Subgroups — C₇×#23174 #13200390

Order 1792.

1144 conjugacy classes of subgroups.

The subgroup lattice has 564 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, complemented
2	C₂×C₇×#435	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₇×#127	C₂	1	1	yes	maximal, characteristic
5	C₇×#1955	C₂	1	2	yes	maximal, complemented
6	C₇×#1954	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1954	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1870	C₂	1	2	yes	maximal, complemented
9	C₇×#1869	C₂	1	2	yes	maximal, complemented
10	C₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
13	C₂³×C₇×C₈	C₂²	1	2	yes	—
14	C₂×C₇×#127	C₄	1	1	yes	characteristic
15	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
17	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
19	C₇×#436	—	2	4	no	—
20	C₇×#435	—	2	8	no	—
21	C₇×C₈×D₄	—	2	4	no	—
22	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
30	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂×C₇×#97	C₂²	1	2	yes	—
33	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
35	C₇×#390	C₄	1	2	yes	—
36	C₇×#390	C₄	1	1	yes	characteristic
37	C₇×#390	C₄	1	1	yes	characteristic
38	C₇×#390	C₄	1	2	yes	—
39	C₇×#384	C₄	1	1	yes	characteristic
40	C₄³×C₇	C₂²	1	1	yes	characteristic
41	C₇×#336	C₂²	1	1	yes	characteristic
42	C₇×#336	C₂²	1	1	yes	characteristic
43	C₇×#336	C₂²	1	2	yes	—
44	#23174	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
47	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
48	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₇×M₁₆	—	2	4	no	—
52	C₂×C₇×M₁₆	—	2	4	no	—
53	C₂×C₇×M₁₆	—	4	2	no	—
54	C₂²×C₇×C₈	—	2	4	no	—
55	C₂²×C₇×C₈	—	2	4	no	—
56	C₂²×C₇×C₈	—	4	2	no	—
57	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
59	C₂²×C₇×C₈	C₂³	1	2	yes	—
60	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₂²×C₇×C₈	C₂³	1	2	yes	—
63	C₇×#127	—	2	4	no	—
64	C₄×C₇×D₄	—	2	2	no	—
65	C₄×C₇×D₄	—	2	4	no	—
66	C₄×C₇×D₄	—	2	2	no	—
67	C₄×C₇×D₄	—	2	2	no	—
68	C₄×C₇×D₄	—	2	4	no	—
69	C₄×C₇×D₄	—	2	2	no	—
70	C₄×C₇×D₄	—	4	2	no	—
71	C₄×C₇×D₄	—	4	2	no	—
72	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
73	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
76	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
77	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
78	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
79	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
80	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
81	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
82	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
89	C₇×C₄⋊C₈	—	2	2	no	—
90	C₇×C₄⋊C₈	—	2	2	no	—
91	C₇×C₄⋊C₈	D₄	1	4	yes	complemented
92	C₇×C₄⋊C₈	D₄	1	4	yes	complemented
93	C₇×C₂²⋊C₈	—	2	4	no	—
94	C₇×C₂²⋊C₈	—	2	4	no	—
95	C₇×C₂²⋊C₈	—	2	4	no	—
96	C₇×C₂²⋊C₈	—	2	4	no	—
97	C₇×C₂²⋊C₈	—	4	2	no	—
98	C₇×C₂²⋊C₈	—	4	2	no	—
99	C₇×#97	—	2	4	no	—
100	C₄×C₇×C₈	—	2	2	no	—
101	C₄×C₇×C₈	—	2	2	no	—
102	C₂×#436	C₂×C₇	1	1	yes	characteristic, complemented
103	C₂×#435	C₂×C₇	1	2	yes	complemented
104	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic, complemented
105	C₄×#127	C₂×C₇	1	1	yes	characteristic
106	#1955	C₂×C₇	1	2	yes	complemented
