Group #13106406 — subgroups

Subgroups — #13106406

Order 1600.

688 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 200 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×Q₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×Dic₅	C₂	1	2	yes	maximal, complemented
3	#11716992	C₂	1	3	yes	maximal
4	#11716967	C₂	1	3	yes	maximal
5	#11716962	C₂	1	6	yes	maximal, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	Q₈×C₅⋊D₅	C₄	1	8	yes	complemented
8	C₂×C₄×C₅⋊D₅	C₂²	1	3	yes	—
9	C₂×#6045	C₂²	1	3	yes	—
10	C₅×C₄⋊Dic₅	C₂²	1	6	yes	—
11	C₄×C₅×Dic₅	C₂²	1	6	yes	—
12	#87781	C₂²	1	6	yes	—
13	#87779	C₂²	1	3	yes	—
14	#87779	C₂²	1	1	yes	characteristic
15	Dic₅²	C₂²	1	6	yes	—
16	#65906	—	5	2	no	maximal
17	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
18	C₅²×Q₈	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₅²	C₂³	1	3	yes	—
20	C₂×#1039	C₂³	1	3	yes	—
21	C₄×C₅⋊D₅	C₂×C₄	1	12	yes	—
22	#6045	C₂×C₄	1	12	yes	—
23	C₂×C₅×Dic₅	C₂³	1	6	yes	—
24	C₂×C₅×Dic₅	C₂³	1	2	yes	—
25	C₂×Q₈×D₅	—	5	2	no	—
26	C₂×Q₈×D₅	—	10	2	no	—
27	C₄×C₅×Q₈	—	5	2	no	—
28	Q₈×Dic₅	D₅	1	2	yes	complemented
29	#4038	—	5	6	no	—
30	#4033	—	5	6	no	—
31	#4018	—	5	6	no	—
32	C₄×Dic₁₀	—	5	6	no	—
33	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
34	C₂×C₅⋊D₅	—	2	2	no	—
35	C₂×C₅⋊D₅	C₂²×C₄	1	2	yes	—
36	C₄×C₅²	C₂²×C₄	1	6	yes	—
37	#1039	C₂²×C₄	1	6	yes	—
38	C₅×Dic₅	—	2	2	no	—
39	C₅×Dic₅	—	2	6	no	—
40	C₂×C₅×Q₈	—	2	2	no	—
41	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
42	Q₈×D₅	—	5	16	no	—
43	Q₈×D₅	—	10	16	no	—
44	C₂×C₄×D₅	—	5	6	no	—
45	C₂×C₄×D₅	—	10	6	no	—
46	C₂×Dic₁₀	—	5	6	no	—
47	C₂×Dic₁₀	—	10	6	no	—
48	C₅×C₄⋊C₄	—	5	6	no	—
49	C₄²×C₅	—	5	6	no	—
50	#685	—	5	2	no	—
51	#685	—	5	6	no	—
52	C₄⋊Dic₅	C₂×D₅	1	6	yes	—
53	#683	—	5	6	no	—
54	#683	—	5	6	no	—
55	C₄×Dic₅	—	5	12	no	—
56	C₄×Dic₅	C₂×D₅	1	6	yes	—
57	#519	—	25	1	no	Sylow
58	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
59	C₂×C₅²	2₋¹⁺⁴	1	2	yes	—
60	C₅⋊D₅	—	2	2	no	—
61	C₂²×D₅	—	5	2	no	—
62	C₂²×D₅	—	10	2	no	—
63	C₅×Q₈	—	2	8	no	—
64	C₅×Q₈	C₄×D₅	1	8	yes	—
65	C₂×C₄×C₅	—	2	6	no	—
66	C₂×C₄×C₅	—	5	6	no	—
67	C₂×C₄×C₅	—	5	2	no	—
68	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
69	C₂×Dic₅	—	5	6	no	—
70	C₂×Dic₅	—	10	6	no	—
71	C₂×Dic₅	C₂²×D₅	1	6	yes	—
72	C₂×Dic₅	C₂²×D₅	1	2	yes	—
73	C₄×D₅	—	5	24	no	—
74	C₄×D₅	—	10	24	no	—
75	Dic₁₀	—	5	24	no	—
76	Dic₁₀	—	10	24	no	—
77	C₂²×Q₈	—	25	1	no	—
78	C₄×Q₈	—	5	2	no	—
79	C₄×Q₈	—	25	6	no	—
80	#117	—	25	3	no	—
81	#117	—	25	3	no	—
82	C₅²	#519	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₅	—	2	2	no	—
84	C₂²×C₅	C₂³×D₅	1	2	yes	—
85	C₂×D₅	—	5	4	no	—
86	C₂×D₅	—	10	4	no	—
87	C₂×D₅	—	10	8	no	—
88	C₂×D₅	—	10	2	no	—
89	C₂×D₅	—	10	2	no	—
90	C₄×C₅	—	2	12	no	—
91	C₄×C₅	—	10	6	no	—
92	C₄×C₅	—	10	2	no	—
93	C₄×C₅	C₂×C₄×D₅	1	12	yes	—
94	Dic₅	—	2	2	no	—
95	Dic₅	—	2	6	no	—
96	Dic₅	—	5	12	no	—
97	Dic₅	—	10	12	no	—
98	C₂×Q₈	—	25	8	no	—
99	C₂×Q₈	—	25	3	no	—
100	C₂×Q₈	D₅²	1	1	yes	characteristic
101	C₂²×C₄	—	25	3	no	—
102	C₄⋊C₄	—	5	6	no	—
103	C₄⋊C₄	—	25	6	no	—
104	C₂²⋊C₄	—	25	1	no	—
105	C₂²⋊C₄	—	25	3	no	—
106	C₄²	—	5	6	no	—
107	C₄²	—	25	6	no	—
108	C₂×C₅	—	2	4	no	—
109	C₂×C₅	—	2	2	no	—
110	C₂×C₅	C₂²×C₄×D₅	1	2	yes	—
111	C₂×C₅	#4147	1	2	yes	—
112	C₂×C₅	#4150	1	2	yes	—
113	D₅	—	10	8	no	—
114	D₅	—	10	4	no	—
115	C₂³	—	25	1	no	—
116	Q₈	—	25	12	no	—
117	Q₈	#6036	1	4	yes	—
118	C₂×C₄	—	5	2	no	—
119	C₂×C₄	—	5	6	no	—
120	C₂×C₄	—	25	3	no	—
121	C₂×C₄	—	25	12	no	—
122	C₂×C₄	C₂×D₅²	1	3	yes	—
123	C₅	—	2	2	no	—
124	C₅	#65906	1	2	yes	complemented
125	C₂²	—	25	2	no	—
126	C₂²	—	50	2	no	—
127	C₂²	C₂²×D₅²	1	1	yes	characteristic
128	C₄	—	10	2	no	—
129	C₄	—	10	6	no	—
130	C₄	—	25	6	no	—
131	C₄	C₂×#6036	1	6	yes	—
132	C₂	—	50	2	no	—
133	C₂	#11717480	1	2	yes	—
134	C₂	C₂²×#6036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₅⋊D₅

