Group #13027794 — subgroups

Subgroups — #13027794

Order 1280.

206 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 100 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×#64842	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#10599865	C₂	1	1	yes	maximal, characteristic
4	#10598655	C₂	1	2	yes	maximal, complemented
5	C₄⋊#3928	C₂	1	2	yes	maximal, complemented
6	C₂×#3930	C₄	1	1	yes	characteristic
7	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
8	#64978	C₄	1	1	yes	characteristic
9	C₂×#3928	C₂²	1	2	yes	—
10	C₂×#3928	C₂²	1	2	yes	—
11	#64842	—	2	2	no	—
12	D₅⋊C₃₂	—	2	2	no	—
13	C₅×#346	C₄	1	1	yes	characteristic
14	#64722	C₄	1	1	yes	characteristic
15	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
16	C₄×#672	C₂²	1	1	yes	characteristic
17	#13628	—	5	1	no	maximal, Sylow
18	C₂×#676	C₈	1	1	yes	characteristic
19	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
20	#4018	C₈	1	1	yes	characteristic
21	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	#3947	C₈	1	1	yes	characteristic
25	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂×#672	C₂×C₄	1	1	yes	characteristic
27	C₂×#672	C₂³	1	1	yes	characteristic
28	#3935	C₈	1	1	yes	characteristic
29	#3930	—	2	2	no	—
30	D₅×C₁₆	D₄	1	2	yes	—
31	D₅×C₁₆	Q₈	1	2	yes	—
32	#3928	—	2	2	no	—
33	#3928	—	2	2	no	—
34	C₂×M₆₄	—	5	1	no	—
35	C₂²×C₃₂	—	5	1	no	—
36	#2108	—	5	1	no	—
37	C₄⋊C₃₂	—	5	2	no	—
38	C₄⋊C₃₂	—	5	2	no	—
39	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
40	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
43	#685	C₁₆	1	2	yes	—
44	#683	C₁₆	1	2	yes	—
45	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
46	C₂×#168	C₂×C₈	1	1	yes	characteristic
47	C₂×#168	C₂×C₈	1	1	yes	characteristic
48	#676	—	2	2	no	—
49	C₈×D₅	C₄⋊C₄	1	2	yes	—
50	C₈×D₅	M₁₆	1	2	yes	—
51	C₅×C₁₆	—	2	1	no	—
52	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
53	#672	—	2	1	no	—
54	#672	C₂×D₄	1	1	yes	characteristic
55	#672	C₂×Q₈	1	1	yes	characteristic
56	C₂×M₃₂	—	5	1	no	—
57	C₂²×C₁₆	—	5	1	no	—
58	#405	—	5	1	no	—
59	M₆₄	—	10	2	no	—
60	C₂×C₃₂	—	5	2	no	—
61	C₂×C₃₂	—	5	2	no	—
62	C₂×C₃₂	—	10	2	no	—
63	#346	—	5	1	no	—
64	#346	AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₄×C₁₆	—	5	1	no	—
66	C₄×C₁₆	—	5	1	no	—
67	C₂²×D₅	C₂×C₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
69	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
70	C₂×Dic₅	C₂×C₁₆	1	2	yes	—
71	C₂×Dic₅	C₂×C₁₆	1	1	yes	characteristic
72	C₄×D₅	C₄⋊C₈	1	2	yes	—
73	C₄×D₅	M₃₂	1	2	yes	—
74	C₅×C₈	—	2	1	no	—
75	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
77	#168	—	2	1	no	—
78	#168	C₄⋊C₈	1	2	yes	—
79	C₂×M₁₆	—	5	1	no	—
80	C₂²×C₈	—	5	1	no	—
81	#117	—	5	1	no	—
82	M₃₂	—	10	2	no	—
83	C₂×C₁₆	—	5	1	no	—
84	C₂×C₁₆	—	5	2	no	—
85	C₂×C₁₆	—	5	2	no	—
86	C₂×C₁₆	—	5	1	no	—
87	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
89	#97	—	5	1	no	—
90	#97	—	5	1	no	—
91	C₄×C₈	—	5	1	no	—
92	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₃₂	—	10	2	no	—
94	C₃₂	—	10	2	no	—
95	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
96	C₂×D₅	—	2	2	no	—
97	C₂×D₅	C₄⋊C₁₆	1	2	yes	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
101	Dic₅	—	2	2	no	—
102	Dic₅	C₄⋊C₁₆	1	2	yes	—
103	C₂²×C₄	—	5	1	no	—
104	M₁₆	—	10	2	no	—
105	C₂×C₈	—	5	1	no	—
106	C₂×C₈	—	5	2	no	—
107	C₂×C₈	—	5	2	no	—
108	C₂×C₈	—	5	1	no	—
109	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
110	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₄⋊C₄	—	5	2	no	—
112	C₂²⋊C₄	—	5	2	no	—
113	C₄²	—	5	1	no	—
114	C₄²	D₅⋊C₈	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	5	1	no	—
117	C₁₆	—	5	1	no	—
118	C₁₆	—	10	1	no	—
119	C₁₆	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
120	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
121	C₂×C₅	#2258	1	2	yes	—
122	D₅	—	2	2	no	—
123	C₂³	—	5	1	no	—
124	C₂×C₄	—	5	1	no	—
125	C₂×C₄	—	5	2	no	—
126	C₂×C₄	—	5	2	no	—
127	C₂×C₄	—	5	2	no	—
128	C₂×C₄	D₅⋊C₁₆	1	2	yes	—
129	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	5	2	no	—
132	C₈	—	10	1	no	—
133	C₈	D₅⋊M₁₆	1	1	yes	characteristic
134	C₈	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₅	#13628	1	1	yes	characteristic, complemented, Sylow
136	C₂²	—	5	2	no	—
137	C₂²	—	10	2	no	—
138	C₂²	C₂×D₅⋊C₁₆	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	5	2	no	—
141	C₄	—	10	2	no	—
142	C₄	D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
143	C₄	D₅⋊M₃₂	1	1	yes	characteristic
144	C₂	—	10	2	no	—
145	C₂	D₅⋊(C₄⋊C₁₆)	1	1	yes	characteristic
146	C₂	#10604111	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64842

