Group #13103091 — subgroups

Subgroups — #13103091

Order 1600.

744 conjugacy classes of subgroups; only the 319 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 319 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	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#1039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#1039	C₂	1	1	yes	maximal, characteristic
4	#11716698	C₂	1	4	yes	maximal
5	C₂×C₅²×Q₈	—	2	2	no	—
6	C₂×C₅²×Q₈	—	2	2	no	—
7	C₂×C₅²×Q₈	C₄	1	4	yes	complemented
8	C₂²×C₄×C₅²	C₂²	1	2	yes	—
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×#1039	C₂²	1	2	yes	—
11	C₂²×#1039	C₂²	1	2	yes	—
12	C₄⋊#1039	—	2	2	no	—
13	C₄×#1039	—	2	2	no	—
14	#65519	—	5	6	no	maximal
15	C₂³×C₅²	C₂³	1	1	yes	characteristic
16	C₅²×Q₈	—	2	8	no	—
17	C₂×C₄×C₅²	—	2	2	no	—
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅²	D₄	1	2	yes	—
22	C₂×C₄×C₅²	D₄	1	2	yes	—
23	C₂×#1039	—	2	2	no	—
24	C₂×#1039	—	2	2	no	—
25	C₂×#1039	—	2	2	no	—
26	C₂×#1039	—	2	4	no	—
27	C₂×#1039	Q₈	1	4	yes	complemented
28	C₂²×C₅×Q₈	D₅	1	6	yes	—
29	C₂×C₄⋊Dic₅	—	5	6	no	—
30	C₂×C₄×Dic₅	—	5	6	no	—
31	#3963	—	5	24	no	—
32	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₅²	C₂×Q₈	1	2	yes	—
36	C₂²×C₅²	#41	1	2	yes	—
37	C₄×C₅²	—	2	4	no	—
38	C₄×C₅²	C₂²⋊C₄	1	4	yes	—
39	#1039	—	2	4	no	—
40	#1039	—	4	2	no	—
41	C₂×C₅×Q₈	—	2	12	no	—
42	C₂×C₅×Q₈	—	2	12	no	—
43	C₂×C₅×Q₈	Dic₅	1	24	yes	complemented
44	C₂²×C₄×C₅	C₂×D₅	1	12	yes	—
45	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
46	C₂²×Dic₅	—	5	12	no	—
47	C₂²×Dic₅	—	5	12	no	—
48	C₄⋊Dic₅	—	10	12	no	—
49	C₄×Dic₅	—	10	12	no	—
50	#391	—	25	1	no	Sylow
51	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₅²	C₄×Q₈	1	2	yes	—
53	C₂×C₅²	C₂²⋊Q₈	1	2	yes	—
54	C₂×C₅²	#124	1	1	yes	characteristic
55	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
56	C₂³×C₅	C₂²×D₅	1	6	yes	—
57	C₅×Q₈	—	2	48	no	—
58	C₂×C₄×C₅	—	2	12	no	—
59	C₂×C₄×C₅	—	2	12	no	—
60	C₂×C₄×C₅	C₂×Dic₅	1	24	yes	—
61	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
62	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
63	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
64	C₂×Dic₅	—	5	24	no	—
65	C₂×Dic₅	—	10	12	no	—
66	C₂×Dic₅	—	10	12	no	—
67	C₂×Dic₅	—	10	24	no	—
68	C₂×Dic₅	—	10	12	no	—
69	C₂²×Q₈	C₅⋊D₅	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	25	1	no	—
71	C₂×C₄²	—	25	1	no	—
72	#95	—	25	4	no	—
73	C₅²	#391	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₅	Q₈×D₅	1	12	yes	—
75	C₂²×C₅	#713	1	12	yes	—
76	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
77	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
78	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
79	C₄×C₅	—	2	24	no	—
80	C₄×C₅	C₂²⋊Dic₅	1	24	yes	—
81	Dic₅	—	10	24	no	—
82	Dic₅	—	20	12	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	#1039	1	4	yes	complemented
86	C₂²×C₄	—	25	2	no	—
87	C₂²×C₄	—	25	2	no	—
88	C₂²×C₄	C₂×C₅⋊D₅	1	2	yes	—
89	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
90	C₄⋊C₄	—	50	2	no	—
91	C₄²	—	50	2	no	—
92	C₂×C₅	#4090	1	6	yes	—
93	C₂×C₅	Q₈×Dic₅	1	12	yes	—
94	C₂×C₅	#4092	1	12	yes	—
95	C₂×C₅	#4093	1	6	yes	—
96	C₂×C₅	C₂×C₂²⋊Dic₅	1	6	yes	—
97	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	25	4	no	—
102	C₂×C₄	—	50	2	no	—
103	C₂×C₄	—	50	2	no	—
104	C₂×C₄	—	50	4	no	—
105	C₂×C₄	—	50	2	no	—
106	C₂×C₄	C₂×#1039	1	4	yes	—
107	C₂×C₄	C₂×#1039	1	2	yes	—
108	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
109	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
110	C₅	#65519	1	6	yes	complemented
111	C₂²	Q₈×C₅⋊D₅	1	2	yes	—
112	C₂²	#87904	1	2	yes	—
113	C₂²	C₂²×#1039	1	1	yes	characteristic
114	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
115	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	50	4	no	—
118	C₄	—	100	2	no	—
119	C₄	C₂²⋊#1039	1	4	yes	—
120	C₂	#11717264	1	1	yes	characteristic
121	C₂	Q₈×#1039	1	2	yes	—
122	C₂	#11717266	1	2	yes	—
123	C₂	#11717267	1	1	yes	characteristic
124	C₂	C₂×C₂²⋊#1039	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅²×Q₈

