Subgroups — C₇×#8261 #13185477

Order 1792.

502 conjugacy classes of subgroups.

The subgroup lattice has 248 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	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#322	C₂	1	2	yes	maximal
5	C₇×#1273	C₂	1	2	yes	maximal
6	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
7	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₄³×C₇	C₂²	1	1	yes	characteristic
15	C₇×#336	C₄	1	2	yes	—
16	C₇×#336	C₄	1	2	yes	—
17	C₇×#322	—	2	4	no	—
18	#8261	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
30	C₇×C₄⋊C₈	—	2	4	no	—
31	C₇×C₄⋊C₈	C₈	1	8	yes	complemented
32	C₄×C₇×C₈	—	2	4	no	—
33	C₄×C₇×C₈	—	2	4	no	—
34	C₄×C₇×C₈	—	2	2	no	—
35	C₄×C₇×C₈	—	2	2	no	—
36	C₄×C₇×C₈	—	2	1	no	—
37	C₄×C₇×C₈	—	2	1	no	—
38	C₄×C₇×C₈	—	2	2	no	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₄×C₇×C₈	D₄	1	2	yes	—
41	C₄×C₇×C₈	Q₈	1	2	yes	—
42	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
43	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
44	C₄²×C₈	C₂×C₇	1	1	yes	characteristic
45	C₂×#322	C₂×C₇	1	2	yes	—
46	#1273	C₂×C₇	1	2	yes	—
47	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	—	2	4	no	—
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	2	no	—
59	C₂×C₇×C₈	—	2	4	no	—
60	C₂×C₇×C₈	—	2	4	no	—
61	C₂×C₇×C₈	—	2	4	no	—
62	C₂×C₇×C₈	—	2	4	no	—
63	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
64	C₂×C₇×C₈	C₂×C₈	1	8	yes	—
65	C₂×C₇×C₈	M₁₆	1	4	yes	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	C₂×C₈	1	4	yes	—
70	C₄²×C₇	M₁₆	1	4	yes	—
71	C₄²×C₇	M₁₆	1	4	yes	—
72	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₇	#41	1	1	yes	characteristic
75	C₄²×C₇	#41	1	1	yes	characteristic
76	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
77	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
78	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
79	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
82	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
84	C₄³	C₂²×C₇	1	1	yes	characteristic
85	#336	C₄×C₇	1	2	yes	—
86	#336	C₄×C₇	1	2	yes	—
87	#322	—	2	4	no	—
88	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₇	#97	1	2	yes	—
98	C₂×C₄×C₇	#97	1	2	yes	—
99	C₂×C₄×C₇	#97	1	2	yes	—
100	C₂×C₄×C₇	#97	1	2	yes	—
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₇	#131	1	2	yes	—
112	C₂×C₄×C₇	#131	1	2	yes	—
113	C₇×C₈	—	2	4	no	—
114	C₇×C₈	—	2	4	no	—
115	C₇×C₈	—	2	8	no	—
116	C₇×C₈	—	4	4	no	—
117	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
118	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
119	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
120	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
121	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
123	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
127	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	C₇×C₈	1	8	yes	complemented
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	C₇×D₄	1	2	yes	—
139	C₄×C₈	C₇×Q₈	1	2	yes	—
140	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
142	C₂²×C₇	C₂×#97	1	1	yes	characteristic
143	C₂²×C₇	C₂×#97	1	1	yes	characteristic
144	C₂²×C₇	C₄×M₁₆	1	1	yes	characteristic
145	C₂²×C₇	#405	1	1	yes	characteristic
146	C₂²×C₇	#405	1	1	yes	characteristic
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	2	2	no	—
