Group #28317890 — subgroups

Subgroups — C₄²×D₅₃⋊C₈ #28317890

Order 13568.

2046 conjugacy classes of subgroups; only the 1152 normal subgroups are shown.

normal subgroups only

This group has 2046 conjugacy classes of subgroups — too many to draw. Only its 1152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₃⋊C₈	C₂	1	6	yes	maximal
2	C₄²×#88324	C₂	1	8	yes	maximal, complemented
3	C₄³×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₅₃⋊C₈	C₂²	1	4	yes	—
5	C₄×D₅₃⋊C₈	C₄	1	48	yes	direct product factor
6	C₂×C₄×#88324	C₂²	1	24	yes	—
7	C₂×C₄²×D₅₃	C₄	1	4	yes	—
8	C₂×C₄²×D₅₃	C₂²	1	3	yes	—
9	C₄²×Dic₅₃	C₄	1	3	yes	—
10	C₄²×Dic₅₃	C₂²	1	4	yes	—
11	C₄³×C₅₃	C₄	1	1	yes	characteristic
12	C₂×D₅₃⋊C₈	C₂×C₄	1	48	yes	—
13	C₂²×#88324	C₂³	1	8	yes	—
14	C₄×#88324	C₂×C₄	1	96	yes	complemented
15	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
16	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
19	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
20	C₄²×D₅₃	C₈	1	32	yes	complemented
21	C₄²×D₅₃	C₂×C₄	1	24	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
24	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
27	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
28	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×D₅₃	C₄²	1	24	yes	—
30	C₂×C₄×D₅₃	C₂×C₈	1	48	yes	—
31	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
32	C₂×#88324	C₂²×C₄	1	48	yes	—
33	D₅₃⋊C₈	C₄²	1	64	yes	direct product factor
34	C₄²×C₅₃	C₄²	1	12	yes	—
35	C₄²×C₅₃	C₂×C₈	1	16	yes	complemented
36	C₄×Dic₅₃	C₄²	1	12	yes	—
37	C₄×Dic₅₃	C₂×C₈	1	48	yes	complemented
38	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
39	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
40	C₂²×D₅₃	C₂×C₄²	1	6	yes	—
41	C₂²×D₅₃	C₂²×C₈	1	8	yes	—
42	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
43	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
44	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	—
45	C₂×Dic₅₃	C₂×C₄²	1	12	yes	—
46	C₂×Dic₅₃	C₂²×C₈	1	24	yes	—
47	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
48	C₄×D₅₃	C₄×C₈	1	96	yes	complemented
49	C₄×D₅₃	C₂×C₄²	1	16	yes	—
50	#88324	C₂×C₄²	1	32	yes	complemented
51	C₂×C₄²×C₈	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	C₂²×C₄²	1	3	yes	—
53	C₂²×C₅₃	C₂³×C₈	1	4	yes	—
54	C₂×D₅₃	C₄³	1	4	yes	—
55	C₂×D₅₃	C₂×C₄×C₈	1	24	yes	—
56	C₄×C₅₃	C₄³	1	4	yes	—
57	C₄×C₅₃	C₂×C₄×C₈	1	24	yes	complemented
58	Dic₅₃	C₂×C₄×C₈	1	24	yes	complemented
59	Dic₅₃	C₂²×C₄²	1	4	yes	—
60	C₂²×C₄×C₈	—	53	6	no	—
61	C₂×C₄³	—	53	1	no	—
62	C₄²×C₈	—	53	8	no	—
63	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
64	C₂×C₅₃	C₂²×C₄×C₈	1	6	yes	—
65	D₅₃	C₄²×C₈	1	8	yes	complemented
66	C₂³×C₈	—	53	4	no	—
67	C₂²×C₄²	—	53	3	no	—
68	C₂²×C₄²	—	53	4	no	—
69	C₂×C₄×C₈	—	53	24	no	—
70	C₂×C₄×C₈	—	53	48	no	—
71	C₄³	—	53	4	no	—
72	C₄³	—	53	3	no	—
73	C₄³	#6157	1	1	yes	characteristic
74	C₅₃	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
75	C₂³×C₄	—	53	1	no	—
76	C₂³×C₄	—	53	6	no	—
77	C₂²×C₈	—	53	8	no	—
78	C₂²×C₈	—	53	48	no	—
79	C₂×C₄²	—	53	3	no	—
80	C₂×C₄²	—	53	12	no	—
81	C₂×C₄²	—	53	6	no	—
82	C₂×C₄²	—	53	24	no	—
83	C₂×C₄²	—	53	32	no	—
84	C₂×C₄²	C₂×#6157	1	3	yes	—
85	C₂×C₄²	C₂×#6157	1	4	yes	—
86	C₄×C₈	—	53	96	no	—
87	C₂⁴	—	53	1	no	—
88	C₂²×C₄	—	53	6	no	—
89	C₂²×C₄	—	53	1	no	—
90	C₂²×C₄	—	53	24	no	—
91	C₂²×C₄	—	53	12	no	—
92	C₂²×C₄	—	53	48	no	—
93	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
94	C₂²×C₄	C₂²×#6157	1	6	yes	—
95	C₂×C₈	—	53	48	no	—
96	C₂×C₈	—	53	64	no	—
97	C₄²	—	53	24	no	—
98	C₄²	—	53	12	no	—
99	C₄²	—	53	48	no	—
100	C₄²	D₅₃⋊C₈	1	16	yes	direct product factor
101	C₄²	C₄×#6157	1	12	yes	—
102	C₂³	—	53	6	no	—
103	C₂³	—	53	8	no	—
104	C₂³	C₂³×#6157	1	1	yes	characteristic
105	C₂×C₄	—	53	12	no	—
106	C₂×C₄	—	53	6	no	—
107	C₂×C₄	—	53	24	no	—
108	C₂×C₄	—	53	16	no	—
109	C₂×C₄	—	53	96	no	—
110	C₂×C₄	C₂×C₄×#6157	1	6	yes	—
111	C₂×C₄	C₂×C₄×#6157	1	12	yes	—
112	C₂×C₄	C₂×D₅₃⋊C₈	1	24	yes	—
113	C₈	—	53	32	no	—
114	C₂²	—	53	4	no	—
115	C₂²	—	53	24	no	—
116	C₂²	C₂²×C₄×#6157	1	3	yes	—
117	C₂²	C₂²×D₅₃⋊C₈	1	4	yes	—
118	C₄	—	53	4	no	—
119	C₄	—	53	24	no	—
120	C₄	C₄²×#6157	1	4	yes	—
121	C₄	C₄×D₅₃⋊C₈	1	24	yes	direct product factor
122	C₂	—	53	8	no	—
123	C₂	C₂×C₄²×#6157	1	1	yes	characteristic
124	C₂	C₂×C₄×D₅₃⋊C₈	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₅₃⋊C₈

