Group #12014029 — subgroups

Subgroups — #12014029

Order 1280.

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

normal subgroups only

This group has 360 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×#1709	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602106	C₂	1	2	yes	maximal, complemented
3	#10602083	C₂	1	2	yes	maximal, complemented
4	#10598855	C₂	1	1	yes	maximal, characteristic, complemented
5	#10597928	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#494	C₂²	1	1	yes	characteristic
7	C₅×#493	C₂²	1	1	yes	characteristic
8	C₅×#486	—	2	2	no	—
9	C₂×#3960	—	2	4	no	—
10	C₂×#3958	—	2	4	no	—
11	#64984	—	2	2	no	—
12	#64984	C₂²	1	2	yes	—
13	C₅×C₈²	C₂²	1	1	yes	characteristic
14	C₄×#672	C₂²	1	2	yes	—
15	#12519	—	5	1	no	maximal, Sylow
16	C₂×C₅×Q₁₆	—	2	2	no	—
17	C₂×C₅×D₈	—	2	2	no	—
18	C₅×C₄⋊Q₈	—	2	1	no	—
19	C₅×#127	—	2	1	no	—
20	C₂×D₄₀	—	2	2	no	—
21	C₂×D₄₀	—	2	2	no	—
22	C₂×D₄₀	—	4	4	no	—
23	#4020	—	2	2	no	—
24	C₅×#102	—	4	4	no	—
25	C₄×C₅×C₈	D₄	1	2	yes	—
26	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
27	#3960	—	4	8	no	—
28	#3958	—	4	8	no	—
29	C₂×#672	D₄	1	4	yes	—
30	#2248	—	5	2	no	—
31	#2246	—	5	2	no	—
32	#1712	—	5	1	no	—
33	#1709	D₅	1	1	yes	characteristic, complemented
34	#1311	—	5	1	no	—
35	C₂×C₅×Q₈	—	4	1	no	—
36	C₂×C₅×D₄	—	4	1	no	—
37	C₂×C₅×D₄	—	4	2	no	—
38	C₂×D₂₀	—	4	2	no	—
39	C₂×D₂₀	—	4	4	no	—
40	C₅×Q₁₆	—	4	4	no	—
41	C₅×D₈	—	4	4	no	—
42	C₂×C₅×C₈	D₈	1	4	yes	—
43	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
44	C₅×C₄⋊C₄	—	4	2	no	—
45	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
46	D₄₀	—	4	4	no	—
47	D₄₀	—	4	4	no	—
48	D₄₀	—	8	8	no	—
49	#672	D₈	1	8	yes	complemented
50	C₂×SD₃₂	—	10	4	no	—
51	C₂×D₁₆	—	10	4	no	—
52	#494	C₂×D₅	1	1	yes	characteristic
53	#493	—	5	2	no	—
54	#493	—	10	2	no	—
55	#493	C₂×D₅	1	1	yes	characteristic
56	#486	—	2	2	no	—
57	C₄×C₁₆	—	5	2	no	—
58	C₈²	C₂×D₅	1	1	yes	characteristic
59	C₂³×C₅	—	8	1	no	—
60	C₂²×D₅	—	8	2	no	—
61	C₅×Q₈	—	8	2	no	—
62	C₅×D₄	—	8	2	no	—
63	C₅×D₄	—	8	4	no	—
64	C₂×C₄×C₅	—	8	1	no	—
65	C₂×C₄×C₅	#127	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
67	D₂₀	—	8	2	no	—
68	D₂₀	—	8	2	no	—
69	D₂₀	—	8	8	no	—
70	C₅×C₈	—	2	4	no	—
71	C₅×C₈	C₂×D₈	1	4	yes	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	—	10	2	no	—
75	C₂×D₈	—	10	2	no	—
76	C₂×D₈	—	20	4	no	—
77	C₄⋊Q₈	—	2	1	no	—
78	#127	—	2	1	no	—
79	#127	—	10	2	no	—
80	SD₃₂	—	20	8	no	—
81	D₁₆	—	20	8	no	—
82	C₂×C₁₆	—	5	4	no	—
83	#102	—	4	4	no	—
84	C₄×C₈	C₂²⋊D₅	1	2	yes	—
85	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
86	C₂²×C₅	—	16	1	no	—
87	C₂²×C₅	—	16	2	no	—
88	C₂²×C₅	#493	1	1	yes	characteristic
89	C₂×D₅	—	16	2	no	—
90	C₂×D₅	—	16	4	no	—
91	C₄×C₅	—	16	1	no	—
92	C₄×C₅	#493	1	2	yes	—
93	C₄×C₅	#509	1	4	yes	—
94	C₂×Q₈	—	4	1	no	—
95	C₂×D₄	—	4	2	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂×D₄	—	20	4	no	—
98	C₂×D₄	—	20	2	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	4	4	no	—
101	D₈	—	20	4	no	—
102	D₈	—	20	4	no	—
103	D₈	—	40	8	no	—
104	C₂×C₈	#686	1	4	yes	—
105	C₂×C₈	D₄×D₅	1	2	yes	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
108	C₁₆	—	5	8	no	—
109	C₂×C₅	—	32	1	no	—
110	C₂×C₅	#1712	1	1	yes	characteristic
111	C₂×C₅	#2250	1	2	yes	—
112	D₅	—	32	2	no	—
113	C₂³	—	8	1	no	—
114	C₂³	—	40	2	no	—
115	Q₈	—	8	2	no	—
116	D₄	—	8	2	no	—
117	D₄	—	8	4	no	—
118	D₄	—	40	2	no	—
119	D₄	—	40	2	no	—
120	D₄	—	40	8	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	C₂×#686	1	2	yes	—
123	C₂×C₄	#4086	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₈	D₅×D₈	1	4	yes	—
126	C₅	#12519	1	1	yes	characteristic, complemented, Sylow
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	80	2	no	—
130	C₂²	—	80	4	no	—
131	C₂²	#65362	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#65440	1	2	yes	—
134	C₄	#65482	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	160	2	no	—
137	C₂	#10602069	1	1	yes	characteristic
138	C₂	#10602114	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1709

