Group #12014101 — subgroups

Subgroups — #12014101

Order 1280.

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

normal subgroups only

This group has 250 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×#1707	C₂	1	1	yes	maximal, characteristic
2	#10602177	C₂	1	2	yes	maximal
3	#10602103	C₂	1	2	yes	maximal
4	#10598858	C₂	1	1	yes	maximal, characteristic
5	#10597929	C₂	1	1	yes	maximal, characteristic
6	C₅×#500	C₂²	1	1	yes	characteristic
7	C₅×#498	—	2	1	no	—
8	C₅×#494	C₂²	1	1	yes	characteristic
9	C₅×#487	—	2	1	no	—
10	#64988	—	2	1	no	—
11	#64971	—	2	1	no	—
12	#64971	C₂²	1	2	yes	—
13	C₅×C₈²	C₂²	1	1	yes	characteristic
14	#64784	—	2	4	no	—
15	#64709	—	2	4	no	—
16	C₄×#672	C₂²	1	2	yes	—
17	#12534	—	5	1	no	maximal, Sylow
18	C₂×C₅×Q₁₆	—	2	2	no	—
19	C₅×C₄⋊Q₈	—	2	1	no	—
20	C₅×C₄⋊Q₈	—	2	1	no	—
21	C₂×Dic₂₀	—	4	2	no	—
22	#4015	—	2	2	no	—
23	C₅×#107	—	2	2	no	—
24	C₅×#106	—	4	2	no	—
25	C₅×#103	—	4	2	no	—
26	C₄×C₅×C₈	D₄	1	1	yes	characteristic
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
29	#3950	—	2	2	no	—
30	#3950	—	2	2	no	—
31	#3950	—	4	2	no	—
32	C₂×#672	D₄	1	4	yes	—
33	#2252	—	5	2	no	—
34	#2241	—	5	2	no	—
35	#1713	—	5	1	no	—
36	#1707	D₅	1	1	yes	characteristic
37	#1312	—	5	1	no	—
38	C₂×C₅×Q₈	—	4	1	no	—
39	C₂×C₅×Q₈	—	4	1	no	—
40	C₂×Dic₁₀	—	4	2	no	—
41	C₅×Q₁₆	—	4	4	no	—
42	C₂×C₅×C₈	D₈	1	2	yes	—
43	C₂×C₅×C₈	D₈	1	2	yes	—
44	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
45	C₅×C₄⋊C₄	—	4	1	no	—
46	C₅×C₄⋊C₄	—	4	1	no	—
47	C₅×C₄⋊C₄	—	4	1	no	—
48	C₅×C₄⋊C₄	—	4	1	no	—
49	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
50	C₄⋊Dic₅	—	4	2	no	—
51	C₄⋊Dic₅	—	4	2	no	—
52	Dic₂₀	—	8	4	no	—
53	#672	—	2	4	no	—
54	#500	—	5	2	no	—
55	#500	—	10	1	no	—
56	#500	C₂×D₅	1	1	yes	characteristic
57	#498	—	2	1	no	—
58	#494	—	10	1	no	—
59	#494	C₂×D₅	1	1	yes	characteristic
60	#487	—	2	1	no	—
61	#366	—	10	4	no	—
62	#358	—	10	4	no	—
63	C₄×C₁₆	—	5	2	no	—
64	C₈²	C₂×D₅	1	1	yes	characteristic
65	C₅×Q₈	—	8	2	no	—
66	C₅×Q₈	—	8	2	no	—
67	C₂×C₄×C₅	—	8	1	no	—
68	C₂×C₄×C₅	—	8	1	no	—
69	C₂×C₄×C₅	#127	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
72	C₂×Dic₅	—	8	2	no	—
73	Dic₁₀	—	8	4	no	—
74	C₅×C₈	—	2	2	no	—
75	C₅×C₈	Q₃₂	1	4	yes	—
76	C₅×C₈	#137	1	4	yes	—
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×Q₁₆	—	20	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	2	1	no	—
81	C₄⋊Q₈	—	10	2	no	—
82	C₂×C₁₆	—	5	4	no	—
83	#107	—	2	2	no	—
84	#107	—	10	2	no	—
85	#107	—	10	2	no	—
86	#107	—	20	2	no	—
87	#106	—	4	2	no	—
88	#103	—	4	2	no	—
89	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
92	C₂²×C₅	#493	1	1	yes	characteristic
93	C₄×C₅	—	16	1	no	—
94	C₄×C₅	—	16	1	no	—
95	C₄×C₅	#497	1	2	yes	—
96	C₄×C₅	C₂×Q₃₂	1	2	yes	—
97	C₄×C₅	#509	1	2	yes	—
98	Dic₅	—	16	2	no	—
99	C₂×Q₈	—	4	1	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂×Q₈	—	20	2	no	—
102	Q₁₆	—	4	4	no	—
103	Q₁₆	—	40	4	no	—
104	C₂×C₈	#686	1	2	yes	—
105	C₂×C₈	#686	1	2	yes	—
106	C₂×C₈	D₄×D₅	1	2	yes	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	20	2	no	—
112	C₄⋊C₄	—	20	2	no	—
113	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
114	C₁₆	—	10	4	no	—
115	C₂×C₅	#1722	1	1	yes	characteristic
116	C₂×C₅	#2247	1	1	yes	characteristic
117	C₂×C₅	#2250	1	1	yes	characteristic
118	Q₈	—	8	2	no	—
119	Q₈	—	8	2	no	—
120	Q₈	—	40	4	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	40	2	no	—
124	C₂×C₄	C₂×#686	1	1	yes	characteristic
125	C₂×C₄	C₂×#686	1	1	yes	characteristic
126	C₂×C₄	#4086	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	#3961	1	4	yes	—
129	C₈	#4061	1	4	yes	—
130	C₅	#12534	1	1	yes	characteristic, complemented, Sylow
131	C₂²	#65362	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	—	16	1	no	—
134	C₄	—	80	2	no	—
135	C₄	#65456	1	2	yes	—
136	C₄	C₂×#3961	1	2	yes	—
137	C₄	#65482	1	2	yes	—
138	C₂	#10602056	1	1	yes	characteristic
139	C₂	#10602105	1	1	yes	characteristic
140	C₂	#10602114	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1707

