Subgroups — C₇×#22762 #13199978

Order 1792.

902 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#1935	C₂	1	2	yes	maximal, complemented
5	C₇×#1920	C₂	1	2	yes	maximal, complemented
6	C₇×#1877	C₂	1	2	yes	maximal
7	C₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1762	C₂	1	2	yes	maximal
12	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
15	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
17	C₄×C₇×SD₁₆	C₄	1	8	yes	complemented
18	C₂×C₇×#106	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×#103	C₂²	1	2	yes	—
21	C₂×C₇×#103	C₂²	1	1	yes	characteristic
22	C₂×C₇×#102	C₂²	1	2	yes	—
23	C₂×C₇×#102	C₂²	1	1	yes	characteristic
24	C₂×C₇×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#391	C₂²	1	2	yes	—
27	C₇×#389	C₂²	1	2	yes	—
28	C₇×#387	—	2	2	no	—
29	C₇×#385	—	2	2	no	—
30	C₇×#384	C₂²	1	1	yes	characteristic
31	C₇×#384	C₂²	1	1	yes	characteristic
32	C₇×#382	—	2	1	no	—
33	C₇×#380	—	2	1	no	—
34	C₄×C₇×C₄⋊C₄	—	2	1	no	—
35	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
36	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₇×#340	C₂²	1	2	yes	—
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#336	C₂²	1	2	yes	—
42	#22762	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₇	—	2	1	no	—
46	C₂³×C₄×C₇	—	2	2	no	—
47	C₂×C₇×SD₁₆	—	2	2	no	—
48	C₂×C₇×SD₁₆	—	2	2	no	—
49	C₂×C₇×SD₁₆	C₂×C₄	1	4	yes	—
50	C₂²×C₇×C₈	C₂³	1	2	yes	—
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₇×Q₈	—	2	2	no	—
54	C₄×C₇×Q₈	C₂×C₄	1	4	yes	complemented
55	C₄×C₇×D₄	—	2	4	no	—
56	C₄×C₇×D₄	—	2	2	no	—
57	C₄×C₇×D₄	—	4	2	no	—
58	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
59	C₂×C₇×C₄⋊C₄	—	2	1	no	—
60	C₂×C₇×C₄⋊C₄	—	2	1	no	—
61	C₂×C₇×C₄⋊C₄	—	2	2	no	—
62	C₂×C₇×C₄⋊C₄	—	2	1	no	—
63	C₂×C₇×C₄⋊C₄	—	2	1	no	—
64	C₂×C₇×C₄⋊C₄	—	2	1	no	—
65	C₂×C₇×C₄⋊C₄	—	4	4	no	—
66	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
67	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
70	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
73	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
74	C₂×C₄²×C₇	—	2	1	no	—
75	C₂×C₄²×C₇	—	2	2	no	—
76	C₂×C₄²×C₇	C₂³	1	2	yes	—
77	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
78	C₇×#106	C₂×C₄	1	4	yes	complemented
79	C₇×C₄⋊C₈	—	2	2	no	—
80	C₇×#103	—	4	2	no	—
81	C₇×#103	C₂×C₄	1	4	yes	complemented
82	C₇×#102	—	2	4	no	—
83	C₇×#102	C₂×C₄	1	4	yes	—
84	C₇×#97	—	4	2	no	—
85	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
86	C₇×#95	—	2	2	no	—
87	C₇×#95	—	2	2	no	—
88	C₇×#95	—	2	2	no	—
89	C₇×#95	—	2	2	no	—
90	C₇×#95	D₄	1	2	yes	—
91	C₂×C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic
92	#2355	C₂×C₇	1	1	yes	characteristic, complemented
93	D₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
94	#1935	C₂×C₇	1	2	yes	complemented
95	#1920	C₂×C₇	1	2	yes	complemented
96	#1877	C₂×C₇	1	2	yes	—
97	#1834	C₂×C₇	1	1	yes	characteristic, complemented
98	#1803	C₂×C₇	1	1	yes	characteristic, complemented
99	#1802	C₂×C₇	1	1	yes	characteristic
100	#1770	C₂×C₇	1	1	yes	characteristic, complemented
101	#1762	C₂×C₇	1	2	yes	—
102	C₂⁴×C₇	—	2	1	no	—
103	C₂×C₇×Q₈	—	2	1	no	—
104	C₂×C₇×Q₈	—	2	1	no	—
105	C₂×C₇×Q₈	—	2	1	no	—
106	C₂×C₇×Q₈	—	2	1	no	—
107	C₂×C₇×Q₈	—	2	1	no	—
108	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
109	C₂×C₇×D₄	—	2	1	no	—
110	C₂×C₇×D₄	—	2	1	no	—
111	C₂×C₇×D₄	—	2	1	no	—
112	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
113	C₂×C₇×D₄	#41	1	2	yes	—
114	C₂×C₇×D₄	#41	1	2	yes	—
115	C₂²×C₄×C₇	—	2	1	no	—
