes	characteristic
42	C₇×#336	C₂²	1	2	yes	—
43	#22701	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₇×M₁₆	—	2	4	no	—
47	C₂×C₇×M₁₆	—	2	4	no	—
48	C₂×C₇×M₁₆	—	4	2	no	—
49	C₂²×C₇×C₈	—	2	4	no	—
50	C₂²×C₇×C₈	—	2	4	no	—
51	C₂²×C₇×C₈	—	4	2	no	—
52	C₂²×C₇×C₈	C₂³	1	2	yes	—
53	C₂²×C₇×C₈	C₂³	1	2	yes	—
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₇×D₄	D₄	1	4	yes	—
59	C₄×C₇×D₄	Q₈	1	4	yes	—
60	C₇×#117	—	4	2	no	—
61	C₇×#117	—	4	2	no	—
62	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
64	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
65	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
66	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
70	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
71	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
80	C₇×C₄⋊C₈	—	2	2	no	—
81	C₇×C₄⋊C₈	—	2	2	no	—
82	C₇×C₄⋊C₈	—	2	2	no	—
83	C₇×C₄⋊C₈	—	2	2	no	—
84	C₇×C₄⋊C₈	—	4	2	no	—
85	C₇×C₄⋊C₈	—	4	2	no	—
86	C₇×C₂²⋊C₈	—	2	4	no	—
87	C₇×C₂²⋊C₈	—	2	4	no	—
88	C₄×C₇×C₈	—	2	2	no	—
89	C₄×C₇×C₈	—	2	2	no	—
90	C₇×#95	C₂×C₄	1	2	yes	—
91	C₇×#95	C₂×C₄	1	2	yes	—
92	C₇×#95	C₂×C₄	1	2	yes	—
93	C₂×#436	C₂×C₇	1	1	yes	characteristic
94	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
95	#2293	C₂×C₇	1	1	yes	characteristic
96	#1939	C₂×C₇	1	1	yes	characteristic, complemented
97	#1939	C₂×C₇	1	1	yes	characteristic, complemented
98	#1851	C₂×C₇	1	2	yes	complemented
99	#1850	C₂×C₇	1	2	yes	complemented
100	#1831	C₂×C₇	1	1	yes	characteristic, complemented
101	#1798	C₂×C₇	1	2	yes	complemented
102	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
103	#1729	C₂×C₇	1	2	yes	complemented
104	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
105	C₂×C₇×D₄	—	2	2	no	—
106	C₂×C₇×D₄	—	2	2	no	—
107	C₂×C₇×D₄	C₄⋊C₄	1	4	yes	—
108	C₂²×C₄×C₇	—	2	2	no	—
109	C₂²×C₄×C₇	—	2	2	no	—
110	C₂²×C₄×C₇	—	2	2	no	—
111	C₂²×C₄×C₇	—	2	2	no	—
112	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
122	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
123	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
124	C₇×M₁₆	—	4	8	no	—
125	C₂×C₇×C₈	—	2	1	no	—
126	C₂×C₇×C₈	—	2	1	no	—
127	C₂×C₇×C₈	—	2	1	no	—
128	C₂×C₇×C₈	—	2	1	no	—
129	C₂×C₇×C₈	—	2	1	no	—
130	C₂×C₇×C₈	—	2	1	no	—
131	C₂×C₇×C₈	—	2	2	no	—
132	C₂×C₇×C₈	—	2	2	no	—
133	C₂×C₇×C₈	—	2	1	no	—
134	C₂×C₇×C₈	—	2	1	no	—
135	C₂×C₇×C₈	—	2	2	no	—
136	C₂×C₇×C₈	—	2	2	no	—
137	C₂×C₇×C₈	—	2	1	no	—
138	C₂×C₇×C₈	—	2	1	no	—
139	C₂×C₇×C₈	—	2	2	no	—
140	C₂×C₇×C₈	—	2	2	no	—
141	C₂×C₇×C₈	—	4	8	no	—
142	C₂×C₇×C₈	#41	1	2	yes	—
143	C₂×C₇×C₈	#41	1	2	yes	—
144	C₇×C₄⋊C₄	—	2	2	no	—
145	C₇×C₄⋊C₄	—	2	2	no	—
146	C₇×C₄⋊C₄	—	4	2	no	—
147	C₇×C₄⋊C₄	—	4	2	no	—
148	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
149	C₇×C₂²⋊C₄	—	2	4	no	—
150	C₇×C₂²⋊C₄	—	4	2	no	—
151	C₇×C₂²⋊C₄	—	4	2	no	—
152	C₇×C₂²⋊C₄	—	4	2	no	—
153	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
154	C₄²×C₇	—	2	2	no	—
155	C₄²×C₇	—	2	2	no	—
156	C₄²×C₇	—	2	2	no	—
157	C₄²×C₇	—	2	1	no	—
158	C₄²×C₇	—	2	1	no	—
159	C₄²×C₇	—	2	2	no	—
160	C₄²×C₇	—	2	2	no	—
161	C₄²×C₇	—	2	1	no	—
162	C₄²×C₇	—	2	1	no	—
