Subgroups — #89278

Order 448.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₈	C₄	1	1	yes	characteristic
5	C₂×C₄×D₇	—	2	1	no	—
6	C₂×C₇×C₈	—	2	1	no	—
7	C₄²×C₇	C₂²	1	1	yes	characteristic
8	(C₂×D₇)⋊C₄	—	2	1	no	—
9	C₄⋊Dic₇	C₄	1	1	yes	characteristic
10	C₂×C₇⋊C₈	—	2	1	no	—
11	D₄⋊C₈	—	7	1	no	maximal, Sylow
12	C₂²×D₇	—	2	1	no	—
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×Dic₇	—	2	1	no	—
17	D₂₈	—	2	1	no	—
18	D₂₈	C₈	1	2	yes	complemented
19	C₄×D₇	—	4	1	no	—
20	C₄×D₇	—	4	1	no	—
21	C₇×C₈	—	4	1	no	—
22	C₇⋊C₈	—	2	2	no	—
23	C₄×D₄	—	7	1	no	—
24	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
25	C₄×C₈	—	7	1	no	—
26	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₇	—	2	2	no	—
28	C₂×D₇	—	4	1	no	—
29	C₂×D₇	—	4	1	no	—
30	C₄×C₇	—	2	1	no	—
31	C₄×C₇	C₂×C₈	1	1	yes	characteristic
32	C₄×C₇	M₁₆	1	1	yes	characteristic
33	C₄×C₇	D₈	1	1	yes	characteristic
34	C₄×C₇	SD₁₆	1	1	yes	characteristic
35	Dic₇	—	4	1	no	—
36	C₂×D₄	—	7	1	no	—
37	C₂²×C₄	—	14	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	14	1	no	—
40	C₄⋊C₄	—	7	1	no	—
41	C₂²⋊C₄	—	14	1	no	—
42	C₄²	C₂×D₇	1	1	yes	characteristic
43	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₇	#102	1	1	yes	characteristic
45	C₂×C₇	#104	1	1	yes	characteristic
46	D₇	—	4	2	no	—
47	C₂³	—	14	1	no	—
48	D₄	—	7	2	no	—
49	D₄	—	14	1	no	—
50	C₂×C₄	—	14	1	no	—
51	C₂×C₄	—	28	1	no	—
52	C₂×C₄	—	28	1	no	—
53	C₂×C₄	C₄×D₇	1	1	yes	characteristic
54	C₂×C₄	D₂₈	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₈	—	14	2	no	—
58	C₇	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	14	2	no	—
60	C₂²	—	28	1	no	—
61	C₂²	—	28	1	no	—
62	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	28	1	no	—
65	C₄	C₈×D₇	1	1	yes	characteristic
66	C₄	#1131	1	1	yes	characteristic
67	C₄	#1141	1	1	yes	characteristic
68	C₄	#1143	1	1	yes	characteristic
69	C₂	—	28	2	no	—
70	C₂	#6383	1	1	yes	characteristic
71	C₂	(C₂×D₇)⋊C₈	1	1	yes	characteristic
72	C₂	#6399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₈
Frattini	C₄²	C₂×D₇
Fitting	C₇×C₄⋊C₈	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂×D₂₈	C₄
cosocle	C₄²×C₇	C₂²
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	D₄⋊C₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₇×C₄⋊C₈	C₂	semidirect
3. C₄×C₇⋊C₈	C₂	semidirect
18. D₂₈	C₈	semidirect
24. C₄⋊C₈	D₇	semidirect
58. C₇	D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₂₈	2
C₇×C₄⋊C₈	2
C₄×C₇⋊C₈	2
D₄⋊C₈	7