Group #13102955 — subgroups

Subgroups — #13102955

Order 1600.

432 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#87812	C₂	1	2	yes	maximal
3	C₄×#6045	C₂	1	1	yes	maximal, characteristic
4	#11716702	C₂	1	2	yes	maximal
5	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅²×Q₈	C₂²	1	2	yes	—
7	C₂×#6045	C₂²	1	1	yes	characteristic
8	C₅²×C₄⋊C₄	—	2	1	no	—
9	C₅²×C₄⋊C₄	—	2	1	no	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	#87812	—	2	4	no	—
12	C₄⋊#1039	C₂²	1	1	yes	characteristic
13	#87806	—	2	1	no	—
14	C₄×#1039	—	2	1	no	—
15	C₂×#6029	C₂²	1	2	yes	—
16	#65379	—	5	6	no	maximal
17	C₅²×Q₈	—	2	2	no	—
18	C₅²×Q₈	—	2	2	no	—
19	C₂×C₄×C₅²	—	2	2	no	—
20	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₂×#1039	—	2	1	no	—
24	C₂×#1039	—	2	1	no	—
25	#6045	—	2	1	no	—
26	#6045	D₄	1	2	yes	—
27	#6029	—	2	2	no	—
28	C₅×C₄⋊Q₈	D₅	1	6	yes	—
29	C₂×#689	—	5	12	no	—
30	C₄×Dic₁₀	—	5	6	no	—
31	#3967	—	5	12	no	—
32	C₄⋊#168	—	5	6	no	—
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅²	—	2	1	no	—
35	C₄×C₅²	—	4	2	no	—
36	C₄×C₅²	Q₁₆	1	2	yes	complemented
37	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
38	C₄×C₅²	#41	1	1	yes	characteristic
39	#1039	—	2	2	no	—
40	#1039	—	4	1	no	—
41	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
42	C₂×Dic₁₀	—	5	6	no	—
43	C₅×C₄⋊C₄	—	2	6	no	—
44	C₅×C₄⋊C₄	—	2	6	no	—
45	C₄²×C₅	C₂×D₅	1	6	yes	—
46	#689	—	10	24	no	—
47	C₄⋊Dic₅	—	5	6	no	—
48	#683	—	10	6	no	—
49	C₄×Dic₅	—	10	6	no	—
50	C₂×#168	—	5	12	no	—
51	#462	—	25	1	no	Sylow
52	C₂×C₅²	#121	1	1	yes	characteristic
53	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₅²	#137	1	1	yes	characteristic
55	C₅×Q₈	—	2	12	no	—
56	C₅×Q₈	—	2	12	no	—
57	C₂×C₄×C₅	—	2	12	no	—
58	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
59	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
60	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
61	C₂×Dic₅	—	10	6	no	—
62	C₂×Dic₅	—	10	6	no	—
63	Dic₁₀	—	5	12	no	—
64	Dic₁₀	—	10	6	no	—
65	#168	—	10	12	no	—
66	C₂×Q₁₆	—	25	2	no	—
67	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
68	C₄×Q₈	—	25	1	no	—
69	C₄⋊C₈	—	25	1	no	—
70	#103	—	25	2	no	—
71	C₅²	#462	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
73	C₄×C₅	—	2	6	no	—
74	C₄×C₅	—	4	12	no	—
75	C₄×C₅	#689	1	12	yes	complemented
76	C₄×C₅	D₄×D₅	1	6	yes	—
77	C₄×C₅	#711	1	6	yes	—
78	Dic₅	—	10	12	no	—
79	Dic₅	—	20	6	no	—
80	C₂×Q₈	—	25	1	no	—
81	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
82	Q₁₆	—	50	4	no	—
83	C₂×C₈	—	25	2	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	25	1	no	—
87	C₄⋊C₄	—	50	1	no	—
88	C₄²	—	50	1	no	—
89	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
90	C₂×C₅	#4084	1	6	yes	—
91	C₂×C₅	#4088	1	6	yes	—
92	C₂×C₅	C₂×#689	1	6	yes	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	25	2	no	—
96	Q₈	—	50	1	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	50	1	no	—
99	C₂×C₄	—	50	1	no	—
100	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
102	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
103	C₈	—	50	2	no	—
104	C₅	#65379	1	6	yes	complemented
105	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	4	2	no	—
108	C₄	—	50	2	no	—
109	C₄	—	100	1	no	—
110	C₄	#87812	1	2	yes	complemented
111	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
112	C₄	#87902	1	1	yes	characteristic
113	C₂	#11717258	1	1	yes	characteristic
114	C₂	#11717262	1	1	yes	characteristic
115	C₂	C₂×#87812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#87812

C₂

yes

maximal

C₄×#6045

C₂

yes

maximal, characteristic

#11716702

C₂

yes

maximal

C₄⋊#6029

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

C₂²

yes

—

C₂×#6045

C₂²

yes

characteristic

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

—

C₄²×C₅²

C₂²

yes

characteristic

#87812

—

C₄⋊#1039

C₂²

yes

—

—

C₂²

yes

—

#65379

—

maximal

C₅²×Q₈

—

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

—

—

D₄

yes

—

#6029

—

C₅×C₄⋊Q₈

D₅

yes

—

C₂×#689

—

C₄×Dic₁₀

—

#3967

—

C₄⋊#168

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

Q₁₆

yes

complemented

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₅

yes

—

#689

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

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

—

C₂×Q₁₆

—

C₄⋊Q₈

C₅⋊D₅

yes

characteristic

C₄×Q₈

—

C₄⋊C₈

—

#103

—

C₅²

#462

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#689

yes

complemented

C₄×C₅

D₄×D₅

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

—

Q₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

#4084

yes

—

C₂×C₅

#4088

yes

—

C₂×C₅

C₂×#689

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

101

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

102

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

103

C₈

—

104

C₅

#65379

yes

complemented

105

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

100

—

110

C₄

#87812

yes

complemented

111

C₄

D₄×C₅⋊D₅

yes

characteristic

112

C₄

#87902

yes

characteristic

113

C₂

#11717258

yes

characteristic

114

C₂

#11717262

yes

characteristic

115

C₂

C₂×#87812

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×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#462
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₄⋊Q₈

C₅⋊D₅

hypocenter

C₅²

#462

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
36. C₄×C₅²	Q₁₆	semidirect
71. C₅²	#462	semidirect
75. C₄×C₅	#689	semidirect
104. C₅	#65379	semidirect
110. C₄	#87812	semidirect

H ≅ G/N

type

36. C₄×C₅²

Q₁₆

semidirect

71. C₅²

#462

semidirect

75. C₄×C₅

#689

semidirect

104. C₅

#65379

semidirect

110. C₄

#87812

semidirect

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

subgroup	index
C₅²×C₄⋊Q₈	2
C₂×#87812	2
C₄×#6045	2
#11716702	2
C₄⋊#6029	2
#65379	5

subgroup

index

C₅²×C₄⋊Q₈

Subgroups — #13102955

Conjugacy classes of subgroups

Complements

Maximal subgroups