Group #12014028 — subgroups

Subgroups — #12014028

Order 1280.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#10599854	C₂	1	2	yes	maximal, complemented
2	#10599853	C₂	1	2	yes	maximal, complemented
3	#10598855	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598853	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#3932	—	2	4	no	—
7	C₂×D₈₀	—	2	4	no	—
8	#64988	C₂²	1	1	yes	characteristic
9	#64984	—	2	2	no	—
10	#64984	C₂²	1	1	yes	characteristic
11	#64984	C₂²	1	2	yes	—
12	#64983	—	2	2	no	—
13	C₄×C₅×C₁₆	C₂²	1	2	yes	—
14	C₅×C₈²	C₂²	1	1	yes	characteristic
15	#12519	—	5	1	no	maximal, Sylow
16	C₂×Dic₂₀	—	2	2	no	—
17	C₂×D₄₀	—	2	2	no	—
18	C₂×D₄₀	—	2	2	no	—
19	C₂×D₄₀	—	2	2	no	—
20	C₂×D₄₀	—	4	4	no	—
21	#4020	—	2	1	no	—
22	#4020	—	2	2	no	—
23	#4015	—	2	1	no	—
24	C₂×C₅×C₁₆	D₄	1	4	yes	—
25	C₄×C₅×C₈	D₄	1	2	yes	—
26	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
27	#3953	—	4	4	no	—
28	#3932	—	4	8	no	—
29	D₈₀	—	4	8	no	—
30	#2248	—	5	2	no	—
31	#2246	—	5	2	no	—
32	#1712	—	5	1	no	—
33	#1709	—	5	1	no	—
34	#1311	D₅	1	1	yes	characteristic, complemented
35	C₂×D₂₀	—	4	1	no	—
36	C₂×D₂₀	—	4	2	no	—
37	C₂×D₂₀	—	4	2	no	—
38	C₂×D₂₀	—	4	4	no	—
39	C₂×Dic₁₀	—	4	1	no	—
40	C₂×C₅×C₈	D₈	1	4	yes	—
41	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
42	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅	—	4	2	no	—
44	Dic₂₀	—	4	4	no	—
45	D₄₀	—	4	4	no	—
46	D₄₀	—	4	4	no	—
47	D₄₀	—	4	4	no	—
48	D₄₀	—	8	8	no	—
49	C₅×C₁₆	D₈	1	8	yes	complemented
50	C₂×SD₃₂	—	10	4	no	—
51	C₂×D₁₆	—	10	4	no	—
52	#494	—	5	1	no	—
53	#493	—	5	1	no	—
54	#493	—	5	2	no	—
55	#493	—	10	2	no	—
56	#486	—	10	2	no	—
57	C₄×C₁₆	C₂×D₅	1	2	yes	—
58	C₈²	C₂×D₅	1	1	yes	characteristic
59	C₂²×D₅	—	8	1	no	—
60	C₂²×D₅	—	8	2	no	—
61	C₂×C₄×C₅	#127	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
63	C₂×Dic₅	—	8	1	no	—
64	D₂₀	—	8	2	no	—
65	D₂₀	—	8	2	no	—
66	D₂₀	—	8	2	no	—
67	D₂₀	—	8	8	no	—
68	D₂₀	—	8	4	no	—
69	Dic₁₀	—	8	2	no	—
70	C₅×C₈	—	2	4	no	—
71	C₅×C₈	C₂×D₈	1	4	yes	—
72	C₂×Q₁₆	—	10	2	no	—
73	C₂×D₈	—	10	2	no	—
74	C₂×D₈	—	10	2	no	—
75	C₂×D₈	—	10	2	no	—
76	C₂×D₈	—	20	4	no	—
77	C₄⋊Q₈	—	10	1	no	—
78	#127	—	10	1	no	—
79	#127	—	10	2	no	—
80	SD₃₂	—	20	8	no	—
81	D₁₆	—	20	8	no	—
82	C₂×C₁₆	D₂₀	1	4	yes	—
83	#102	—	20	4	no	—
84	C₄×C₈	D₂₀	1	2	yes	—
85	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
86	C₂²×C₅	#493	1	1	yes	characteristic
87	C₂×D₅	—	16	1	no	—
88	C₂×D₅	—	16	2	no	—
89	C₂×D₅	—	16	4	no	—
90	C₂×D₅	—	16	2	no	—
91	C₄×C₅	#493	1	2	yes	—
92	C₄×C₅	#509	1	4	yes	—
93	Dic₅	—	16	1	no	—
94	C₂×Q₈	—	20	1	no	—
95	C₂×D₄	—	20	1	no	—
96	C₂×D₄	—	20	2	no	—
97	C₂×D₄	—	20	2	no	—
98	C₂×D₄	—	20	4	no	—
99	Q₁₆	—	20	4	no	—
100	D₈	—	20	4	no	—
101	D₈	—	20	4	no	—
102	D₈	—	20	4	no	—
103	D₈	—	40	8	no	—
104	C₂×C₈	D₄₀	1	4	yes	—
105	C₂×C₈	C₂×D₂₀	1	2	yes	—
106	C₄⋊C₄	—	20	2	no	—
107	C₄²	C₂×D₂₀	1	1	yes	characteristic
108	C₁₆	D₄₀	1	8	yes	complemented
109	C₂×C₅	#1712	1	1	yes	characteristic
110	C₂×C₅	#2250	1	2	yes	—
111	D₅	—	32	2	no	—
112	D₅	—	32	1	no	—
113	C₂³	—	40	1	no	—
114	C₂³	—	40	2	no	—
115	Q₈	—	40	2	no	—
116	D₄	—	40	8	no	—
117	D₄	—	40	4	no	—
118	D₄	—	40	2	no	—
119	D₄	—	40	2	no	—
120	D₄	—	40	2	no	—
121	C₂×C₄	—	40	1	no	—
122	C₂×C₄	#4020	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄₀	1	2	yes	—
124	C₈	—	2	4	no	—
125	C₈	C₂×D₄₀	1	4	yes	—
126	C₅	#12519	1	1	yes	characteristic, complemented, Sylow
127	C₂²	—	80	4	no	—
128	C₂²	—	80	2	no	—
129	C₂²	—	80	1	no	—
130	C₂²	—	80	2	no	—
131	C₂²	#64984	1	1	yes	characteristic
132	C₄	—	80	1	no	—
133	C₄	#64984	1	2	yes	—
134	C₄	#65197	1	4	yes	—
135	C₂	—	160	2	no	—
136	C₂	—	160	1	no	—
137	C₂	#10598855	1	1	yes	characteristic
138	C₂	#10599874	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10599854