C₂

yes

maximal, characteristic, complemented

#10602106

C₂

yes

maximal, complemented

#10602083

C₂

yes

maximal, complemented

#10598855

C₂

yes

maximal, characteristic, complemented

#10597928

C₂

yes

maximal, characteristic, complemented

C₅×#494

C₂²

yes

characteristic

C₅×#493

C₂²

yes

—

—

—

—

C₂²

yes

—

C₅×C₈²

C₂²

yes

characteristic

C₄×#672

C₂²

yes

—

#12519

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₂×C₅×D₈

—

C₅×C₄⋊Q₈

—

C₅×#127

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

C₅×#102

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

D₄

yes

—

—

—

—

D₅

yes

characteristic, complemented

#1311

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×Q₁₆

—

C₅×D₈

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

D₄₀

—

D₄₀

—

D₄₀

—

#672

D₈

yes

complemented

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₅

yes

—

—

C₂×D₅

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₂³×C₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

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

—

#102

—

C₄×C₈

C₂²⋊D₅

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#493

yes

—

C₄×C₅

#509

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

D₈

—

101

D₈

—

102

D₈

—

103

D₈

—

104

C₂×C₈

#686

yes

—

105

C₂×C₈

D₄×D₅

yes

—

106

C₄⋊C₄

—

107

C₄²

C₂×C₂²⋊D₅

yes

characteristic

108

C₁₆

—

109

C₂×C₅

—

110

C₂×C₅

#1712

yes

characteristic

111

C₂×C₅

#2250

yes

—

112

D₅

—

113

C₂³

—

114

C₂³

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×#686

yes

—

123

C₂×C₄

#4086

yes

characteristic

124

C₈

—

125

C₈

D₅×D₈

yes

—

126

C₅

#12519

yes

characteristic, complemented, Sylow

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#65362

yes

characteristic

132

C₄

—

133

C₄

#65440

yes

—

134

C₄

#65482

yes

—

135

C₂

—

136

C₂

—

160

—

137

C₂

#10602069

yes

characteristic

138

C₂

#10602114

yes

—

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1709	C₂
socle	C₂²×C₅	#493
hypercenter	#1709	D₅
hypocenter	C₅	#12519
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#65362

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 ◁ #1709

N	H ≅ G/N	type
1. C₅×#1709	C₂	semidirect
2. #10602106	C₂	semidirect
3. #10602083	C₂	semidirect
4. #10598855	C₂	semidirect
5. #10597928	C₂	semidirect
33. #1709	D₅	semidirect
49. #672	D₈	semidirect
126. C₅	#12519	semidirect

H ≅ G/N

type

1. C₅×#1709

C₂

semidirect

2. #10602106

C₂

semidirect

3. #10602083

C₂

semidirect

4. #10598855

C₂

semidirect

5. #10597928

C₂

semidirect

33. #1709

D₅

semidirect

49. #672

D₈

semidirect

126. C₅

#12519

semidirect

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

subgroup	index
C₅×#1709	2
#10602106	2
#10602083	2
#10598855	2
#10597928	2
#12519	5

subgroup

index

Subgroups — #12014029

Conjugacy classes of subgroups

Complements

Maximal subgroups