Group #13104926 — subgroups

Subgroups — #13104926

Order 1600.

258 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 70 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	Q₈×#1039	C₂	1	1	yes	maximal, characteristic
2	#11716897	C₂	1	2	yes	maximal
3	#11716895	C₂	1	2	yes	maximal
4	C₄⋊#6034	C₂	1	1	yes	maximal, characteristic
5	C₄×#6034	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅²×Q₈	C₄	1	1	yes	characteristic
7	C₂×#6034	C₂²	1	1	yes	characteristic
8	C₂×#6034	C₂²	1	2	yes	—
9	C₂×#6034	C₂²	1	1	yes	characteristic
10	C₄⋊#1039	C₄	1	2	yes	—
11	C₄⋊#1039	C₄	1	1	yes	characteristic
12	C₄×#1039	C₂²	1	1	yes	characteristic
13	C₄×#1039	C₂²	1	2	yes	—
14	#65788	—	5	2	no	maximal
15	C₅²×Q₈	—	2	2	no	—
16	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	1	yes	characteristic
19	C₂×#1039	C₂×C₄	1	2	yes	—
20	C₂×#1039	C₂³	1	1	yes	characteristic
21	#6034	—	2	1	no	—
22	#6034	—	2	2	no	—
23	#6034	Q₈	1	2	yes	complemented
24	Q₈×Dic₅	—	5	2	no	—
25	Q₈×Dic₅	—	10	2	no	—
26	#4004	—	5	4	no	—
27	#4002	—	5	4	no	—
28	C₄⋊#170	—	5	2	no	—
29	C₄×#170	—	5	2	no	—
30	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₅²	—	2	2	no	—
32	C₄×C₅²	M₁₆	1	2	yes	—
33	#1039	—	2	1	no	—
34	#1039	—	2	2	no	—
35	#1039	C₂×Q₈	1	1	yes	characteristic
36	#1039	#41	1	1	yes	characteristic
37	C₂×C₅×Q₈	—	2	2	no	—
38	C₂×C₅×Q₈	AGL(1,𝔽₅)	1	2	yes	—
39	C₂×#170	—	5	2	no	—
40	C₂×#170	—	5	2	no	—
41	C₂×#170	—	5	4	no	—
42	C₄⋊Dic₅	—	5	4	no	—
43	C₄⋊Dic₅	—	5	2	no	—
44	C₄⋊Dic₅	—	10	4	no	—
45	C₄⋊Dic₅	—	10	2	no	—
46	C₄×Dic₅	—	5	2	no	—
47	C₄×Dic₅	—	5	4	no	—
48	C₄×Dic₅	—	10	2	no	—
49	C₄×Dic₅	—	10	4	no	—
50	#446	—	25	1	no	Sylow
51	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
52	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₅²	#131	1	1	yes	characteristic
54	C₅×Q₈	—	2	8	no	—
55	C₅×Q₈	—	2	4	no	—
56	C₂×C₄×C₅	—	2	4	no	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	4	yes	—
59	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
60	C₂×Dic₅	—	5	2	no	—
61	C₂×Dic₅	—	5	2	no	—
62	C₂×Dic₅	—	5	4	no	—
63	C₂×Dic₅	—	10	2	no	—
64	C₂×Dic₅	—	10	2	no	—
65	C₂×Dic₅	—	10	4	no	—
66	#170	—	5	4	no	—
67	#170	—	10	2	no	—
68	#170	—	10	4	no	—
69	C₄×Q₈	—	25	1	no	—
70	C₄⋊C₈	—	25	1	no	—
71	C₄⋊C₈	—	25	2	no	—
72	#97	—	25	2	no	—
73	C₄×C₈	—	25	1	no	—
74	C₅²	#446	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₅	—	2	2	no	—
76	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
77	C₄×C₅	—	2	8	no	—
78	C₄×C₅	—	2	4	no	—
79	C₄×C₅	—	2	4	no	—
80	C₄×C₅	#704	1	4	yes	—
81	Dic₅	—	5	2	no	—
82	Dic₅	—	5	2	no	—
83	Dic₅	—	10	2	no	—
84	Dic₅	—	10	2	no	—
85	Dic₅	—	10	2	no	—
86	Dic₅	—	10	4	no	—
87	Dic₅	—	20	2	no	—
88	Dic₅	—	20	4	no	—
89	C₂×Q₈	#1044	1	1	yes	characteristic
90	C₂×C₈	—	25	1	no	—
91	C₂×C₈	—	25	1	no	—
92	C₂×C₈	—	25	2	no	—
93	C₄⋊C₄	—	25	2	no	—
94	C₄⋊C₄	—	25	1	no	—
95	C₄²	—	25	1	no	—
96	C₄²	—	25	2	no	—
97	C₂×C₅	—	2	2	no	—
98	C₂×C₅	—	2	2	no	—
99	C₂×C₅	—	2	2	no	—
100	C₂×C₅	#4133	1	2	yes	—
101	C₂×C₅	Q₈×AGL(1,𝔽₅)	1	2	yes	—
102	C₂×C₅	C₂×#704	1	2	yes	—
103	Q₈	—	2	2	no	—
104	C₂×C₄	—	25	1	no	—
105	C₂×C₄	—	25	1	no	—
106	C₂×C₄	—	25	2	no	—
107	C₂×C₄	C₂×#1044	1	2	yes	—
108	C₂×C₄	C₂×#1044	1	1	yes	characteristic
109	C₈	—	25	2	no	—
110	C₈	—	50	1	no	—
111	C₈	—	50	2	no	—
112	C₅	—	2	2	no	—
113	C₅	#65788	1	2	yes	complemented
114	C₂²	C₂²×#1044	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	25	1	no	—
117	C₄	—	25	1	no	—
118	C₄	—	50	1	no	—
119	C₄	—	50	2	no	—
120	C₄	#87867	1	2	yes	—
121	C₂	#11717454	1	1	yes	characteristic
122	C₂	Q₈×#1044	1	1	yes	characteristic
123	C₂	C₂×#87867	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#1039

