Group #13104829 — subgroups

Subgroups — #13104829

Order 1600.

336 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₂×#87867	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6034	C₂	1	1	yes	maximal, characteristic
3	#11717197	C₂	1	1	yes	maximal, characteristic
4	#11716895	C₂	1	2	yes	maximal, complemented
5	C₄×#6034	C₂	1	2	yes	maximal, complemented
6	C₂²×#1039	C₂²	1	1	yes	characteristic
7	#87867	C₄	1	4	yes	complemented
8	C₂×#6034	C₄	1	4	yes	complemented
9	C₂×#6034	C₂²	1	1	yes	characteristic
10	C₂×#6034	C₂²	1	2	yes	—
11	C₂×#6034	C₂²	1	1	yes	characteristic
12	C₅²×C₂²⋊C₄	C₄	1	1	yes	characteristic
13	C₂²⋊#1039	C₄	1	1	yes	characteristic
14	#87806	C₄	1	2	yes	—
15	C₄×#1039	C₂²	1	2	yes	—
16	#65691	—	5	2	no	maximal
17	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
19	C₂×#1039	C₂×C₄	1	2	yes	—
20	C₂×#1039	C₂×C₄	1	1	yes	characteristic
21	C₂×#1039	C₂×C₄	1	2	yes	—
22	C₂×#1039	C₂×C₄	1	2	yes	—
23	C₂×#1039	C₂³	1	1	yes	characteristic
24	#6034	C₂×C₄	1	2	yes	—
25	#6034	C₂×C₄	1	2	yes	—
26	#6034	C₂×C₄	1	4	yes	—
27	C₂×#704	—	5	2	no	—
28	C₂²×#170	—	5	2	no	—
29	#4023	—	5	2	no	—
30	#4023	—	10	2	no	—
31	#4002	—	5	4	no	—
32	C₄×#170	—	5	4	no	—
33	C₂²×C₅²	—	2	2	no	—
34	C₂²×C₅²	C₄²	1	2	yes	—
35	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₅²	—	2	2	no	—
37	#1039	—	2	2	no	—
38	#1039	C₄²	1	2	yes	—
39	#1039	C₂²×C₄	1	1	yes	characteristic
40	#1039	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅	—	5	2	no	—
42	C₂²×Dic₅	—	10	2	no	—
43	#704	—	5	8	no	—
44	C₂×#170	—	5	8	no	—
45	C₂×#170	—	5	2	no	—
46	C₂×#170	—	5	2	no	—
47	C₂×#170	—	5	4	no	—
48	C₅×C₂²⋊C₄	—	2	2	no	—
49	C₅×C₂²⋊C₄	AGL(1,𝔽₅)	1	2	yes	—
50	C₂²⋊Dic₅	—	5	2	no	—
51	C₂²⋊Dic₅	—	10	2	no	—
52	#683	—	5	4	no	—
53	#683	—	10	4	no	—
54	C₄×Dic₅	—	5	4	no	—
55	C₄×Dic₅	—	10	4	no	—
56	#405	—	25	1	no	Sylow
57	C₂×C₅²	—	2	2	no	—
58	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₅²	#131	1	2	yes	—
60	C₂³×C₅	—	2	2	no	—
61	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
62	C₂×C₄×C₅	—	2	4	no	—
63	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	4	yes	—
64	C₂×Dic₅	—	5	4	no	—
65	C₂×Dic₅	—	5	4	no	—
66	C₂×Dic₅	—	5	2	no	—
67	C₂×Dic₅	—	5	2	no	—
68	C₂×Dic₅	—	5	4	no	—
69	C₂×Dic₅	—	10	2	no	—
70	C₂×Dic₅	—	10	4	no	—
71	C₂×Dic₅	—	10	4	no	—
72	C₂×Dic₅	—	10	2	no	—
73	C₂×Dic₅	—	10	4	no	—
74	#170	—	5	4	no	—
75	#170	—	5	4	no	—
76	#170	—	5	8	no	—
77	C₂×M₁₆	—	25	1	no	—
78	C₂²×C₈	—	25	1	no	—
79	#117	—	25	1	no	—
80	#97	—	25	2	no	—
81	C₄×C₈	—	25	2	no	—
82	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₅	—	2	4	no	—
84	C₂²×C₅	—	2	4	no	—
85	C₂²×C₅	—	2	2	no	—
86	C₂²×C₅	—	4	4	no	—
87	C₂²×C₅	C₄×AGL(1,𝔽₅)	1	4	yes	—
88	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
89	C₄×C₅	—	2	4	no	—
90	C₄×C₅	—	4	4	no	—
91	Dic₅	—	5	4	no	—
92	Dic₅	—	5	2	no	—
93	Dic₅	—	5	2	no	—
94	Dic₅	—	10	4	no	—
95	Dic₅	—	10	2	no	—
96	Dic₅	—	10	2	no	—
97	Dic₅	—	10	4	no	—
98	Dic₅	—	20	4	no	—
99	C₂²×C₄	—	25	1	no	—
100	M₁₆	—	25	4	no	—
101	C₂×C₈	—	25	1	no	—
102	C₂×C₈	—	25	2	no	—
103	C₂×C₈	—	25	1	no	—
104	C₂×C₈	—	25	4	no	—
105	C₄⋊C₄	—	25	2	no	—
106	C₂²⋊C₄	—	25	1	no	—
107	C₂²⋊C₄	#1044	1	1	yes	characteristic
108	C₄²	—	25	2	no	—
109	C₂×C₅	—	2	4	no	—
110	C₂×C₅	—	2	2	no	—
111	C₂×C₅	—	2	4	no	—
112	C₂×C₅	—	4	4	no	—
113	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
114	C₂×C₅	#4131	1	4	yes	—
115	C₂³	C₂×#1044	1	1	yes	characteristic
116	C₂×C₄	—	25	2	no	—
117	C₂×C₄	—	25	2	no	—
118	C₂×C₄	—	25	1	no	—
119	C₂×C₄	—	25	1	no	—
120	C₂×C₄	—	25	2	no	—
121	C₂×C₄	C₂×#1044	1	2	yes	—
122	C₈	—	25	2	no	—
123	C₈	—	25	2	no	—
124	C₈	—	25	4	no	—
125	C₅	—	2	2	no	—
126	C₅	#65691	1	2	yes	complemented
127	C₂²	—	2	2	no	—
128	C₂²	C₄×#1044	1	2	yes	—
129	C₂²	C₂²×#1044	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	25	1	no	—
132	C₄	—	25	1	no	—
133	C₄	—	25	2	no	—
134	C₄	—	50	2	no	—
135	C₂	—	2	2	no	—
136	C₂	C₂×C₄×#1044	1	1	yes	characteristic
137	C₂	#11717452	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87867

