Group #13104154 — subgroups

Subgroups — #13104154

Order 1600.

232 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 37 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	#11717389	C₂	1	1	yes	maximal, characteristic
2	#11717385	C₂	1	2	yes	maximal, complemented
3	#11717380	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716914	C₂	1	2	yes	maximal, complemented
5	C₈×#1044	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅²⋊C₂	—	2	2	no	—
7	#87877	C₂²	1	2	yes	—
8	#87872	C₂²	1	2	yes	—
9	C₄×#1044	C₂²	1	1	yes	characteristic
10	#87862	C₂²	1	1	yes	characteristic
11	#87838	—	2	4	no	—
12	#87836	—	2	2	no	—
13	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
14	C₅×#677	—	2	2	no	—
15	#87772	—	2	2	no	—
16	C₂×D₅²	—	2	2	no	—
17	C₂×#1044	D₄	1	2	yes	—
18	D₅²⋊C₂	—	4	4	no	—
19	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
20	C₅×D₂₀	—	2	2	no	—
21	C₅×Dic₁₀	—	2	2	no	—
22	#6039	—	2	2	no	—
23	#6037	—	2	2	no	—
24	#6036	—	2	2	no	—
25	#6034	D₄	1	2	yes	complemented
26	C₅²×C₈	D₄	1	1	yes	characteristic, complemented
27	#6029	D₄	1	1	yes	characteristic, complemented
28	D₅×SD₁₆	—	10	2	no	—
29	C₈×AGL(1,𝔽₅)	—	10	1	no	—
30	C₂×C₅⋊D₅	C₂×D₄	1	1	yes	characteristic
31	C₂×C₅×D₅	—	4	2	no	—
32	D₅²	—	4	4	no	—
33	#1044	SD₁₆	1	4	yes	complemented
34	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
35	#1039	C₂×D₄	1	1	yes	characteristic
36	C₅×Dic₅	—	4	2	no	—
37	Q₈×D₅	—	10	2	no	—
38	D₄×D₅	—	10	2	no	—
39	C₄×AGL(1,𝔽₅)	—	10	1	no	—
40	D₅⋊C₈	—	10	1	no	—
41	C₅×SD₁₆	—	10	2	no	—
42	#688	—	10	2	no	—
43	#687	—	10	2	no	—
44	#677	—	2	2	no	—
45	C₈×D₅	—	10	1	no	—
46	C₈×D₅	—	10	2	no	—
47	#492	—	25	1	no	maximal, Sylow
48	C₂×C₅²	#127	1	1	yes	characteristic
49	C₅⋊D₅	C₂×SD₁₆	1	2	yes	—
50	C₅×D₅	—	8	2	no	—
51	C₂²×D₅	—	20	2	no	—
52	C₂×AGL(1,𝔽₅)	—	10	2	no	—
53	C₅×Q₈	—	10	2	no	—
54	C₅×D₄	—	10	2	no	—
55	C₂²⋊D₅	—	20	2	no	—
56	D₂₀	—	2	2	no	—
57	C₄×D₅	—	10	1	no	—
58	C₄×D₅	—	10	2	no	—
59	C₄×D₅	—	20	2	no	—
60	Dic₁₀	—	2	2	no	—
61	Dic₁₀	—	20	2	no	—
62	#170	—	10	2	no	—
63	C₅×C₈	—	2	1	no	—
64	C₅×C₈	—	2	2	no	—
65	#168	—	10	1	no	—
66	#168	—	10	2	no	—
67	C₂×SD₁₆	—	25	2	no	—
68	C₂×SD₁₆	—	25	2	no	—
69	C₄⋊Q₈	—	25	1	no	—
70	#127	—	25	1	no	—
71	C₄×C₈	—	25	1	no	—
72	C₅²	#492	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₅	—	20	2	no	—
74	C₂×D₅	—	4	2	no	—
75	C₂×D₅	—	10	2	no	—
76	C₂×D₅	—	10	1	no	—
77	C₂×D₅	—	40	4	no	—
78	AGL(1,𝔽₅)	—	10	4	no	—
79	C₄×C₅	—	2	2	no	—
80	C₄×C₅	—	2	1	no	—
81	C₄×C₅	—	20	2	no	—
82	Dic₅	—	4	2	no	—
83	Dic₅	—	10	1	no	—
84	Dic₅	—	10	2	no	—
85	C₂×Q₈	—	25	2	no	—
86	C₂×D₄	—	25	2	no	—
87	C₂×D₄	—	50	2	no	—
88	SD₁₆	—	10	2	no	—
89	SD₁₆	—	50	2	no	—
90	SD₁₆	—	50	4	no	—
91	C₂×C₈	—	25	1	no	—
92	C₂×C₈	—	25	1	no	—
93	C₄⋊C₄	—	50	2	no	—
94	C₄²	—	25	1	no	—
95	C₂×C₅	—	2	2	no	—
96	C₂×C₅	—	2	1	no	—
97	C₂×C₅	—	40	2	no	—
98	D₅	—	8	2	no	—
99	D₅	—	10	2	no	—
100	D₅	—	10	4	no	—
101	C₂³	—	50	2	no	—
102	Q₈	—	10	2	no	—
103	Q₈	—	50	2	no	—
104	D₄	—	10	2	no	—
105	D₄	—	50	2	no	—
106	D₄	—	100	4	no	—
107	C₂×C₄	—	25	2	no	—
108	C₂×C₄	—	25	1	no	—
109	C₂×C₄	—	50	2	no	—
110	C₈	—	25	2	no	—
111	C₈	—	25	1	no	—
112	C₈	D₅²⋊C₂	1	1	yes	characteristic, complemented
113	C₅	—	2	2	no	—
114	C₅	—	2	1	no	—
115	C₂²	—	20	2	no	—
116	C₂²	—	25	1	no	—
117	C₂²	—	100	4	no	—
118	C₄	—	20	2	no	—
119	C₄	—	25	4	no	—
120	C₄	—	25	1	no	—
121	C₄	C₂×D₅²⋊C₂	1	1	yes	characteristic
122	C₂	—	25	2	no	—
123	C₂	—	40	2	no	—
124	C₂	#11717389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717389

