Group #12015716 — subgroups

Subgroups — #12015716

Order 1280.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	C₅×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×#64731	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#10600536	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₅	C₄	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₂×#3944	C₂²	1	2	yes	—
10	C₄²×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₄×#168	C₄	1	1	yes	characteristic
12	C₅×C₄⋊C₁₆	—	2	2	no	—
13	#64731	—	2	8	no	—
14	C₄⋊#672	—	2	2	no	—
15	#12900	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅	—	2	2	no	—
20	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
21	C₂²×#168	C₂×C₄	1	2	yes	—
22	C₅×M₃₂	—	2	4	no	—
23	C₂×C₅×C₁₆	—	2	2	no	—
24	C₄×C₅×C₈	D₄	1	2	yes	—
25	C₄×C₅×C₈	Q₈	1	2	yes	—
26	C₈×Dic₅	—	2	4	no	—
27	C₈×Dic₅	D₄	1	4	yes	—
28	C₈×Dic₅	Q₈	1	4	yes	—
29	#3944	—	2	4	no	—
30	C₂×#672	—	2	2	no	—
31	C₄×#168	C₈	1	2	yes	—
32	C₄×#168	C₈	1	2	yes	—
33	C₂×#364	—	5	4	no	—
34	#2150	—	5	1	no	—
35	#2150	D₅	1	1	yes	characteristic
36	C₄²×C₈	—	5	1	no	—
37	C₂²×C₄×C₅	M₁₆	1	2	yes	—
38	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₅	—	2	2	no	—
40	C₂×C₅×C₈	—	2	2	no	—
41	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
43	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
46	C₄²×C₅	C₄⋊C₄	1	2	yes	—
47	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
49	C₄×Dic₅	—	2	8	no	—
50	C₄×Dic₅	—	2	2	no	—
51	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
52	C₂×#168	C₄⋊C₄	1	4	yes	—
53	C₂×#168	C₂×C₈	1	4	yes	—
54	C₂×#168	M₁₆	1	4	yes	—
55	C₅×C₁₆	—	4	2	no	—
56	#672	—	4	2	no	—
57	C₂×M₃₂	—	5	2	no	—
58	C₂×M₃₂	C₂×D₅	1	2	yes	—
59	C₂×C₄×C₈	—	5	2	no	—
60	C₂×C₄×C₈	—	5	1	no	—
61	C₂×C₄×C₈	—	5	2	no	—
62	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
63	C₄³	—	5	1	no	—
64	#364	—	10	8	no	—
65	C₄⋊C₁₆	—	2	2	no	—
66	C₄⋊C₁₆	—	10	2	no	—
67	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅	—	2	4	no	—
69	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
70	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
71	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
75	C₂×Dic₅	—	2	8	no	—
76	C₂×Dic₅	—	2	4	no	—
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	#127	1	1	yes	characteristic
79	C₅×C₈	C₄⋊Q₈	1	2	yes	—
80	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
81	#168	C₄⋊C₈	1	8	yes	—
82	C₂²×C₈	—	5	2	no	—
83	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
84	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
85	C₂×C₄²	—	5	2	no	—
86	C₂×C₄²	—	10	2	no	—
87	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
88	M₃₂	—	2	4	no	—
89	M₃₂	—	10	4	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	10	2	no	—
92	C₄×C₈	—	5	4	no	—
93	C₄×C₈	—	5	4	no	—
94	C₄×C₈	—	5	2	no	—
95	C₄×C₈	—	5	2	no	—
96	C₄×C₈	—	10	4	no	—
97	C₄×C₈	Dic₁₀	1	2	yes	—
98	C₄×C₈	C₂²⋊D₅	1	2	yes	—
99	C₂²×C₅	—	2	2	no	—
100	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₅	#423	1	2	yes	—
102	C₄×C₅	—	2	2	no	—
103	C₄×C₅	#364	1	4	yes	complemented
104	C₄×C₅	#384	1	1	yes	characteristic
105	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
106	C₄×C₅	#423	1	1	yes	characteristic
107	Dic₅	—	2	8	no	—
108	C₂²×C₄	—	10	2	no	—
109	C₂²×C₄	#676	1	2	yes	—
110	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	5	4	no	—
113	C₂×C₈	—	5	4	no	—
114	C₂×C₈	—	5	4	no	—
115	C₂×C₈	#683	1	2	yes	—
116	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
117	C₂×C₈	D₄×D₅	1	2	yes	—
118	C₂×C₈	Q₈×D₅	1	2	yes	—
119	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
120	C₄²	—	5	4	no	—
121	C₄²	—	10	8	no	—
122	C₄²	—	10	2	no	—
123	C₄²	C₈×D₅	1	1	yes	characteristic
124	C₄²	C₈×D₅	1	1	yes	characteristic
125	C₄²	#683	1	2	yes	—
126	C₁₆	—	4	2	no	—
127	C₁₆	—	20	2	no	—
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	#1842	1	1	yes	characteristic
130	C₂×C₅	C₂×#364	1	2	yes	—
131	C₂³	C₂×#676	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	10	8	no	—
134	C₂×C₄	—	10	4	no	—
135	C₂×C₄	#3946	1	2	yes	—
136	C₂×C₄	#3946	1	2	yes	—
137	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
138	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
139	C₂×C₄	D₅×M₁₆	1	2	yes	—
140	C₂×C₄	C₂×#683	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	5	8	no	—
143	C₈	#4034	1	2	yes	—
144	C₈	#4086	1	1	yes	characteristic
145	C₈	#4090	1	1	yes	characteristic
146	C₅	#12900	1	1	yes	characteristic, complemented, Sylow
147	C₂²	—	2	2	no	—
148	C₂²	#65126	1	2	yes	—
149	C₂²	C₂×#3946	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	10	8	no	—
152	C₄	#64731	1	4	yes	complemented
153	C₄	D₅×C₄⋊C₈	1	2	yes	—
154	C₄	#65265	1	1	yes	characteristic
155	C₄	#65407	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#10601238	1	1	yes	characteristic
158	C₂	C₂×#64731	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2150

