Group #12015692 — subgroups

Subgroups — #12015692

Order 1280.

274 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 139 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×#1848	C₂	1	1	yes	maximal, characteristic
2	#10600585	C₂	1	1	yes	maximal, characteristic
3	C₂×#64681	C₂	1	1	yes	maximal, characteristic
4	#10598219	C₂	1	4	yes	maximal
5	C₂×C₅×#108	—	2	1	no	—
6	C₂×C₅×#108	C₂²	1	2	yes	—
7	C₅×#423	—	2	1	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3951	—	2	1	no	—
10	C₂×#3951	C₂²	1	2	yes	—
11	C₂²×#672	C₂²	1	2	yes	—
12	#65213	—	2	1	no	—
13	#64681	C₄	1	4	yes	—
14	#12892	—	5	1	no	maximal, Sylow
15	C₂×C₅×M₁₆	—	2	2	no	—
16	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
19	C₂×#681	—	2	2	no	—
20	C₅×#108	—	2	4	no	—
21	C₅×#108	—	4	2	no	—
22	C₅×C₄⋊C₈	—	4	2	no	—
23	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅×C₈	Q₈	1	2	yes	—
25	#3951	—	2	4	no	—
26	#3951	—	4	2	no	—
27	C₂×#672	—	2	2	no	—
28	C₂×#672	—	2	2	no	—
29	C₂×#672	C₂×C₄	1	4	yes	—
30	C₄⋊#168	—	4	2	no	—
31	C₂×#346	—	5	1	no	—
32	#1848	—	5	1	no	—
33	#1848	D₅	1	1	yes	characteristic
34	#1381	—	5	4	no	—
35	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
36	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	C₅×M₁₆	—	4	2	no	—
38	C₅×M₁₆	—	4	2	no	—
39	C₂×C₅×C₈	—	4	2	no	—
40	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
41	C₂×C₅×C₈	SD₁₆	1	2	yes	—
42	C₂×C₅×C₈	SD₁₆	1	2	yes	—
43	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₅×C₈	#41	1	2	yes	—
46	C₂×C₅×C₈	#41	1	2	yes	—
47	C₄²×C₅	C₄⋊C₄	1	2	yes	—
48	C₄²×C₅	D₈	1	2	yes	—
49	#681	—	4	2	no	—
50	#681	—	4	2	no	—
51	C₂×#168	—	4	2	no	—
52	#672	—	2	4	no	—
53	C₂²×C₁₆	—	5	2	no	—
54	C₂×#108	—	2	1	no	—
55	C₂×#108	—	5	2	no	—
56	C₂×#108	—	10	1	no	—
57	C₂×#108	C₂×D₅	1	2	yes	—
58	#423	—	2	1	no	—
59	#423	—	10	1	no	—
60	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
61	#346	—	5	4	no	—
62	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅	—	4	2	no	—
64	C₂×C₄×C₅	#107	1	4	yes	—
65	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅	#124	1	1	yes	characteristic
67	C₂×C₄×C₅	#124	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
71	C₅×C₈	—	8	2	no	—
72	C₅×C₈	#106	1	4	yes	—
73	C₅×C₈	#117	1	2	yes	—
74	C₅×C₈	#125	1	2	yes	—
75	#168	—	8	2	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂×M₁₆	—	10	2	no	—
78	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
79	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
80	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
81	C₂×C₁₆	—	5	4	no	—
82	C₂×C₁₆	—	10	2	no	—
83	C₂×C₁₆	—	10	2	no	—
84	#108	—	2	4	no	—
85	#108	—	4	2	no	—
86	#108	—	10	4	no	—
87	#108	—	20	2	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	20	2	no	—
90	C₄×C₈	Dic₁₀	1	2	yes	—
91	C₄×C₈	C₄×D₅	1	2	yes	—
92	C₂²×C₅	—	2	2	no	—
93	C₂²×C₅	C₂×#107	1	1	yes	characteristic
94	C₂²×C₅	#487	1	1	yes	characteristic
95	C₂²×C₅	#487	1	1	yes	characteristic
96	C₄×C₅	—	2	4	no	—
97	C₄×C₅	#382	1	1	yes	characteristic
98	C₄×C₅	C₂×#106	1	1	yes	characteristic
99	C₄×C₅	#486	1	1	yes	characteristic
100	C₄×C₅	#486	1	1	yes	characteristic
101	C₂²×C₄	#689	1	2	yes	—
102	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
103	M₁₆	—	4	2	no	—
104	M₁₆	—	4	2	no	—
105	M₁₆	—	20	2	no	—
106	M₁₆	—	20	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	20	2	no	—
109	C₂×C₈	#683	1	2	yes	—
110	C₂×C₈	#687	1	2	yes	—
111	C₂×C₈	#688	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
113	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
114	C₂×C₈	#711	1	2	yes	—
115	C₂×C₈	#713	1	2	yes	—
116	C₄²	#683	1	2	yes	—
117	C₄²	#686	1	2	yes	—
118	C₁₆	—	10	4	no	—
119	C₂×C₅	—	2	1	no	—
120	C₂×C₅	—	2	1	no	—
121	C₂×C₅	#1834	1	1	yes	characteristic
122	C₂×C₅	#2161	1	2	yes	—
123	C₂³	C₂×#689	1	1	yes	characteristic
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	#3939	1	4	yes	—
126	C₂×C₄	C₂×#683	1	1	yes	characteristic
127	C₂×C₄	C₂×#686	1	1	yes	characteristic
128	C₂×C₄	C₂×#687	1	1	yes	characteristic
129	C₂×C₄	#4082	1	1	yes	characteristic
130	C₂×C₄	C₂×#688	1	1	yes	characteristic
131	C₂×C₄	#4093	1	1	yes	characteristic
132	C₈	—	8	2	no	—
133	C₈	—	40	2	no	—
134	C₈	#3940	1	4	yes	—
135	C₈	#4036	1	2	yes	—
136	C₈	#4038	1	2	yes	—
137	C₅	#12892	1	1	yes	characteristic, complemented, Sylow
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#3939	1	1	yes	characteristic
140	C₂²	#65370	1	1	yes	characteristic
141	C₂²	#65382	1	1	yes	characteristic
142	C₄	—	2	4	no	—
143	C₄	C₂×#3940	1	1	yes	characteristic
144	C₄	#65266	1	1	yes	characteristic
145	C₄	#65359	1	1	yes	characteristic
146	C₄	#65377	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#10600367	1	1	yes	characteristic
150	C₂	#10601476	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1848