C₂

yes

maximal, complemented

#10599853

C₂

yes

maximal, complemented

#10598855

C₂

yes

maximal, characteristic, complemented

#10598853

C₂

yes

maximal, characteristic, complemented

C₅×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#3932

—

C₂×D₈₀

—

#64988

C₂²

yes

characteristic

#64984

—

#64984

C₂²

yes

characteristic

#64984

C₂²

yes

—

#64983

—

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

characteristic

#12519

—

maximal, Sylow

C₂×Dic₂₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

#4020

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

characteristic

#3953

—

#3932

—

D₈₀

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

Dic₂₀

—

D₄₀

—

D₄₀

—

D₄₀

—

D₄₀

—

C₅×C₁₆

D₈

yes

complemented

C₂×SD₃₂

—

C₂×D₁₆

—

—

—

—

—

—

C₄×C₁₆

C₂×D₅

yes

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₂₀

yes

—

#102

—

C₄×C₈

D₂₀

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#493

yes

—

C₄×C₅

#509

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

D₈

—

101

D₈

—

102

D₈

—

103

D₈

—

104

C₂×C₈

D₄₀

yes

—

105

C₂×C₈

C₂×D₂₀

yes

—

106

C₄⋊C₄

—

107

C₄²

C₂×D₂₀

yes

characteristic

108

C₁₆

D₄₀

yes

complemented

109

C₂×C₅

#1712

yes

characteristic

110

C₂×C₅

#2250

yes

—

111

D₅

—

112

D₅

—

113

C₂³

—

114

C₂³

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

#4020

yes

characteristic

123

C₂×C₄

C₂×D₄₀

yes

—

124

C₈

—

125

C₈

C₂×D₄₀

yes

—

126

C₅

#12519

yes

characteristic, complemented, Sylow

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#64984

yes

characteristic

132

C₄

—

133

C₄

#64984

yes

—

134

C₄

#65197

yes

—

135

C₂

—

160

—

136

C₂

—

160

—

137

C₂

#10598855

yes

characteristic

138

C₂

#10599874

yes

—

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1311	C₂
socle	C₂²×C₅	#493
hypercenter	#1311	D₅
hypocenter	C₅	#12519
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#64984

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 ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #10599854	C₂	semidirect
2. #10599853	C₂	semidirect
3. #10598855	C₂	semidirect
4. #10598853	C₂	semidirect
5. C₅×#1311	C₂	semidirect
34. #1311	D₅	semidirect
49. C₅×C₁₆	D₈	semidirect
108. C₁₆	D₄₀	semidirect
126. C₅	#12519	semidirect

H ≅ G/N

type

1. #10599854

C₂

semidirect

2. #10599853

C₂

semidirect

3. #10598855

C₂

semidirect

4. #10598853

C₂

semidirect

5. C₅×#1311

C₂

semidirect

34. #1311

D₅

semidirect

49. C₅×C₁₆

D₈

semidirect

108. C₁₆

D₄₀

semidirect

126. C₅

#12519

semidirect

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

subgroup	index
#10599854	2
#10599853	2
#10598855	2
#10598853	2
C₅×#1311	2
#12519	5

subgroup

index

Subgroups — #12014028

Conjugacy classes of subgroups

Complements

Maximal subgroups