C₂

yes

maximal, characteristic

C₂²×#6034

C₂

yes

maximal, characteristic

#11717197

C₂

yes

maximal, characteristic

#11716895

C₂

yes

maximal, complemented

C₄×#6034

C₂

yes

maximal, complemented

C₂²×#1039

C₂²

yes

characteristic

#87867

C₄

yes

complemented

C₂×#6034

C₄

yes

complemented

C₂×#6034

C₂²

yes

characteristic

C₂×#6034

C₂²

yes

—

C₂×#6034

C₂²

yes

characteristic

C₅²×C₂²⋊C₄

C₄

yes

characteristic

C₂²⋊#1039

C₄

yes

characteristic

#87806

C₄

yes

—

C₄×#1039

C₂²

yes

—

#65691

—

maximal

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂³

yes

characteristic

#6034

C₂×C₄

yes

—

#6034

C₂×C₄

yes

—

#6034

C₂×C₄

yes

—

C₂×#704

—

C₂²×#170

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

C₄²

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

—

#1039

—

#1039

C₄²

yes

—

#1039

C₂²×C₄

yes

characteristic

#1039

C₂²×C₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

—

—

—

—

—

C₅×C₂²⋊C₄

—

C₅×C₂²⋊C₄

AGL(1,𝔽₅)

yes

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

#405

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄²

yes

characteristic

C₂×C₅²

#131

yes

—

C₂³×C₅

—

C₂³×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

#170

—

#170

—

#170

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

#97

—

C₄×C₈

—

C₅²

#405

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₄×AGL(1,𝔽₅)

yes

—

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

#1044

yes

characteristic

108

C₄²

—

109

C₂×C₅

—

110

C₂×C₅

—

111

C₂×C₅

—

112

C₂×C₅

—

113

C₂×C₅

C₂×C₄×AGL(1,𝔽₅)

yes

—

114

C₂×C₅

#4131

yes

—

115

C₂³

C₂×#1044

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×#1044

yes

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₅

—

126

C₅

#65691

yes

complemented

127

C₂²

—

128

C₂²

C₄×#1044

yes

—

129

C₂²

C₂²×#1044

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₂

—

136

C₂

C₂×C₄×#1044

yes

characteristic

137

C₂

#11717452

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1044
commutator	C₂×C₅²	C₂×C₄²
Fitting	C₅²×C₂²⋊C₄	C₄
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₂²⋊C₄	#1044
hypocenter	C₅²	#405
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#1044

commutator

C₂×C₅²

C₂×C₄²

Fitting

C₅²×C₂²⋊C₄

C₄

socle

C₂²×C₅²

C₂²×C₄

prime radical

C₂³×C₅²

C₂×C₄

C₂³

C₂²⋊C₄

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂²⋊C₄

N	H ≅ G/N	type
4. #11716895	C₂	semidirect
5. C₄×#6034	C₂	semidirect
7. #87867	C₄	semidirect
8. C₂×#6034	C₄	semidirect
82. C₅²	#405	semidirect
126. C₅	#65691	semidirect

H ≅ G/N

type

4. #11716895

C₂

semidirect

5. C₄×#6034

C₂

semidirect

7. #87867

C₄

semidirect

8. C₂×#6034

C₄

semidirect

82. C₅²

#405

semidirect

126. C₅

#65691

semidirect

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

subgroup	index
C₂×#87867	2
C₂²×#6034	2
#11717197	2
#11716895	2
C₄×#6034	2
#65691	5

subgroup

index

C₂²×#6034

Subgroups — #13104829

Conjugacy classes of subgroups

Complements

Maximal subgroups