107	#1954	C₂×C₇	1	1	yes	characteristic, complemented
108	#1954	C₂×C₇	1	1	yes	characteristic, complemented
109	#1870	C₂×C₇	1	2	yes	complemented
110	#1869	C₂×C₇	1	2	yes	complemented
111	#1767	C₂×C₇	1	1	yes	characteristic, complemented
112	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
113	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
114	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
115	C₂×C₇×D₄	—	2	4	no	—
116	C₂×C₇×D₄	—	2	4	no	—
117	C₂×C₇×D₄	—	2	2	no	—
118	C₂×C₇×D₄	—	2	4	no	—
119	C₂×C₇×D₄	—	2	2	no	—
120	C₂×C₇×D₄	—	2	2	no	—
121	C₂×C₇×D₄	—	2	2	no	—
122	C₂×C₇×D₄	—	2	4	no	—
123	C₂×C₇×D₄	—	2	2	no	—
124	C₂×C₇×D₄	—	4	2	no	—
125	C₂×C₇×D₄	—	4	2	no	—
126	C₂×C₇×D₄	M₁₆	1	4	yes	complemented
127	C₂²×C₄×C₇	—	2	2	no	—
128	C₂²×C₄×C₇	—	2	2	no	—
129	C₂²×C₄×C₇	—	2	2	no	—
130	C₂²×C₄×C₇	—	2	2	no	—
131	C₂²×C₄×C₇	—	2	2	no	—
132	C₂²×C₄×C₇	—	2	2	no	—
133	C₂²×C₄×C₇	—	2	2	no	—
134	C₂²×C₄×C₇	—	2	2	no	—
135	C₂²×C₄×C₇	—	2	2	no	—
136	C₂²×C₄×C₇	—	2	2	no	—
137	C₂²×C₄×C₇	—	2	2	no	—
138	C₂²×C₄×C₇	—	2	2	no	—
139	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
148	C₇×M₁₆	—	4	8	no	—
149	C₂×C₇×C₈	—	2	1	no	—
150	C₂×C₇×C₈	—	2	1	no	—
151	C₂×C₇×C₈	—	2	2	no	—
152	C₂×C₇×C₈	—	2	1	no	—
153	C₂×C₇×C₈	—	2	2	no	—
154	C₂×C₇×C₈	—	2	1	no	—
155	C₂×C₇×C₈	—	2	1	no	—
156	C₂×C₇×C₈	—	2	1	no	—
157	C₂×C₇×C₈	—	2	2	no	—
158	C₂×C₇×C₈	—	2	1	no	—
159	C₂×C₇×C₈	—	2	1	no	—
160	C₂×C₇×C₈	—	2	2	no	—
161	C₂×C₇×C₈	—	2	1	no	—
162	C₂×C₇×C₈	—	2	1	no	—
163	C₂×C₇×C₈	—	2	2	no	—
164	C₂×C₇×C₈	—	4	8	no	—
165	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
166	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
167	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
168	C₇×C₄⋊C₄	—	2	4	no	—
169	C₇×C₄⋊C₄	—	2	2	no	—
170	C₇×C₄⋊C₄	—	2	2	no	—
171	C₇×C₄⋊C₄	—	4	2	no	—
172	C₇×C₂²⋊C₄	—	2	4	no	—
173	C₇×C₂²⋊C₄	—	2	4	no	—
174	C₇×C₂²⋊C₄	—	2	4	no	—
175	C₇×C₂²⋊C₄	—	2	4	no	—
176	C₇×C₂²⋊C₄	—	4	2	no	—
177	C₇×C₂²⋊C₄	—	4	2	no	—
178	C₄²×C₇	—	2	2	no	—
179	C₄²×C₇	—	2	2	no	—
180	C₄²×C₇	—	2	2	no	—
181	C₄²×C₇	—	2	1	no	—
182	C₄²×C₇	—	2	1	no	—
183	C₄²×C₇	—	4	1	no	—
184	C₄²×C₇	C₂×D₄	1	2	yes	—
185	C₄²×C₇	C₂×D₄	1	2	yes	—
186	C₄²×C₇	#41	1	2	yes	—
187	C₄²×C₇	#41	1	2	yes	—
188	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
189	C₂³×C₈	C₂²×C₇	1	2	yes	—
190	C₂×#127	C₄×C₇	1	1	yes	characteristic
191	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
192	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
193	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
194	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
195	#436	—	2	4	no	—
196	#435	—	2	8	no	—
197	C₈×D₄	—	2	4	no	—
198	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
202	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
203	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
204	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
205	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
206	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
207	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
208	C₂×#97	C₂²×C₇	1	2	yes	—
209	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
211	#390	C₄×C₇	1	2	yes	—
212	#390	C₄×C₇	1	1	yes	characteristic
