Group #11746970 — subgroups

Subgroups — #11746970

Order 960.

180 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 68 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×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#4016	C₂	1	1	yes	maximal, characteristic
3	#91087	C₂	1	1	yes	maximal, characteristic
4	#90989	C₂	1	2	yes	maximal
5	#90986	C₂	1	2	yes	maximal
6	#65369	—	3	1	no	maximal, Hall
7	C₂×C₅×Dic₆	—	2	1	no	—
8	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊#168	C₂²	1	2	yes	—
10	C₅×C₄⋊Dic₃	—	2	1	no	—
11	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₃×#683	—	2	1	no	—
15	C₃×#683	—	2	1	no	—
16	#5064	—	5	1	no	maximal, Hall
17	C₅×C₄⋊Q₈	—	3	1	no	—
18	#4016	D₃	1	1	yes	characteristic
19	#3940	—	3	2	no	—
20	#3939	—	3	2	no	—
21	#3935	—	3	1	no	—
22	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×Dic₃	—	2	2	no	—
26	C₅×Dic₆	—	4	2	no	—
27	C₂×C₃×Dic₅	—	2	1	no	—
28	C₂×C₃×Dic₅	—	2	1	no	—
29	C₃⋊#168	Q₈	1	4	yes	complemented
30	C₃×#125	—	5	1	no	—
31	#869	D₅	1	1	yes	characteristic
32	#808	—	5	2	no	—
33	#807	—	5	2	no	—
34	#803	—	5	1	no	—
35	C₂×C₅×Q₈	—	6	1	no	—
36	C₅×C₄⋊C₄	—	3	2	no	—
37	C₅×C₄⋊C₄	—	6	1	no	—
38	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
40	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
41	#683	—	2	1	no	—
42	#683	—	2	1	no	—
43	C₂×#168	—	3	2	no	—
44	#501	—	15	1	no	Sylow
45	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
46	C₃×C₄×C₅	—	2	1	no	—
47	C₃×C₄×C₅	—	2	1	no	—
48	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
49	C₃×Dic₅	—	4	1	no	—
50	C₃×Dic₅	—	4	1	no	—
51	C₅×Dic₃	—	4	2	no	—
52	C₂×Dic₆	—	2	1	no	—
53	C₃×C₄⋊C₄	—	5	1	no	—
54	C₃×C₄⋊C₄	—	5	1	no	—
55	C₃×C₄⋊C₄	—	10	1	no	—
56	C₃×C₄⋊C₄	—	10	1	no	—
57	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
58	C₄⋊Dic₃	—	2	1	no	—
59	C₄⋊Dic₃	C₂×D₅	1	2	yes	—
60	C₂×C₃⋊C₈	—	5	2	no	—
61	C₅×Q₈	—	12	2	no	—
62	C₂×C₄×C₅	—	6	2	no	—
63	C₂×C₄×C₅	#67	1	1	yes	characteristic
64	C₂×C₄×C₅	#67	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
66	C₂×Dic₅	—	2	1	no	—
67	C₂×Dic₅	—	2	1	no	—
68	#168	—	3	4	no	—
69	C₄⋊Q₈	—	3	1	no	—
70	#125	—	5	1	no	—
71	#107	—	15	2	no	—
72	#106	—	15	2	no	—
73	#97	—	15	1	no	—
74	C₂×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×C₃×C₅	#137	1	1	yes	characteristic
77	C₂×C₃×C₄	—	10	1	no	—
78	C₂×C₃×C₄	—	10	1	no	—
79	C₂×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
82	C₂×Dic₃	—	2	2	no	—
83	Dic₆	—	4	2	no	—
84	C₃⋊C₈	—	5	4	no	—
85	C₂²×C₅	C₂×#67	1	1	yes	characteristic
86	C₄×C₅	—	2	1	no	—
87	C₄×C₅	—	2	1	no	—
88	C₄×C₅	—	12	2	no	—
89	C₄×C₅	D₃×Q₈	1	2	yes	—
90	Dic₅	—	4	1	no	—
91	Dic₅	—	4	1	no	—
92	C₂×Q₈	—	6	1	no	—
93	C₂×C₈	—	15	2	no	—
94	C₄⋊C₄	—	3	2	no	—
95	C₄⋊C₄	—	5	1	no	—
96	C₄⋊C₄	—	5	1	no	—
97	C₄⋊C₄	—	6	1	no	—
98	C₄⋊C₄	—	10	1	no	—
99	C₄⋊C₄	—	10	1	no	—
100	C₄²	D₃×D₅	1	1	yes	characteristic
101	C₃×C₅	#501	1	1	yes	characteristic, complemented, Hall
102	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
103	C₃×C₄	—	2	1	no	—
104	C₃×C₄	—	2	1	no	—
105	C₃×C₄	—	20	1	no	—
106	C₃×C₄	—	20	1	no	—
107	C₃×C₄	Q₈×D₅	1	2	yes	—
108	Dic₃	—	4	2	no	—
109	C₂×C₅	#944	1	1	yes	characteristic
110	C₂×C₅	#949	1	1	yes	characteristic
111	C₂×C₅	#951	1	1	yes	characteristic
112	Q₈	—	12	2	no	—
113	C₂×C₄	—	6	2	no	—
114	C₂×C₄	—	10	1	no	—
115	C₂×C₄	—	10	1	no	—
116	C₂×C₄	#1201	1	1	yes	characteristic
117	C₂×C₄	#1201	1	1	yes	characteristic
118	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
119	C₈	—	15	4	no	—
120	C₂×C₃	#4078	1	1	yes	characteristic
121	C₂×C₃	#4088	1	1	yes	characteristic
122	C₂×C₃	#4090	1	1	yes	characteristic
123	C₅	#5064	1	1	yes	characteristic, complemented, Sylow
124	C₂²	C₂×#1201	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	12	2	no	—
128	C₄	—	20	1	no	—
129	C₄	—	20	1	no	—
130	C₄	D₁₅⋊Q₈	1	2	yes	—
131	C₃	#65369	1	1	yes	characteristic, complemented, Sylow
132	C₂	#91299	1	1	yes	characteristic
133	C₂	#91307	1	1	yes	characteristic
134	C₂	#91385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#869

