Group #13101033 — subgroups

Subgroups — #13101033

Order 1600.

240 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 62 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	#11717037	C₂	1	1	yes	maximal, characteristic
2	#11716985	C₂	1	1	yes	maximal, characteristic
3	C₈×#1039	C₂	1	1	yes	maximal, characteristic
4	C₅×#3948	C₂	1	2	yes	maximal
5	#11716606	C₂	1	2	yes	maximal
6	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
7	C₂×#6039	—	2	1	no	—
8	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
9	C₄×#1039	C₂²	1	1	yes	characteristic
10	C₂×#6029	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
12	#87781	—	2	1	no	—
13	#87781	—	2	1	no	—
14	#87781	—	2	1	no	—
15	#65081	—	5	2	no	maximal
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	D₄	1	1	yes	characteristic
18	C₂×#1039	D₄	1	1	yes	characteristic
19	C₂×C₅×Dic₅	—	2	2	no	—
20	C₂×C₅×Dic₅	—	2	2	no	—
21	C₅×Dic₁₀	—	2	2	no	—
22	C₅×Dic₁₀	—	2	2	no	—
23	#6039	—	4	2	no	—
24	C₅²×C₈	—	2	1	no	—
25	#6029	—	2	1	no	—
26	#4090	—	5	2	no	—
27	#4034	—	5	2	no	—
28	C₅×#103	—	5	2	no	—
29	#3967	—	5	2	no	—
30	#3948	D₅	1	2	yes	—
31	C₈×Dic₅	—	5	2	no	—
32	C₈×Dic₅	—	10	2	no	—
33	#3942	—	5	2	no	—
34	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
35	C₄×C₅²	#41	1	1	yes	characteristic
36	C₄×C₅²	#41	1	1	yes	characteristic
37	#1039	SD₁₆	1	2	yes	—
38	#1039	Q₁₆	1	2	yes	—
39	C₅×Dic₅	—	4	2	no	—
40	C₅×Dic₅	—	4	2	no	—
41	C₂×C₅×Q₈	—	5	2	no	—
42	C₂×Dic₁₀	—	10	2	no	—
43	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
46	C₅×C₄⋊C₄	—	5	2	no	—
47	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
48	#683	—	10	2	no	—
49	#683	—	10	2	no	—
50	#683	—	10	2	no	—
51	C₄×Dic₅	—	5	2	no	—
52	C₄×Dic₅	—	10	2	no	—
53	C₂×#168	—	5	2	no	—
54	C₂×#168	—	10	2	no	—
55	#487	—	25	1	no	Sylow
56	C₂×C₅²	#124	1	1	yes	characteristic
57	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
59	C₅×Q₈	—	10	2	no	—
60	C₅×Q₈	—	10	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	10	2	no	—
63	C₂×C₄×C₅	—	10	2	no	—
64	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
65	C₂×Dic₅	—	2	2	no	—
66	C₂×Dic₅	—	2	2	no	—
67	C₂×Dic₅	—	5	2	no	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	10	2	no	—
70	C₂×Dic₅	—	10	2	no	—
71	Dic₁₀	—	2	2	no	—
72	Dic₁₀	—	2	2	no	—
73	Dic₁₀	—	20	4	no	—
74	C₅×C₈	—	2	4	no	—
75	C₅×C₈	—	2	2	no	—
76	#168	—	10	4	no	—
77	#168	—	10	2	no	—
78	C₄⋊Q₈	—	25	1	no	—
79	C₄⋊Q₈	—	25	1	no	—
80	#103	—	5	2	no	—
81	#103	—	25	2	no	—
82	C₄×C₈	—	25	1	no	—
83	C₅²	#487	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₅	—	2	2	no	—
85	C₂²×C₅	D₄×D₅	1	2	yes	—
86	C₄×C₅	—	2	2	no	—
87	C₄×C₅	—	2	2	no	—
88	C₄×C₅	—	20	2	no	—
89	C₄×C₅	—	20	2	no	—
90	C₄×C₅	#709	1	2	yes	—
91	C₄×C₅	#711	1	2	yes	—
92	Dic₅	—	4	2	no	—
93	Dic₅	—	4	2	no	—
94	Dic₅	—	5	4	no	—
95	Dic₅	—	5	4	no	—
96	Dic₅	—	10	4	no	—
97	Dic₅	—	10	4	no	—
98	C₂×Q₈	—	5	2	no	—
99	C₂×Q₈	—	50	1	no	—
100	C₂×C₈	—	25	1	no	—
101	C₂×C₈	D₅²	1	1	yes	characteristic
102	C₄⋊C₄	—	5	2	no	—
103	C₄⋊C₄	—	50	1	no	—
104	C₄⋊C₄	—	50	1	no	—
105	C₄⋊C₄	—	50	1	no	—
106	C₄²	—	25	1	no	—
107	C₂×C₅	—	2	2	no	—
108	C₂×C₅	—	2	2	no	—
109	C₂×C₅	—	2	2	no	—
110	C₂×C₅	#4031	1	2	yes	—
111	C₂×C₅	D₅×SD₁₆	1	2	yes	—
112	C₂×C₅	D₅×Q₁₆	1	2	yes	—
113	Q₈	—	10	2	no	—
114	Q₈	—	10	2	no	—
115	Q₈	—	100	2	no	—
116	C₂×C₄	—	10	2	no	—
117	C₂×C₄	—	10	2	no	—
118	C₂×C₄	—	25	1	no	—
119	C₂×C₄	—	25	1	no	—
120	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	50	1	no	—
123	C₅	—	2	2	no	—
124	C₅	#65081	1	2	yes	complemented
125	C₂²	#87877	1	1	yes	characteristic
126	C₄	—	20	2	no	—
127	C₄	—	20	2	no	—
128	C₄	—	25	2	no	—
129	C₄	—	25	2	no	—
130	C₄	#87873	1	1	yes	characteristic
131	C₄	#87880	1	1	yes	characteristic
132	C₂	#11716914	1	1	yes	characteristic
133	C₂	#11716920	1	1	yes	characteristic
134	C₂	#11717033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717037

