Group #13104912 — subgroups

Subgroups — #13104912

Order 1600.

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

normal subgroups only

This group has 338 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	C₂×#87867	C₂	1	2	yes	maximal, complemented
2	D₄×#1039	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#6034	C₂	1	2	yes	maximal, complemented
4	C₄⋊#6034	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6034	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×D₄	C₄	1	1	yes	characteristic
7	C₂²×#1039	C₂²	1	2	yes	—
8	#87867	—	2	4	no	—
9	C₂×#6034	C₂²	1	1	yes	characteristic
10	C₂×#6034	C₂²	1	2	yes	—
11	C₂×#6034	C₂²	1	1	yes	characteristic
12	C₂²⋊#1039	C₄	1	2	yes	—
13	C₄⋊#1039	C₄	1	1	yes	characteristic
14	C₄×#1039	C₂²	1	1	yes	characteristic
15	#65774	—	5	2	no	maximal
16	C₂³×C₅²	C₂×C₄	1	2	yes	—
17	C₅²×D₄	—	2	2	no	—
18	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×#1039	—	2	2	no	—
20	C₂×#1039	—	2	2	no	—
21	C₂×#1039	C₂×C₄	1	1	yes	characteristic
22	C₂×#1039	C₂×C₄	1	2	yes	—
23	C₂×#1039	C₂³	1	1	yes	characteristic
24	#6034	—	2	2	no	—
25	#6034	—	2	1	no	—
26	#6034	D₄	1	2	yes	complemented
27	C₂×#704	—	5	4	no	—
28	D₄×Dic₅	—	5	2	no	—
29	D₄×Dic₅	—	10	2	no	—
30	C₂²⋊#170	—	5	4	no	—
31	C₄⋊#170	—	5	2	no	—
32	C₄×#170	—	5	2	no	—
33	C₂²×C₅²	—	2	2	no	—
34	C₂²×C₅²	—	2	2	no	—
35	C₂²×C₅²	—	2	2	no	—
36	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅²	M₁₆	1	2	yes	complemented
38	#1039	—	2	2	no	—
39	#1039	—	2	1	no	—
40	#1039	C₂×D₄	1	1	yes	characteristic
41	#1039	#41	1	1	yes	characteristic
42	C₂×C₅×D₄	—	2	2	no	—
43	C₂×C₅×D₄	AGL(1,𝔽₅)	1	2	yes	—
44	C₂²×Dic₅	—	5	4	no	—
45	C₂²×Dic₅	—	10	4	no	—
46	#704	—	10	8	no	—
47	C₂×#170	—	5	2	no	—
48	C₂×#170	—	5	4	no	—
49	C₂×#170	—	5	2	no	—
50	C₂²⋊Dic₅	—	5	4	no	—
51	C₂²⋊Dic₅	—	10	4	no	—
52	C₄⋊Dic₅	—	5	2	no	—
53	C₄⋊Dic₅	—	10	2	no	—
54	C₄×Dic₅	—	5	2	no	—
55	C₄×Dic₅	—	10	2	no	—
56	#436	—	25	1	no	Sylow
57	C₂×C₅²	—	4	2	no	—
58	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
59	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₅²	#131	1	1	yes	characteristic
61	C₂³×C₅	—	2	4	no	—
62	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	4	yes	—
63	C₅×D₄	—	2	8	no	—
64	C₅×D₄	—	2	4	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
67	C₂×Dic₅	—	5	2	no	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	5	4	no	—
70	C₂×Dic₅	—	10	2	no	—
71	C₂×Dic₅	—	10	2	no	—
72	C₂×Dic₅	—	10	4	no	—
73	C₂×Dic₅	—	10	4	no	—
74	C₂×Dic₅	—	10	4	no	—
75	C₂×Dic₅	—	20	4	no	—
76	C₂×Dic₅	—	20	4	no	—
77	#170	—	5	4	no	—
78	#170	—	10	2	no	—
79	#170	—	10	4	no	—
80	C₂×M₁₆	—	25	2	no	—
81	C₄×D₄	—	25	1	no	—
82	C₄⋊C₈	—	25	1	no	—
83	C₂²⋊C₈	—	25	2	no	—
84	C₄×C₈	—	25	1	no	—
85	C₅²	#436	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₅	—	2	8	no	—
87	C₂²×C₅	—	2	4	no	—
88	C₂²×C₅	—	2	4	no	—
89	C₂²×C₅	—	2	4	no	—
90	C₂²×C₅	—	2	2	no	—
91	C₂²×C₅	—	4	4	no	—
92	C₂²×C₅	—	4	4	no	—
93	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
94	C₄×C₅	—	2	4	no	—
95	C₄×C₅	#704	1	4	yes	complemented
96	Dic₅	—	5	2	no	—
97	Dic₅	—	5	2	no	—
98	Dic₅	—	10	2	no	—
99	Dic₅	—	10	2	no	—
100	Dic₅	—	10	2	no	—
101	Dic₅	—	10	4	no	—
102	Dic₅	—	20	2	no	—
103	Dic₅	—	20	4	no	—
104	C₂×D₄	#1044	1	1	yes	characteristic
105	C₂²×C₄	—	25	2	no	—
106	M₁₆	—	50	4	no	—
107	C₂×C₈	—	25	1	no	—
108	C₂×C₈	—	25	1	no	—
109	C₂×C₈	—	25	2	no	—
110	C₄⋊C₄	—	25	1	no	—
111	C₂²⋊C₄	—	25	2	no	—
112	C₄²	—	25	1	no	—
113	C₂×C₅	—	2	2	no	—
114	C₂×C₅	—	2	2	no	—
115	C₂×C₅	—	2	2	no	—
116	C₂×C₅	—	4	8	no	—
117	C₂×C₅	—	4	4	no	—
118	C₂×C₅	#4131	1	2	yes	—
119	C₂×C₅	D₄×AGL(1,𝔽₅)	1	2	yes	—
120	C₂×C₅	C₂×#704	1	2	yes	—
121	C₂³	C₂×#1044	1	2	yes	—
122	D₄	—	2	2	no	—
123	C₂×C₄	—	25	1	no	—
124	C₂×C₄	—	25	1	no	—
125	C₂×C₄	—	25	2	no	—
126	C₂×C₄	—	50	2	no	—
127	C₂×C₄	—	50	2	no	—
128	C₂×C₄	C₂×#1044	1	1	yes	characteristic
129	C₈	—	25	2	no	—
130	C₈	—	50	2	no	—
131	C₈	—	50	1	no	—
132	C₅	—	2	2	no	—
133	C₅	#65774	1	2	yes	complemented
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂²×#1044	1	1	yes	characteristic
138	C₄	—	25	1	no	—
139	C₄	—	25	1	no	—
140	C₄	—	50	1	no	—
141	C₄	—	50	2	no	—
142	C₄	#87867	1	2	yes	complemented
143	C₂	—	4	2	no	—
144	C₂	#11717452	1	1	yes	characteristic
145	C₂	D₄×#1044	1	1	yes	characteristic
146	C₂	C₂×#87867	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87867

