Group #12013959 — subgroups

Subgroups — #12013959

Order 1280.

262 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2252	C₂	1	1	yes	maximal, characteristic
2	#10602175	C₂	1	2	yes	maximal
3	#10602153	C₂	1	2	yes	maximal
4	#10599859	C₂	1	1	yes	maximal, characteristic
5	C₁₆×#168	C₂	1	1	yes	maximal, characteristic
6	C₅×#500	C₂²	1	2	yes	—
7	#65370	—	2	2	no	—
8	#65368	—	2	2	no	—
9	C₂×Dic₄₀	—	2	2	no	—
10	#64988	C₂²	1	2	yes	—
11	C₅×#366	—	2	2	no	—
12	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	#64712	—	2	4	no	—
14	C₄×#672	C₂²	1	1	yes	characteristic
15	C₈×#168	C₂²	1	1	yes	characteristic
16	#12504	—	5	1	no	maximal, Sylow
17	C₅×C₄⋊Q₈	—	2	2	no	—
18	C₂×Dic₂₀	—	2	2	no	—
19	C₂×Dic₂₀	—	2	2	no	—
20	#4015	—	2	2	no	—
21	C₂×C₅×C₁₆	D₄	1	2	yes	—
22	C₅×#107	—	2	2	no	—
23	C₅×#107	—	2	2	no	—
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	#3967	—	4	4	no	—
26	C₂×#672	D₄	1	2	yes	—
27	#3940	—	4	4	no	—
28	C₄×#168	D₄	1	1	yes	characteristic
29	C₄×#168	D₄	1	1	yes	characteristic
30	Dic₄₀	—	4	4	no	—
31	#2252	D₅	1	1	yes	characteristic
32	#2247	—	5	1	no	—
33	#2241	—	5	2	no	—
34	#1707	—	5	2	no	—
35	C₈×C₁₆	—	5	1	no	—
36	C₂×C₅×Q₈	—	4	2	no	—
37	C₂×Dic₁₀	—	4	2	no	—
38	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
40	C₅×C₄⋊C₄	—	4	2	no	—
41	C₅×C₄⋊C₄	—	4	2	no	—
42	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅	—	4	2	no	—
44	C₄⋊Dic₅	—	4	2	no	—
45	C₂×#168	D₈	1	2	yes	—
46	C₂×#168	D₈	1	2	yes	—
47	Dic₂₀	—	4	4	no	—
48	Dic₂₀	—	4	4	no	—
49	C₅×C₁₆	SD₁₆	1	4	yes	—
50	#672	—	2	2	no	—
51	C₂×Q₃₂	—	10	2	no	—
52	#500	C₂×D₅	1	2	yes	—
53	#498	—	10	2	no	—
54	#494	—	5	2	no	—
55	#487	—	10	2	no	—
56	#366	—	2	2	no	—
57	#358	—	10	4	no	—
58	C₄×C₁₆	—	5	1	no	—
59	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
60	C₈²	—	5	1	no	—
61	C₅×Q₈	—	8	4	no	—
62	C₂×C₄×C₅	—	8	2	no	—
63	C₂×C₄×C₅	#127	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
66	C₂×Dic₅	—	8	2	no	—
67	Dic₁₀	—	8	2	no	—
68	Dic₁₀	—	8	2	no	—
69	C₅×C₈	C₂×SD₁₆	1	2	yes	—
70	C₅×C₈	#135	1	2	yes	—
71	#168	—	2	2	no	—
72	#168	Q₃₂	1	4	yes	complemented
73	C₂×Q₁₆	—	10	2	no	—
74	C₂×Q₁₆	—	10	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	10	2	no	—
77	Q₃₂	—	20	4	no	—
78	C₂×C₁₆	—	5	2	no	—
79	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
80	#107	—	2	2	no	—
81	#107	—	2	2	no	—
82	#106	—	20	4	no	—
83	#103	—	20	4	no	—
84	C₄×C₈	—	5	1	no	—
85	C₄×C₈	—	5	1	no	—
86	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
87	C₂²×C₅	#493	1	1	yes	characteristic
88	C₄×C₅	—	16	2	no	—
89	C₄×C₅	#492	1	1	yes	characteristic
90	C₄×C₅	#495	1	1	yes	characteristic
91	C₄×C₅	C₂×Q₃₂	1	2	yes	—
92	C₄×C₅	#508	1	2	yes	—
93	Dic₅	—	16	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	20	2	no	—
96	Q₁₆	—	20	4	no	—
97	Q₁₆	—	20	4	no	—
98	C₂×C₈	—	5	2	no	—
99	C₂×C₈	—	5	2	no	—
100	C₂×C₈	D₄×D₅	1	1	yes	characteristic
101	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	20	2	no	—
105	C₄⋊C₄	—	20	2	no	—
106	C₄²	D₄×D₅	1	1	yes	characteristic
107	C₁₆	—	10	2	no	—
108	C₁₆	#688	1	4	yes	—
109	C₂×C₅	#1706	1	1	yes	characteristic
110	C₂×C₅	#2247	1	1	yes	characteristic
111	C₂×C₅	#2249	1	1	yes	characteristic
112	Q₈	—	8	4	no	—
113	Q₈	—	40	2	no	—
114	Q₈	—	40	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	40	2	no	—
117	C₂×C₄	D₅×D₈	1	1	yes	characteristic
118	C₂×C₄	D₅×D₈	1	1	yes	characteristic
119	C₂×C₄	#4086	1	1	yes	characteristic
120	C₈	—	5	4	no	—
121	C₈	—	10	2	no	—
122	C₈	#4065	1	2	yes	—
123	C₈	C₂×#688	1	2	yes	—
124	C₅	#12504	1	1	yes	characteristic, complemented, Sylow
125	C₂²	#65440	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	80	2	no	—
128	C₄	#65201	1	2	yes	—
129	C₄	D₅×Q₃₂	1	2	yes	—
130	C₄	#65374	1	1	yes	characteristic
131	C₄	#65480	1	1	yes	characteristic
132	C₂	#10602152	1	1	yes	characteristic
133	C₂	#10602258	1	1	yes	characteristic
134	C₂	#10602292	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2252