C₂

yes

maximal, characteristic

#11716985

C₂

yes

maximal, characteristic

C₈×#1039

C₂

yes

maximal, characteristic

C₅×#3948

C₂

yes

maximal

#11716606

C₂

yes

maximal

C₂×C₅×Dic₁₀

C₂²

yes

—

C₂×#6039

—

C₂×C₅²×C₈

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

—

—

—

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

—

#6039

—

C₅²×C₈

—

—

—

—

—

—

D₅

yes

—

C₈×Dic₅

—

C₈×Dic₅

—

#3942

—

C₂²×C₅²

C₂×D₄

yes

C₄×C₅²

yes

C₄×C₅²

yes

SD₁₆

yes

—

#1039

Q₁₆

yes

—

C₅×Dic₅

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₅×C₄⋊C₄

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#103

—

#103

—

C₄×C₈

—

C₅²

#487

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#709

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×C₈

—

101

C₂×C₈

D₅²

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₂×C₅

—

108

C₂×C₅

—

109

C₂×C₅

—

110

C₂×C₅

#4031

yes

—

111

C₂×C₅

D₅×SD₁₆

yes

—

112

C₂×C₅

D₅×Q₁₆

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

100

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×D₅²

yes

characteristic

121

C₈

—

122

C₈

—

123

C₅

—

124

C₅

#65081

yes

complemented

125

C₂²

#87877

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#87873

yes

characteristic

131

C₄

#87880

yes

characteristic

132

C₂

#11716914

yes

characteristic

133

C₂

#11716920

yes

characteristic

134

C₂

#11717033

yes

characteristic

	subgroup	G/H
center	C₂²	#87877
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₂×C₅²×C₈	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#487
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

#87877

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₂×C₅²×C₈

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₂×C₈

D₅²

hypocenter

C₅²

#487

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
83. C₅²	#487	semidirect
124. C₅	#65081	semidirect

H ≅ G/N

type

83. C₅²

#487

semidirect

124. C₅

#65081

semidirect

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

subgroup	index
#11717037	2
#11716985	2
C₈×#1039	2
C₅×#3948	2
#11716606	2
#65081	5

subgroup

index

Subgroups — #13101033

Conjugacy classes of subgroups

Complements

Maximal subgroups