C₂

yes

maximal, complemented

D₄×#1039

C₂

yes

maximal, characteristic

C₂²⋊#6034

C₂

yes

maximal, complemented

C₄⋊#6034

C₂

yes

maximal, characteristic, complemented

C₄×#6034

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

C₄

yes

characteristic

C₂²×#1039

C₂²

yes

—

#87867

—

C₂×#6034

C₂²

yes

characteristic

C₂×#6034

C₂²

yes

—

C₂×#6034

C₂²

yes

characteristic

C₂²⋊#1039

C₄

yes

—

C₄⋊#1039

C₄

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

#65774

—

maximal

C₂³×C₅²

C₂×C₄

yes

—

C₅²×D₄

—

C₂×C₄×C₅²

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂³

yes

—

—

D₄

yes

complemented

C₂×#704

—

D₄×Dic₅

—

D₄×Dic₅

—

C₂²⋊#170

—

C₄⋊#170

—

C₄×#170

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

M₁₆

yes

—

—

C₂×D₄

yes

yes

C₂×C₅×D₄

—

C₂×C₅×D₄

AGL(1,𝔽₅)

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

—

—

—

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#436

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₄×D₄

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

characteristic

C₂×C₅²

#131

yes

characteristic

C₂³×C₅

—

C₂³×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₅×D₄

—

C₅×D₄

—

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

—

C₄×D₄

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₅²

#436

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

—

C₄×C₅

—

C₄×C₅

#704

yes

complemented

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

100

Dic₅

—

101

Dic₅

—

102

Dic₅

—

103

Dic₅

—

104

C₂×D₄

#1044

yes

characteristic

105

C₂²×C₄

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₂×C₅

—

114

C₂×C₅

—

115

C₂×C₅

—

116

C₂×C₅

—

117

C₂×C₅

—

118

C₂×C₅

#4131

yes

—

119

C₂×C₅

D₄×AGL(1,𝔽₅)

yes

—

120

C₂×C₅

C₂×#704

yes

—

121

C₂³

C₂×#1044

yes

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×#1044

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₅

—

133

C₅

#65774

yes

complemented

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×#1044

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#87867

yes

complemented

143

C₂

—

144

C₂

#11717452

yes

characteristic

145

C₂

D₄×#1044

yes

characteristic

146

C₂

C₂×#87867

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1044
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting, prime radical	C₂×C₅²×D₄	C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₂×D₄	#1044
hypocenter	C₅²	#436
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#1044

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting, prime radical

C₂×C₅²×D₄

C₄

C₂³

C₂×D₄

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. C₂×#87867	C₂	semidirect
3. C₂²⋊#6034	C₂	semidirect
4. C₄⋊#6034	C₂	semidirect
5. C₄×#6034	C₂	semidirect
26. #6034	D₄	semidirect
37. C₄×C₅²	M₁₆	semidirect
85. C₅²	#436	semidirect
95. C₄×C₅	#704	semidirect
133. C₅	#65774	semidirect
142. C₄	#87867	semidirect

H ≅ G/N

type

1. C₂×#87867

C₂

semidirect

3. C₂²⋊#6034

C₂

semidirect

4. C₄⋊#6034

C₂

semidirect

5. C₄×#6034

C₂

semidirect

26. #6034

D₄

semidirect

37. C₄×C₅²

M₁₆

semidirect

85. C₅²

#436

semidirect

95. C₄×C₅

#704

semidirect

133. C₅

#65774

semidirect

142. C₄

#87867

semidirect

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

subgroup	index
C₂×#87867	2
D₄×#1039	2
C₂²⋊#6034	2
C₄⋊#6034	2
C₄×#6034	2
#65774	5

subgroup

index

C₂²⋊#6034

Subgroups — #13104912

Conjugacy classes of subgroups

Complements

Maximal subgroups