Group #12011302 — subgroups

Subgroups — #12011302

Order 1280.

206 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×Dic₄₀	C₂	1	1	yes	maximal, characteristic
2	#10599357	C₂	1	2	yes	maximal
3	C₅×#1372	C₂	1	1	yes	maximal, characteristic
4	#10597989	C₂	1	1	yes	maximal, characteristic
5	#10597963	C₂	1	2	yes	maximal
6	C₂×Dic₄₀	C₄	1	1	yes	characteristic
7	#65120	—	2	2	no	—
8	C₄×Dic₂₀	C₂²	1	2	yes	—
9	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₅×#335	C₂²	1	2	yes	—
11	#64724	C₄	1	1	yes	characteristic
12	#64723	C₄	1	2	yes	—
13	#64704	—	2	2	no	—
14	C₄×#672	C₂²	1	1	yes	characteristic
15	C₈×#168	C₂²	1	1	yes	characteristic
16	#12085	—	5	1	no	maximal, Sylow
17	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₀	—	2	2	no	—
19	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₅×C₄⋊C₈	—	2	2	no	—
22	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
23	#3950	C₂×C₄	1	2	yes	—
24	#3948	—	2	2	no	—
25	#3947	—	4	2	no	—
26	#3946	—	4	2	no	—
27	C₂×#672	—	2	1	no	—
28	C₄×#168	D₄	1	1	yes	characteristic
29	C₄×#168	D₄	1	1	yes	characteristic
30	Dic₄₀	—	2	2	no	—
31	C₄×Q₃₂	—	5	1	no	—
32	#1591	—	5	2	no	—
33	#1372	D₅	1	1	yes	characteristic
34	#1332	—	5	2	no	—
35	#1311	—	5	1	no	—
36	C₂×Dic₁₀	—	2	2	no	—
37	C₂×C₅×C₈	—	4	2	no	—
38	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₅×C₈	#41	1	1	yes	characteristic
40	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₅	—	2	2	no	—
42	#683	—	4	2	no	—
43	C₄×Dic₅	—	4	2	no	—
44	C₂×#168	—	2	1	no	—
45	C₂×#168	D₈	1	2	yes	—
46	Dic₂₀	—	2	2	no	—
47	Dic₂₀	—	2	2	no	—
48	C₅×C₁₆	—	2	1	no	—
49	C₅×C₁₆	M₁₆	1	2	yes	—
50	#672	—	2	2	no	—
51	C₂×Q₃₂	—	5	1	no	—
52	#446	—	10	2	no	—
53	C₄×Q₁₆	—	5	2	no	—
54	#366	—	5	1	no	—
55	#358	—	5	2	no	—
56	C₄×C₁₆	—	5	1	no	—
57	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
58	#335	C₂×D₅	1	2	yes	—
59	Q₈⋊C₈	—	10	2	no	—
60	C₈²	—	5	1	no	—
61	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅	#135	1	1	yes	characteristic
64	C₂×Dic₅	—	4	2	no	—
65	C₂×Dic₅	—	4	2	no	—
66	Dic₁₀	—	4	2	no	—
67	Dic₁₀	—	4	2	no	—
68	C₅×C₈	—	2	1	no	—
69	C₅×C₈	—	8	2	no	—
70	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
71	C₅×C₈	#131	1	1	yes	characteristic
72	#168	—	2	2	no	—
73	#168	—	4	1	no	—
74	C₂×Q₁₆	—	5	2	no	—
75	C₄×Q₈	—	10	2	no	—
76	Q₃₂	—	10	2	no	—
77	C₂×C₁₆	—	10	1	no	—
78	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
79	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
80	#107	—	5	2	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	20	2	no	—
83	#103	—	10	2	no	—
84	#97	—	20	2	no	—
85	C₄×C₈	—	5	1	no	—
86	C₄×C₈	—	5	1	no	—
87	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
88	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
89	C₄×C₅	—	2	1	no	—
90	C₄×C₅	#436	1	1	yes	characteristic
91	C₄×C₅	#443	1	1	yes	characteristic
92	C₄×C₅	#510	1	2	yes	—
93	Dic₅	—	8	2	no	—
94	Dic₅	—	8	2	no	—
95	C₂×Q₈	—	10	2	no	—
96	Q₁₆	—	10	2	no	—
97	Q₁₆	—	10	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	5	2	no	—
100	C₂×C₈	—	10	1	no	—
101	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
102	C₂×C₈	#713	1	1	yes	characteristic
103	C₄⋊C₄	—	10	2	no	—
104	C₄⋊C₄	—	20	2	no	—
105	C₄²	—	20	2	no	—
106	C₄²	D₄×D₅	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	—	10	2	no	—
109	C₁₆	#676	1	2	yes	—
110	C₂×C₅	#1589	1	1	yes	characteristic
111	C₂×C₅	#2176	1	1	yes	characteristic
112	C₂×C₅	#2179	1	1	yes	characteristic
113	Q₈	—	20	2	no	—
114	Q₈	—	20	2	no	—
115	C₂×C₄	—	20	2	no	—
116	C₂×C₄	—	20	2	no	—
117	C₂×C₄	#4039	1	1	yes	characteristic
118	C₂×C₄	D₅×D₈	1	1	yes	characteristic
119	C₂×C₄	#4065	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	8	2	no	—
122	C₈	—	10	2	no	—
123	C₈	—	20	1	no	—
124	C₈	C₂×#676	1	1	yes	characteristic
125	C₈	#4053	1	1	yes	characteristic
126	C₅	#12085	1	1	yes	characteristic, complemented, Sylow
127	C₂²	#65161	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	40	2	no	—
130	C₄	—	40	2	no	—
131	C₄	#65127	1	1	yes	characteristic
132	C₄	#65184	1	1	yes	characteristic
133	C₄	#65204	1	2	yes	—
134	C₂	#10599356	1	1	yes	characteristic
135	C₂	#10600131	1	1	yes	characteristic
136	C₂	#10600151	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₄₀