149	C₄×C₇	—	2	2	no	—
150	C₄×C₇	#322	1	4	yes	complemented
151	C₄×C₇	#423	1	2	yes	—
152	C₄×C₇	#432	1	2	yes	—
153	C₄×C₇	C₈×D₄	1	2	yes	—
154	C₄×C₇	#436	1	2	yes	—
155	C₄×C₇	C₈×Q₈	1	2	yes	—
156	C₄×C₇	#446	1	2	yes	—
157	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
158	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
159	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
160	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
174	C₂×C₈	C₂×C₇×C₈	1	8	yes	—
175	C₂×C₈	C₇×M₁₆	1	4	yes	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₂×C₇×C₈	1	4	yes	—
180	C₄²	C₇×M₁₆	1	4	yes	—
181	C₄²	C₇×M₁₆	1	4	yes	—
182	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
183	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
184	C₄²	C₇×#41	1	1	yes	characteristic
185	C₄²	C₇×#41	1	1	yes	characteristic
186	C₂×C₇	C₂×#322	1	1	yes	characteristic
187	C₂×C₇	C₈×M₁₆	1	1	yes	characteristic
188	C₂×C₇	#1451	1	1	yes	characteristic
189	C₂×C₇	#1452	1	1	yes	characteristic
190	C₂×C₇	#1767	1	1	yes	characteristic
191	C₂×C₇	C₈×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₇	#1770	1	1	yes	characteristic
193	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	C₄×C₇×C₈	1	2	yes	—
201	C₂×C₄	C₄×C₇×C₈	1	2	yes	—
202	C₂×C₄	C₇×#97	1	2	yes	—
203	C₂×C₄	C₇×#97	1	2	yes	—
204	C₂×C₄	C₇×#97	1	2	yes	—
205	C₂×C₄	C₇×#97	1	2	yes	—
206	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₇×#117	1	1	yes	characteristic
208	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
209	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
210	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
211	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
213	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
214	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
215	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
216	C₂×C₄	C₇×#131	1	2	yes	—
217	C₂×C₄	C₇×#131	1	2	yes	—
218	C₈	—	2	4	no	—
219	C₈	—	2	4	no	—
220	C₈	—	2	8	no	—
221	C₈	—	4	4	no	—
222	C₇	#8261	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
224	C₂²	C₂×C₄×C₇×C₈	1	1	yes	characteristic
225	C₂²	C₂×C₇×#97	1	1	yes	characteristic
226	C₂²	C₂×C₇×#97	1	1	yes	characteristic
227	C₂²	C₄×C₇×M₁₆	1	1	yes	characteristic
228	C₂²	C₇×#405	1	1	yes	characteristic
229	C₂²	C₇×#405	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	C₇×#322	1	4	yes	complemented
234	C₄	C₇×#423	1	2	yes	—
235	C₄	C₇×#432	1	2	yes	—
236	C₄	C₇×C₈×D₄	1	2	yes	—
237	C₄	C₇×#436	1	2	yes	—
238	C₄	C₇×C₈×Q₈	1	2	yes	—
239	C₄	C₇×#446	1	2	yes	—
240	C₂	C₂×C₇×#322	1	1	yes	characteristic
241	C₂	C₇×C₈×M₁₆	1	1	yes	characteristic
242	C₂	C₇×#1451	1	1	yes	characteristic
243	C₂	C₇×#1452	1	1	yes	characteristic
244	C₂	C₇×#1767	1	1	yes	characteristic
245	C₂	C₇×C₈×C₄⋊C₄	1	1	yes	characteristic
246	C₂	C₇×#1770	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₇	C₂³
commutator	C₂²	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
18. #8261	C₇	direct (N × H)
31. C₇×C₄⋊C₈	C₈	semidirect
129. C₄⋊C₈	C₇×C₈	semidirect
150. C₄×C₇	#322	semidirect
222. C₇	#8261	direct (N × H)
233. C₄	C₇×#322	semidirect

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
C₄²×C₇×C₈	2
C₂×C₇×#322	2
C₇×#1273	2
#8261	7

Subgroups — C7×#8261 #13185477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8261 #13185477