C₂

yes

maximal, characteristic

C₂×C₄⋊#1039

C₂

yes

maximal, characteristic

C₂×C₄×#1039

C₂

yes

maximal, characteristic

#11716698

C₂

yes

maximal

C₂×C₅²×Q₈

—

C₂×C₅²×Q₈

—

C₂×C₅²×Q₈

C₄

yes

complemented

C₂²×C₄×C₅²

C₂²

yes

—

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂²×#1039

C₂²

yes

—

C₂²×#1039

C₂²

yes

—

—

—

—

C₂³×C₅²

C₂³

yes

characteristic

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

D₄

yes

—

—

—

—

—

Q₈

yes

complemented

C₂²×C₅×Q₈

D₅

yes

—

C₂×C₄⋊Dic₅

—

C₂×C₄×Dic₅

—

#3963

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂²×C₅²

C₂×Q₈

yes

—

C₂²×C₅²

#41

yes

—

C₄×C₅²

—

C₄×C₅²

C₂²⋊C₄

yes

—

#1039

—

#1039

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

Dic₅

yes

complemented

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

#391

—

Sylow

C₂×C₅²

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₅²

C₄×Q₈

yes

—

C₂×C₅²

C₂²⋊Q₈

yes

—

C₂×C₅²

#124

yes

characteristic

C₂×C₅²

C₄⋊Q₈

yes

characteristic

C₂³×C₅

C₂²×D₅

yes

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂²×Q₈

C₅⋊D₅

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

#95

—

C₅²

#391

yes

characteristic, complemented, Sylow

C₂²×C₅

Q₈×D₅

yes

—

C₂²×C₅

#713

yes

—

C₂²×C₅

C₂²×Dic₅

yes

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

C₂²⋊Dic₅

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#1039

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₅⋊D₅

yes

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₂×C₅

#4090

yes

—

C₂×C₅

Q₈×Dic₅

yes

—

C₂×C₅

#4092

yes

—

C₂×C₅

#4093

yes

—

C₂×C₅

C₂×C₂²⋊Dic₅

yes

—

C₂³

C₂²×C₅⋊D₅

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×#1039

yes

—

107

C₂×C₄

C₂×#1039

yes

—

108

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

109

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

110

C₅

#65519

yes

complemented

111

C₂²

Q₈×C₅⋊D₅

yes

—

112

C₂²

#87904

yes

—

113

C₂²

C₂²×#1039

yes

characteristic

114

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

115

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

100

—

119

C₄

C₂²⋊#1039

yes

—

120

C₂

#11717264

yes

characteristic

121

C₂

Q₈×#1039

yes

—

122

C₂

#11717266

yes

—

123

C₂

#11717267

yes

characteristic

124

C₂

C₂×C₂²⋊#1039

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator	C₂²×C₅²	C₂²×C₄
Fitting	C₂²×C₅²×Q₈	C₂
socle, prime radical, cosocle	C₂³×C₅²	C₂³
hypercenter	C₂²×Q₈	C₅⋊D₅
hypocenter	C₅²	#391
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂²×C₅⋊D₅

commutator

C₂²×C₅²

C₂²×C₄

Fitting

C₂²×C₅²×Q₈

C₂

socle, prime radical, cosocle

C₂³×C₅²

C₂³

hypercenter

C₂²×Q₈

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
7. C₂×C₅²×Q₈	C₄	semidirect
27. C₂×#1039	Q₈	semidirect
43. C₂×C₅×Q₈	Dic₅	semidirect
73. C₅²	#391	semidirect
85. C₂×Q₈	#1039	semidirect
110. C₅	#65519	semidirect

H ≅ G/N

type

7. C₂×C₅²×Q₈

C₄

semidirect

27. C₂×#1039

Q₈

semidirect

43. C₂×C₅×Q₈

Dic₅

semidirect

73. C₅²

#391

semidirect

85. C₂×Q₈

#1039

semidirect

110. C₅

#65519

semidirect

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

subgroup	index
C₂²×C₅²×Q₈	2
C₂×C₄⋊#1039	2
C₂×C₄×#1039	2
#11716698	2
#65519	5

subgroup

index

C₂²×C₅²×Q₈

C₂×C₄⋊#1039

C₂×C₄×#1039

#11716698

#65519

Subgroups — #13103091

Conjugacy classes of subgroups

Complements

Maximal subgroups