Group #13102951 — subgroups

Subgroups — #13102951

Order 1600.

432 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
3	#11716676	C₂	1	2	yes	maximal
4	#11716674	C₂	1	2	yes	maximal, complemented
5	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×Q₈	—	2	1	no	—
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅⋊D₅	—	2	1	no	—
9	C₅²×C₄⋊C₄	—	2	1	no	—
10	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	#87808	—	2	1	no	—
13	C₄⋊#1039	C₂²	1	1	yes	characteristic
14	C₂×#6029	C₂²	1	2	yes	—
15	#65376	—	5	6	no	maximal
16	C₂²×C₅⋊D₅	—	2	1	no	—
17	C₅²×Q₈	—	4	2	no	—
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×#1039	—	2	1	no	—
23	C₄⋊(C₅⋊D₅)	—	2	1	no	—
24	C₄⋊(C₅⋊D₅)	Q₈	1	2	yes	complemented
25	C₄×C₅⋊D₅	—	4	2	no	—
26	#6029	—	2	2	no	—
27	C₅×C₄⋊Q₈	D₅	1	6	yes	complemented
28	C₄×D₂₀	—	5	6	no	—
29	#3941	—	5	12	no	—
30	#3939	—	5	12	no	—
31	C₄⋊#168	—	5	6	no	—
32	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
33	C₂×C₅⋊D₅	—	2	2	no	—
34	C₂×C₅⋊D₅	—	4	1	no	—
35	C₂×C₅⋊D₅	—	4	1	no	—
36	C₄×C₅²	—	2	1	no	—
37	C₄×C₅²	—	4	2	no	—
38	C₄×C₅²	D₈	1	2	yes	—
39	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₅²	#41	1	1	yes	characteristic
41	#1039	—	4	1	no	—
42	C₂×C₅×Q₈	—	2	6	no	—
43	C₂×D₂₀	—	5	6	no	—
44	C₂×C₄×D₅	—	10	6	no	—
45	C₅×C₄⋊C₄	—	2	6	no	—
46	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
47	C₄²×C₅	C₂×D₅	1	6	yes	—
48	#685	—	10	6	no	—
49	C₄⋊Dic₅	—	5	6	no	—
50	C₂×#168	—	5	12	no	—
51	#474	—	25	1	no	Sylow
52	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
54	C₂×C₅²	#137	1	1	yes	characteristic
55	C₅⋊D₅	—	4	2	no	—
56	C₂²×D₅	—	10	6	no	—
57	C₅×Q₈	—	4	12	no	—
58	C₂×C₄×C₅	—	2	12	no	—
59	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
60	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
61	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
62	C₂×Dic₅	—	10	6	no	—
63	D₂₀	—	5	12	no	—
64	D₂₀	—	10	6	no	—
65	C₄×D₅	—	20	12	no	—
66	#168	—	10	12	no	—
67	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
68	C₄×D₄	—	25	1	no	—
69	#107	—	25	2	no	—
70	C₄⋊C₈	—	25	1	no	—
71	#102	—	25	2	no	—
72	C₅²	#474	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
74	C₂×D₅	—	10	12	no	—
75	C₂×D₅	—	20	6	no	—
76	C₂×D₅	—	20	6	no	—
77	C₄×C₅	—	2	6	no	—
78	C₄×C₅	—	4	12	no	—
79	C₄×C₅	#686	1	12	yes	—
80	C₄×C₅	Q₈×D₅	1	6	yes	—
81	C₄×C₅	#713	1	6	yes	—
82	Dic₅	—	20	6	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×D₄	—	25	1	no	—
85	C₂²×C₄	—	50	1	no	—
86	C₂×C₈	—	25	2	no	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	25	1	no	—
89	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
90	C₂²⋊C₄	—	50	1	no	—
91	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
92	C₂×C₅	C₂×#686	1	6	yes	—
93	C₂×C₅	#4088	1	6	yes	—
94	C₂×C₅	#4092	1	6	yes	—
95	D₅	—	20	12	no	—
96	C₂³	—	50	1	no	—
97	Q₈	—	4	2	no	—
98	D₄	—	25	2	no	—
99	D₄	—	50	1	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	50	1	no	—
102	C₂×C₄	—	100	2	no	—
103	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
104	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
106	C₈	—	50	2	no	—
107	C₅	#65376	1	6	yes	complemented
108	C₂²	—	50	2	no	—
109	C₂²	—	100	1	no	—
110	C₂²	—	100	1	no	—
111	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	100	1	no	—
115	C₄	#87809	1	2	yes	—
116	C₄	Q₈×C₅⋊D₅	1	1	yes	characteristic
117	C₄	#87904	1	1	yes	characteristic
118	C₂	—	100	2	no	—
119	C₂	C₂×#87809	1	1	yes	characteristic
120	C₂	#11717262	1	1	yes	characteristic
121	C₂	#11717266	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₄×C₄⋊(C₅⋊D₅)

