Group #12015719 — subgroups

Subgroups — #12015719

Order 1280.

342 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 159 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	#10600542	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×#3947	C₄	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₂×#3944	C₂²	1	2	yes	—
10	#65221	C₄	1	1	yes	characteristic
11	C₂×#3935	C₄	1	1	yes	characteristic
12	C₅×C₄⋊C₁₆	—	2	2	no	—
13	#64731	—	2	8	no	—
14	C₄⋊#672	—	2	2	no	—
15	#12901	—	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	C₂×C₄	1	2	yes	—
20	C₂²×#168	C₂×C₄	1	2	yes	—
21	C₅×M₃₂	—	2	4	no	—
22	C₂×C₅×C₁₆	—	2	2	no	—
23	C₄×C₅×C₈	D₄	1	2	yes	—
24	C₄×C₅×C₈	Q₈	1	2	yes	—
25	#3963	—	2	2	no	—
26	#3947	—	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	#3935	C₈	1	4	yes	—
32	C₂×#364	—	5	4	no	—
33	#2150	—	5	1	no	—
34	#2150	D₅	1	1	yes	characteristic
35	#1726	—	5	1	no	—
36	C₂²×C₄×C₅	M₁₆	1	2	yes	—
37	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₅	—	2	2	no	—
39	C₂×C₅×C₈	—	2	2	no	—
40	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
41	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
42	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
45	C₄²×C₅	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅	C₂×C₈	1	2	yes	—
47	C₄×Dic₅	—	2	2	no	—
48	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
49	C₂×#168	C₄⋊C₄	1	4	yes	—
50	C₂×#168	C₂×C₈	1	4	yes	—
51	C₂×#168	M₁₆	1	4	yes	—
52	C₅×C₁₆	—	4	2	no	—
53	#672	—	4	2	no	—
54	C₂×M₃₂	—	5	2	no	—
55	C₂×M₃₂	C₂×D₅	1	2	yes	—
56	C₂×#97	—	5	1	no	—
57	C₂×#97	—	5	2	no	—
58	C₂×C₄×C₈	—	5	2	no	—
59	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
60	#376	—	5	1	no	—
61	#364	—	10	8	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₄⋊C₁₆	—	10	2	no	—
64	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅	—	4	2	no	—
66	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
67	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
71	C₂×Dic₅	—	2	4	no	—
72	C₂×Dic₅	—	4	4	no	—
73	C₅×C₈	—	2	2	no	—
74	C₅×C₈	#127	1	1	yes	characteristic
75	C₅×C₈	C₄⋊Q₈	1	2	yes	—
76	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
77	#168	C₄⋊C₈	1	8	yes	—
78	C₂²×C₈	—	5	2	no	—
79	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
81	C₂×C₄²	—	5	2	no	—
82	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
83	M₃₂	—	2	4	no	—
84	M₃₂	—	10	4	no	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	10	2	no	—
87	#97	—	5	4	no	—
88	#97	—	10	4	no	—
89	C₄×C₈	—	5	4	no	—
90	C₄×C₈	—	5	4	no	—
91	C₄×C₈	Dic₁₀	1	2	yes	—
92	C₄×C₈	C₂²⋊D₅	1	2	yes	—
93	#95	—	10	2	no	—
94	C₂²×C₅	—	2	2	no	—
95	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₂²×C₅	#423	1	2	yes	—
97	C₄×C₅	—	2	4	no	—
98	C₄×C₅	#384	1	1	yes	characteristic
99	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
100	C₄×C₅	#423	1	1	yes	characteristic
101	Dic₅	—	4	4	no	—
102	C₂²×C₄	—	10	2	no	—
103	C₂²×C₄	#676	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	5	4	no	—
107	C₂×C₈	—	5	4	no	—
108	C₂×C₈	—	5	4	no	—
109	C₂×C₈	#683	1	2	yes	—
110	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
111	C₂×C₈	D₄×D₅	1	2	yes	—
112	C₂×C₈	Q₈×D₅	1	2	yes	—
113	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
114	C₄²	—	5	4	no	—
115	C₄²	—	10	2	no	—
116	C₄²	C₈×D₅	1	2	yes	—
117	C₄²	#683	1	2	yes	—
118	C₁₆	—	4	2	no	—
119	C₁₆	—	20	2	no	—
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	#1842	1	1	yes	characteristic
122	C₂×C₅	#2153	1	2	yes	—
123	C₂³	C₂×#676	1	1	yes	characteristic
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	10	4	no	—
126	C₂×C₄	—	20	4	no	—
127	C₂×C₄	#3946	1	4	yes	—
128	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
130	C₂×C₄	D₅×M₁₆	1	2	yes	—
131	C₂×C₄	C₂×#683	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	5	8	no	—
134	C₈	#4034	1	2	yes	—
135	C₈	#4086	1	1	yes	characteristic
136	C₈	#4090	1	1	yes	characteristic
137	C₅	#12901	1	1	yes	characteristic, complemented, Sylow
138	C₂²	—	2	2	no	—
139	C₂²	#65126	1	2	yes	—
140	C₂²	C₂×#3946	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	20	4	no	—
143	C₄	D₅×C₄⋊C₈	1	2	yes	—
144	C₄	#65265	1	1	yes	characteristic
145	C₄	#65407	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#10601238	1	1	yes	characteristic
148	C₂	#10602216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2150

