Subgroups — #89473

Order 448.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#97	C₂	1	1	yes	maximal, characteristic
3	#6389	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
7	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₄	C₄	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₇	C₄	1	1	yes	characteristic
13	Dic₇⋊C₄	C₄	1	1	yes	characteristic
14	Dic₇⋊C₄	C₄	1	1	yes	characteristic
15	C₄×Dic₇	C₂²	1	1	yes	characteristic
16	C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#446	—	7	1	no	maximal, Sylow
20	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	C₂³	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	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
27	Dic₁₄	—	2	2	no	—
28	C₇×C₈	—	2	1	no	—
29	C₇×C₈	Q₈	1	2	yes	complemented
30	C₇⋊C₈	—	2	1	no	—
31	C₇⋊C₈	—	2	1	no	—
32	C₄×Q₈	—	7	1	no	—
33	C₄⋊C₈	—	7	1	no	—
34	C₄⋊C₈	—	7	1	no	—
35	C₄⋊C₈	—	7	1	no	—
36	#97	—	7	1	no	—
37	#97	D₇	1	1	yes	characteristic
38	C₄×C₈	—	7	1	no	—
39	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₇	—	2	1	no	—
41	C₄×C₇	—	2	1	no	—
42	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
43	C₄×C₇	#41	1	1	yes	characteristic
44	Dic₇	—	2	1	no	—
45	Dic₇	—	2	1	no	—
46	Dic₇	—	2	1	no	—
47	Dic₇	M₁₆	1	2	yes	—
48	C₂×Q₈	—	7	1	no	—
49	C₂×C₈	—	7	1	no	—
50	C₂×C₈	—	7	1	no	—
51	C₂×C₈	C₂×D₇	1	1	yes	characteristic
52	C₂×C₈	C₂×D₇	1	1	yes	characteristic
53	C₄⋊C₄	—	7	1	no	—
54	C₄⋊C₄	—	7	1	no	—
55	C₄⋊C₄	—	7	1	no	—
56	C₄²	—	7	1	no	—
57	C₄²	—	7	1	no	—
58	C₄²	C₂×D₇	1	1	yes	characteristic
59	C₂×C₇	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₇	#131	1	1	yes	characteristic
62	Q₈	—	14	2	no	—
63	C₂×C₄	—	7	1	no	—
64	C₂×C₄	—	7	1	no	—
65	C₂×C₄	—	7	1	no	—
66	C₂×C₄	—	7	1	no	—
67	C₂×C₄	C₄×D₇	1	1	yes	characteristic
68	C₂×C₄	C₄×D₇	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	14	1	no	—
72	C₈	—	14	1	no	—
73	C₈	Dic₁₄	1	2	yes	complemented
74	C₇	#446	1	1	yes	characteristic, complemented, Sylow
75	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	7	2	no	—
79	C₄	—	14	1	no	—
80	C₄	—	14	1	no	—
81	C₄	—	14	1	no	—
82	C₄	C₂×Dic₁₄	1	1	yes	characteristic
83	C₄	#1157	1	1	yes	characteristic
84	C₂	C₄×Dic₁₄	1	1	yes	characteristic
85	C₂	D₇×M₁₆	1	1	yes	characteristic
86	C₂	#6470	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle, prime radical	C₂²×C₇	C₂²×C₄
Fitting	C₇×#97	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#97	D₇
hypocenter	C₇	#446
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
29. C₇×C₈	Q₈	semidirect
73. C₈	Dic₁₄	semidirect
74. C₇	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₁₄	2
C₇×#97	2
#6389	2
Dic₇⋊C₈	2
C₈×Dic₇	2
C₄⋊(C₇⋊C₈)	2
#446	7