163	C₄²×C₇	—	4	1	no	—
164	C₄²×C₇	C₂×D₄	1	2	yes	—
165	C₄²×C₇	C₂×Q₈	1	2	yes	—
166	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
167	C₂³×C₈	C₂²×C₇	1	2	yes	—
168	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
169	C₂×#117	C₂²×C₇	1	2	yes	—
170	#436	—	2	4	no	—
171	C₈×D₄	—	2	4	no	—
172	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
173	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
174	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
177	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
178	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
179	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
182	#390	C₄×C₇	1	1	yes	characteristic
183	#390	C₄×C₇	1	1	yes	characteristic
184	#389	C₄×C₇	1	1	yes	characteristic
185	#389	C₄×C₇	1	1	yes	characteristic
186	#388	C₄×C₇	1	2	yes	—
187	#388	C₄×C₇	1	2	yes	—
188	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
189	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
190	C₄³	C₂²×C₇	1	1	yes	characteristic
191	#336	C₂²×C₇	1	1	yes	characteristic
192	#336	C₂²×C₇	1	2	yes	—
193	#336	C₂²×C₇	1	2	yes	—
194	#336	C₂²×C₇	1	2	yes	—
195	#336	C₂²×C₇	1	2	yes	—
196	#336	C₂²×C₇	1	1	yes	characteristic
197	C₂³×C₇	—	2	2	no	—
198	C₂³×C₇	—	2	2	no	—
199	C₂³×C₇	—	2	2	no	—
200	C₂³×C₇	—	2	2	no	—
201	C₂³×C₇	—	2	2	no	—
202	C₂³×C₇	—	2	2	no	—
203	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
204	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
205	C₇×D₄	—	2	8	no	—
206	C₂×C₄×C₇	—	2	1	no	—
207	C₂×C₄×C₇	—	2	2	no	—
208	C₂×C₄×C₇	—	2	2	no	—
209	C₂×C₄×C₇	—	2	1	no	—
210	C₂×C₄×C₇	—	2	1	no	—
211	C₂×C₄×C₇	—	2	1	no	—
212	C₂×C₄×C₇	—	2	4	no	—
213	C₂×C₄×C₇	—	2	4	no	—
214	C₂×C₄×C₇	—	2	4	no	—
215	C₂×C₄×C₇	—	2	4	no	—
216	C₂×C₄×C₇	—	2	1	no	—
217	C₂×C₄×C₇	—	2	1	no	—
218	C₂×C₄×C₇	—	2	1	no	—
219	C₂×C₄×C₇	—	2	2	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	2	1	no	—
222	C₂×C₄×C₇	—	2	1	no	—
223	C₂×C₄×C₇	—	2	1	no	—
224	C₂×C₄×C₇	—	2	1	no	—
225	C₂×C₄×C₇	—	2	2	no	—
226	C₂×C₄×C₇	—	2	2	no	—
227	C₂×C₄×C₇	—	2	2	no	—
228	C₂×C₄×C₇	—	2	1	no	—
229	C₂×C₄×C₇	—	2	1	no	—
230	C₂×C₄×C₇	—	2	2	no	—
231	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
232	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
233	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
234	C₂×C₄×C₇	#131	1	2	yes	—
235	C₂×C₄×C₇	#131	1	2	yes	—
236	C₂×C₄×C₇	#131	1	2	yes	—
237	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
239	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
241	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
243	C₇×C₈	—	2	2	no	—
244	C₇×C₈	—	2	2	no	—
245	C₇×C₈	—	4	1	no	—
246	C₇×C₈	—	4	1	no	—
247	C₇×C₈	—	4	2	no	—
248	C₇×C₈	—	4	2	no	—
249	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
250	C₂³×C₄	C₂³×C₇	1	2	yes	—
251	C₂×M₁₆	—	2	4	no	—
252	C₂×M₁₆	—	2	4	no	—
253	C₂×M₁₆	—	4	2	no	—
254	C₂²×C₈	—	2	4	no	—
255	C₂²×C₈	—	2	4	no	—
256	C₂²×C₈	—	4	2	no	—
257	C₂²×C₈	C₂³×C₇	1	2	yes	—
258	C₂²×C₈	C₂³×C₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
263	C₄×D₄	C₇×D₄	1	4	yes	—
264	C₄×D₄	C₇×Q₈	1	4	yes	—