C₂

yes

maximal, characteristic

#10602177

C₂

yes

maximal

#10602103

C₂

yes

maximal

#10598858

C₂

yes

maximal, characteristic

#10597929

C₂

yes

maximal, characteristic

C₅×#500

C₂²

yes

characteristic

C₅×#498

—

C₅×#494

C₂²

yes

—

—

—

C₂²

yes

—

C₅×C₈²

C₂²

yes

—

—

C₂²

yes

—

#12534

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

—

—

—

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

—

D₄

yes

—

—

—

—

D₅

yes

characteristic

#1312

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

—

—

—

C₂×D₅

yes

—

—

C₂×D₅

yes

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

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

—

C₅×C₈

—

C₅×C₈

Q₃₂

yes

—

C₅×C₈

#137

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#497

yes

—

C₄×C₅

C₂×Q₃₂

yes

—

C₄×C₅

#509

yes

—

Dic₅

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

Q₁₆

—

103

Q₁₆

—

104

C₂×C₈

#686

yes

—

105

C₂×C₈

#686

yes

—

106

C₂×C₈

D₄×D₅

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂×C₂²⋊D₅

yes

characteristic

114

C₁₆

—

115

C₂×C₅

#1722

yes

characteristic

116

C₂×C₅

#2247

yes

characteristic

117

C₂×C₅

#2250

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×#686

yes

characteristic

125

C₂×C₄

C₂×#686

yes

characteristic

126

C₂×C₄

#4086

yes

characteristic

127

C₈

—

128

C₈

#3961

yes

—

129

C₈

#4061

yes

—

130

C₅

#12534

yes

characteristic, complemented, Sylow

131

C₂²

#65362

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#65456

yes

—

136

C₄

C₂×#3961

yes

—

137

C₄

#65482

yes

—

138

C₂

#10602056

yes

characteristic

139

C₂

#10602105

yes

characteristic

140

C₂

#10602114

yes

characteristic

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1707	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#1707	D₅
hypocenter	C₅	#12534
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

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 ◁ #1707

N	H ≅ G/N	type
130. C₅	#12534	semidirect

H ≅ G/N

type

130. C₅

#12534

semidirect

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

subgroup	index
C₅×#1707	2
#10602177	2
#10602103	2
#10598858	2
#10597929	2
#12534	5

subgroup

index

Subgroups — #12014101

Conjugacy classes of subgroups

Complements

Maximal subgroups