Subgroups — #89924

Order 448.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₇	C₂	1	2	yes	maximal, complemented
3	#6406	C₂	1	2	yes	maximal, complemented
4	#6391	C₂	1	1	yes	maximal, characteristic, complemented
5	#6389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₂²	1	2	yes	—
7	C₂²×Dic₇	—	2	2	no	—
8	C₇×D₈	C₄	1	4	yes	complemented
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂²⋊Dic₇	—	2	2	no	—
11	C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	#442	—	7	1	no	maximal, Sylow
15	C₂³×C₇	—	2	2	no	—
16	C₇×D₄	—	2	2	no	—
17	C₇×D₄	C₂×C₄	1	4	yes	complemented
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₇	—	2	2	no	—
20	C₂×Dic₇	—	4	4	no	—
21	C₂×Dic₇	D₄	1	2	yes	—
22	C₇×C₈	C₂×C₄	1	2	yes	complemented
23	C₇⋊C₈	—	2	1	no	—
24	C₂×D₈	D₇	1	1	yes	characteristic
25	C₄×D₄	—	7	2	no	—
26	#106	—	7	1	no	—
27	#102	—	7	2	no	—
28	#97	—	7	1	no	—
29	C₂²×C₇	—	2	4	no	—
30	C₂²×C₇	—	4	2	no	—
31	C₂²×C₇	—	4	2	no	—
32	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₇	#41	1	1	yes	characteristic
35	Dic₇	—	2	2	no	—
36	Dic₇	—	4	2	no	—
37	C₂×D₄	C₂×D₇	1	2	yes	—
38	C₂²×C₄	—	14	2	no	—
39	D₈	Dic₇	1	4	yes	complemented
40	C₂×C₈	—	7	1	no	—
41	C₂×C₈	C₂×D₇	1	1	yes	characteristic
42	C₄⋊C₄	—	7	2	no	—
43	C₂²⋊C₄	—	14	2	no	—
44	C₄²	—	7	1	no	—
45	C₂×C₇	—	4	4	no	—
46	C₂×C₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
48	C₂³	—	2	2	no	—
49	D₄	—	2	2	no	—
50	D₄	C₂×Dic₇	1	4	yes	complemented
51	C₂×C₄	—	7	2	no	—
52	C₂×C₄	—	14	2	no	—
53	C₂×C₄	—	28	4	no	—
54	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
55	C₈	—	14	1	no	—
56	C₈	C₂×Dic₇	1	2	yes	complemented
57	C₇	#442	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₇	1	1	yes	characteristic
62	C₄	—	14	2	no	—
63	C₄	—	28	2	no	—
64	C₄	#1159	1	1	yes	characteristic
65	C₄	C₂²×Dic₇	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	#6474	1	2	yes	—
68	C₂	D₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₂×C₇×D₈	C₂
socle	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×D₈	D₇
hypocenter	C₇	#442
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×Dic₇	C₂	semidirect
3. #6406	C₂	semidirect
4. #6391	C₂	semidirect
5. #6389	C₂	semidirect
8. C₇×D₈	C₄	semidirect
17. C₇×D₄	C₂×C₄	semidirect
22. C₇×C₈	C₂×C₄	semidirect
39. D₈	Dic₇	semidirect
50. D₄	C₂×Dic₇	semidirect
56. C₈	C₂×Dic₇	semidirect
57. C₇	#442	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₈	2
D₄×Dic₇	2
#6406	2
#6391	2
#6389	2
#442	7