213	#390	C₄×C₇	1	1	yes	characteristic
214	#390	C₄×C₇	1	2	yes	—
215	#384	C₄×C₇	1	1	yes	characteristic
216	C₄³	C₂²×C₇	1	1	yes	characteristic
217	#336	C₂²×C₇	1	1	yes	characteristic
218	#336	C₂²×C₇	1	2	yes	—
219	#336	C₂²×C₇	1	1	yes	characteristic
220	C₂³×C₇	—	2	2	no	—
221	C₂³×C₇	—	2	2	no	—
222	C₂³×C₇	—	2	2	no	—
223	C₂³×C₇	—	2	2	no	—
224	C₂³×C₇	—	2	2	no	—
225	C₂³×C₇	—	2	2	no	—
226	C₂³×C₇	—	2	2	no	—
227	C₂³×C₇	—	2	2	no	—
228	C₂³×C₇	—	2	2	no	—
229	C₂³×C₇	—	2	2	no	—
230	C₂³×C₇	—	2	2	no	—
231	C₂³×C₇	—	2	2	no	—
232	C₂³×C₇	—	2	2	no	—
233	C₂³×C₇	C₂×M₁₆	1	4	yes	—
234	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
235	C₇×D₄	—	2	8	no	—
236	C₇×D₄	—	4	8	no	—
237	C₇×D₄	—	4	8	no	—
238	C₇×D₄	—	4	4	no	—
239	C₂×C₄×C₇	—	2	1	no	—
240	C₂×C₄×C₇	—	2	1	no	—
241	C₂×C₄×C₇	—	2	1	no	—
242	C₂×C₄×C₇	—	2	1	no	—
243	C₂×C₄×C₇	—	2	1	no	—
244	C₂×C₄×C₇	—	2	1	no	—
245	C₂×C₄×C₇	—	2	1	no	—
246	C₂×C₄×C₇	—	2	1	no	—
247	C₂×C₄×C₇	—	2	2	no	—
248	C₂×C₄×C₇	—	2	2	no	—
249	C₂×C₄×C₇	—	2	2	no	—
250	C₂×C₄×C₇	—	2	2	no	—
251	C₂×C₄×C₇	—	2	1	no	—
252	C₂×C₄×C₇	—	2	2	no	—
253	C₂×C₄×C₇	—	2	2	no	—
254	C₂×C₄×C₇	—	2	1	no	—
255	C₂×C₄×C₇	—	2	1	no	—
256	C₂×C₄×C₇	—	4	2	no	—
257	C₂×C₄×C₇	—	4	2	no	—
258	C₂×C₄×C₇	—	4	2	no	—
259	C₂×C₄×C₇	—	4	2	no	—
260	C₂×C₄×C₇	—	4	2	no	—
261	C₂×C₄×C₇	—	4	2	no	—
262	C₂×C₄×C₇	—	4	2	no	—
263	C₂×C₄×C₇	—	4	2	no	—
264	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
265	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
266	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
267	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
268	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
269	C₂×C₄×C₇	#131	1	2	yes	—
270	C₂×C₄×C₇	#131	1	2	yes	—
271	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
275	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
277	C₇×C₈	—	2	2	no	—
278	C₇×C₈	—	2	4	no	—
279	C₇×C₈	—	2	2	no	—
280	C₇×C₈	—	4	1	no	—
281	C₇×C₈	—	4	1	no	—
282	C₇×C₈	—	4	2	no	—
283	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
284	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
285	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
286	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₇	1	2	yes	—
288	C₂³×C₄	C₂³×C₇	1	2	yes	—
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	4	2	no	—
292	C₂²×C₈	—	2	4	no	—
293	C₂²×C₈	—	2	4	no	—
294	C₂²×C₈	—	4	2	no	—
295	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₇	1	2	yes	—
298	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₇	1	2	yes	—
300	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
301	#127	—	2	4	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	—	2	4	no	—
304	C₄×D₄	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	2	4	no	—
307	C₄×D₄	—	2	2	no	—
308	C₄×D₄	—	4	2	no	—
309	C₄×D₄	—	4	2	no	—
310	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
311	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
314	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
315	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
316	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
317	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
318	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