C₂

yes

maximal, characteristic

#11716897

C₂

yes

maximal

#11716895

C₂

yes

maximal

C₄⋊#6034

C₂

yes

maximal, characteristic

C₄×#6034

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

C₄

yes

characteristic

C₂×#6034

C₂²

yes

characteristic

C₂×#6034

C₂²

yes

—

C₂×#6034

C₂²

yes

characteristic

C₄⋊#1039

C₄

yes

—

C₄⋊#1039

C₄

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

—

#65788

—

maximal

C₅²×Q₈

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂³

yes

—

—

Q₈

yes

complemented

Q₈×Dic₅

—

Q₈×Dic₅

—

—

—

—

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

M₁₆

yes

—

#1039

—

#1039

—

#1039

C₂×Q₈

yes

yes

C₂×C₅×Q₈

—

C₂×C₅×Q₈

AGL(1,𝔽₅)

yes

—

C₂×#170

—

C₂×#170

—

C₂×#170

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#446

—

Sylow

C₂×C₅²

C₄×Q₈

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

characteristic

C₂×C₅²

#131

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

#170

—

#170

—

#170

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₅²

#446

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#704

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

#1044

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

100

C₂×C₅

#4133

yes

—

101

C₂×C₅

Q₈×AGL(1,𝔽₅)

yes

—

102

C₂×C₅

C₂×#704

yes

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×#1044

yes

—

108

C₂×C₄

C₂×#1044

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₅

—

113

C₅

#65788

yes

complemented

114

C₂²

C₂²×#1044

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#87867

yes

—

121

C₂

#11717454

yes

characteristic

122

C₂

Q₈×#1044

yes

characteristic

123

C₂

C₂×#87867

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1044
commutator, socle, prime radical	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₅²×Q₈	C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₂×Q₈	#1044
hypocenter	C₅²	#446
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂²×#1044

commutator, socle, prime radical

C₂²×C₅²

C₂²×C₄

Fitting

C₂×C₅²×Q₈

C₄

C₂³

C₂×Q₈

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
23. #6034	Q₈	semidirect
74. C₅²	#446	semidirect
113. C₅	#65788	semidirect

H ≅ G/N

type

23. #6034

Q₈

semidirect

74. C₅²

#446

semidirect

113. C₅

#65788

semidirect

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

subgroup	index
Q₈×#1039	2
#11716897	2
#11716895	2
C₄⋊#6034	2
C₄×#6034	2
#65788	5

subgroup

index

Subgroups — #13104926

Conjugacy classes of subgroups

Complements

Maximal subgroups