Group #13102927 — subgroups

Subgroups — #13102927

Order 1600.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₄×#6045	C₂	1	1	yes	maximal, characteristic
3	#11716677	C₂	1	1	yes	maximal, characteristic
4	#11716677	C₂	1	1	yes	maximal, characteristic
5	#11716675	C₂	1	1	yes	maximal, characteristic
6	#11716674	C₂	1	1	yes	maximal, characteristic
7	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic
8	C₂×#6045	C₂²	1	1	yes	characteristic
9	C₅²×C₄⋊C₄	—	2	1	no	—
10	C₅²×C₄⋊C₄	—	2	1	no	—
11	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₄²×C₅²	C₂²	1	1	yes	characteristic
14	C₄⋊#1039	C₂²	1	1	yes	characteristic
15	#87806	—	2	1	no	—
16	C₄×#1039	—	2	1	no	—
17	C₂×#6029	C₂²	1	1	yes	characteristic
18	C₂×#6029	C₂²	1	1	yes	characteristic
19	#65352	—	5	6	no	maximal
20	C₂×C₄×C₅²	—	2	1	no	—
21	C₂×C₄×C₅²	—	2	1	no	—
22	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
25	C₂×#1039	—	2	1	no	—
26	C₂×#1039	—	2	1	no	—
27	#6045	—	2	1	no	—
28	#6045	Q₈	1	2	yes	—
29	#6029	—	2	1	no	—
30	#6029	—	2	1	no	—
31	C₅×#125	D₅	1	6	yes	—
32	C₄×Dic₁₀	—	5	6	no	—
33	#3942	—	5	6	no	—
34	#3942	—	5	6	no	—
35	#3940	—	5	6	no	—
36	#3939	—	5	6	no	—
37	C₄⋊#168	—	5	6	no	—
38	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅²	—	2	1	no	—
40	C₄×C₅²	—	2	1	no	—
41	C₄×C₅²	—	4	1	no	—
42	C₄×C₅²	—	4	1	no	—
43	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
44	C₄×C₅²	#41	1	1	yes	characteristic
45	#1039	—	2	2	no	—
46	#1039	—	4	1	no	—
47	C₂×Dic₁₀	—	5	6	no	—
48	C₅×C₄⋊C₄	—	2	6	no	—
49	C₅×C₄⋊C₄	—	2	6	no	—
50	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
51	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
52	C₄²×C₅	C₂×D₅	1	6	yes	—
53	C₄⋊Dic₅	—	5	6	no	—
54	#683	—	10	6	no	—
55	C₄×Dic₅	—	10	6	no	—
56	C₂×#168	—	5	6	no	—
57	C₂×#168	—	5	6	no	—
58	#479	—	25	1	no	Sylow
59	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₅²	#135	1	1	yes	characteristic
61	C₂×C₅²	#137	1	1	yes	characteristic
62	C₂×C₄×C₅	—	2	6	no	—
63	C₂×C₄×C₅	—	2	6	no	—
64	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
65	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
66	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
67	C₂×Dic₅	—	10	6	no	—
68	C₂×Dic₅	—	10	6	no	—
69	Dic₁₀	—	5	12	no	—
70	Dic₁₀	—	10	6	no	—
71	#168	—	10	6	no	—
72	#168	—	10	6	no	—
73	#125	C₅⋊D₅	1	1	yes	characteristic
74	C₄×Q₈	—	25	1	no	—
75	#107	—	25	1	no	—
76	#106	—	25	1	no	—
77	C₄⋊C₈	—	25	1	no	—
78	#103	—	25	1	no	—
79	#103	—	25	1	no	—
80	C₅²	#479	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
82	C₄×C₅	—	2	6	no	—
83	C₄×C₅	—	2	6	no	—
84	C₄×C₅	—	4	6	no	—
85	C₄×C₅	—	4	6	no	—
86	C₄×C₅	Q₈×D₅	1	6	yes	—
87	C₄×C₅	#713	1	6	yes	—
88	Dic₅	—	10	12	no	—
89	Dic₅	—	20	6	no	—
90	C₂×Q₈	—	25	1	no	—
91	C₂×C₈	—	25	1	no	—
92	C₂×C₈	—	25	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	25	1	no	—
96	C₄⋊C₄	—	50	1	no	—
97	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
98	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
99	C₄²	—	50	1	no	—
100	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
101	C₂×C₅	#4092	1	6	yes	—
102	C₂×C₅	#4096	1	6	yes	—
103	C₂×C₅	#4097	1	6	yes	—
104	Q₈	—	25	2	no	—
105	Q₈	—	50	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	50	1	no	—
109	C₂×C₄	—	50	1	no	—
110	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
111	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
113	C₈	—	50	1	no	—
114	C₈	—	50	1	no	—
115	C₅	#65352	1	6	yes	complemented
116	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	1	no	—
120	C₄	—	4	1	no	—
121	C₄	—	50	2	no	—
122	C₄	—	100	1	no	—
123	C₄	Q₈×C₅⋊D₅	1	1	yes	characteristic
124	C₄	#87904	1	1	yes	characteristic
125	C₂	#11717266	1	1	yes	characteristic
126	C₂	#11717270	1	1	yes	characteristic
127	C₂	#11717271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#125

C₂

yes

maximal, characteristic

C₄×#6045

C₂

yes

maximal, characteristic

#11716677

C₂

yes

maximal, characteristic

#11716677

C₂

yes

maximal, characteristic

#11716675

C₂

yes

maximal, characteristic

#11716674

C₂

yes

maximal, characteristic

C₄⋊#6029

C₂

yes

maximal, characteristic

C₂×#6045

C₂²

yes

characteristic

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

C₄⋊#1039

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#65352

—

maximal

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

—

—

Q₈

yes

—

#6029

—

#6029

—

C₅×#125

D₅

yes

—

C₄×Dic₁₀

—

—

—

—

—

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

C₄×C₅²

yes

—

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

—

—

—

C₂×C₅²

C₂²⋊Q₈

yes

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

#168

—

#168

—

#125

C₅⋊D₅

yes

characteristic

C₄×Q₈

—

#107

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

C₅²

#479

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

Q₈×D₅

yes

—

C₄×C₅

#713

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

C₂×C₅⋊D₅

yes

characteristic

C₄²

—

100

C₄²

C₂×C₅⋊D₅

yes

characteristic

101

C₂×C₅

#4092

yes

—

102

C₂×C₅

#4096

yes

—

103

C₂×C₅

#4097

yes

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

111

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

112

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

113

C₈

—

114

C₈

—

115

C₅

#65352

yes

complemented

116

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

100

—

123

C₄

Q₈×C₅⋊D₅

yes

characteristic

124

C₄

#87904

yes

characteristic

125

C₂

#11717266

yes

characteristic

126

C₂

#11717270

yes

characteristic

127

C₂

#11717271

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×#125	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#479
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#125

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
80. C₅²	#479	semidirect
115. C₅	#65352	semidirect

H ≅ G/N

type

80. C₅²

#479

semidirect

115. C₅

#65352

semidirect

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

subgroup	index
C₅²×#125	2
C₄×#6045	2
#11716677	2
#11716675	2
#11716674	2
C₄⋊#6029	2
#65352	5

subgroup

index

C₅²×#125

Subgroups — #13102927

Conjugacy classes of subgroups

Complements

Maximal subgroups