Subgroups — #89621

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₈	C₂	1	2	yes	maximal
2	#6432	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
4	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#6390	C₂	1	2	yes	maximal
6	C₂×Dic₁₄	C₂²	1	2	yes	—
7	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	2	yes	—
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₇	—	2	1	no	—
11	C₄⋊Dic₇	—	2	1	no	—
12	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
13	Dic₇⋊C₄	—	2	1	no	—
14	C₄×Dic₇	—	2	1	no	—
15	Dic₂₈	—	2	4	no	—
16	#462	—	7	1	no	maximal, Sylow
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	1	no	—
21	C₂×Dic₇	—	2	1	no	—
22	C₂×Dic₇	—	2	2	no	—
23	Dic₁₄	—	2	1	no	—
24	Dic₁₄	—	2	2	no	—
25	Dic₁₄	—	2	2	no	—
26	Dic₁₄	D₄	1	2	yes	—
27	C₇×C₈	—	2	2	no	—
28	C₂×Q₁₆	—	7	2	no	—
29	C₄⋊Q₈	—	7	1	no	—
30	C₄×Q₈	—	7	1	no	—
31	C₄⋊C₈	D₇	1	1	yes	characteristic
32	#103	—	7	2	no	—
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₇	—	2	1	no	—
35	C₄×C₇	Q₁₆	1	2	yes	complemented
36	C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	C₄×C₇	#41	1	1	yes	characteristic
38	Dic₇	—	2	2	no	—
39	Dic₇	—	4	2	no	—
40	Dic₇	—	4	1	no	—
41	C₂×Q₈	—	7	1	no	—
42	C₂×Q₈	—	7	2	no	—
43	Q₁₆	—	14	4	no	—
44	C₂×C₈	C₂×D₇	1	2	yes	—
45	C₄⋊C₄	—	7	1	no	—
46	C₄⋊C₄	—	14	1	no	—
47	C₄⋊C₄	—	14	1	no	—
48	C₄⋊C₄	—	14	1	no	—
49	C₄²	—	14	1	no	—
50	C₄²	C₂×D₇	1	1	yes	characteristic
51	C₂×C₇	#121	1	1	yes	characteristic
52	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₇	#137	1	1	yes	characteristic
54	Q₈	—	7	2	no	—
55	Q₈	—	14	1	no	—
56	Q₈	—	14	2	no	—
57	Q₈	—	14	2	no	—
58	C₂×C₄	—	14	1	no	—
59	C₂×C₄	—	14	1	no	—
60	C₂×C₄	—	14	2	no	—
61	C₂×C₄	D₂₈	1	1	yes	characteristic
62	C₂×C₄	D₂₈	1	1	yes	characteristic
63	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₇	#462	1	1	yes	characteristic, complemented, Sylow
66	C₂²	C₂×D₂₈	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	14	2	no	—
69	C₄	—	28	2	no	—
70	C₄	—	28	1	no	—
71	C₄	Dic₂₈	1	2	yes	complemented
72	C₄	D₄×D₇	1	1	yes	characteristic
73	C₄	#1161	1	1	yes	characteristic
74	C₂	#6458	1	1	yes	characteristic
75	C₂	C₂×Dic₂₈	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×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#462
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₇	Q₁₆	semidirect
65. C₇	#462	semidirect
71. C₄	Dic₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₈	2
#6432	2
C₄×Dic₁₄	2
C₇×C₄⋊C₈	2
#6390	2
#462	7