Group #13102991 — subgroups

Subgroups — #13102991

Order 1600.

632 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 195 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×#87900	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6029	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716687	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊(C₅⋊D₅)	C₄	1	2	yes	—
7	#87900	—	2	2	no	—
8	#87900	—	2	2	no	—
9	C₂×C₄⋊(C₅⋊D₅)	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
11	C₂×#6045	C₄	1	1	yes	characteristic
12	C₅²×M₁₆	—	2	2	no	—
13	C₂×C₅²×C₈	C₂²	1	2	yes	—
14	C₂×#6029	—	2	2	no	—
15	C₂×#6029	C₂²	1	2	yes	—
16	#65418	—	5	6	no	maximal
17	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅⋊D₅	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅²	D₄	1	2	yes	—
21	C₂×C₄×C₅²	D₄	1	2	yes	—
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₂²⋊(C₅⋊D₅)	—	2	4	no	—
24	C₂×#1039	C₂×C₄	1	2	yes	—
25	C₄⋊(C₅⋊D₅)	—	2	2	no	—
26	C₄×C₅⋊D₅	—	2	2	no	—
27	C₄×C₅⋊D₅	—	2	2	no	—
28	#6045	—	2	2	no	—
29	C₅²×C₈	—	2	2	no	—
30	#6029	—	2	2	no	—
31	C₂×#709	—	5	6	no	—
32	C₂×C₅×M₁₆	D₅	1	6	yes	complemented
33	C₂²×#168	—	5	6	no	—
34	#3952	—	5	24	no	—
35	C₂²×C₅²	—	2	2	no	—
36	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₅⋊D₅	—	2	2	no	—
39	C₂×C₅⋊D₅	—	2	4	no	—
40	C₄×C₅²	C₂²⋊C₄	1	2	yes	—
41	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
43	#1039	—	2	2	no	—
44	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
45	C₂×C₂²⋊D₅	—	5	12	no	—
46	#709	—	10	12	no	—
47	#709	—	10	12	no	—
48	C₂×D₂₀	—	5	6	no	—
49	C₂×C₄×D₅	—	5	12	no	—
50	C₂×Dic₁₀	—	5	6	no	—
51	C₅×M₁₆	—	2	12	no	—
52	C₂×C₅×C₈	C₂×D₅	1	12	yes	—
53	C₂×#168	—	5	12	no	—
54	C₂×#168	—	10	12	no	—
55	#408	—	25	1	no	Sylow
56	C₂×C₅²	—	2	2	no	—
57	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₅²	#131	1	2	yes	—
59	C₅⋊D₅	—	4	2	no	—
60	C₂³×C₅	C₄×D₅	1	6	yes	—
61	C₂²×D₅	—	5	12	no	—
62	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
63	C₂×C₄×C₅	D₂₀	1	12	yes	—
64	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
65	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
66	C₂²⋊D₅	—	10	24	no	—
67	C₂×Dic₅	—	5	12	no	—
68	D₂₀	—	10	12	no	—
69	C₄×D₅	—	10	12	no	—
70	C₄×D₅	—	10	12	no	—
71	Dic₁₀	—	10	12	no	—
72	C₅×C₈	—	2	12	no	—
73	#168	—	10	12	no	—
74	C₂×#41	—	25	1	no	—
75	C₂×M₁₆	C₅⋊D₅	1	1	yes	characteristic, complemented
76	C₂²×C₈	—	25	1	no	—
77	C₂²⋊C₈	—	25	4	no	—
78	C₅²	#408	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₅	—	2	12	no	—
80	C₂²×C₅	#685	1	12	yes	—
81	C₂²×C₅	C₂×C₄×D₅	1	6	yes	—
82	C₂×D₅	—	10	24	no	—
83	C₂×D₅	—	10	12	no	—
84	C₄×C₅	#685	1	12	yes	—
85	C₄×C₅	C₂×D₂₀	1	6	yes	—
86	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
87	Dic₅	—	10	12	no	—
88	#41	—	50	2	no	—
89	#41	—	50	2	no	—
90	C₂×Q₈	—	25	1	no	—
91	C₂×D₄	—	25	2	no	—
92	C₂×D₄	—	25	1	no	—
93	C₂²×C₄	—	25	2	no	—
94	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
95	M₁₆	—	2	2	no	—
96	C₂×C₈	—	25	2	no	—
97	C₂×C₈	—	50	2	no	—
98	C₂×C₈	C₂×C₅⋊D₅	1	2	yes	—
99	C₂×C₅	—	2	12	no	—
100	C₂×C₅	#4053	1	12	yes	—
101	C₂×C₅	C₂×#685	1	6	yes	—
102	D₅	—	20	12	no	—
103	C₂³	—	25	2	no	—
104	C₂³	C₄×C₅⋊D₅	1	1	yes	characteristic
105	Q₈	—	50	2	no	—
106	D₄	—	50	4	no	—
107	D₄	—	50	2	no	—
108	C₂×C₄	—	25	2	no	—
109	C₂×C₄	—	50	2	no	—
110	C₂×C₄	—	50	2	no	—
111	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
112	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
113	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
114	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	—	50	2	no	—
117	C₅	#65418	1	6	yes	complemented
118	C₂²	—	2	2	no	—
119	C₂²	—	50	4	no	—
120	C₂²	—	50	2	no	—
121	C₂²	#87808	1	2	yes	—
122	C₂²	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
123	C₄	—	50	2	no	—
124	C₄	#87808	1	2	yes	—
125	C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
126	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	100	2	no	—
129	C₂	#11717227	1	2	yes	—
130	C₂	C₂×#87808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87900