C₂

yes

maximal, characteristic

C₅×Q₈×Dic₅

C₂

yes

maximal, complemented

#11716992

C₂

yes

maximal

#11716967

C₂

yes

maximal

#11716962

C₂

yes

maximal, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

Q₈×C₅⋊D₅

C₄

yes

complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₂×#6045

C₂²

yes

—

C₅×C₄⋊Dic₅

C₂²

yes

—

C₄×C₅×Dic₅

C₂²

yes

—

#87781

C₂²

yes

—

#87779

C₂²

yes

—

#87779

C₂²

yes

characteristic

Dic₅²

C₂²

yes

—

#65906

—

maximal

C₂²×C₅⋊D₅

C₂³

yes

characteristic

C₅²×Q₈

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×#1039

C₂³

yes

—

C₄×C₅⋊D₅

C₂×C₄

yes

—

#6045

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂³

yes

—

C₂×C₅×Dic₅

C₂³

yes

—

C₂×Q₈×D₅

—

C₂×Q₈×D₅

—

C₄×C₅×Q₈

—

Q₈×Dic₅

D₅

yes

—

—

—

C₄×Dic₁₀

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₂²×C₄

yes

—

C₄×C₅²

C₂²×C₄

yes

—

#1039

C₂²×C₄

yes

—

C₅×Dic₅

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

Q₈×D₅

—

Q₈×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

#685

—

#685

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₂×D₅

yes

—

#519

—

Sylow

C₂×C₅²

C₂³×C₄

yes

characteristic

C₂×C₅²

2₋¹⁺⁴

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

C₄×D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂²×D₅

yes

—

C₂×Dic₅

C₂²×D₅

yes

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₅²

#519

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₄×D₅

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

D₅²

yes

characteristic

101

C₂²×C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₂×C₅

—

109

C₂×C₅

—

110

C₂×C₅

C₂²×C₄×D₅

yes

—

111

C₂×C₅

#4147

yes

—

112

C₂×C₅

#4150

yes

—

113

D₅

—

114

D₅

—

115

C₂³

—

116

Q₈

—

117

Q₈

#6036

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×D₅²

yes

—

123

C₅

—

124

C₅

#65906

yes

complemented

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₅²

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×#6036

yes

—

132

C₂

—

133

C₂

#11717480

yes

—

134

C₂

C₂²×#6036

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator	C₂×C₅²	C₂³×C₄
Fitting	C₂×C₅²×Q₈	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂²×C₅⋊D₅	C₂³
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#519
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂²×D₅²

commutator

C₂×C₅²

C₂³×C₄

Fitting

C₂×C₅²×Q₈

C₂²

socle, cosocle

C₂²×C₅²

C₂⁴

prime radical

C₂²×C₅⋊D₅

C₂³

hypercenter

C₂×Q₈

D₅²

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
2. C₅×Q₈×Dic₅	C₂	semidirect
5. #11716962	C₂	semidirect
7. Q₈×C₅⋊D₅	C₄	semidirect
28. Q₈×Dic₅	D₅	semidirect
82. C₅²	#519	semidirect
124. C₅	#65906	semidirect

H ≅ G/N

type

2. C₅×Q₈×Dic₅

C₂

semidirect

5. #11716962

C₂

semidirect

7. Q₈×C₅⋊D₅

C₄

semidirect

28. Q₈×Dic₅

D₅

semidirect

82. C₅²

#519

semidirect

124. C₅

#65906

semidirect

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

subgroup	index
C₂×Q₈×C₅⋊D₅	2
C₅×Q₈×Dic₅	2
#11716992	2
#11716967	2
#11716962	2
#65906	5

subgroup

index

C₂×Q₈×C₅⋊D₅

C₅×Q₈×Dic₅

Subgroups — #13106406

Conjugacy classes of subgroups

Complements

Maximal subgroups