C₂

yes

maximal, characteristic

C₂×#64731

C₂

yes

maximal

C₄×C₈×Dic₅

C₂

yes

maximal, characteristic

#10600536

C₂

yes

maximal, characteristic

C₂×C₅×M₃₂

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×C₈×Dic₅

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₂×#3944

C₂²

yes

—

C₄²×Dic₅

C₄

yes

characteristic

C₂×C₄×#168

C₄

yes

characteristic

C₅×C₄⋊C₁₆

—

—

—

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

—

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

C₈×Dic₅

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

#3944

—

C₂×#672

—

C₄×#168

C₈

yes

—

C₄×#168

C₈

yes

—

C₂×#364

—

#2150

—

#2150

D₅

yes

characteristic

C₄²×C₈

—

C₂²×C₄×C₅

M₁₆

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

C₂×Q₈

yes

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₄⋊C₄

yes

—

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₂×C₈

yes

—

C₂×#168

M₁₆

yes

—

C₅×C₁₆

—

#672

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

C₄³

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂³×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

#127

yes

characteristic

C₅×C₈

C₄⋊Q₈

yes

—

C₅×C₈

C₄⋊Q₈

yes

characteristic

#168

C₄⋊C₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅

yes

characteristic

C₂²×C₈

C₂²×D₅

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄×D₅

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀

yes

—

C₄×C₈

C₂²⋊D₅

yes

—

C₂²×C₅

—

100

C₂²×C₅

C₂×C₄⋊C₈

yes

characteristic

101

C₂²×C₅

#423

yes

—

102

C₄×C₅

—

103

C₄×C₅

#364

yes

complemented

104

C₄×C₅

#384

yes

characteristic

105

C₄×C₅

C₂×C₄⋊C₈

yes

—

106

C₄×C₅

#423

yes

characteristic

107

Dic₅

—

108

C₂²×C₄

—

109

C₂²×C₄

#676

yes

—

110

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

#683

yes

—

116

C₂×C₈

C₂×Dic₁₀

yes

characteristic

117

C₂×C₈

D₄×D₅

yes

—

118

C₂×C₈

Q₈×D₅

yes

—

119

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₈×D₅

yes

characteristic

124

C₄²

C₈×D₅

yes

characteristic

125

C₄²

#683

yes

—

126

C₁₆

—

127

C₁₆

—

128

C₂×C₅

—

129

C₂×C₅

#1842

yes

characteristic

130

C₂×C₅

C₂×#364

yes

—

131

C₂³

C₂×#676

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

#3946

yes

—

136

C₂×C₄

#3946

yes

—

137

C₂×C₄

D₅×C₄⋊C₄

yes

—

138

C₂×C₄

C₂×C₈×D₅

yes

characteristic

139

C₂×C₄

D₅×M₁₆

yes

—

140

C₂×C₄

C₂×#683

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

#4034

yes

—

144

C₈

#4086

yes

characteristic

145

C₈

#4090

yes

characteristic

146

C₅

#12900

yes

characteristic, complemented, Sylow

147

C₂²

—

148

C₂²

#65126

yes

—

149

C₂²

C₂×#3946

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

#64731

yes

complemented

153

C₄

D₅×C₄⋊C₈

yes

—

154

C₄

#65265

yes

characteristic

155

C₄

#65407

yes

characteristic

156

C₂

—

157

C₂

#10601238

yes

characteristic

158

C₂

C₂×#64731

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2150	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2150	D₅
hypocenter	C₅	#12900
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅

Fitting

C₅×#2150

C₂

socle

C₂²×C₅

C₂×C₄⋊C₈

prime radical

C₂³×C₅

C₂×M₁₆

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
103. C₄×C₅	#364	semidirect
146. C₅	#12900	semidirect
152. C₄	#64731	semidirect

H ≅ G/N

type

103. C₄×C₅

#364

semidirect

146. C₅

#12900

semidirect

152. C₄

#64731

semidirect

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

subgroup	index
C₅×#2150	2
C₂×#64731	2
C₄×C₈×Dic₅	2
#10600536	2
#12900	5

subgroup

index

C₄×C₈×Dic₅

Subgroups — #12015716

Conjugacy classes of subgroups

Complements

Maximal subgroups