Group #13102868 — subgroups

Subgroups — #13102868

Order 1600.

600 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 195 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×#6045	C₂	1	1	yes	maximal, characteristic
2	C₅²×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×#87804	C₂	1	1	yes	maximal, characteristic
4	#11716677	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₂²×#1039	—	2	1	no	—
7	C₂×#6045	—	2	1	no	—
8	C₂×#6045	—	2	2	no	—
9	C₂×#6045	C₄	1	4	yes	complemented
10	C₂×#6045	C₂²	1	2	yes	—
11	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
12	C₅²×C₂²⋊C₄	—	2	1	no	—
13	C₄²×C₅²	—	2	1	no	—
14	#87804	—	2	2	no	—
15	C₂×#6029	C₂²	1	2	yes	—
16	#65293	—	5	6	no	maximal
17	C₂³×C₅²	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅²	D₄	1	2	yes	—
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₂×#1039	—	2	4	no	—
24	C₂×#1039	—	2	2	no	—
25	#6045	—	2	4	no	—
26	#6045	—	2	2	no	—
27	#6045	C₂×C₄	1	4	yes	—
28	#6029	—	2	2	no	—
29	C₂²×Dic₁₀	—	5	6	no	—
30	C₅×#117	D₅	1	6	yes	—
31	C₂×#681	—	5	6	no	—
32	#3942	—	5	24	no	—
33	C₂²×C₅²	—	2	2	no	—
34	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
36	C₄×C₅²	—	4	2	no	—
37	C₄×C₅²	C₂²⋊C₄	1	2	yes	—
38	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
40	#1039	—	2	4	no	—
41	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
42	C₂²×Dic₅	—	10	6	no	—
43	C₂×Dic₁₀	—	5	24	no	—
44	C₂×Dic₁₀	—	5	12	no	—
45	C₂×Dic₁₀	—	10	12	no	—
46	C₂×Dic₁₀	—	10	6	no	—
47	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
48	C₅×C₂²⋊C₄	—	2	6	no	—
49	C₄²×C₅	—	2	6	no	—
50	#681	—	10	12	no	—
51	C₂×#168	—	5	12	no	—
52	#419	—	25	1	no	Sylow
53	C₂×C₅²	—	2	2	no	—
54	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₅²	#137	1	2	yes	—
56	C₂³×C₅	C₂²⋊D₅	1	6	yes	—
57	C₂×C₄×C₅	—	2	12	no	—
58	C₂×C₄×C₅	C₄×D₅	1	12	yes	—
59	C₂×C₄×C₅	D₂₀	1	12	yes	—
60	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
61	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
62	C₂×Dic₅	—	10	24	no	—
63	C₂×Dic₅	—	10	12	no	—
64	Dic₁₀	—	5	24	no	—
65	Dic₁₀	—	10	24	no	—
66	Dic₁₀	—	10	12	no	—
67	#168	—	10	12	no	—
68	C₂²×Q₈	—	25	1	no	—
69	C₂×M₁₆	—	25	1	no	—
70	#117	C₅⋊D₅	1	1	yes	characteristic
71	#103	—	25	4	no	—
72	C₅²	#419	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₅	—	2	12	no	—
74	C₂²×C₅	#685	1	12	yes	—
75	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
76	C₄×C₅	—	4	12	no	—
77	C₄×C₅	#685	1	12	yes	—
78	C₄×C₅	C₂×C₄×D₅	1	6	yes	—
79	C₄×C₅	C₂×D₂₀	1	6	yes	—
80	Dic₅	—	10	24	no	—
81	C₂×Q₈	—	25	4	no	—
82	C₂×Q₈	—	25	2	no	—
83	C₂×Q₈	—	50	2	no	—
84	C₂×Q₈	—	50	1	no	—
85	C₂²×C₄	—	50	1	no	—
86	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
87	M₁₆	—	50	2	no	—
88	C₂×C₈	—	25	2	no	—
89	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
90	C₂²⋊C₄	—	2	1	no	—
91	C₄²	—	2	1	no	—
92	C₂×C₅	—	2	12	no	—
93	C₂×C₅	C₂×#685	1	6	yes	—
94	C₂×C₅	#4097	1	12	yes	—
95	C₂³	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
96	Q₈	—	25	4	no	—
97	Q₈	—	50	4	no	—
98	Q₈	—	50	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	50	4	no	—
101	C₂×C₄	—	50	2	no	—
102	C₂×C₄	C₄×C₅⋊D₅	1	2	yes	—
103	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
104	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
106	C₈	—	50	2	no	—
107	C₅	#65293	1	6	yes	complemented
108	C₂²	—	2	2	no	—
109	C₂²	#87808	1	2	yes	—
110	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
111	C₄	—	4	2	no	—
112	C₄	—	50	4	no	—
113	C₄	#87808	1	2	yes	—
114	C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
115	C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
116	C₂	—	2	2	no	—
117	C₂	C₂×#87808	1	1	yes	characteristic
118	C₂	#11717271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6045