116	C₂²×C₄×C₇	—	2	1	no	—
117	C₂²×C₄×C₇	—	2	1	no	—
118	C₂²×C₄×C₇	—	2	4	no	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	—	2	2	no	—
121	C₂²×C₄×C₇	—	4	4	no	—
122	C₂²×C₄×C₇	—	4	4	no	—
123	C₂²×C₄×C₇	—	4	2	no	—
124	C₂²×C₄×C₇	—	4	2	no	—
125	C₂²×C₄×C₇	—	8	2	no	—
126	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
131	C₇×SD₁₆	—	2	8	no	—
132	C₂×C₇×C₈	—	2	1	no	—
133	C₂×C₇×C₈	—	2	1	no	—
134	C₂×C₇×C₈	—	2	1	no	—
135	C₂×C₇×C₈	—	2	1	no	—
136	C₂×C₇×C₈	—	2	2	no	—
137	C₂×C₇×C₈	—	2	2	no	—
138	C₂×C₇×C₈	—	2	2	no	—
139	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
140	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
141	C₇×C₄⋊C₄	—	2	4	no	—
142	C₇×C₄⋊C₄	—	2	2	no	—
143	C₇×C₄⋊C₄	—	2	2	no	—
144	C₇×C₄⋊C₄	—	2	1	no	—
145	C₇×C₄⋊C₄	—	2	1	no	—
146	C₇×C₄⋊C₄	—	2	2	no	—
147	C₇×C₄⋊C₄	—	2	2	no	—
148	C₇×C₄⋊C₄	—	2	2	no	—
149	C₇×C₄⋊C₄	—	2	2	no	—
150	C₇×C₄⋊C₄	—	4	4	no	—
151	C₇×C₄⋊C₄	—	4	2	no	—
152	C₇×C₄⋊C₄	—	4	2	no	—
153	C₇×C₄⋊C₄	—	8	1	no	—
154	C₇×C₄⋊C₄	—	8	1	no	—
155	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
156	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
157	C₇×C₂²⋊C₄	—	2	4	no	—
158	C₇×C₂²⋊C₄	—	4	4	no	—
159	C₄²×C₇	—	2	4	no	—
160	C₄²×C₇	—	2	2	no	—
161	C₄²×C₇	—	2	2	no	—
162	C₄²×C₇	—	8	2	no	—
163	C₄²×C₇	C₂²×C₄	1	2	yes	—
164	C₄²×C₇	#41	1	2	yes	—
165	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
166	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
167	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
168	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
169	C₂²×C₄⋊C₄	—	2	1	no	—
170	C₄×SD₁₆	C₄×C₇	1	8	yes	complemented
171	C₂×#106	C₂²×C₇	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
173	C₂×#103	C₂²×C₇	1	2	yes	—
174	C₂×#103	C₂²×C₇	1	1	yes	characteristic
175	C₂×#102	C₂²×C₇	1	2	yes	—
176	C₂×#102	C₂²×C₇	1	1	yes	characteristic
177	C₂×#97	C₂²×C₇	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
179	#391	C₂²×C₇	1	2	yes	—
180	#389	C₂²×C₇	1	2	yes	—
181	#387	—	2	2	no	—
182	#385	—	2	2	no	—
183	#384	C₂²×C₇	1	1	yes	characteristic
184	#384	C₂²×C₇	1	1	yes	characteristic
185	#382	—	2	1	no	—
186	#380	—	2	1	no	—
187	C₄×C₄⋊C₄	—	2	1	no	—
188	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
189	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
190	#340	C₂²×C₇	1	2	yes	—
191	#340	C₂²×C₇	1	2	yes	—
192	#340	C₂²×C₇	1	2	yes	—
193	#340	C₂²×C₇	1	2	yes	—
194	#336	C₂²×C₇	1	2	yes	—
195	C₂³×C₇	—	2	2	no	—
196	C₂³×C₇	—	2	2	no	—
197	C₂³×C₇	—	2	2	no	—
198	C₂³×C₇	—	4	1	no	—
199	C₂³×C₇	—	4	1	no	—
200	C₂³×C₇	—	4	1	no	—
201	C₂³×C₇	—	4	1	no	—
202	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
203	C₇×Q₈	—	2	2	no	—
204	C₇×Q₈	—	2	2	no	—
205	C₇×Q₈	—	4	1	no	—
206	C₇×Q₈	—	4	1	no	—
207	C₇×D₄	—	2	2	no	—
208	C₇×D₄	—	2	2	no	—
209	C₇×D₄	—	2	2	no	—
210	C₇×D₄	#117	1	4	yes	—
211	C₂×C₄×C₇	—	2	1	no	—
212	C₂×C₄×C₇	—	2	1	no	—
213	C₂×C₄×C₇	—	2	2	no	—
214	C₂×C₄×C₇	—	2	2	no	—
215	C₂×C₄×C₇	—	2	2	no	—
216	C₂×C₄×C₇	—	2	2	no	—
217	C₂×C₄×C₇	—	2	2	no	—
218	C₂×C₄×C₇	—	2	2	no	—
219	C₂×C₄×C₇	—	2	2	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	2	2	no	—
222	C₂×C₄×C₇	—	4	1	no	—
223	C₂×C₄×C₇	—	4	1	no	—
224	C₂×C₄×C₇	—	4	1	no	—
225	C₂×C₄×C₇	—	4	1	no	—
226	C₂×C₄×C₇	—	4	1	no	—
227	C₂×C₄×C₇	—	4	1	no	—
228	C₂×C₄×C₇	—	4	8	no	—
229	C₂×C₄×C₇	—	4	2	no	—
230	C₂×C₄×C₇	—	4	2	no	—
231	C₂×C₄×C₇	—	4	2	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	2	no	—
234	C₂×C₄×C₇	—	4	2	no	—
235	C₂×C₄×C₇	—	8	8	no	—
236	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