C₂

yes

maximal, characteristic

C₂×C₅²×M₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×#6029

C₂

yes

maximal, characteristic, complemented

#11716687

C₂

yes

maximal, complemented

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂×C₂²⋊(C₅⋊D₅)

C₄

yes

—

#87900

—

#87900

—

C₂×C₄⋊(C₅⋊D₅)

C₄

yes

characteristic

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₂×#6045

C₄

yes

characteristic

C₅²×M₁₆

—

C₂×C₅²×C₈

C₂²

yes

—

C₂×#6029

—

C₂×#6029

C₂²

yes

—

#65418

—

maximal

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂²×C₅⋊D₅

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

C₂×C₄

yes

—

C₄⋊(C₅⋊D₅)

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

—

#6045

—

C₅²×C₈

—

#6029

—

C₂×#709

—

C₂×C₅×M₁₆

D₅

yes

complemented

C₂²×#168

—

#3952

—

C₂²×C₅²

—

C₂²×C₅²

C₂²⋊C₄

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

C₂²⋊C₄

yes

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂×C₂²⋊D₅

—

#709

—

#709

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₂×C₅×C₈

C₂×D₅

yes

—

—

—

—

C₂×C₅²

—

C₂×C₅²

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₅²

#131

yes

—

C₅⋊D₅

—

C₂³×C₅

C₄×D₅

yes

—

C₂²×D₅

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₂×#41

—

C₂×M₁₆

C₅⋊D₅

yes

characteristic, complemented

C₂²×C₈

—

C₂²⋊C₈

—

C₅²

#408

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

#685

yes

—

C₂²×C₅

C₂×C₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#685

yes

—

C₄×C₅

C₂×D₂₀

yes

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₅⋊D₅

yes

—

C₂×C₅

—

100

C₂×C₅

#4053

yes

—

101

C₂×C₅

C₂×#685

yes

—

102

D₅

—

103

C₂³

—

104

C₂³

C₄×C₅⋊D₅

yes

characteristic

105

Q₈

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₄×C₅⋊D₅

yes

characteristic

112

C₂×C₄

C₄⋊(C₅⋊D₅)

yes

—

113

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

114

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

115

C₈

—

116

C₈

—

117

C₅

#65418

yes

complemented

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#87808

yes

—

122

C₂²

C₂×C₄×C₅⋊D₅

yes

characteristic

123

C₄

—

124

C₄

#87808

yes

—

125

C₄

C₂×C₄⋊(C₅⋊D₅)

yes

characteristic

126

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

127

C₂

—

128

C₂

—

100

—

129

C₂

#11717227

yes

—

130

C₂

C₂×#87808

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₅⋊D₅
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₅²×M₁₆	C₂
prime radical	C₂×#87900	C₂
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×M₁₆	C₅⋊D₅
hypocenter	C₅²	#408
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×C₅⋊D₅

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₂×C₅²×M₁₆

C₂

prime radical

C₂×#87900

C₂

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂×M₁₆

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. C₂×C₅²×M₁₆	C₂	semidirect
3. C₂²×#6029	C₂	semidirect
4. #11716687	C₂	semidirect
32. C₂×C₅×M₁₆	D₅	semidirect
75. C₂×M₁₆	C₅⋊D₅	semidirect
78. C₅²	#408	semidirect
117. C₅	#65418	semidirect

H ≅ G/N

type

2. C₂×C₅²×M₁₆

C₂

semidirect

3. C₂²×#6029

C₂

semidirect

4. #11716687

C₂

semidirect

32. C₂×C₅×M₁₆

D₅

semidirect

75. C₂×M₁₆

C₅⋊D₅

semidirect

78. C₅²

#408

semidirect

117. C₅

#65418

semidirect

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

subgroup	index
C₂×#87900	2
C₂×C₅²×M₁₆	2
C₂²×#6029	2
#11716687	2
#65418	5

subgroup

index

C₂×#87900

C₂×C₅²×M₁₆

C₂²×#6029

#11716687

#65418

Subgroups — #13102991

Conjugacy classes of subgroups

Complements

Maximal subgroups