Subgroups — #89345

Order 448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂²×Dic₇	C₄	1	2	yes	—
6	C₂×C₇×C₈	—	2	1	no	—
7	C₂×C₇×C₈	—	2	1	no	—
8	C₂×C₇×C₈	C₄	1	2	yes	complemented
9	C₄×Dic₇	—	2	1	no	—
10	C₄×Dic₇	—	2	1	no	—
11	C₂×C₇⋊C₈	—	2	1	no	—
12	C₂×C₇⋊C₈	—	2	1	no	—
13	C₂×C₇⋊C₈	C₄	1	2	yes	complemented
14	#336	—	7	1	no	maximal, Sylow
15	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
22	C₂×Dic₇	—	2	2	no	—
23	C₂×Dic₇	—	2	2	no	—
24	C₂×Dic₇	C₈	1	4	yes	complemented
25	C₇×C₈	—	2	2	no	—
26	C₇⋊C₈	—	2	2	no	—
27	C₂²×C₈	—	7	1	no	—
28	C₂²×C₈	D₇	1	1	yes	characteristic
29	C₂×C₄²	—	7	1	no	—
30	C₂²×C₇	C₄²	1	1	yes	characteristic
31	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₇	C₂×C₈	1	2	yes	—
34	C₂²×C₇	M₁₆	1	2	yes	—
35	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
39	Dic₇	—	2	4	no	—
40	C₂²×C₄	—	7	2	no	—
41	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	7	2	no	—
45	C₂×C₈	—	14	1	no	—
46	C₂×C₈	—	14	1	no	—
47	C₂×C₈	Dic₇	1	2	yes	complemented
48	C₄²	—	14	1	no	—
49	C₄²	—	14	1	no	—
50	C₂×C₇	#95	1	1	yes	characteristic
51	C₂×C₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₇	#97	1	1	yes	characteristic
53	C₂×C₇	C₂²⋊C₈	1	2	yes	—
54	C₂×C₇	C₄⋊C₈	1	2	yes	—
55	C₂³	C₄×D₇	1	1	yes	characteristic
56	C₂×C₄	—	7	4	no	—
57	C₂×C₄	—	14	2	no	—
58	C₂×C₄	—	14	2	no	—
59	C₂×C₄	Dic₁₄	1	1	yes	characteristic
60	C₂×C₄	C₄×D₇	1	1	yes	characteristic
61	C₂×C₄	D₂₈	1	1	yes	characteristic
62	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₈	—	14	2	no	—
67	C₇	#336	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₈×D₇	1	2	yes	—
69	C₂²	#1131	1	2	yes	—
70	C₂²	C₄×Dic₇	1	1	yes	characteristic
71	C₂²	Dic₇⋊C₄	1	1	yes	characteristic
72	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
73	C₄	—	14	4	no	—
74	C₄	Dic₇⋊C₄	1	1	yes	characteristic
75	C₄	C₄⋊Dic₇	1	1	yes	characteristic
76	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
77	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
78	C₂	C₈×Dic₇	1	1	yes	characteristic
79	C₂	Dic₇⋊C₈	1	2	yes	—
80	C₂	#6389	1	1	yes	characteristic
81	C₂	(C₂×D₇)⋊C₈	1	2	yes	—
82	C₂	C₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₇
commutator	C₂×C₇	C₄×C₈
Fitting	C₂²×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	#336
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×C₇×C₈	C₄	semidirect
13. C₂×C₇⋊C₈	C₄	semidirect
24. C₂×Dic₇	C₈	semidirect
47. C₂×C₈	Dic₇	semidirect
67. C₇	#336	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
C₂×C₄×Dic₇	2
C₂²×C₇⋊C₈	2
#336	7