319	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
320	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
321	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
322	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
323	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	C₇×D₄	1	4	yes	complemented
330	C₄⋊C₈	C₇×D₄	1	4	yes	complemented
331	C₂²⋊C₈	—	2	4	no	—
332	C₂²⋊C₈	—	2	4	no	—
333	C₂²⋊C₈	—	2	4	no	—
334	C₂²⋊C₈	—	2	4	no	—
335	C₂²⋊C₈	—	4	2	no	—
336	C₂²⋊C₈	—	4	2	no	—
337	#97	—	2	4	no	—
338	C₄×C₈	—	2	2	no	—
339	C₄×C₈	—	2	2	no	—
340	C₂²×C₇	—	2	4	no	—
341	C₂²×C₇	—	2	4	no	—
342	C₂²×C₇	—	2	4	no	—
343	C₂²×C₇	—	4	2	no	—
344	C₂²×C₇	—	4	2	no	—
345	C₂²×C₇	—	4	2	no	—
346	C₂²×C₇	—	4	2	no	—
347	C₂²×C₇	—	4	2	no	—
348	C₂²×C₇	—	4	2	no	—
349	C₂²×C₇	—	4	2	no	—
350	C₂²×C₇	—	4	2	no	—
351	C₂²×C₇	—	4	2	no	—
352	C₂²×C₇	—	4	2	no	—
353	C₂²×C₇	—	4	2	no	—
354	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
355	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
356	C₂²×C₇	#518	1	1	yes	characteristic
357	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
358	C₂²×C₇	C₂×#131	1	1	yes	characteristic
359	C₂²×C₇	C₂×#131	1	1	yes	characteristic
360	C₂²×C₇	#568	1	1	yes	characteristic
361	C₄×C₇	—	2	2	no	—
362	C₄×C₇	—	2	2	no	—
363	C₄×C₇	—	2	2	no	—
364	C₄×C₇	—	2	2	no	—
365	C₄×C₇	—	4	2	no	—
366	C₄×C₇	—	4	2	no	—
367	C₄×C₇	—	4	1	no	—
368	C₄×C₇	#435	1	4	yes	complemented
369	C₄×C₇	D₄²	1	1	yes	characteristic
370	C₄×C₇	#550	1	1	yes	characteristic
371	C₄×C₇	#550	1	1	yes	characteristic
372	C₄×C₇	#559	1	1	yes	characteristic
373	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
374	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂×D₄	C₇×M₁₆	1	4	yes	complemented
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
400	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
401	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
402	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
403	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
404	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
405	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
406	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
407	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
408	M₁₆	—	4	8	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	4	8	no	—
425	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
426	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
427	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	4	2	no	—
432	C₂²⋊C₄	—	2	4	no	—
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	4	2	no	—
437	C₂²⋊C₄	—	4	2	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	4	1	no	—
444	C₄²	C₂×C₇×D₄	1	2	yes	—
445	C₄²	C₂×C₇×D₄	1	2	yes	—
446	C₄²	C₇×#41	1	2	yes	—
447	C₄²	C₇×#41	1	2	yes	—
448	C₂×C₇	—	4	4	no	—
449	C₂×C₇	—	8	2	no	—
450	C₂×C₇	#2366	1	1	yes	characteristic
451	C₂×C₇	C₂×#435	1	1	yes	characteristic
452	C₂×C₇	#2931	1	1	yes	characteristic
453	C₂×C₇	D₄×M₁₆	1	1	yes	characteristic
454	C₂×C₇	#2935	1	1	yes	characteristic
455	C₂×C₇	#2974	1	1	yes	characteristic
456	C₂×C₇	#2981	1	1	yes	characteristic
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	C₂×C₇×M₁₆	1	4	yes	—
