Group #12011303 — subgroups

Subgroups — #12011303

Order 1280.

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

normal subgroups only

This group has 206 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	C₄×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	#10601698	C₂	1	2	yes	maximal
3	#10598256	C₂	1	2	yes	maximal
4	#10597992	C₂	1	1	yes	maximal, characteristic
5	#10597989	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×Q₃₂	C₄	1	1	yes	characteristic
7	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
8	#65313	—	2	2	no	—
9	C₅×#366	C₄	1	1	yes	characteristic
10	C₅×#358	C₄	1	2	yes	—
11	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	Q₈⋊#168	—	2	2	no	—
13	C₄×#672	C₂²	1	1	yes	characteristic
14	#64678	C₂²	1	2	yes	—
15	C₈×#168	C₂²	1	1	yes	characteristic
16	#12085	—	5	1	no	maximal, Sylow
17	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
18	C₄×C₅×Q₈	—	2	2	no	—
19	C₅×Q₃₂	—	2	2	no	—
20	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₅×#107	C₂×C₄	1	2	yes	—
23	C₅×#103	—	2	2	no	—
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×#672	—	2	1	no	—
26	C₄⋊#168	—	2	2	no	—
27	C₄⋊#168	—	4	2	no	—
28	#3935	—	4	2	no	—
29	C₄×#168	D₄	1	1	yes	characteristic
30	C₄×#168	D₄	1	1	yes	characteristic
31	C₄×Q₃₂	D₅	1	1	yes	characteristic
32	#1591	—	5	2	no	—
33	#1372	—	5	1	no	—
34	#1332	—	5	2	no	—
35	#1311	—	5	1	no	—
36	C₂×C₅×Q₈	—	2	2	no	—
37	C₅×Q₁₆	—	2	2	no	—
38	C₅×Q₁₆	—	2	2	no	—
39	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₅×C₈	#41	1	1	yes	characteristic
41	C₅×C₄⋊C₄	—	2	2	no	—
42	C₅×C₄⋊C₄	—	4	2	no	—
43	C₄²×C₅	—	4	2	no	—
44	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂×#168	—	2	1	no	—
46	C₂×#168	—	4	2	no	—
47	C₂×#168	D₈	1	2	yes	—
48	C₅×C₁₆	—	2	1	no	—
49	C₅×C₁₆	M₁₆	1	2	yes	—
50	#672	—	2	2	no	—
51	C₂×Q₃₂	Dic₅	1	1	yes	characteristic
52	#446	—	10	2	no	—
53	C₄×Q₁₆	C₂×D₅	1	2	yes	—
54	#366	Dic₅	1	1	yes	characteristic
55	#358	Dic₅	1	2	yes	—
56	C₄×C₁₆	—	5	1	no	—
57	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
58	#335	—	5	2	no	—
59	Q₈⋊C₈	—	10	2	no	—
60	C₈²	—	5	1	no	—
61	C₅×Q₈	—	4	2	no	—
62	C₅×Q₈	—	4	2	no	—
63	C₂×C₄×C₅	—	4	2	no	—
64	C₂×C₄×C₅	—	4	2	no	—
65	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅	#135	1	1	yes	characteristic
68	C₅×C₈	—	2	1	no	—
69	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
70	C₅×C₈	#131	1	1	yes	characteristic
71	#168	—	2	2	no	—
72	#168	—	4	1	no	—
73	#168	—	8	2	no	—
74	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
75	C₄×Q₈	—	2	2	no	—
76	Q₃₂	—	2	2	no	—
77	C₂×C₁₆	—	10	1	no	—
78	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
79	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
80	#107	C₂×Dic₅	1	2	yes	—
81	C₄⋊C₈	—	10	2	no	—
82	C₄⋊C₈	—	20	2	no	—
83	#103	—	2	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	—	8	2	no	—
91	C₄×C₅	—	8	2	no	—
92	C₄×C₅	#436	1	1	yes	characteristic
93	C₄×C₅	#443	1	1	yes	characteristic
94	C₄×C₅	#510	1	2	yes	—
95	C₂×Q₈	—	2	2	no	—
96	Q₁₆	—	2	2	no	—
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	5	2	no	—
99	C₂×C₈	—	10	1	no	—
100	C₂×C₈	—	20	2	no	—
101	C₂×C₈	#711	1	1	yes	characteristic
102	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄²	—	4	2	no	—
106	C₄²	D₄×D₅	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	—	10	2	no	—
109	C₁₆	#681	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	—	4	2	no	—
114	Q₈	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	D₅×D₈	1	1	yes	characteristic
118	C₂×C₄	#4058	1	1	yes	characteristic
119	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	10	2	no	—
122	C₈	—	20	1	no	—
123	C₈	—	40	2	no	—
124	C₈	C₂×#681	1	1	yes	characteristic
125	C₈	#4094	1	1	yes	characteristic
126	C₅	#12085	1	1	yes	characteristic, complemented, Sylow
127	C₂²	D₈×Dic₅	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	8	2	no	—
130	C₄	—	8	2	no	—
131	C₄	#65207	1	2	yes	—
132	C₄	#65301	1	1	yes	characteristic
133	C₄	#65485	1	1	yes	characteristic
134	C₂	#10601674	1	1	yes	characteristic
135	C₂	#10602289	1	1	yes	characteristic
136	C₂	#10602305	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×Q₃₂

