Subgroups — #89486

Order 448.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6438	C₂	1	1	yes	maximal, characteristic
2	#6432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	2	yes	maximal
5	#6390	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₈	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₄	—	2	1	no	—
8	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	(C₂×D₇)⋊C₄	—	2	1	no	—
12	(C₂×D₇)⋊C₄	—	2	1	no	—
13	C₄⋊Dic₇	—	2	1	no	—
14	C₄⋊Dic₇	C₂²	1	2	yes	—
15	#489	—	7	1	no	maximal, Sylow
16	C₂²×D₇	—	2	1	no	—
17	C₂×C₄×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₇	—	2	2	no	—
21	C₂×Dic₇	—	2	1	no	—
22	D₂₈	—	2	2	no	—
23	Dic₁₄	—	2	2	no	—
24	Dic₁₄	—	4	2	no	—
25	C₇×C₈	—	2	2	no	—
26	C₄⋊Q₈	—	7	1	no	—
27	#124	—	7	1	no	—
28	#103	—	7	2	no	—
29	#102	—	7	2	no	—
30	#97	D₇	1	1	yes	characteristic, complemented
31	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂×D₇	—	4	1	no	—
33	C₂×D₇	—	4	1	no	—
34	C₂×D₇	—	4	1	no	—
35	C₄×C₇	—	2	1	no	—
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	#41	1	2	yes	—
38	Dic₇	—	4	1	no	—
39	Dic₇	—	4	2	no	—
40	C₂×Q₈	—	7	1	no	—
41	C₂×Q₈	—	14	1	no	—
42	C₂×D₄	—	7	1	no	—
43	C₂×C₈	C₂×D₇	1	2	yes	—
44	C₄⋊C₄	—	7	2	no	—
45	C₄⋊C₄	—	14	1	no	—
46	C₂²⋊C₄	—	14	1	no	—
47	C₂²⋊C₄	—	14	1	no	—
48	C₄²	C₂×D₇	1	1	yes	characteristic
49	C₂×C₇	#124	1	1	yes	characteristic
50	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₂×C₇	#137	1	1	yes	characteristic
52	D₇	—	8	1	no	—
53	C₂³	—	14	1	no	—
54	Q₈	—	14	2	no	—
55	Q₈	—	28	2	no	—
56	D₄	—	14	2	no	—
57	C₂×C₄	—	14	1	no	—
58	C₂×C₄	—	14	2	no	—
59	C₂×C₄	D₂₈	1	1	yes	characteristic
60	C₂×C₄	D₂₈	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₇	#489	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	28	1	no	—
65	C₂²	—	28	1	no	—
66	C₂²	—	28	1	no	—
67	C₂²	C₂×D₂₈	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	28	1	no	—
71	C₄	—	28	2	no	—
72	C₄	#1157	1	2	yes	—
73	C₂	—	56	1	no	—
74	C₂	#6438	1	1	yes	characteristic
75	C₂	#6471	1	1	yes	characteristic
76	C₂	#6472	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#97	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₂₈	C₂²
hypercenter	#97	D₇
hypocenter	C₇	#489
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. #6432	C₂	semidirect
3. C₇×#97	C₂	semidirect
5. #6390	C₂	semidirect
30. #97	D₇	semidirect
63. C₇	#489	semidirect

Maximal subgroups

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

subgroup	index
#6438	2
#6432	2
C₇×#97	2
#6395	2
#6390	2
#489	7