471	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
472	D₄	—	2	8	no	—
473	D₄	—	4	8	no	—
474	D₄	—	4	8	no	—
475	D₄	—	4	4	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
502	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
503	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
504	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
505	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
506	C₂×C₄	C₇×#131	1	2	yes	—
507	C₂×C₄	C₇×#131	1	2	yes	—
508	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
509	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
510	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
511	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
512	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
513	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
514	C₈	—	2	2	no	—
515	C₈	—	2	4	no	—
516	C₈	—	2	2	no	—
517	C₈	—	4	1	no	—
518	C₈	—	4	1	no	—
519	C₈	—	4	2	no	—
520	C₇	#23174	1	1	yes	characteristic, direct product factor, Sylow
521	C₂²	—	2	4	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
536	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
537	C₂²	C₇×#518	1	1	yes	characteristic
538	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
539	C₂²	C₂×C₇×#131	1	1	yes	characteristic
540	C₂²	C₂×C₇×#131	1	1	yes	characteristic
541	C₂²	C₇×#568	1	1	yes	characteristic
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	2	no	—
549	C₄	C₇×#435	1	4	yes	complemented
550	C₄	C₇×D₄²	1	1	yes	characteristic
551	C₄	C₇×#550	1	1	yes	characteristic
552	C₄	C₇×#550	1	1	yes	characteristic
553	C₄	C₇×#559	1	1	yes	characteristic
554	C₂	—	4	4	no	—
555	C₂	—	8	2	no	—
556	C₂	C₇×#2366	1	1	yes	characteristic
557	C₂	C₂×C₇×#435	1	1	yes	characteristic
558	C₂	C₇×#2931	1	1	yes	characteristic
559	C₂	C₇×D₄×M₁₆	1	1	yes	characteristic
560	C₂	C₇×#2935	1	1	yes	characteristic
561	C₂	C₇×#2974	1	1	yes	characteristic
562	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	C₂×C₇×#127	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
1. C₂×C₇×#436	C₂	semidirect
2. C₂×C₇×#435	C₂	semidirect
3. C₂×C₇×C₈×D₄	C₂	semidirect
5. C₇×#1955	C₂	semidirect
6. C₇×#1954	C₂	semidirect
7. C₇×#1954	C₂	semidirect
8. C₇×#1870	C₂	semidirect
9. C₇×#1869	C₂	semidirect
10. C₇×#1767	C₂	semidirect
11. C₄×C₇×C₄⋊C₈	C₂	semidirect
44. #23174	C₇	direct (N × H)
91. C₇×C₄⋊C₈	D₄	semidirect
92. C₇×C₄⋊C₈	D₄	semidirect
102. C₂×#436	C₂×C₇	semidirect
103. C₂×#435	C₂×C₇	semidirect
104. C₂×C₈×D₄	C₂×C₇	semidirect
106. #1955	C₂×C₇	semidirect
107. #1954	C₂×C₇	semidirect
108. #1954	C₂×C₇	semidirect
109. #1870	C₂×C₇	semidirect
110. #1869	C₂×C₇	semidirect
111. #1767	C₂×C₇	semidirect
112. C₄×C₄⋊C₈	C₂×C₇	semidirect
126. C₂×C₇×D₄	M₁₆	semidirect
329. C₄⋊C₈	C₇×D₄	semidirect
330. C₄⋊C₈	C₇×D₄	semidirect
368. C₄×C₇	#435	semidirect
386. C₂×D₄	C₇×M₁₆	semidirect
520. C₇	#23174	direct (N × H)
549. C₄	C₇×#435	semidirect

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×#435	2
C₂×C₇×C₈×D₄	2
C₄×C₇×#127	2
C₇×#1955	2
C₇×#1954	2
C₇×#1870	2
C₇×#1869	2
C₇×#1767	2
C₄×C₇×C₄⋊C₈	2
#23174	7

Subgroups — C7×#23174 #13200390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23174 #13200390