Group #11993066 — subgroups

Subgroups — #11993066

Order 1280.

486 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 235 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×#1844	C₂	1	1	yes	maximal, characteristic
2	#10600557	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic
4	#10600369	C₂	1	4	yes	maximal
5	C₂×C₅×#106	—	2	2	no	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×#384	C₂²	1	2	yes	—
8	C₂×#3949	—	2	2	no	—
9	C₂×#3940	—	2	8	no	—
10	#65224	C₂²	1	2	yes	—
11	C₂×C₄×#168	C₂²	1	2	yes	—
12	C₈×#168	C₄	1	4	yes	complemented
13	#8813	—	5	1	no	maximal, Sylow
14	C₂²×C₅×C₈	D₄	1	2	yes	—
15	C₂×C₅×C₄⋊C₄	—	2	2	no	—
16	C₂×C₅×C₄⋊C₄	—	2	4	no	—
17	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅	—	2	2	no	—
19	C₂×C₄⋊Dic₅	—	2	4	no	—
20	C₂²×#168	D₄	1	4	yes	—
21	C₅×#106	—	4	4	no	—
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅×C₈	Q₈	1	2	yes	—
24	#3949	—	4	4	no	—
25	#3940	—	4	16	no	—
26	C₄×#168	C₂×C₄	1	4	yes	—
27	C₄×#168	Q₈	1	4	yes	—
28	#1844	—	5	1	no	—
29	#1844	—	5	4	no	—
30	#1844	D₅	1	1	yes	characteristic
31	C₂×C₈²	—	5	1	no	—
32	C₂²×C₄×C₅	—	4	2	no	—
33	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
34	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅	—	4	2	no	—
36	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₅×C₈	SD₁₆	1	4	yes	—
38	C₂×C₅×C₈	SD₁₆	1	4	yes	—
39	C₅×C₄⋊C₄	—	4	8	no	—
40	C₅×C₄⋊C₄	—	4	4	no	—
41	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅	C₂×Q₈	1	2	yes	—
44	C₄⋊Dic₅	—	4	8	no	—
45	C₄⋊Dic₅	—	4	4	no	—
46	C₂×#168	C₄⋊C₄	1	8	yes	—
47	C₂×#168	SD₁₆	1	16	yes	—
48	C₂×#106	—	2	2	no	—
49	C₂×#106	—	10	8	no	—
50	C₂×#106	—	10	2	no	—
51	C₂×C₄×C₈	—	5	2	no	—
52	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
53	#384	—	5	2	no	—
54	#384	C₂×D₅	1	2	yes	—
55	C₈²	—	5	4	no	—
56	C₂³×C₅	#127	1	1	yes	characteristic
57	C₂×C₄×C₅	—	8	2	no	—
58	C₂×C₄×C₅	—	8	4	no	—
59	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×SD₁₆	1	8	yes	—
64	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
65	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
66	C₂×Dic₅	—	8	2	no	—
67	C₂×Dic₅	—	8	4	no	—
68	C₅×C₈	#106	1	8	yes	complemented
69	#168	#106	1	16	yes	complemented
70	C₂²×C₈	—	5	4	no	—
71	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
72	C₂×C₄⋊C₄	—	2	4	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	—	10	4	no	—
75	C₂×C₄⋊C₄	—	10	2	no	—
76	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
77	#106	—	4	4	no	—
78	#106	—	20	16	no	—
79	#106	—	20	4	no	—
80	C₄×C₈	—	5	4	no	—
81	C₄×C₈	—	5	4	no	—
82	C₄×C₈	Dic₁₀	1	2	yes	—
83	C₄×C₈	C₄×D₅	1	2	yes	—
84	C₂²×C₅	#384	1	1	yes	characteristic
85	C₂²×C₅	#492	1	1	yes	characteristic
86	C₂²×C₅	#492	1	1	yes	characteristic
87	C₂²×C₅	#492	1	4	yes	—
88	C₄×C₅	—	16	2	no	—
89	C₄×C₅	C₂×#106	1	4	yes	—
90	C₄×C₅	C₂×#106	1	2	yes	—
91	C₄×C₅	#498	1	2	yes	—
92	C₄×C₅	#498	1	4	yes	—
