Group #12013971 — subgroups

Subgroups — #12013971

Order 1280.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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×#2254	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602174	C₂	1	2	yes	maximal, complemented
3	#10602152	C₂	1	2	yes	maximal, complemented
4	#10599853	C₂	1	1	yes	maximal, characteristic
5	#10597989	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#500	C₂²	1	2	yes	—
7	#65377	—	2	2	no	—
8	#65374	—	2	2	no	—
9	C₂×D₈₀	—	2	2	no	—
10	#64984	C₂²	1	2	yes	—
11	C₅×#367	—	2	2	no	—
12	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	#64711	—	2	4	no	—
14	C₄×#672	C₂²	1	1	yes	characteristic
15	C₈×#168	C₂²	1	1	yes	characteristic
16	#12506	—	5	1	no	maximal, Sylow
17	C₅×C₄⋊Q₈	—	2	2	no	—
18	C₂×#688	—	4	4	no	—
19	C₂×D₄₀	—	2	2	no	—
20	C₂×D₄₀	—	2	2	no	—
21	#4020	—	2	2	no	—
22	C₂×C₅×C₁₆	D₄	1	2	yes	—
23	C₅×#107	—	2	2	no	—
24	C₅×#107	—	2	2	no	—
25	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂×#672	D₄	1	2	yes	—
27	#3941	—	4	4	no	—
28	C₄×#168	D₄	1	2	yes	—
29	D₈₀	—	4	4	no	—
30	#2254	D₅	1	1	yes	characteristic, complemented
31	#2246	—	5	1	no	—
32	#2240	—	5	2	no	—
33	#1706	—	5	2	no	—
34	#1311	—	5	1	no	—
35	C₂×C₅×Q₈	—	4	2	no	—
36	C₂×D₂₀	—	4	2	no	—
37	C₂×D₂₀	—	4	2	no	—
38	C₂×D₂₀	—	4	2	no	—
39	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
41	C₅×C₄⋊C₄	—	4	2	no	—
42	C₅×C₄⋊C₄	—	4	2	no	—
43	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
44	#688	—	8	8	no	—
45	C₂×#168	D₈	1	4	yes	—
46	D₄₀	—	4	4	no	—
47	D₄₀	—	4	4	no	—
48	C₅×C₁₆	SD₁₆	1	4	yes	complemented
49	#672	—	2	2	no	—
50	C₂×D₁₆	—	10	2	no	—
51	#500	C₂×D₅	1	2	yes	—
52	#493	—	5	2	no	—
53	#492	—	10	2	no	—
54	#486	—	10	2	no	—
55	#367	—	2	2	no	—
56	#357	—	10	4	no	—
57	C₄×C₁₆	—	5	1	no	—
58	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
59	C₈²	—	5	1	no	—
60	C₂²×D₅	—	8	2	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	2	yes	—
65	D₂₀	—	8	2	no	—
66	D₂₀	—	8	2	no	—
67	D₂₀	—	8	4	no	—
68	D₂₀	—	8	4	no	—
69	C₅×C₈	C₂×SD₁₆	1	2	yes	—
70	C₅×C₈	#135	1	2	yes	—
71	#168	—	2	4	no	—
72	C₂×SD₁₆	—	20	4	no	—
73	C₂×D₈	—	10	2	no	—
74	C₂×D₈	—	10	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	#127	—	10	2	no	—
77	D₁₆	—	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	#102	—	20	4	no	—
83	C₄×C₈	—	5	2	no	—
84	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
85	C₂²×C₅	#493	1	1	yes	characteristic
86	C₂×D₅	—	16	2	no	—
87	C₂×D₅	—	16	2	no	—
88	C₂×D₅	—	16	2	no	—
89	C₄×C₅	—	16	2	no	—
90	C₄×C₅	#492	1	1	yes	characteristic
91	C₄×C₅	#495	1	1	yes	characteristic
92	C₄×C₅	#509	1	4	yes	—
93	C₂×Q₈	—	4	2	no	—
94	C₂×D₄	—	20	2	no	—
95	C₂×D₄	—	20	2	no	—
96	C₂×D₄	—	20	2	no	—
97	SD₁₆	—	40	8	no	—
98	D₈	—	20	4	no	—
99	D₈	—	20	4	no	—
100	C₂×C₈	—	5	4	no	—
101	C₂×C₈	D₄×D₅	1	1	yes	characteristic
102	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄²	D₄×D₅	1	1	yes	characteristic
106	C₁₆	—	10	2	no	—
107	C₁₆	#688	1	4	yes	complemented
108	C₂×C₅	#1706	1	1	yes	characteristic
109	C₂×C₅	#2250	1	2	yes	—
110	D₅	—	32	2	no	—
111	C₂³	—	40	2	no	—
112	Q₈	—	8	4	no	—
113	D₄	—	40	2	no	—
114	D₄	—	40	4	no	—
115	D₄	—	40	4	no	—
116	D₄	—	40	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	D₅×D₈	1	2	yes	—
119	C₂×C₄	#4086	1	1	yes	characteristic
120	C₈	—	10	4	no	—
121	C₈	#4065	1	2	yes	—
122	C₈	C₂×#688	1	2	yes	—
123	C₅	#12506	1	1	yes	characteristic, complemented, Sylow
124	C₂²	—	80	2	no	—
125	C₂²	—	80	2	no	—
126	C₂²	—	80	2	no	—
127	C₂²	#65440	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#65203	1	4	yes	—
130	C₄	#65374	1	1	yes	characteristic
131	C₄	#65480	1	1	yes	characteristic
132	C₂	—	160	2	no	—
133	C₂	#10602152	1	1	yes	characteristic
134	C₂	#10602274	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2254