237	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
238	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
239	C₂×C₄×C₇	#137	1	2	yes	—
240	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
241	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
245	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
246	C₇×C₈	—	2	2	no	—
247	C₇×C₈	—	2	2	no	—
248	C₇×C₈	—	4	2	no	—
249	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
250	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
251	C₂³×C₄	—	2	1	no	—
252	C₂³×C₄	—	2	2	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×SD₁₆	—	2	2	no	—
255	C₂×SD₁₆	C₂×C₄×C₇	1	4	yes	—
256	C₂²×C₈	C₂³×C₇	1	2	yes	—
257	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
259	C₄×Q₈	—	2	2	no	—
260	C₄×Q₈	C₂×C₄×C₇	1	4	yes	complemented
261	C₄×D₄	—	2	4	no	—
262	C₄×D₄	—	2	2	no	—
263	C₄×D₄	—	4	2	no	—
264	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	4	4	no	—
272	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
277	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
278	C₂×C₂²⋊C₄	—	2	1	no	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₄²	—	2	1	no	—
281	C₂×C₄²	—	2	2	no	—
282	C₂×C₄²	C₂³×C₇	1	2	yes	—
283	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
284	#106	C₂×C₄×C₇	1	4	yes	complemented
285	C₄⋊C₈	—	2	2	no	—
286	#103	—	4	2	no	—
287	#103	C₂×C₄×C₇	1	4	yes	complemented
288	#102	—	2	4	no	—
289	#102	C₂×C₄×C₇	1	4	yes	—
290	#97	—	4	2	no	—
291	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
292	#95	—	2	2	no	—
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	C₇×D₄	1	2	yes	—
297	C₂²×C₇	—	2	4	no	—
298	C₂²×C₇	—	4	2	no	—
299	C₂²×C₇	—	4	2	no	—
300	C₂²×C₇	—	4	2	no	—
301	C₂²×C₇	—	4	2	no	—
302	C₂²×C₇	—	4	2	no	—
303	C₂²×C₇	—	4	2	no	—
304	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
305	C₂²×C₇	#546	1	1	yes	characteristic
306	C₂²×C₇	#548	1	1	yes	characteristic
307	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂²×C₇	C₂×#137	1	1	yes	characteristic
309	C₂²×C₇	#577	1	1	yes	characteristic
310	C₂²×C₇	#577	1	1	yes	characteristic
311	C₄×C₇	—	2	2	no	—
312	C₄×C₇	—	4	2	no	—
313	C₄×C₇	—	4	2	no	—
314	C₄×C₇	—	4	2	no	—
315	C₄×C₇	—	4	2	no	—
316	C₄×C₇	—	4	2	no	—
317	C₄×C₇	—	8	2	no	—
318	C₄×C₇	—	8	2	no	—
319	C₄×C₇	#440	1	4	yes	—
320	C₄×C₇	C₂×#117	1	1	yes	characteristic
321	C₄×C₇	#520	1	1	yes	characteristic
322	C₄×C₇	#552	1	1	yes	characteristic
323	C₄×C₇	#553	1	1	yes	characteristic
324	C₂⁴	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	1	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
335	C₂×D₄	C₇×#41	1	2	yes	—
336	C₂×D₄	C₇×#41	1	2	yes	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	4	4	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	—	4	2	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	—	8	2	no	—
348	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
349	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
350	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
351	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
352	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
353	SD₁₆	—	2	8	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
362	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	4	4	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	8	1	no	—
376	C₄⋊C₄	—	8	1	no	—
377	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
378	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₄²	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	8	2	no	—