C₂

yes

maximal, characteristic

#10601698

C₂

yes

maximal

#10598256

C₂

yes

maximal

#10597992

C₂

yes

maximal, characteristic

#10597989

C₂

yes

maximal, characteristic

C₂×C₅×Q₃₂

C₄

yes

characteristic

C₄×C₅×Q₁₆

C₂²

yes

—

#65313

—

C₅×#366

C₄

yes

characteristic

C₅×#358

C₄

yes

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

Q₈⋊#168

—

C₄×#672

C₂²

yes

characteristic

#64678

C₂²

yes

—

C₈×#168

C₂²

yes

characteristic

#12085

—

maximal, Sylow

C₂×C₅×Q₁₆

C₂×C₄

yes

—

C₄×C₅×Q₈

—

C₅×Q₃₂

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₅×#107

C₂×C₄

yes

—

C₅×#103

—

C₄×C₅×C₈

C₂³

yes

—

—

—

—

D₄

yes

characteristic

C₄×#168

D₄

yes

characteristic

C₄×Q₃₂

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

#41

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×D₄

yes

—

—

D₈

yes

—

C₅×C₁₆

—

C₅×C₁₆

M₁₆

yes

—

#672

—

C₂×Q₃₂

Dic₅

yes

characteristic

#446

—

C₄×Q₁₆

C₂×D₅

yes

—

#366

Dic₅

yes

characteristic

#358

Dic₅

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

characteristic

C₅×C₈

—

C₅×C₈

C₂×M₁₆

yes

C₅×C₈

yes

—

—

—

C₂×Q₁₆

C₂×Dic₅

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅

yes

characteristic

C₂×C₁₆

C₂×Dic₅

yes

characteristic

#107

C₂×Dic₅

yes

—

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

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

#711

yes

characteristic

102

C₂×C₈

C₂²×Dic₅

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

D₄×D₅

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₁₆

#681

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

D₅×D₈

yes

characteristic

118

C₂×C₄

#4058

yes

characteristic

119

C₂×C₄

D₄×Dic₅

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#681

yes

characteristic

125

C₈

#4094

yes

characteristic

126

C₅

#12085

yes

characteristic, complemented, Sylow

127

C₂²

D₈×Dic₅

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#65207

yes

—

132

C₄

#65301

yes

characteristic

133

C₄

#65485

yes

characteristic

134

C₂

#10601674

yes

characteristic

135

C₂

#10602289

yes

characteristic

136

C₂

#10602305

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₈
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₄×C₅×Q₃₂	C₂
socle, prime radical	C₂²×C₅	C₄×D₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₄×Q₃₂	D₅
hypocenter	C₅	#12085
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₅×D₈

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅

Fitting

C₄×C₅×Q₃₂

C₂

socle, prime radical

C₂²×C₅

C₄×D₈

cosocle

C₄×C₅×C₈

C₂³

hypercenter

C₄×Q₃₂

D₅

hypocenter

C₅

#12085

LCS

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

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×C₅×Q₃₂	2
#10601698	2
#10598256	2
#10597992	2
#10597989	2
#12085	5

subgroup

index

C₄×C₅×Q₃₂

Subgroups — #12011303

Conjugacy classes of subgroups

Complements

Maximal subgroups