Group #13102950 — subgroups

Subgroups — #13102950

Order 1600.

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

normal subgroups only

This group has 480 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	C₂×#87811	C₂	1	2	yes	maximal
3	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
4	#11716702	C₂	1	2	yes	maximal, complemented
5	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×Q₈	C₂²	1	2	yes	—
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	—	2	1	no	—
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	#87811	—	2	4	no	—
13	#87808	—	2	1	no	—
14	C₄⋊#1039	C₂²	1	1	yes	characteristic
15	C₂×#6029	C₂²	1	2	yes	—
16	#65372	—	5	6	no	maximal
17	C₂²×C₅⋊D₅	—	2	1	no	—
18	C₅²×Q₈	—	2	2	no	—
19	C₅²×Q₈	—	2	2	no	—
20	C₂×C₄×C₅²	—	2	2	no	—
21	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
24	C₂×#1039	—	2	1	no	—
25	C₄⋊(C₅⋊D₅)	—	2	1	no	—
26	C₄⋊(C₅⋊D₅)	D₄	1	2	yes	—
27	C₄×C₅⋊D₅	—	4	2	no	—
28	#6029	—	2	2	no	—
29	C₅×C₄⋊Q₈	D₅	1	6	yes	complemented
30	C₂×#688	—	5	12	no	—
31	C₄×D₂₀	—	5	6	no	—
32	#3967	—	5	12	no	—
33	C₄⋊#168	—	5	6	no	—
34	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
35	C₂×C₅⋊D₅	—	2	2	no	—
36	C₂×C₅⋊D₅	—	4	1	no	—
37	C₂×C₅⋊D₅	—	4	1	no	—
38	C₄×C₅²	—	2	1	no	—
39	C₄×C₅²	—	4	2	no	—
40	C₄×C₅²	SD₁₆	1	2	yes	complemented
41	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅²	#41	1	1	yes	characteristic
43	#1039	—	4	1	no	—
44	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
45	C₂×D₂₀	—	5	6	no	—
46	C₂×C₄×D₅	—	10	6	no	—
47	C₅×C₄⋊C₄	—	2	6	no	—
48	C₅×C₄⋊C₄	—	2	6	no	—
49	C₄²×C₅	C₂×D₅	1	6	yes	—
50	#688	—	10	24	no	—
51	#685	—	10	6	no	—
52	C₄⋊Dic₅	—	5	6	no	—
53	C₂×#168	—	5	12	no	—
54	#461	—	25	1	no	Sylow
55	C₂×C₅²	#121	1	1	yes	characteristic
56	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₅²	#137	1	1	yes	characteristic
58	C₅⋊D₅	—	4	2	no	—
59	C₂²×D₅	—	10	6	no	—
60	C₅×Q₈	—	2	12	no	—
61	C₅×Q₈	—	2	12	no	—
62	C₂×C₄×C₅	—	2	12	no	—
63	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
64	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
65	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
66	C₂×Dic₅	—	10	6	no	—
67	D₂₀	—	5	12	no	—
68	D₂₀	—	10	6	no	—
69	C₄×D₅	—	20	12	no	—
70	#168	—	10	12	no	—
71	C₂×SD₁₆	—	25	2	no	—
72	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
73	C₄×D₄	—	25	1	no	—
74	C₄⋊C₈	—	25	1	no	—
75	#103	—	25	2	no	—
76	C₅²	#461	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
78	C₂×D₅	—	10	12	no	—
79	C₂×D₅	—	20	6	no	—
80	C₂×D₅	—	20	6	no	—
81	C₄×C₅	—	2	6	no	—
82	C₄×C₅	—	4	12	no	—
83	C₄×C₅	#688	1	12	yes	complemented
84	C₄×C₅	D₄×D₅	1	6	yes	—
85	C₄×C₅	#711	1	6	yes	—
86	Dic₅	—	20	6	no	—
87	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
88	C₂×D₄	—	25	1	no	—
89	C₂²×C₄	—	50	1	no	—
90	SD₁₆	—	50	4	no	—
91	C₂×C₈	—	25	2	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	25	1	no	—
95	C₂²⋊C₄	—	50	1	no	—
96	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
97	C₂×C₅	#4084	1	6	yes	—
98	C₂×C₅	C₂×#688	1	6	yes	—
99	C₂×C₅	#4088	1	6	yes	—
100	D₅	—	20	12	no	—
101	C₂³	—	50	1	no	—
102	Q₈	—	2	2	no	—
103	Q₈	—	2	2	no	—
104	D₄	—	25	2	no	—
105	D₄	—	50	1	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	50	1	no	—
108	C₂×C₄	—	100	2	no	—
109	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
111	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
112	C₈	—	50	2	no	—
113	C₅	#65372	1	6	yes	complemented
114	C₂²	—	50	2	no	—
115	C₂²	—	100	1	no	—
116	C₂²	—	100	1	no	—
117	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	4	2	no	—
120	C₄	—	100	1	no	—
121	C₄	#87811	1	2	yes	complemented
122	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
123	C₄	#87902	1	1	yes	characteristic
124	C₂	—	100	2	no	—
125	C₂	#11717258	1	1	yes	characteristic
126	C₂	C₂×#87811	1	1	yes	characteristic
127	C₂	#11717262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×#87811