C₂

yes

maximal, characteristic

#11716676

C₂

yes

maximal

#11716674

C₂

yes

maximal, complemented

C₄⋊#6029

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×C₄×C₅⋊D₅

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

#87808

—

C₄⋊#1039

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

#65376

—

maximal

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

Q₈

yes

complemented

C₄×C₅⋊D₅

—

#6029

—

C₅×C₄⋊Q₈

D₅

yes

complemented

C₄×D₂₀

—

#3941

—

#3939

—

C₄⋊#168

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

D₈

yes

—

C₄×C₅²

C₂×Q₈

yes

C₄×C₅²

yes

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

#685

—

C₄⋊Dic₅

—

C₂×#168

—

#474

—

Sylow

C₂×C₅²

C₂²⋊Q₈

yes

characteristic

C₂×C₅²

C₂×D₈

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₅⋊D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

C₄×D₅

—

#168

—

C₄⋊Q₈

C₅⋊D₅

yes

characteristic, complemented

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₅²

#474

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#686

yes

—

C₄×C₅

Q₈×D₅

yes

—

C₄×C₅

#713

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₂²⋊C₄

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

C₂×#686

yes

—

C₂×C₅

#4088

yes

—

C₂×C₅

#4092

yes

—

D₅

—

C₂³

—

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

100

—

103

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

104

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

105

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

106

C₈

—

107

C₅

#65376

yes

complemented

108

C₂²

—

109

C₂²

—

100

—

110

C₂²

—

100

—

111

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

100

—

115

C₄

#87809

yes

—

116

C₄

Q₈×C₅⋊D₅

yes

characteristic

117

C₄

#87904

yes

characteristic

118

C₂

—

100

—

119

C₂

C₂×#87809

yes

characteristic

120

C₂

#11717262

yes

characteristic

121

C₂

#11717266

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#474
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×C₄⋊Q₈

C₂

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×C₄⋊(C₅⋊D₅)

C₂²

hypercenter

C₄⋊Q₈

C₅⋊D₅

hypocenter

C₅²

#474

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₅²×C₄⋊Q₈	C₂	semidirect
4. #11716674	C₂	semidirect
5. C₄⋊#6029	C₂	semidirect
24. C₄⋊(C₅⋊D₅)	Q₈	semidirect
27. C₅×C₄⋊Q₈	D₅	semidirect
67. C₄⋊Q₈	C₅⋊D₅	semidirect
72. C₅²	#474	semidirect
107. C₅	#65376	semidirect

H ≅ G/N

type

1. C₅²×C₄⋊Q₈

C₂

semidirect

4. #11716674

C₂

semidirect

5. C₄⋊#6029

C₂

semidirect

24. C₄⋊(C₅⋊D₅)

Q₈

semidirect

27. C₅×C₄⋊Q₈

D₅

semidirect

67. C₄⋊Q₈

C₅⋊D₅

semidirect

72. C₅²

#474

semidirect

107. C₅

#65376

semidirect

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

subgroup	index
C₅²×C₄⋊Q₈	2
C₄×C₄⋊(C₅⋊D₅)	2
#11716676	2
#11716674	2
C₄⋊#6029	2
#65376	5

subgroup

index

C₅²×C₄⋊Q₈

C₄×C₄⋊(C₅⋊D₅)

Subgroups — #13102951

Conjugacy classes of subgroups

Complements

Maximal subgroups