C₂

yes

maximal, characteristic

#10599357

C₂

yes

maximal

C₅×#1372

C₂

yes

maximal, characteristic

#10597989

C₂

yes

maximal, characteristic

#10597963

C₂

yes

maximal

C₂×Dic₄₀

C₄

yes

characteristic

#65120

—

C₄×Dic₂₀

C₂²

yes

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

C₅×#335

C₂²

yes

—

#64724

C₄

yes

characteristic

#64723

C₄

yes

—

#64704

—

C₄×#672

C₂²

yes

characteristic

C₈×#168

C₂²

yes

characteristic

#12085

—

maximal, Sylow

C₂×Dic₂₀

C₂×C₄

yes

—

C₄×Dic₁₀

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂³

yes

characteristic

#3950

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#168

D₄

yes

characteristic

Dic₄₀

—

C₄×Q₃₂

—

#1591

—

#1372

D₅

yes

characteristic

#1332

—

#1311

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

#41

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

D₈

yes

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

—

C₅×C₁₆

M₁₆

yes

—

#672

—

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

#335

C₂×D₅

yes

—

Q₈⋊C₈

—

C₈²

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×M₁₆

yes

C₅×C₈

yes

—

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅

yes

characteristic

C₂×C₁₆

C₄×D₅

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

C₄×D₈

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×C₄×D₅

yes

characteristic

102

C₂×C₈

#713

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

D₄×D₅

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₁₆

#676

yes

—

110

C₂×C₅

#1589

yes

characteristic

111

C₂×C₅

#2176

yes

characteristic

112

C₂×C₅

#2179

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#4039

yes

characteristic

118

C₂×C₄

D₅×D₈

yes

characteristic

119

C₂×C₄

#4065

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#676

yes

characteristic

125

C₈

#4053

yes

characteristic

126

C₅

#12085

yes

characteristic, complemented, Sylow

127

C₂²

#65161

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#65127

yes

characteristic

132

C₄

#65184

yes

characteristic

133

C₄

#65204

yes

—

134

C₂

#10599356

yes

characteristic

135

C₂

#10600131

yes

characteristic

136

C₂

#10600151

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₈
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1372	C₂
socle, prime radical	C₂²×C₅	C₄×D₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	#1372	D₅
hypocenter	C₅	#12085
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂×C₄

D₅×D₈

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅

Fitting

C₅×#1372

C₂

socle, prime radical

C₂²×C₅

C₄×D₈

cosocle

C₄×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
126. C₅	#12085	semidirect

H ≅ G/N

type

126. C₅

#12085

semidirect

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

subgroup	index
C₄×Dic₄₀	2
#10599357	2
C₅×#1372	2
#10597989	2
#10597963	2
#12085	5

subgroup

index

C₄×Dic₄₀

Subgroups — #12011302

Conjugacy classes of subgroups

Complements

Maximal subgroups