C₂

yes

maximal

C₄×C₄⋊(C₅⋊D₅)

C₂

yes

maximal, characteristic

#11716702

C₂

yes

maximal, complemented

C₄⋊#6029

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×C₄×C₅⋊D₅

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

—

C₄²×C₅²

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

#65372

—

maximal

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

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)

D₄

yes

—

C₄×C₅⋊D₅

—

#6029

—

C₅×C₄⋊Q₈

D₅

yes

complemented

C₂×#688

—

C₄×D₂₀

—

#3967

—

C₄⋊#168

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

SD₁₆

yes

complemented

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₅

yes

—

#688

—

#685

—

C₄⋊Dic₅

—

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₅⋊D₅

—

C₂²×D₅

—

C₅×Q₈

—

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

—

C₂×SD₁₆

—

C₄⋊Q₈

C₅⋊D₅

yes

characteristic, complemented

C₄×D₄

—

C₄⋊C₈

—

#103

—

C₅²

#461

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#688

yes

complemented

C₄×C₅

D₄×D₅

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

C₂×Q₈

C₂×C₅⋊D₅

yes

—

C₂×D₄

—

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

#4084

yes

—

C₂×C₅

C₂×#688

yes

—

C₂×C₅

#4088

yes

—

100

D₅

—

101

C₂³

—

102

Q₈

—

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

100

—

109

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

110

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

111

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

112

C₈

—

113

C₅

#65372

yes

complemented

114

C₂²

—

115

C₂²

—

100

—

116

C₂²

—

100

—

117

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

100

—

121

C₄

#87811

yes

complemented

122

C₄

D₄×C₅⋊D₅

yes

characteristic

123

C₄

#87902

yes

characteristic

124

C₂

—

100

—

125

C₂

#11717258

yes

characteristic

126

C₂

C₂×#87811

yes

characteristic

127

C₂

#11717262

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	#461
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

#461

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. #11716702	C₂	semidirect
5. C₄⋊#6029	C₂	semidirect
29. C₅×C₄⋊Q₈	D₅	semidirect
40. C₄×C₅²	SD₁₆	semidirect
72. C₄⋊Q₈	C₅⋊D₅	semidirect
76. C₅²	#461	semidirect
83. C₄×C₅	#688	semidirect
113. C₅	#65372	semidirect
121. C₄	#87811	semidirect

H ≅ G/N

type

1. C₅²×C₄⋊Q₈

C₂

semidirect

4. #11716702

C₂

semidirect

5. C₄⋊#6029

C₂

semidirect

29. C₅×C₄⋊Q₈

D₅

semidirect

40. C₄×C₅²

SD₁₆

semidirect

72. C₄⋊Q₈

C₅⋊D₅

semidirect

76. C₅²

#461

semidirect

83. C₄×C₅

#688

semidirect

113. C₅

#65372

semidirect

121. C₄

#87811

semidirect

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

subgroup	index
C₅²×C₄⋊Q₈	2
C₂×#87811	2
C₄×C₄⋊(C₅⋊D₅)	2
#11716702	2
C₄⋊#6029	2
#65372	5

subgroup

index

C₅²×C₄⋊Q₈

C₄×C₄⋊(C₅⋊D₅)

Subgroups — #13102950

Conjugacy classes of subgroups

Complements

Maximal subgroups