C₂

yes

maximal, characteristic

C₃×#4016

C₂

yes

maximal, characteristic

#91087

C₂

yes

maximal, characteristic

#90989

C₂

yes

maximal

#90986

C₂

yes

maximal

#65369

—

maximal, Hall

C₂×C₅×Dic₆

—

C₃×C₄²×C₅

C₂²

yes

characteristic

C₂×C₃⋊#168

C₂²

yes

—

C₅×C₄⋊Dic₃

—

C₅×C₄⋊Dic₃

C₂²

yes

—

C₃×C₄⋊Dic₅

C₂²

yes

characteristic

C₃×C₄⋊Dic₅

C₂²

yes

—

—

—

C₅×C₄⋊Q₈

—

#4016

D₃

yes

—

—

—

C₂×C₃×C₄×C₅

D₄

yes

characteristic

C₂×C₃×C₄×C₅

D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂³

yes

characteristic

C₂×C₅×Dic₃

—

C₅×Dic₆

—

C₂×C₃×Dic₅

—

C₂×C₃×Dic₅

—

C₃⋊#168

Q₈

yes

complemented

C₃×#125

—

#869

D₅

yes

—

—

—

C₂×C₅×Q₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₃

yes

characteristic

C₄⋊Dic₅

C₂×D₃

yes

characteristic

C₄⋊Dic₅

C₂×D₃

yes

—

—

—

—

C₂²×C₃×C₅

C₂×D₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×Q₈

yes

—

C₃×Dic₅

—

C₃×Dic₅

—

C₅×Dic₃

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₅

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×D₅

yes

—

C₂×C₃⋊C₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂²×D₃

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

#168

—

C₄⋊Q₈

—

—

—

—

—

C₂×C₃×C₅

C₄⋊Q₈

yes

C₂×C₃×C₅

yes

C₂×C₃×C₅

yes

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊D₅

yes

characteristic

C₂×C₃×C₄

C₂²⋊D₅

yes

characteristic

C₂×C₃×C₄

C₂²×D₅

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

—

C₂²×C₅

C₂×#67

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

D₃×Q₈

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

D₃×D₅

yes

characteristic

101

C₃×C₅

#501

yes

characteristic, complemented, Hall

102

C₂²×C₃

C₂×C₂²⋊D₅

yes

characteristic

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

Q₈×D₅

yes

—

108

Dic₃

—

109

C₂×C₅

#944

yes

characteristic

110

C₂×C₅

#949

yes

characteristic

111

C₂×C₅

#951

yes

characteristic

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#1201

yes

characteristic

117

C₂×C₄

#1201

yes

characteristic

118

C₂×C₄

C₂×D₃×D₅

yes

characteristic

119

C₈

—

120

C₂×C₃

#4078

yes

characteristic

121

C₂×C₃

#4088

yes

characteristic

122

C₂×C₃

#4090

yes

characteristic

123

C₅

#5064

yes

characteristic, complemented, Sylow

124

C₂²

C₂×#1201

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

D₁₅⋊Q₈

yes

—

131

C₃

#65369

yes

characteristic, complemented, Sylow

132

C₂

#91299

yes

characteristic

133

C₂

#91307

yes

characteristic

134

C₂

#91385

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#1201
commutator, cosocle	C₂×C₃×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂×D₃×D₅
Fitting	C₃×C₄²×C₅	C₂²
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#501
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#1201

commutator, cosocle

C₂×C₃×C₄×C₅

C₂³

Frattini

C₂×C₄

C₂×D₃×D₅

Fitting

C₃×C₄²×C₅

C₂²

socle, prime radical

C₂²×C₃×C₅

C₂×D₄

hypercenter

C₄²

D₃×D₅

hypocenter

C₃×C₅

#501

LCS

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
29. C₃⋊#168	Q₈	semidirect
101. C₃×C₅	#501	semidirect
123. C₅	#5064	semidirect
131. C₃	#65369	semidirect

H ≅ G/N

type

29. C₃⋊#168

Q₈

semidirect

101. C₃×C₅

#501

semidirect

123. C₅

#5064

semidirect

131. C₃

#65369

semidirect

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

subgroup	index
C₅×#869	2
C₃×#4016	2
#91087	2
#90989	2
#90986	2
#65369	3
#5064	5

subgroup

index

Subgroups — #11746970

Conjugacy classes of subgroups

Complements

Maximal subgroups