Subgroups — #89615

Order 448.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₆	C₂	1	2	yes	maximal, complemented
2	#6437	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#6395	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₈	—	2	1	no	—
7	C₂×D₂₈	—	2	1	no	—
8	C₂×D₂₈	C₂²	1	2	yes	—
9	C₂×D₂₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₇	—	2	1	no	—
11	C₂×C₇×C₈	C₂²	1	2	yes	—
12	C₄²×C₇	C₂²	1	1	yes	characteristic
13	(C₂×D₇)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	D₅₆	—	2	4	no	—
16	#459	—	7	1	no	maximal, Sylow
17	C₂²×D₇	—	2	1	no	—
18	C₂²×D₇	—	2	2	no	—
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₇	—	2	1	no	—
23	D₂₈	—	2	2	no	—
24	D₂₈	—	2	1	no	—
25	D₂₈	—	2	2	no	—
26	D₂₈	—	4	2	no	—
27	D₂₈	—	4	2	no	—
28	D₂₈	D₄	1	2	yes	complemented
29	C₄×D₇	—	4	2	no	—
30	C₇×C₈	—	2	2	no	—
31	C₂×D₈	—	7	2	no	—
32	#127	—	7	1	no	—
33	C₄×D₄	—	7	1	no	—
34	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
35	#102	—	7	2	no	—
36	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂×D₇	—	2	2	no	—
38	C₂×D₇	—	4	1	no	—
39	C₂×D₇	—	4	1	no	—
40	C₂×D₇	—	4	2	no	—
41	C₂×D₇	—	4	2	no	—
42	C₂×D₇	—	4	2	no	—
43	C₄×C₇	—	2	1	no	—
44	C₄×C₇	D₈	1	2	yes	complemented
45	C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₄×C₇	#41	1	1	yes	characteristic
47	Dic₇	—	4	1	no	—
48	C₂×D₄	—	7	1	no	—
49	C₂×D₄	—	7	2	no	—
50	C₂×D₄	—	14	1	no	—
51	C₂×D₄	—	14	1	no	—
52	C₂²×C₄	—	14	1	no	—
53	D₈	—	14	4	no	—
54	C₂×C₈	C₂×D₇	1	2	yes	—
55	C₄⋊C₄	—	7	1	no	—
56	C₂²⋊C₄	—	14	1	no	—
57	C₄²	C₂×D₇	1	1	yes	characteristic
58	C₂×C₇	#121	1	1	yes	characteristic
59	C₂×C₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	D₇	—	4	2	no	—
62	D₇	—	8	2	no	—
63	C₂³	—	14	1	no	—
64	C₂³	—	14	2	no	—
65	D₄	—	7	2	no	—
66	D₄	—	14	2	no	—
67	D₄	—	14	1	no	—
68	D₄	—	14	2	no	—
69	D₄	—	28	2	no	—
70	D₄	—	28	2	no	—
71	C₂×C₄	—	14	1	no	—
72	C₂×C₄	—	28	2	no	—
73	C₂×C₄	D₂₈	1	1	yes	characteristic
74	C₂×C₄	D₂₈	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₇	#459	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	14	2	no	—
79	C₂²	—	28	1	no	—
80	C₂²	—	28	1	no	—
81	C₂²	—	28	2	no	—
82	C₂²	—	28	2	no	—
83	C₂²	—	28	2	no	—
84	C₂²	C₂×D₂₈	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	28	1	no	—
87	C₄	D₅₆	1	2	yes	complemented
88	C₄	D₄×D₇	1	1	yes	characteristic
89	C₄	#1161	1	1	yes	characteristic
90	C₂	—	28	2	no	—
91	C₂	—	56	2	no	—
92	C₂	#6458	1	1	yes	characteristic
93	C₂	C₂×D₅₆	1	1	yes	characteristic
94	C₂	#6471	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	C₂²×C₇	C₂×D₄
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#459
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
1. C₂×D₅₆	C₂	semidirect
2. #6437	C₂	semidirect
3. C₄×D₂₈	C₂	semidirect
4. C₇×C₄⋊C₈	C₂	semidirect
5. #6395	C₂	semidirect
28. D₂₈	D₄	semidirect
34. C₄⋊C₈	D₇	semidirect
44. C₄×C₇	D₈	semidirect
77. C₇	#459	semidirect
87. C₄	D₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₆	2
#6437	2
C₄×D₂₈	2
C₇×C₄⋊C₈	2
#6395	2
#459	7