265	#117	—	4	2	no	—
266	#117	—	4	2	no	—
267	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
268	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
269	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
272	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
273	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
274	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
275	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
276	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
277	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
278	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
279	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
280	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₇	1	2	yes	—
283	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	2	2	no	—
287	C₄⋊C₈	—	2	2	no	—
288	C₄⋊C₈	—	2	2	no	—
289	C₄⋊C₈	—	4	2	no	—
290	C₄⋊C₈	—	4	2	no	—
291	C₂²⋊C₈	—	2	4	no	—
292	C₂²⋊C₈	—	2	4	no	—
293	C₄×C₈	—	2	2	no	—
294	C₄×C₈	—	2	2	no	—
295	#95	C₂×C₄×C₇	1	2	yes	—
296	#95	C₂×C₄×C₇	1	2	yes	—
297	#95	C₂×C₄×C₇	1	2	yes	—
298	C₂²×C₇	—	2	4	no	—
299	C₂²×C₇	—	2	4	no	—
300	C₂²×C₇	—	2	4	no	—
301	C₂²×C₇	—	4	2	no	—
302	C₂²×C₇	—	4	2	no	—
303	C₂²×C₇	—	4	2	no	—
304	C₂²×C₇	—	4	2	no	—
305	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
306	C₂²×C₇	C₄×#41	1	1	yes	characteristic
307	C₂²×C₇	#518	1	1	yes	characteristic
308	C₂²×C₇	C₂×#131	1	1	yes	characteristic
309	C₂²×C₇	C₂×#131	1	1	yes	characteristic
310	C₂²×C₇	C₂×#131	1	1	yes	characteristic
311	C₂²×C₇	#568	1	1	yes	characteristic
312	C₄×C₇	—	2	4	no	—
313	C₄×C₇	—	2	2	no	—
314	C₄×C₇	—	2	2	no	—
315	C₄×C₇	—	4	1	no	—
316	C₄×C₇	—	4	1	no	—
317	C₄×C₇	—	4	2	no	—
318	C₄×C₇	—	4	1	no	—
319	C₄×C₇	—	4	2	no	—
320	C₄×C₇	#424	1	4	yes	—
321	C₄×C₇	#546	1	1	yes	characteristic
322	C₄×C₇	#546	1	1	yes	characteristic
323	C₄×C₇	#554	1	1	yes	characteristic
324	C₄×C₇	#554	1	1	yes	characteristic
325	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	C₇×C₄⋊C₄	1	4	yes	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
334	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
335	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
336	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
337	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
338	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
339	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
340	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
341	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
342	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
343	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
344	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
345	M₁₆	—	4	8	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	4	8	no	—
363	C₂×C₈	C₇×#41	1	2	yes	—
364	C₂×C₈	C₇×#41	1	2	yes	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
370	C₂²⋊C₄	—	2	4	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₂²⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	1	no	—
383	C₄²	—	2	1	no	—
384	C₄²	—	4	1	no	—
