Subgroups — C₇×#8266 #13185482

Order 1792.

414 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1564	C₂	1	2	yes	maximal
3	C₇×#1450	C₂	1	2	yes	maximal
4	C₇×#1276	C₂	1	2	yes	maximal
5	C₇×#431	C₂²	1	4	yes	—
6	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	C₇×#335	C₄	1	4	yes	complemented
12	C₇×#334	C₄	1	4	yes	complemented
13	C₇×#322	—	2	2	no	—
14	C₇×C₈²	—	2	2	no	—
15	#8266	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇	—	2	2	no	—
20	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
22	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
23	C₇×C₂²⋊C₈	—	2	4	no	—
24	C₄×C₇×C₈	—	2	4	no	—
25	C₄×C₇×C₈	—	2	4	no	—
26	C₄×C₇×C₈	C₈	1	8	yes	complemented
27	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₇×C₈	D₄	1	1	yes	characteristic
30	C₄×C₇×C₈	D₄	1	1	yes	characteristic
31	C₄×C₇×C₈	Q₈	1	2	yes	—
32	C₄²×C₈	C₂×C₇	1	1	yes	characteristic
33	#1564	C₂×C₇	1	2	yes	—
34	#1450	C₂×C₇	1	2	yes	—
35	#1276	C₂×C₇	1	2	yes	—
36	C₂²×C₄×C₇	—	2	2	no	—
37	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇	#41	1	2	yes	—
39	C₂×C₇×C₈	—	2	2	no	—
40	C₂×C₇×C₈	—	2	8	no	—
41	C₂×C₇×C₈	C₄²	1	2	yes	—
42	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
43	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
44	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
47	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
48	C₂×C₇×C₈	M₁₆	1	4	yes	—
49	C₄²×C₇	—	2	8	no	—
50	C₄²×C₇	—	2	2	no	—
51	C₄²×C₇	C₂×C₈	1	4	yes	—
52	C₄²×C₇	C₂²×C₄	1	2	yes	—
53	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
55	#431	C₂²×C₇	1	4	yes	—
56	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
57	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
58	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
60	C₄³	C₂²×C₇	1	1	yes	characteristic
61	#335	C₄×C₇	1	4	yes	complemented
62	#334	C₄×C₇	1	4	yes	complemented
63	#322	—	2	2	no	—
64	C₈²	—	2	2	no	—
65	C₂³×C₇	#117	1	1	yes	characteristic
66	C₂×C₄×C₇	—	2	8	no	—
67	C₂×C₄×C₇	—	2	4	no	—
68	C₂×C₄×C₇	—	2	4	no	—
69	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
75	C₇×C₈	—	2	2	no	—
76	C₇×C₈	—	4	8	no	—
77	C₇×C₈	C₄×C₈	1	2	yes	—
78	C₇×C₈	C₄×C₈	1	2	yes	—
79	C₇×C₈	C₄⋊C₈	1	4	yes	—
80	C₇×C₈	C₄⋊C₈	1	4	yes	—
81	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
83	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
84	C₂×C₄²	—	2	2	no	—
85	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
86	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
87	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
88	C₂²⋊C₈	—	2	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	C₇×C₈	1	8	yes	complemented
92	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
93	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
94	C₄×C₈	C₇×D₄	1	1	yes	characteristic
95	C₄×C₈	C₇×D₄	1	1	yes	characteristic
96	C₄×C₈	C₇×Q₈	1	2	yes	—
97	C₂²×C₇	—	2	2	no	—
98	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₇	#431	1	2	yes	—
100	C₄×C₇	—	2	8	no	—
101	C₄×C₇	—	2	2	no	—
102	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
104	C₄×C₇	C₂×C₄⋊C₈	1	2	yes	—
105	C₄×C₇	#443	1	4	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
108	C₂²×C₄	C₇×#41	1	2	yes	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	C₄²×C₇	1	2	yes	—
112	C₂×C₈	C₄²×C₇	1	1	yes	characteristic
113	C₂×C₈	C₄²×C₇	1	1	yes	characteristic
114	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
115	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
116	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
117	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
118	C₂×C₈	C₇×M₁₆	1	4	yes	—
119	C₄²	—	2	8	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂×C₇×C₈	1	4	yes	—
122	C₄²	C₂²×C₄×C₇	1	2	yes	—
123	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
124	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
125	C₂×C₇	—	2	2	no	—
126	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
127	C₂×C₇	#1772	1	2	yes	—
128	C₂³	C₇×#117	1	1	yes	characteristic
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
135	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
136	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
137	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	4	8	no	—
140	C₈	C₄×C₇×C₈	1	2	yes	—
141	C₈	C₄×C₇×C₈	1	2	yes	—
142	C₈	C₇×C₄⋊C₈	1	4	yes	—
143	C₈	C₇×C₄⋊C₈	1	4	yes	—
144	C₇	#8266	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
147	C₂²	C₇×#431	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×C₇×C₈	1	1	yes	characteristic
151	C₄	C₄×C₇×M₁₆	1	1	yes	characteristic
152	C₄	C₂×C₇×C₄⋊C₈	1	2	yes	—
153	C₄	C₇×#443	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₇×C₄⋊C₈	1	1	yes	characteristic
156	C₂	C₇×#1772	1	2	yes	—

	subgroup	G/H
center	C₄×C₇×C₈	D₄
commutator	C₄	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	C₄×C₄⋊C₄
prime radical	C₂³×C₇	#117
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₇×#335	C₄	semidirect
12. C₇×#334	C₄	semidirect
15. #8266	C₇	direct (N × H)
26. C₄×C₇×C₈	C₈	semidirect
61. #335	C₄×C₇	semidirect
62. #334	C₄×C₇	semidirect
91. C₄×C₈	C₇×C₈	semidirect
144. C₇	#8266	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄²×C₇×C₈	2
C₇×#1564	2
C₇×#1450	2
C₇×#1276	2
#8266	7

Subgroups — C7×#8266 #13185482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8266 #13185482