Subgroups — #89490

Order 448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	C₄	1	1	yes	characteristic
8	C₂²×Dic₇	C₄	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₄×Dic₇	C₄	1	1	yes	characteristic
11	C₄×Dic₇	C₄	1	1	yes	characteristic
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₄×Dic₇	C₂²	1	1	yes	characteristic
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#431	—	7	1	no	maximal, Sylow
16	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₇	—	2	2	no	—
21	C₂×Dic₇	C₈	1	4	yes	complemented
22	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₇×C₈	—	2	2	no	—
27	C₇⋊C₈	—	2	2	no	—
28	C₂×C₄²	—	7	1	no	—
29	C₄⋊C₈	—	7	2	no	—
30	C₂²⋊C₈	—	7	1	no	—
31	C₂²⋊C₈	D₇	1	1	yes	characteristic
32	C₄×C₈	—	7	2	no	—
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	—	2	1	no	—
35	C₂²×C₇	C₂×C₈	1	2	yes	—
36	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₇	—	2	1	no	—
38	C₄×C₇	#41	1	2	yes	—
39	Dic₇	—	2	1	no	—
40	Dic₇	C₂×C₈	1	2	yes	—
41	Dic₇	C₂×C₈	1	2	yes	—
42	Dic₇	M₁₆	1	2	yes	—
43	C₂²×C₄	—	7	1	no	—
44	C₂²×C₄	—	7	1	no	—
45	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
46	C₂×C₈	—	7	2	no	—
47	C₂×C₈	C₂×D₇	1	2	yes	—
48	C₄²	—	7	1	no	—
49	C₄²	—	7	1	no	—
50	C₄²	—	7	1	no	—
51	C₄²	—	7	1	no	—
52	C₂×C₇	—	2	2	no	—
53	C₂×C₇	#117	1	1	yes	characteristic
54	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
56	C₂³	C₄×D₇	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	7	4	no	—
59	C₂×C₄	—	7	1	no	—
60	C₂×C₄	—	7	1	no	—
61	C₂×C₄	—	7	1	no	—
62	C₂×C₄	—	7	1	no	—
63	C₂×C₄	—	14	2	no	—
64	C₂×C₄	C₄×D₇	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	14	2	no	—
68	C₇	#431	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	C₈×D₇	1	2	yes	—
72	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	7	2	no	—
75	C₄	—	7	2	no	—
76	C₄	—	7	2	no	—
77	C₄	—	14	1	no	—
78	C₄	#1159	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	#6440	1	1	yes	characteristic
81	C₂	C₂×C₈×D₇	1	1	yes	characteristic
82	C₂	D₇×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂²×C₈
Fitting	C₇×C₂²⋊C₈	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²⋊C₈	D₇
hypocenter	C₇	#431
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. Dic₇⋊C₈	C₂	semidirect
5. C₈×Dic₇	C₂	semidirect
21. C₂×Dic₇	C₈	semidirect
68. C₇	#431	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₇	2
C₇×C₂²⋊C₈	2
C₂²⋊(C₇⋊C₈)	2
Dic₇⋊C₈	2
C₈×Dic₇	2
#431	7