C₂

yes

maximal, characteristic

#11717385

C₂

yes

maximal, complemented

#11717380

C₂

yes

maximal, characteristic, complemented

#11716914

C₂

yes

maximal, complemented

C₈×#1044

C₂

yes

maximal, characteristic, complemented

C₂×D₅²⋊C₂

—

#87877

C₂²

yes

—

#87872

C₂²

yes

—

C₄×#1044

C₂²

yes

characteristic

#87862

C₂²

yes

characteristic

#87838

—

#87836

—

C₈×C₅⋊D₅

C₂²

yes

characteristic

C₅×#677

—

#87772

—

C₂×D₅²

—

C₂×#1044

D₄

yes

—

D₅²⋊C₂

—

C₄×C₅⋊D₅

C₂³

yes

characteristic

C₅×D₂₀

—

C₅×Dic₁₀

—

—

—

—

D₄

yes

complemented

C₅²×C₈

D₄

yes

characteristic, complemented

#6029

D₄

yes

characteristic, complemented

D₅×SD₁₆

—

C₈×AGL(1,𝔽₅)

—

C₂×C₅⋊D₅

C₂×D₄

yes

characteristic

C₂×C₅×D₅

—

D₅²

—

#1044

SD₁₆

yes

complemented

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

C₂×D₄

yes

characteristic

C₅×Dic₅

—

Q₈×D₅

—

D₄×D₅

—

C₄×AGL(1,𝔽₅)

—

D₅⋊C₈

—

C₅×SD₁₆

—

#688

—

#687

—

#677

—

C₈×D₅

—

C₈×D₅

—

—

C₂×C₅²

yes

C₅⋊D₅

C₂×SD₁₆

yes

—

C₅×D₅

—

C₂²×D₅

—

C₂×AGL(1,𝔽₅)

—

C₅×Q₈

—

C₅×D₄

—

C₂²⋊D₅

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#127

—

C₄×C₈

—

C₅²

#492

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

D₅

—

D₅

—

100

D₅

—

101

C₂³

—

102

Q₈

—

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

100

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₈

—

111

C₈

—

112

C₈

D₅²⋊C₂

yes

characteristic, complemented

113

C₅

—

114

C₅

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

100

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×D₅²⋊C₂

yes

characteristic

122

C₂

—

123

C₂

—

124

C₂

#11717389

yes

characteristic

	subgroup	G/H
center	C₂	#11717389
commutator, cosocle	C₄×C₅⋊D₅	C₂³
Frattini	C₄	C₂×D₅²⋊C₂
Fitting	C₅²×C₈	D₄
socle	C₂×C₅²	#127
prime radical	#11717389	C₂
hypercenter	C₈	D₅²⋊C₂
hypocenter	C₅²	#492
LCS	G ▷ C₄×C₅⋊D₅ ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

subgroup

G/H

center

C₂

#11717389

commutator, cosocle

C₄×C₅⋊D₅

C₂³

Frattini

C₄

C₂×D₅²⋊C₂

Fitting

C₅²×C₈

D₄

C₂×C₅²

C₂

C₈

D₅²⋊C₂

hypocenter

C₅²

#492

LCS

G ▷ C₄×C₅⋊D₅ ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
2. #11717385	C₂	semidirect
3. #11717380	C₂	semidirect
4. #11716914	C₂	semidirect
5. C₈×#1044	C₂	semidirect
25. #6034	D₄	semidirect
26. C₅²×C₈	D₄	semidirect
27. #6029	D₄	semidirect
33. #1044	SD₁₆	semidirect
72. C₅²	#492	semidirect
112. C₈	D₅²⋊C₂	semidirect

H ≅ G/N

type

2. #11717385

C₂

semidirect

3. #11717380

C₂

semidirect

4. #11716914

C₂

semidirect

5. C₈×#1044

C₂

semidirect

25. #6034

D₄

semidirect

26. C₅²×C₈

D₄

semidirect

27. #6029

D₄

semidirect

33. #1044

SD₁₆

semidirect

72. C₅²

#492

semidirect

112. C₈

D₅²⋊C₂

semidirect

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

subgroup	index
#11717389	2
#11717385	2
#11717380	2
#11716914	2
C₈×#1044	2
#492	25

subgroup

index

Subgroups — #13104154

Conjugacy classes of subgroups

Complements

Maximal subgroups