Subgroups — #89619

Order 448.

102 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	#6433	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	1	yes	maximal, characteristic
5	#6395	C₂	1	1	yes	maximal, characteristic
6	#6392	C₂	1	1	yes	maximal, characteristic, complemented
7	#6391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₇	—	2	1	no	—
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₄²×C₇	C₂²	1	1	yes	characteristic
13	(C₂×D₇)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
17	Dic₇⋊C₄	—	2	1	no	—
18	Dic₇⋊C₄	—	2	1	no	—
19	#478	—	7	1	no	maximal, Sylow
20	C₂²×D₇	—	2	1	no	—
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₇	—	2	1	no	—
25	C₂×Dic₇	—	2	1	no	—
26	C₂×Dic₇	—	2	1	no	—
27	D₂₈	—	2	1	no	—
28	D₂₈	Q₈	1	2	yes	—
29	C₄×D₇	—	4	2	no	—
30	C₇×C₈	—	2	1	no	—
31	C₇×C₈	—	2	1	no	—
32	#125	—	7	1	no	—
33	C₄×D₄	—	7	1	no	—
34	#107	—	7	1	no	—
35	#106	—	7	1	no	—
36	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
37	#102	—	7	1	no	—
38	#102	—	7	1	no	—
39	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂×D₇	—	2	2	no	—
41	C₂×D₇	—	4	1	no	—
42	C₂×D₇	—	4	1	no	—
43	C₄×C₇	—	2	1	no	—
44	C₄×C₇	—	2	1	no	—
45	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×C₇	#41	1	1	yes	characteristic
47	Dic₇	—	4	1	no	—
48	Dic₇	—	4	1	no	—
49	Dic₇	—	4	1	no	—
50	C₂×D₄	—	7	1	no	—
51	C₂²×C₄	—	14	1	no	—
52	C₂×C₈	C₂×D₇	1	1	yes	characteristic
53	C₂×C₈	C₂×D₇	1	1	yes	characteristic
54	C₄⋊C₄	—	7	1	no	—
55	C₄⋊C₄	—	7	1	no	—
56	C₄⋊C₄	—	7	1	no	—
57	C₄⋊C₄	—	14	1	no	—
58	C₄⋊C₄	—	14	1	no	—
59	C₂²⋊C₄	—	14	1	no	—
60	C₄²	C₂×D₇	1	1	yes	characteristic
61	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₇	#135	1	1	yes	characteristic
63	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	D₇	—	4	2	no	—
65	C₂³	—	14	1	no	—
66	D₄	—	7	2	no	—
67	D₄	—	14	1	no	—
68	C₂×C₄	—	14	1	no	—
69	C₂×C₄	—	14	1	no	—
70	C₂×C₄	—	14	1	no	—
71	C₂×C₄	—	28	2	no	—
72	C₂×C₄	D₂₈	1	1	yes	characteristic
73	C₂×C₄	D₂₈	1	1	yes	characteristic
74	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	2	1	no	—
77	C₇	#478	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	14	2	no	—
79	C₂²	—	28	1	no	—
80	C₂²	—	28	1	no	—
81	C₂²	C₂×D₂₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	28	1	no	—
85	C₄	—	28	1	no	—
86	C₄	—	28	1	no	—
87	C₄	#1159	1	1	yes	characteristic
88	C₄	Q₈×D₇	1	1	yes	characteristic
89	C₂	—	28	2	no	—
90	C₂	#6460	1	1	yes	characteristic
91	C₂	#6467	1	1	yes	characteristic
92	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
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#478
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
2. #6433	C₂	semidirect
3. C₇×C₄⋊C₈	C₂	semidirect
6. #6392	C₂	semidirect
7. #6391	C₂	semidirect
36. C₄⋊C₈	D₇	semidirect
77. C₇	#478	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
#6433	2
C₇×C₄⋊C₈	2
#6395	2
#6392	2
#6391	2
#478	7