C₂

yes

maximal, characteristic

C₂×D₅⋊C₃₂

C₂

yes

maximal, characteristic

#10599865

C₂

yes

maximal, characteristic

#10598655

C₂

yes

maximal, complemented

C₄⋊#3928

C₂

yes

maximal, complemented

C₂×#3930

C₄

yes

characteristic

C₂×D₅×C₁₆

C₂²

yes

characteristic

#64978

C₄

yes

characteristic

C₂×#3928

C₂²

yes

—

C₂×#3928

C₂²

yes

—

#64842

—

D₅⋊C₃₂

—

C₅×#346

C₄

yes

characteristic

#64722

C₄

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

C₄×#672

C₂²

yes

—

C₈

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

#4018

C₈

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3947

C₈

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

C₂³

yes

characteristic

#3935

C₈

yes

characteristic

#3930

—

D₅×C₁₆

D₄

yes

—

D₅×C₁₆

Q₈

yes

—

#3928

—

#3928

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₂×C₄×D₅

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

#685

C₁₆

yes

—

#683

C₁₆

yes

—

C₄×Dic₅

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

#676

—

C₈×D₅

C₄⋊C₄

yes

—

C₈×D₅

M₁₆

yes

—

C₅×C₁₆

—

C₅×C₁₆

C₄⋊C₄

yes

—

#672

—

#672

C₂×D₄

yes

characteristic

#672

C₂×Q₈

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×D₅

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅

C₂×C₁₆

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×Dic₅

C₂×C₁₆

yes

—

C₂×Dic₅

C₂×C₁₆

yes

characteristic

C₄×D₅

C₄⋊C₈

yes

—

C₄×D₅

M₃₂

yes

—

C₅×C₈

—

C₅×C₈

C₂×C₄⋊C₄

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

characteristic

#168

—

#168

C₄⋊C₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃₂

—

C₃₂

—

C₂²×C₅

C₂²×C₁₆

yes

characteristic

C₂×D₅

—

C₂×D₅

C₄⋊C₁₆

yes

—

C₄×C₅

—

C₄×C₅

C₂×C₄⋊C₈

yes

characteristic

100

C₄×C₅

C₂×M₃₂

yes

characteristic

101

Dic₅

—

102

Dic₅

C₄⋊C₁₆

yes

—

103

C₂²×C₄

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

D₅⋊C₈

yes

characteristic

110

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

D₅⋊C₈

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

C₄⋊AGL(1,𝔽₅)

yes

—

120

C₂×C₅

C₂×C₄⋊C₁₆

yes

characteristic

121

C₂×C₅

#2258

yes

—

122

D₅

—

123

C₂³

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

D₅⋊C₁₆

yes

—

129

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

D₅⋊M₁₆

yes

characteristic

134

C₈

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

135

C₅

#13628

yes

characteristic, complemented, Sylow

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×D₅⋊C₁₆

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

D₅⋊(C₄⋊C₈)

yes

characteristic

143

C₄

D₅⋊M₃₂

yes

characteristic

144

C₂

—

145

C₂

D₅⋊(C₄⋊C₁₆)

yes

characteristic

146

C₂

#10604111

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₁₆
Fitting	C₅×#346	C₄
prime radical	C₂²×D₅	C₂×C₁₆
cosocle	C₂×#672	C₂³
hypercenter	#346	AGL(1,𝔽₅)
hypocenter	C₅	#13628
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂²×C₁₆

Fitting

C₅×#346

C₄

prime radical

C₂²×D₅

C₂×C₁₆

C₂³

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
4. #10598655	C₂	semidirect
5. C₄⋊#3928	C₂	semidirect
135. C₅	#13628	semidirect

H ≅ G/N

type

4. #10598655

C₂

semidirect

5. C₄⋊#3928

C₂

semidirect

135. C₅

#13628

semidirect

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

subgroup	index
C₂×#64842	2
C₂×D₅⋊C₃₂	2
#10599865	2
#10598655	2
C₄⋊#3928	2
#13628	5

subgroup

index

C₂×D₅⋊C₃₂

Subgroups — #13027794

Conjugacy classes of subgroups

Complements

Maximal subgroups