C₂

yes

maximal, characteristic

C₅²×#117

C₂

yes

maximal, characteristic

C₂×#87804

C₂

yes

maximal, characteristic

#11716677

C₂

yes

maximal, complemented

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂²×#1039

—

C₂×#6045

—

C₂×#6045

—

C₂×#6045

C₄

yes

complemented

C₂×#6045

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₅²×C₂²⋊C₄

—

C₄²×C₅²

—

#87804

—

C₂×#6029

C₂²

yes

—

#65293

—

maximal

C₂³×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

—

—

—

—

C₂×C₄

yes

—

#6029

—

C₂²×Dic₁₀

—

C₅×#117

D₅

yes

—

C₂×#681

—

#3942

—

C₂²×C₅²

—

C₂²×C₅²

C₂²⋊C₄

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂²⋊C₄

yes

—

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

—

—

—

C₂×C₅²

—

C₂×C₅²

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₅²

#137

yes

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

#168

—

C₂²×Q₈

—

C₂×M₁₆

—

#117

C₅⋊D₅

yes

characteristic

#103

—

C₅²

#419

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

#685

yes

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

#685

yes

—

C₄×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

C₂×D₂₀

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₂²⋊C₄

—

C₄²

—

C₂×C₅

—

C₂×C₅

C₂×#685

yes

—

C₂×C₅

#4097

yes

—

C₂³

C₂²⋊(C₅⋊D₅)

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₄×C₅⋊D₅

yes

—

103

C₂×C₄

C₄⋊(C₅⋊D₅)

yes

—

104

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

105

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

106

C₈

—

107

C₅

#65293

yes

complemented

108

C₂²

—

109

C₂²

#87808

yes

—

110

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#87808

yes

—

114

C₄

C₂×C₄×C₅⋊D₅

yes

characteristic

115

C₄

C₂×C₄⋊(C₅⋊D₅)

yes

characteristic

116

C₂

—

117

C₂

C₂×#87808

yes

characteristic

118

C₂

#11717271

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#117	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂³×C₅²	D₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	#117	C₅⋊D₅
hypocenter	C₅²	#419
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#117

C₂

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂³×C₅²

D₄

cosocle

C₂×C₄×C₅²

C₂³

C₅⋊D₅

C₅²

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

N	H ≅ G/N	type
4. #11716677	C₂	semidirect
9. C₂×#6045	C₄	semidirect
72. C₅²	#419	semidirect
107. C₅	#65293	semidirect

H ≅ G/N

type

4. #11716677

C₂

semidirect

9. C₂×#6045

C₄

semidirect

72. C₅²

#419

semidirect

107. C₅

#65293

semidirect

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

subgroup	index
C₂²×#6045	2
C₅²×#117	2
C₂×#87804	2
#11716677	2
#65293	5

subgroup

index

C₂²×#6045

C₅²×#117

C₂×#87804

#11716677

#65293

Subgroups — #13102868

Conjugacy classes of subgroups

Complements

Maximal subgroups