C₂

yes

maximal, characteristic

#10600585

C₂

yes

maximal, characteristic

C₂×#64681

C₂

yes

maximal, characteristic

#10598219

C₂

yes

maximal

C₂×C₅×#108

—

C₂×C₅×#108

C₂²

yes

—

C₅×#423

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3951

—

C₂×#3951

C₂²

yes

—

C₂²×#672

C₂²

yes

—

#65213

—

#64681

C₄

yes

—

#12892

—

maximal, Sylow

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

—

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

—

—

—

D₅

yes

characteristic

#1381

—

C₂²×C₄×C₅

Q₁₆

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₂×C₅×C₈

#41

yes

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

D₈

yes

—

—

—

—

—

C₂²×C₁₆

—

—

—

—

C₂×D₅

yes

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#346

—

C₂³×C₅

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

#107

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

C₅×C₈

—

C₅×C₈

#106

yes

—

C₅×C₈

#117

yes

—

C₅×C₈

#125

yes

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₅

yes

characteristic

C₂²×C₈

C₂²×D₅

yes

characteristic

C₂×C₄²

C₂²⋊D₅

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

Dic₁₀

yes

—

C₄×C₈

C₄×D₅

yes

—

C₂²×C₅

—

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

100

C₄×C₅

#486

yes

characteristic

101

C₂²×C₄

#689

yes

—

102

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

#683

yes

—

110

C₂×C₈

#687

yes

—

111

C₂×C₈

#688

yes

—

112

C₂×C₈

C₂×Dic₁₀

yes

characteristic

113

C₂×C₈

C₂×C₄×D₅

yes

characteristic

114

C₂×C₈

#711

yes

—

115

C₂×C₈

#713

yes

—

116

C₄²

#683

yes

—

117

C₄²

#686

yes

—

118

C₁₆

—

119

C₂×C₅

—

120

C₂×C₅

—

121

C₂×C₅

#1834

yes

characteristic

122

C₂×C₅

#2161

yes

—

123

C₂³

C₂×#689

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

#3939

yes

—

126

C₂×C₄

C₂×#683

yes

characteristic

127

C₂×C₄

C₂×#686

yes

characteristic

128

C₂×C₄

C₂×#687

yes

characteristic

129

C₂×C₄

#4082

yes

characteristic

130

C₂×C₄

C₂×#688

yes

characteristic

131

C₂×C₄

#4093

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

#3940

yes

—

135

C₈

#4036

yes

—

136

C₈

#4038

yes

—

137

C₅

#12892

yes

characteristic, complemented, Sylow

138

C₂²

—

139

C₂²

C₂×#3939

yes

characteristic

140

C₂²

#65370

yes

characteristic

141

C₂²

#65382

yes

characteristic

142

C₄

—

143

C₄

C₂×#3940

yes

characteristic

144

C₄

#65266

yes

characteristic

145

C₄

#65359

yes

characteristic

146

C₄

#65377

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#10600367

yes

characteristic

150

C₂

#10601476

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#686
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1848	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1848	D₅
hypocenter	C₅	#12892
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

subgroup

G/H

center

C₂×C₄

C₂×#686

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×Q₁₆

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

N	H ≅ G/N	type
137. C₅	#12892	semidirect

H ≅ G/N

type

137. C₅

#12892

semidirect

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

subgroup	index
C₅×#1848	2
#10600585	2
C₂×#64681	2
#10598219	2
#12892	5

subgroup

index

Subgroups — #12015692

Conjugacy classes of subgroups

Complements

Maximal subgroups