Group #12015723 — subgroups

Subgroups — #12015723

Order 1280.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2153	C₂	1	1	yes	maximal, characteristic
2	#10602216	C₂	1	2	yes	maximal
3	#10602214	C₂	1	2	yes	maximal
4	#10600688	C₂	1	1	yes	maximal, characteristic
5	#10600675	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₅×#405	C₂²	1	1	yes	characteristic
8	#65390	C₂²	1	2	yes	—
9	C₂×#3944	C₂²	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	C₅×#364	—	2	2	no	—
12	#64780	C₄	1	1	yes	characteristic
13	#64777	C₄	1	1	yes	characteristic
14	#64751	C₄	1	2	yes	—
15	#64731	—	2	4	no	—
16	#64721	—	2	4	no	—
17	C₄⋊#672	—	2	2	no	—
18	#12902	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₅×#117	C₂×C₄	1	1	yes	characteristic
22	#4072	C₂×C₄	1	2	yes	—
23	C₂×#681	C₂×C₄	1	2	yes	—
24	C₅×M₃₂	—	2	2	no	—
25	C₅×M₃₂	—	2	2	no	—
26	C₂×C₅×C₁₆	—	2	2	no	—
27	C₅×#97	Q₈	1	2	yes	—
28	C₄×C₅×C₈	D₄	1	2	yes	—
29	#3947	Q₈	1	4	yes	—
30	C₈×Dic₅	D₄	1	4	yes	—
31	#3944	—	2	2	no	—
32	C₂×#672	—	2	1	no	—
33	C₂×#672	—	2	2	no	—
34	C₂×#672	—	2	1	no	—
35	#2153	—	5	2	no	—
36	#2153	D₅	1	1	yes	characteristic
37	#2151	—	5	2	no	—
38	#2151	—	5	1	no	—
39	#1747	—	5	1	no	—
40	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₅×M₁₆	—	2	2	no	—
42	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
44	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
47	C₅×C₄⋊C₄	—	2	1	no	—
48	C₅×C₂²⋊C₄	—	2	1	no	—
49	C₄²×C₅	C₄⋊C₄	1	2	yes	—
50	C₂²⋊Dic₅	—	2	2	no	—
51	C₄⋊Dic₅	—	2	2	no	—
52	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
53	#681	—	2	2	no	—
54	#681	—	2	2	no	—
55	C₂×#168	C₄⋊C₄	1	4	yes	—
56	C₅×C₁₆	—	4	2	no	—
57	#672	—	4	1	no	—
58	#672	—	4	1	no	—
59	C₂×M₃₂	—	5	1	no	—
60	C₂×M₃₂	C₂×D₅	1	2	yes	—
61	C₂²×C₁₆	—	5	1	no	—
62	#405	—	5	2	no	—
63	#405	C₂×D₅	1	1	yes	characteristic
64	#364	—	2	2	no	—
65	#364	—	10	4	no	—
66	C₄⋊C₁₆	—	10	4	no	—
67	C₄⋊C₁₆	—	10	2	no	—
68	#338	—	5	1	no	—
69	H(3,ℤ/4)	—	5	1	no	—
70	H(3,ℤ/4)	—	5	2	no	—
71	C₂³×C₅	#131	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅	#131	1	2	yes	—
76	C₂×C₄×C₅	#131	1	1	yes	characteristic
77	C₂×Dic₅	—	2	4	no	—
78	C₅×C₈	—	2	2	no	—
79	C₅×C₈	#127	1	1	yes	characteristic
80	C₅×C₈	C₄⋊Q₈	1	2	yes	—
81	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
82	#168	—	2	4	no	—
83	C₂×M₁₆	—	5	2	no	—
84	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
85	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
86	#117	—	5	2	no	—
87	#117	C₄×D₅	1	1	yes	characteristic
88	M₃₂	—	2	2	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	10	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	10	2	no	—
93	C₂×C₁₆	—	10	1	no	—
94	C₂×C₁₆	—	10	1	no	—
95	#97	—	5	4	no	—
96	#97	Dic₁₀	1	2	yes	—
97	C₄×C₈	—	5	4	no	—
98	C₄×C₈	C₂²⋊D₅	1	2	yes	—
99	C₂²×C₅	—	4	1	no	—
100	C₂²×C₅	#424	1	2	yes	—
101	C₂²×C₅	#424	1	1	yes	characteristic
102	C₄×C₅	—	4	2	no	—
103	C₄×C₅	#384	1	1	yes	characteristic
104	C₄×C₅	#424	1	2	yes	—
105	C₄×C₅	#424	1	1	yes	characteristic
106	Dic₅	—	4	4	no	—
107	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	10	2	no	—
110	M₁₆	—	10	2	no	—
111	C₂×C₈	—	5	4	no	—
112	C₂×C₈	#683	1	2	yes	—
113	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
114	C₂×C₈	D₄×D₅	1	2	yes	—
115	C₂×C₈	Q₈×D₅	1	2	yes	—
116	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	10	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	10	2	no	—
121	C₄²	—	5	4	no	—
122	C₄²	#683	1	2	yes	—
123	C₁₆	—	4	2	no	—
124	C₁₆	—	20	1	no	—
125	C₁₆	—	20	1	no	—
126	C₂×C₅	—	2	2	no	—
127	C₂×C₅	—	2	1	no	—
128	C₂×C₅	#1843	1	1	yes	characteristic
129	C₂³	#4053	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	10	4	no	—
132	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
133	C₂×C₄	#4047	1	2	yes	—
134	C₂×C₄	#4053	1	1	yes	characteristic
135	C₂×C₄	C₂×#683	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	10	4	no	—
138	C₈	#4034	1	2	yes	—
139	C₈	#4086	1	1	yes	characteristic
140	C₈	#4090	1	1	yes	characteristic
141	C₅	#12902	1	1	yes	characteristic, complemented, Sylow
142	C₂²	—	4	1	no	—
143	C₂²	#65128	1	2	yes	—
144	C₂²	#65408	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	20	4	no	—
147	C₄	#65128	1	2	yes	—
148	C₄	#65265	1	1	yes	characteristic
149	C₄	#65391	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#10601239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2153