C₂

yes

maximal, characteristic, complemented

#10602174

C₂

yes

maximal, complemented

#10602152

C₂

yes

maximal, complemented

#10599853

C₂

yes

maximal, characteristic

#10597989

C₂

yes

maximal, characteristic, complemented

C₅×#500

C₂²

yes

—

#65377

—

#65374

—

C₂×D₈₀

—

#64984

C₂²

yes

—

C₅×#367

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

#64711

—

C₄×#672

C₂²

yes

characteristic

C₈×#168

C₂²

yes

characteristic

#12506

—

maximal, Sylow

C₅×C₄⋊Q₈

—

C₂×#688

—

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

C₂×C₅×C₁₆

D₄

yes

—

C₅×#107

—

C₅×#107

—

C₄×C₅×C₈

C₂³

yes

characteristic

C₂×#672

D₄

yes

—

#3941

—

C₄×#168

D₄

yes

—

D₈₀

—

#2254

D₅

yes

characteristic, complemented

—

—

—

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

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

#688

—

C₂×#168

D₈

yes

—

D₄₀

—

D₄₀

—

C₅×C₁₆

SD₁₆

yes

complemented

#672

—

C₂×D₁₆

—

#500

C₂×D₅

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

C₈²

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

C₂×SD₁₆

yes

—

C₅×C₈

#135

yes

—

#168

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

—

#107

—

#107

—

#102

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

100

C₂×C₈

—

101

C₂×C₈

D₄×D₅

yes

characteristic

102

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

D₄×D₅

yes

characteristic

106

C₁₆

—

107

C₁₆

#688

yes

complemented

108

C₂×C₅

#1706

yes

characteristic

109

C₂×C₅

#2250

yes

—

110

D₅

—

111

C₂³

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

D₅×D₈

yes

—

119

C₂×C₄

#4086

yes

characteristic

120

C₈

—

121

C₈

#4065

yes

—

122

C₈

C₂×#688

yes

—

123

C₅

#12506

yes

characteristic, complemented, Sylow

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#65440

yes

characteristic

128

C₄

—

129

C₄

#65203

yes

—

130

C₄

#65374

yes

characteristic

131

C₄

#65480

yes

characteristic

132

C₂

—

160

—

133

C₂

#10602152

yes

characteristic

134

C₂

#10602274

yes

—

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2254	C₂
socle	C₂²×C₅	#493
prime radical	#10599853	C₂
hypercenter	#2254	D₅
hypocenter	C₅	#12506
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#65440

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
1. C₅×#2254	C₂	semidirect
2. #10602174	C₂	semidirect
3. #10602152	C₂	semidirect
5. #10597989	C₂	semidirect
30. #2254	D₅	semidirect
48. C₅×C₁₆	SD₁₆	semidirect
107. C₁₆	#688	semidirect
123. C₅	#12506	semidirect

H ≅ G/N

type

1. C₅×#2254

C₂

semidirect

2. #10602174

C₂

semidirect

3. #10602152

C₂

semidirect

5. #10597989

C₂

semidirect

30. #2254

D₅

semidirect

48. C₅×C₁₆

SD₁₆

semidirect

107. C₁₆

#688

semidirect

123. C₅

#12506

semidirect

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

subgroup	index
C₅×#2254	2
#10602174	2
#10602152	2
#10599853	2
#10597989	2
#12506	5

subgroup

index

Subgroups — #12013971

Conjugacy classes of subgroups

Complements

Maximal subgroups