C₂

yes

maximal

C₄²×#88324

C₂

yes

maximal, complemented

C₄³×D₅₃

C₂

yes

maximal, characteristic

C₂²×D₅₃⋊C₈

C₂²

yes

—

C₄×D₅₃⋊C₈

C₄

yes

direct product factor

C₂×C₄×#88324

C₂²

yes

—

C₂×C₄²×D₅₃

C₄

yes

—

C₂×C₄²×D₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₄

yes

—

C₄²×Dic₅₃

C₂²

yes

—

C₄³×C₅₃

C₄

yes

characteristic

C₂×D₅₃⋊C₈

C₂×C₄

yes

—

C₂²×#88324

C₂³

yes

—

C₄×#88324

C₂×C₄

yes

complemented

C₂²×C₄×D₅₃

C₂×C₄

yes

—

C₂²×C₄×D₅₃

C₂³

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

—

C₄²×D₅₃

C₈

yes

complemented

C₄²×D₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂³×D₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₂⁴

yes

characteristic

C₂×C₄×D₅₃

C₄²

yes

—

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₄×D₅₃

C₂²×C₄

yes

—

C₂×#88324

C₂²×C₄

yes

—

D₅₃⋊C₈

C₄²

yes

direct product factor

C₄²×C₅₃

C₄²

yes

—

C₄²×C₅₃

C₂×C₈

yes

complemented

C₄×Dic₅₃

C₄²

yes

—

C₄×Dic₅₃

C₂×C₈

yes

complemented

C₄×Dic₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂²×D₅₃

C₂×C₄²

yes

—

C₂²×D₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₂×C₄²

yes

—

C₂×Dic₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₂³×C₄

yes

—

C₄×D₅₃

C₄×C₈

yes

complemented

C₄×D₅₃

C₂×C₄²

yes

—

#88324

C₂×C₄²

yes

complemented

C₂×C₄²×C₈

—

maximal, Sylow

C₂²×C₅₃

C₂²×C₄²

yes

—

C₂²×C₅₃

C₂³×C₈

yes

—

C₂×D₅₃

C₄³

yes

—

C₂×D₅₃

C₂×C₄×C₈

yes

—

C₄×C₅₃

C₄³

yes

—

C₄×C₅₃

C₂×C₄×C₈

yes

complemented

Dic₅₃

C₂×C₄×C₈

yes

complemented

Dic₅₃

C₂²×C₄²

yes

—

C₂²×C₄×C₈

—

C₂×C₄³

—

C₄²×C₈

—

C₂×C₅₃

C₂×C₄³

yes

characteristic

C₂×C₅₃

C₂²×C₄×C₈

yes

—

D₅₃

C₄²×C₈

yes

complemented

C₂³×C₈

—

C₂²×C₄²

—

C₂²×C₄²

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄³

—

C₄³

#6157

yes

characteristic

C₅₃

C₂×C₄²×C₈

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#6157

yes

—

C₂×C₄²

C₂×#6157

yes

—

C₄×C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×#6157

yes

characteristic

C₂²×C₄

C₂²×#6157

yes

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

D₅₃⋊C₈

yes

direct product factor

101

C₄²

C₄×#6157

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

