Subgroups — C₇×#8265 #13185481

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₇×#1273	C₂	1	1	yes	maximal, characteristic
7	C₇×#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₄³×C₇	C₂²	1	1	yes	characteristic
17	C₇×#336	C₄	1	2	yes	—
18	C₇×#336	C₄	1	2	yes	—
19	C₇×#336	C₄	1	2	yes	—
20	C₇×#336	C₄	1	2	yes	—
21	C₇×#322	—	2	2	no	—
22	C₇×#322	—	2	2	no	—
23	#8265	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₇×C₄⋊C₈	—	2	4	no	—
38	C₇×C₄⋊C₈	—	2	4	no	—
39	C₄×C₇×C₈	—	2	1	no	—
40	C₄×C₇×C₈	—	2	1	no	—
41	C₄×C₇×C₈	—	2	1	no	—
42	C₄×C₇×C₈	—	2	1	no	—
43	C₄×C₇×C₈	—	2	1	no	—
44	C₄×C₇×C₈	—	2	1	no	—
45	C₄×C₇×C₈	—	2	1	no	—
46	C₄×C₇×C₈	—	2	1	no	—
47	C₄×C₇×C₈	—	2	1	no	—
48	C₄×C₇×C₈	—	2	1	no	—
49	C₄×C₇×C₈	D₄	1	2	yes	—
50	C₄×C₇×C₈	Q₈	1	2	yes	—
51	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
52	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
53	#1744	C₂×C₇	1	1	yes	characteristic
54	C₂×#322	C₂×C₇	1	1	yes	characteristic
55	C₂×#322	C₂×C₇	1	1	yes	characteristic
56	#1273	C₂×C₇	1	1	yes	characteristic
57	#1273	C₂×C₇	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	—	2	2	no	—
71	C₂×C₇×C₈	—	2	2	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	2	2	no	—
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
82	C₂×C₇×C₈	M₁₆	1	4	yes	—
83	C₄²×C₇	—	2	2	no	—
84	C₄²×C₇	—	2	2	no	—
85	C₄²×C₇	—	2	2	no	—
86	C₄²×C₇	—	2	2	no	—
87	C₄²×C₇	—	2	2	no	—
88	C₄²×C₇	M₁₆	1	4	yes	—
89	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₇	#41	1	1	yes	characteristic
92	C₄²×C₇	#41	1	1	yes	characteristic
93	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
94	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
95	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
101	C₄³	C₂²×C₇	1	1	yes	characteristic
102	#336	C₄×C₇	1	2	yes	—
103	#336	C₄×C₇	1	2	yes	—
104	#336	C₄×C₇	1	2	yes	—
105	#336	C₄×C₇	1	2	yes	—
106	#322	—	2	2	no	—
107	#322	—	2	2	no	—
108	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	1	no	—
119	C₂×C₄×C₇	#97	1	2	yes	—
120	C₂×C₄×C₇	#97	1	2	yes	—
121	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₇	#131	1	2	yes	—
130	C₂×C₄×C₇	#131	1	2	yes	—
131	C₂×C₄×C₇	#131	1	2	yes	—
132	C₂×C₄×C₇	#131	1	2	yes	—
133	C₇×C₈	—	2	4	no	—
134	C₇×C₈	—	2	4	no	—
135	C₇×C₈	—	4	2	no	—
136	C₇×C₈	—	4	2	no	—
137	C₇×C₈	—	4	2	no	—
138	C₇×C₈	—	4	2	no	—
139	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
141	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
142	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
143	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
147	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
148	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
149	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
151	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
152	C₄⋊C₈	—	2	4	no	—
153	C₄⋊C₈	—	2	4	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	C₇×D₄	1	2	yes	—
165	C₄×C₈	C₇×Q₈	1	2	yes	—
166	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₇	C₂×#97	1	1	yes	characteristic
168	C₂²×C₇	C₄×M₁₆	1	1	yes	characteristic
169	C₂²×C₇	#405	1	1	yes	characteristic
170	C₂²×C₇	#405	1	1	yes	characteristic
171	C₂²×C₇	#405	1	1	yes	characteristic
172	C₂²×C₇	#405	1	1	yes	characteristic
173	C₄×C₇	—	2	2	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	4	1	no	—
176	C₄×C₇	—	4	1	no	—
177	C₄×C₇	#424	1	2	yes	—
178	C₄×C₇	#433	1	2	yes	—
179	C₄×C₇	#435	1	2	yes	—
180	C₄×C₇	#436	1	2	yes	—
181	C₄×C₇	#446	1	2	yes	—
182	C₄×C₇	#446	1	2	yes	—
183	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
184	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
185	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
186	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
187	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
207	C₂×C₈	C₇×M₁₆	1	4	yes	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₇×M₁₆	1	4	yes	—
214	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
215	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
216	C₄²	C₇×#41	1	1	yes	characteristic
217	C₄²	C₇×#41	1	1	yes	characteristic
218	C₂×C₇	#1451	1	1	yes	characteristic
219	C₂×C₇	#1452	1	1	yes	characteristic
220	C₂×C₇	#1453	1	1	yes	characteristic
221	C₂×C₇	#1454	1	1	yes	characteristic
222	C₂×C₇	#1768	1	1	yes	characteristic
223	C₂×C₇	#1770	1	1	yes	characteristic
224	C₂×C₇	#1771	1	1	yes	characteristic
225	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	C₇×#97	1	2	yes	—
237	C₂×C₄	C₇×#97	1	2	yes	—
238	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₇×#117	1	1	yes	characteristic
240	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
241	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
242	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
243	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
244	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
245	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
246	C₂×C₄	C₇×#131	1	2	yes	—
247	C₂×C₄	C₇×#131	1	2	yes	—
248	C₂×C₄	C₇×#131	1	2	yes	—
249	C₂×C₄	C₇×#131	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	2	4	no	—
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₇	#8265	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
258	C₂²	C₂×C₇×#97	1	1	yes	characteristic
259	C₂²	C₄×C₇×M₁₆	1	1	yes	characteristic
260	C₂²	C₇×#405	1	1	yes	characteristic
261	C₂²	C₇×#405	1	1	yes	characteristic
262	C₂²	C₇×#405	1	1	yes	characteristic
263	C₂²	C₇×#405	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₇×#424	1	2	yes	—
269	C₄	C₇×#433	1	2	yes	—
270	C₄	C₇×#435	1	2	yes	—
271	C₄	C₇×#436	1	2	yes	—
272	C₄	C₇×#446	1	2	yes	—
273	C₄	C₇×#446	1	2	yes	—
274	C₂	C₇×#1451	1	1	yes	characteristic
275	C₂	C₇×#1452	1	1	yes	characteristic
276	C₂	C₇×#1453	1	1	yes	characteristic
277	C₂	C₇×#1454	1	1	yes	characteristic
278	C₂	C₇×#1768	1	1	yes	characteristic
279	C₂	C₇×#1770	1	1	yes	characteristic
280	C₂	C₇×#1771	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₇	C₂³
commutator	C₂³	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
23. #8265	C₇	direct (N × H)
256. C₇	#8265	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×C₄⋊C₈	2
C₇×#1744	2
C₂×C₇×#322	2
C₇×#1273	2
#8265	7

Subgroups — C7×#8265 #13185481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8265 #13185481