385	C₄²	C₂×C₇×D₄	1	2	yes	—
386	C₄²	C₂×C₇×Q₈	1	2	yes	—
387	C₂×C₇	—	4	4	no	—
388	C₂×C₇	#2349	1	1	yes	characteristic
389	C₂×C₇	C₂×#424	1	1	yes	characteristic
390	C₂×C₇	#2906	1	1	yes	characteristic
391	C₂×C₇	#2966	1	1	yes	characteristic
392	C₂×C₇	#2969	1	1	yes	characteristic
393	C₂×C₇	#2979	1	1	yes	characteristic
394	C₂×C₇	#2981	1	1	yes	characteristic
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
402	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
403	D₄	—	2	8	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
430	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
431	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
432	C₂×C₄	C₇×#131	1	2	yes	—
433	C₂×C₄	C₇×#131	1	2	yes	—
434	C₂×C₄	C₇×#131	1	2	yes	—
435	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
436	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
437	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
438	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
439	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
440	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
441	C₈	—	2	2	no	—
442	C₈	—	2	2	no	—
443	C₈	—	4	1	no	—
444	C₈	—	4	1	no	—
445	C₈	—	4	2	no	—
446	C₈	—	4	2	no	—
447	C₇	#22701	1	1	yes	characteristic, direct product factor, Sylow
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
456	C₂²	C₄×C₇×#41	1	1	yes	characteristic
457	C₂²	C₇×#518	1	1	yes	characteristic
458	C₂²	C₂×C₇×#131	1	1	yes	characteristic
459	C₂²	C₂×C₇×#131	1	1	yes	characteristic
460	C₂²	C₂×C₇×#131	1	1	yes	characteristic
461	C₂²	C₇×#568	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	4	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	1	no	—
469	C₄	—	4	2	no	—
470	C₄	C₇×#424	1	4	yes	—
471	C₄	C₇×#546	1	1	yes	characteristic
472	C₄	C₇×#546	1	1	yes	characteristic
473	C₄	C₇×#554	1	1	yes	characteristic
474	C₄	C₇×#554	1	1	yes	characteristic
475	C₂	—	4	4	no	—
476	C₂	C₇×#2349	1	1	yes	characteristic
477	C₂	C₂×C₇×#424	1	1	yes	characteristic
478	C₂	C₇×#2906	1	1	yes	characteristic
479	C₂	C₇×#2966	1	1	yes	characteristic
480	C₂	C₇×#2969	1	1	yes	characteristic
481	C₂	C₇×#2979	1	1	yes	characteristic
482	C₂	C₇×#2981	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	C₂²×C₇×D₄	C₂×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
4. C₇×#1939	C₂	semidirect
5. C₇×#1939	C₂	semidirect
6. C₇×#1851	C₂	semidirect
7. C₇×#1850	C₂	semidirect
8. C₇×#1831	C₂	semidirect
9. C₇×#1798	C₂	semidirect
10. C₄×C₇×C₄⋊C₈	C₂	semidirect
11. C₇×#1729	C₂	semidirect
43. #22701	C₇	direct (N × H)
96. #1939	C₂×C₇	semidirect
97. #1939	C₂×C₇	semidirect
98. #1851	C₂×C₇	semidirect
99. #1850	C₂×C₇	semidirect
100. #1831	C₂×C₇	semidirect
101. #1798	C₂×C₇	semidirect
102. C₄×C₄⋊C₈	C₂×C₇	semidirect
103. #1729	C₂×C₇	semidirect
447. C₇	#22701	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×#436	2
C₂×C₇×C₈×D₄	2
C₇×#2293	2
C₇×#1939	2
C₇×#1851	2
C₇×#1850	2
C₇×#1831	2
C₇×#1798	2
C₄×C₇×C₄⋊C₈	2
C₇×#1729	2
#22701	7

Subgroups — C7×#22701 #13199917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22701 #13199917