C₂³×#6157

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₄×#6157

yes

—

111

C₂×C₄

C₂×C₄×#6157

yes

—

112

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂²×C₄×#6157

yes

—

117

C₂²

C₂²×D₅₃⋊C₈

yes

—

118

C₄

—

119

C₄

—

120

C₄

C₄²×#6157

yes

—

121

C₄

C₄×D₅₃⋊C₈

yes

direct product factor

122

C₂

—

123

C₂

C₂×C₄²×#6157

yes

characteristic

124

C₂

C₂×C₄×D₅₃⋊C₈

yes

—

	subgroup	G/H
center, hypercenter	C₄³	#6157
commutator, hypocenter	C₅₃	C₂×C₄²×C₈
Frattini	C₂³	C₂³×#6157
Fitting	C₄³×C₅₃	C₄
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×Dic₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄³

subgroup

G/H

center, hypercenter

C₄³

#6157

commutator, hypocenter

C₅₃

C₂×C₄²×C₈

Frattini

C₂³

C₂³×#6157

Fitting

C₄³×C₅₃

C₄

socle

C₂³×C₅₃

C₂³×C₄

prime radical

C₂³×D₅₃

C₂²×C₄

cosocle

C₂²×Dic₅₃

C₂⁴

LCS

G ▷ C₅₃

UCS

C₁ ◁ C₄³

N	H ≅ G/N	type
2. C₄²×#88324	C₂	semidirect
5. C₄×D₅₃⋊C₈	C₄	direct (N × H)
14. C₄×#88324	C₂×C₄	semidirect
20. C₄²×D₅₃	C₈	semidirect
33. D₅₃⋊C₈	C₄²	direct (N × H)
35. C₄²×C₅₃	C₂×C₈	semidirect
37. C₄×Dic₅₃	C₂×C₈	semidirect
48. C₄×D₅₃	C₄×C₈	semidirect
50. #88324	C₂×C₄²	semidirect
57. C₄×C₅₃	C₂×C₄×C₈	semidirect
58. Dic₅₃	C₂×C₄×C₈	semidirect
65. D₅₃	C₄²×C₈	semidirect
74. C₅₃	C₂×C₄²×C₈	semidirect
100. C₄²	D₅₃⋊C₈	direct (N × H)
121. C₄	C₄×D₅₃⋊C₈	direct (N × H)

H ≅ G/N

type

2. C₄²×#88324

C₂

semidirect

5. C₄×D₅₃⋊C₈

C₄

direct (N × H)

14. C₄×#88324

C₂×C₄

semidirect

20. C₄²×D₅₃

C₈

semidirect

33. D₅₃⋊C₈

C₄²

direct (N × H)

35. C₄²×C₅₃

C₂×C₈

semidirect

37. C₄×Dic₅₃

C₂×C₈

semidirect

48. C₄×D₅₃

C₄×C₈

semidirect

50. #88324

C₂×C₄²

semidirect

57. C₄×C₅₃

C₂×C₄×C₈

semidirect

58. Dic₅₃

C₂×C₄×C₈

semidirect

65. D₅₃

C₄²×C₈

semidirect

74. C₅₃

C₂×C₄²×C₈

semidirect

100. C₄²

D₅₃⋊C₈

direct (N × H)

121. C₄

C₄×D₅₃⋊C₈

direct (N × H)

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

subgroup	index
C₂×C₄×D₅₃⋊C₈	2
C₄²×#88324	2
C₄³×D₅₃	2
C₂×C₄²×C₈	53

subgroup

index

C₂×C₄×D₅₃⋊C₈

C₄²×#88324

C₄³×D₅₃

C₂×C₄²×C₈

Subgroups — C₄²×D₅₃⋊C₈ #28317890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C42×D53⋊C8 #28317890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×D₅₃⋊C₈ #28317890