C₂

yes

maximal, characteristic

C₂×#64731

C₂

yes

maximal

#10600542

C₂

yes

maximal, characteristic

#10600536

C₂

yes

maximal, characteristic

C₂×C₅×M₃₂

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3947

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₂×#3944

C₂²

yes

—

#65221

C₄

yes

characteristic

C₂×#3935

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

yes

—

C₂²×#168

C₂×C₄

yes

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

#3963

—

#3947

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

#3944

—

C₂×#672

—

#3935

C₈

yes

—

C₂×#364

—

#2150

—

#2150

D₅

yes

characteristic

#1726

—

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

—

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×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#376

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂³×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

—

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

C₄×D₅

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀

yes

—

C₄×C₈

C₂²⋊D₅

yes

—

#95

—

C₂²×C₅

—

C₂²×C₅

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅

#423

yes

—

C₄×C₅

—

C₄×C₅

#384

yes

characteristic

C₄×C₅

C₂×C₄⋊C₈

yes

—

100

C₄×C₅

#423

yes

characteristic

101

Dic₅

—

102

C₂²×C₄

—

103

C₂²×C₄

#676

yes

—

104

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

#683

yes

—

110

C₂×C₈

C₂×Dic₁₀

yes

characteristic

111

C₂×C₈

D₄×D₅

yes

—

112

C₂×C₈

Q₈×D₅

yes

—

113

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

114

C₄²

—

115

C₄²

—

116

C₄²

C₈×D₅

yes

—

117

C₄²

#683

yes

—

118

C₁₆

—

119

C₁₆

—

120

C₂×C₅

—

121

C₂×C₅

#1842

yes

characteristic

122

C₂×C₅

#2153

yes

—

123

C₂³

C₂×#676

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#3946

yes

—

128

C₂×C₄

D₅×C₄⋊C₄

yes

—

129

C₂×C₄

C₂×C₈×D₅

yes

characteristic

130

C₂×C₄

D₅×M₁₆

yes

—

131

C₂×C₄

C₂×#683

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

#4034

yes

—

135

C₈

#4086

yes

characteristic

136

C₈

#4090

yes

characteristic

137

C₅

#12901

yes

characteristic, complemented, Sylow

138

C₂²

—

139

C₂²

#65126

yes

—

140

C₂²

C₂×#3946

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

D₅×C₄⋊C₈

yes

—

144

C₄

#65265

yes

characteristic

145

C₄

#65407

yes

characteristic

146

C₂

—

147

C₂

#10601238

yes

characteristic

148

C₂

#10602216

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	#12901
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
137. C₅	#12901	semidirect

H ≅ G/N

type

137. C₅

#12901

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
#10600542	2
#10600536	2
#12901	5

subgroup

index

Subgroups — #12015719

Conjugacy classes of subgroups

Complements

Maximal subgroups