385	C₄²	C₂²×C₄×C₇	1	2	yes	—
386	C₄²	C₇×#41	1	2	yes	—
387	C₂×C₇	—	4	4	no	—
388	C₂×C₇	#2352	1	1	yes	characteristic
389	C₂×C₇	C₂×#440	1	1	yes	characteristic
390	C₂×C₇	#2952	1	1	yes	characteristic
391	C₂×C₇	#3324	1	1	yes	characteristic
392	C₂×C₇	#3331	1	1	yes	characteristic
393	C₂×C₇	#3372	1	1	yes	characteristic
394	C₂×C₇	#3373	1	1	yes	characteristic
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	—	4	1	no	—
402	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
403	Q₈	—	2	2	no	—
404	Q₈	—	2	2	no	—
405	Q₈	—	4	1	no	—
406	Q₈	—	4	1	no	—
407	D₄	—	2	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	2	2	no	—
410	D₄	C₇×#117	1	4	yes	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	8	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	8	8	no	—
436	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
437	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
438	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
439	C₂×C₄	C₇×#137	1	2	yes	—
440	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
441	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
442	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
443	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
444	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
445	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
446	C₈	—	2	2	no	—
447	C₈	—	2	2	no	—
448	C₈	—	4	2	no	—
449	C₇	#22762	1	1	yes	characteristic, direct product factor, Sylow
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	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
458	C₂²	C₇×#546	1	1	yes	characteristic
459	C₂²	C₇×#548	1	1	yes	characteristic
460	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
461	C₂²	C₂×C₇×#137	1	1	yes	characteristic
462	C₂²	C₇×#577	1	1	yes	characteristic
463	C₂²	C₇×#577	1	1	yes	characteristic
464	C₄	—	2	2	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	8	2	no	—
471	C₄	—	8	2	no	—
472	C₄	C₇×#440	1	4	yes	—
473	C₄	C₂×C₇×#117	1	1	yes	characteristic
474	C₄	C₇×#520	1	1	yes	characteristic
475	C₄	C₇×#552	1	1	yes	characteristic
476	C₄	C₇×#553	1	1	yes	characteristic
477	C₂	—	4	4	no	—
478	C₂	C₇×#2352	1	1	yes	characteristic
479	C₂	C₂×C₇×#440	1	1	yes	characteristic
480	C₂	C₇×#2952	1	1	yes	characteristic
481	C₂	C₇×#3324	1	1	yes	characteristic
482	C₂	C₇×#3331	1	1	yes	characteristic
483	C₂	C₇×#3372	1	1	yes	characteristic
484	C₂	C₇×#3373	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₇×#2355	C₂	semidirect
4. C₇×#1935	C₂	semidirect
5. C₇×#1920	C₂	semidirect
7. C₇×#1834	C₂	semidirect
8. C₇×#1803	C₂	semidirect
10. C₇×#1770	C₂	semidirect
17. C₄×C₇×SD₁₆	C₄	semidirect
42. #22762	C₇	direct (N × H)
54. C₄×C₇×Q₈	C₂×C₄	semidirect
78. C₇×#106	C₂×C₄	semidirect
81. C₇×#103	C₂×C₄	semidirect
85. C₄×C₇×C₈	C₂×C₄	semidirect
92. #2355	C₂×C₇	semidirect
94. #1935	C₂×C₇	semidirect
95. #1920	C₂×C₇	semidirect
97. #1834	C₂×C₇	semidirect
98. #1803	C₂×C₇	semidirect
100. #1770	C₂×C₇	semidirect
170. C₄×SD₁₆	C₄×C₇	semidirect
260. C₄×Q₈	C₂×C₄×C₇	semidirect
284. #106	C₂×C₄×C₇	semidirect
287. #103	C₂×C₄×C₇	semidirect
291. C₄×C₈	C₂×C₄×C₇	semidirect
449. C₇	#22762	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×C₄×C₇×SD₁₆	2
C₇×#2355	2
C₇×D₄×C₄⋊C₄	2
C₇×#1935	2
C₇×#1920	2
C₇×#1877	2
C₇×#1834	2
C₇×#1803	2
C₇×#1802	2
C₇×#1770	2
C₇×#1762	2
#22762	7

Subgroups — C7×#22762 #13199978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22762 #13199978