C₂

yes

maximal, characteristic

#10602216

C₂

yes

maximal

#10602214

C₂

yes

maximal

#10600688

C₂

yes

maximal, characteristic

#10600675

C₂

yes

maximal, characteristic

C₂×C₅×M₃₂

C₂²

yes

—

C₅×#405

C₂²

yes

characteristic

#65390

C₂²

yes

—

C₂×#3944

C₂²

yes

characteristic

C₂²×#672

C₂²

yes

characteristic

C₅×#364

—

#64780

C₄

yes

characteristic

#64777

C₄

yes

characteristic

#64751

C₄

yes

—

—

—

—

—

C₂×C₅×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₅×#117

C₂×C₄

yes

characteristic

#4072

C₂×C₄

yes

—

C₂×#681

C₂×C₄

yes

—

C₅×M₃₂

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₅×#97

Q₈

yes

—

C₄×C₅×C₈

D₄

yes

—

#3947

Q₈

yes

—

C₈×Dic₅

D₄

yes

—

—

—

—

—

—

D₅

yes

—

—

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₅×M₁₆

—

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

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₄⋊C₄

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

C₄⋊C₄

yes

—

#681

—

#681

—

C₂×#168

C₄⋊C₄

yes

—

C₅×C₁₆

—

#672

—

#672

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₅

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂³×C₅

#131

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

#131

yes

characteristic

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

#127

yes

characteristic

C₅×C₈

C₄⋊Q₈

yes

—

C₅×C₈

C₄⋊Q₈

yes

characteristic

#168

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₅

yes

characteristic

C₂²×C₈

C₂²×D₅

yes

characteristic

#117

—

#117

C₄×D₅

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

Dic₁₀

yes

—

C₄×C₈

—

C₄×C₈

C₂²⋊D₅

yes

—

C₂²×C₅

—

100

C₂²×C₅

#424

yes

—

101

C₂²×C₅

#424

yes

characteristic

102

C₄×C₅

—

103

C₄×C₅

#384

yes

characteristic

104

C₄×C₅

#424

yes

—

105

C₄×C₅

#424

yes

characteristic

106

Dic₅

—

107

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

#683

yes

—

113

C₂×C₈

C₂×Dic₁₀

yes

characteristic

114

C₂×C₈

D₄×D₅

yes

—

115

C₂×C₈

Q₈×D₅

yes

—

116

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

#683

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂×C₅

—

127

C₂×C₅

—

128

C₂×C₅

#1843

yes

characteristic

129

C₂³

#4053

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

D₅×C₄⋊C₄

yes

—

133

C₂×C₄

#4047

yes

—

134

C₂×C₄

#4053

yes

characteristic

135

C₂×C₄

C₂×#683

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

#4034

yes

—

139

C₈

#4086

yes

characteristic

140

C₈

#4090

yes

characteristic

141

C₅

#12902

yes

characteristic, complemented, Sylow

142

C₂²

—

143

C₂²

#65128

yes

—

144

C₂²

#65408

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

#65128

yes

—

148

C₄

#65265

yes

characteristic

149

C₄

#65391

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#10601239

yes

characteristic

	subgroup	G/H
center	C₈	#4086
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2153	C₂
socle	C₂×C₅	#1843
prime radical	C₂³×C₅	#131
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2153	D₅
hypocenter	C₅	#12902
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#4086

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅

C₂

C₂×C₅

C₂³×C₅

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
141. C₅	#12902	semidirect

H ≅ G/N

type

141. C₅

#12902

semidirect

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

subgroup	index
C₅×#2153	2
#10602216	2
#10602214	2
#10600688	2
#10600675	2
#12902	5

subgroup

index

Subgroups — #12015723

Conjugacy classes of subgroups

Complements

Maximal subgroups