C₂

yes

maximal, characteristic

#10602175

C₂

yes

maximal

#10602153

C₂

yes

maximal

#10599859

C₂

yes

maximal, characteristic

C₁₆×#168

C₂

yes

maximal, characteristic

C₅×#500

C₂²

yes

—

#65370

—

#65368

—

C₂×Dic₄₀

—

#64988

C₂²

yes

—

C₅×#366

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

#64712

—

C₄×#672

C₂²

yes

characteristic

C₈×#168

C₂²

yes

characteristic

#12504

—

maximal, Sylow

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₅×#107

—

C₅×#107

—

C₄×C₅×C₈

C₂³

yes

characteristic

#3967

—

C₂×#672

D₄

yes

—

#3940

—

C₄×#168

D₄

yes

characteristic

C₄×#168

D₄

yes

characteristic

Dic₄₀

—

#2252

D₅

yes

—

—

—

C₈×C₁₆

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

D₈

yes

—

C₂×#168

D₈

yes

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

SD₁₆

yes

—

#672

—

C₂×Q₃₂

—

#500

C₂×D₅

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

C₈²

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

C₂×SD₁₆

yes

—

C₅×C₈

#135

yes

—

#168

—

#168

Q₃₂

yes

complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

C₂×Q₃₂

yes

—

C₄×C₅

#508

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

D₄×D₅

yes

characteristic

101

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

D₄×D₅

yes

characteristic

107

C₁₆

—

108

C₁₆

#688

yes

—

109

C₂×C₅

#1706

yes

characteristic

110

C₂×C₅

#2247

yes

characteristic

111

C₂×C₅

#2249

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₅×D₈

yes

characteristic

118

C₂×C₄

D₅×D₈

yes

characteristic

119

C₂×C₄

#4086

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

#4065

yes

—

123

C₈

C₂×#688

yes

—

124

C₅

#12504

yes

characteristic, complemented, Sylow

125

C₂²

#65440

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#65201

yes

—

129

C₄

D₅×Q₃₂

yes

—

130

C₄

#65374

yes

characteristic

131

C₄

#65480

yes

characteristic

132

C₂

#10602152

yes

characteristic

133

C₂

#10602258

yes

characteristic

134

C₂

#10602292

yes

characteristic

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2252	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#2252	D₅
hypocenter	C₅	#12504
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#65440

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
72. #168	Q₃₂	semidirect
124. C₅	#12504	semidirect

H ≅ G/N

type

72. #168

Q₃₂

semidirect

124. C₅

#12504

semidirect

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

subgroup	index
C₅×#2252	2
#10602175	2
#10602153	2
#10599859	2
C₁₆×#168	2
#12504	5

subgroup

index

C₁₆×#168

Subgroups — #12013959

Conjugacy classes of subgroups

Complements

Maximal subgroups