Subgroups — #89667

Order 448.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6470	C₂	1	2	yes	maximal, complemented
2	#6467	C₂	1	1	yes	maximal, characteristic
3	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	#6399	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1157	C₂²	1	2	yes	—
7	C₇×M₁₆	C₂²	1	2	yes	—
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄×Dic₇	C₂²	1	1	yes	characteristic
10	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
11	Dic₂₈	C₄	1	1	yes	characteristic, complemented
12	D₅₆	C₄	1	1	yes	characteristic, complemented
13	#1132	C₄	1	2	yes	complemented
14	#1131	—	2	2	no	—
15	C₈×D₇	—	2	2	no	—
16	#443	—	7	1	no	maximal, Sylow
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂²⋊D₇	—	2	2	no	—
19	C₂×Dic₇	—	2	1	no	—
20	D₂₈	C₂×C₄	1	2	yes	—
21	C₄×D₇	—	2	2	no	—
22	Dic₁₄	C₂×C₄	1	2	yes	—
23	C₇×C₈	—	2	2	no	—
24	C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₇⋊C₈	D₄	1	1	yes	characteristic
27	C₇⋊C₈	D₄	1	1	yes	characteristic
28	#135	—	7	1	no	—
29	#131	—	7	2	no	—
30	#108	D₇	1	1	yes	characteristic, complemented
31	#104	—	7	2	no	—
32	C₄×C₈	—	7	1	no	—
33	C₂²×C₇	#41	1	1	yes	characteristic
34	C₂×D₇	—	2	2	no	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	Dic₇	—	2	2	no	—
38	Dic₇	—	2	2	no	—
39	#41	—	7	2	no	—
40	Q₁₆	—	7	1	no	—
41	SD₁₆	—	7	2	no	—
42	D₈	—	7	1	no	—
43	M₁₆	—	14	2	no	—
44	M₁₆	C₂×D₇	1	2	yes	—
45	C₂×C₈	—	7	1	no	—
46	C₂×C₈	—	14	2	no	—
47	C₂×C₈	C₂×D₇	1	1	yes	characteristic
48	C₄²	—	7	1	no	—
49	C₂×C₇	—	2	1	no	—
50	C₂×C₇	C₄×D₄	1	1	yes	characteristic
51	D₇	—	4	2	no	—
52	Q₈	—	7	2	no	—
53	D₄	—	7	2	no	—
54	D₄	—	14	2	no	—
55	C₂×C₄	—	14	1	no	—
56	C₂×C₄	—	14	2	no	—
57	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	7	1	no	—
60	C₈	—	7	1	no	—
61	C₈	C₄×D₇	1	1	yes	characteristic
62	C₈	C₄×D₇	1	1	yes	characteristic
63	C₇	#443	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	14	2	no	—
65	C₂²	#1161	1	1	yes	characteristic
66	C₄	—	14	2	no	—
67	C₄	—	14	2	no	—
68	C₄	C₂×C₄×D₇	1	1	yes	characteristic
69	C₄	D₄×D₇	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	28	2	no	—
72	C₂	#6456	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	C₄×D₄
prime radical	#6467	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#108	D₇
hypocenter	C₇	#443
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #6470	C₂	semidirect
3. C₇×#108	C₂	semidirect
4. #6399	C₂	semidirect
5. C₈×Dic₇	C₂	semidirect
11. Dic₂₈	C₄	semidirect
12. D₅₆	C₄	semidirect
13. #1132	C₄	semidirect
30. #108	D₇	semidirect
63. C₇	#443	semidirect

Maximal subgroups

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

subgroup	index
#6470	2
#6467	2
C₇×#108	2
#6399	2
C₈×Dic₇	2
#443	7