93	Dic₅	—	16	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	—	20	2	no	—
96	C₂²×C₄	D₄×D₅	1	2	yes	—
97	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
98	C₂×C₈	—	5	8	no	—
99	C₂×C₈	—	5	16	no	—
100	C₂×C₈	#683	1	4	yes	—
101	C₂×C₈	#687	1	4	yes	—
102	C₂×C₈	#688	1	4	yes	—
103	C₄⋊C₄	—	4	8	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	20	8	no	—
106	C₄⋊C₄	—	20	4	no	—
107	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
108	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
109	C₄²	Q₈×D₅	1	2	yes	—
110	C₂×C₅	#1705	1	2	yes	—
111	C₂×C₅	#1705	1	2	yes	—
112	C₂×C₅	#1844	1	2	yes	—
113	C₂×C₅	#1844	1	1	yes	characteristic
114	C₂³	#4086	1	1	yes	characteristic
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	8	4	no	—
117	C₂×C₄	—	40	2	no	—
118	C₂×C₄	—	40	4	no	—
119	C₂×C₄	#4034	1	2	yes	—
120	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
121	C₂×C₄	D₅×SD₁₆	1	8	yes	—
122	C₂×C₄	C₂×#683	1	1	yes	characteristic
123	C₂×C₄	C₂×#687	1	2	yes	—
124	C₂×C₄	C₂×#688	1	2	yes	—
125	C₂×C₄	#4090	1	1	yes	characteristic
126	C₈	—	5	16	no	—
127	C₈	#3940	1	8	yes	complemented
128	C₅	#8813	1	1	yes	characteristic, complemented, Sylow
129	C₂²	#65265	1	1	yes	characteristic
130	C₂²	#65365	1	1	yes	characteristic
131	C₂²	#65374	1	1	yes	characteristic
132	C₂²	#65464	1	4	yes	—
133	C₄	—	16	2	no	—
134	C₄	—	80	2	no	—
135	C₄	#65144	1	4	yes	—
136	C₄	D₅×#106	1	4	yes	—
137	C₄	C₂×#3940	1	2	yes	—
138	C₄	#65368	1	2	yes	—
139	C₂	#10600369	1	1	yes	characteristic
140	C₂	#10601347	1	2	yes	—
141	C₂	#10602047	1	2	yes	—
142	C₂	#10602125	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1844

C₂

yes

maximal, characteristic

#10600557

C₂

yes

maximal, characteristic

C₂×C₈×#168

C₂

yes

maximal, characteristic

#10600369

C₂

yes

maximal

C₂×C₅×#106

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#384

C₂²

yes

—

C₂×#3949

—

C₂×#3940

—

#65224

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

—

C₈×#168

C₄

yes

complemented

#8813

—

maximal, Sylow

C₂²×C₅×C₈

D₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂²×#168

D₄

yes

—

C₅×#106

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

#3949

—

#3940

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

Q₈

yes

—

#1844

—

#1844

—

#1844

D₅

yes

characteristic

C₂×C₈²

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×Q₈

yes

characteristic

C₄²×C₅

C₂×Q₈

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

SD₁₆

yes

—

C₂×#106

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#384

—

#384

C₂×D₅

yes

—

C₈²

—

C₂³×C₅

#127

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

yes

yes

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀

yes

—

C₄×C₈

C₄×D₅

yes

—

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

C₄×C₅

—

C₄×C₅

C₂×#106

yes

—

C₄×C₅

C₂×#106

yes

—

C₄×C₅

#498

yes

—

C₄×C₅

#498

yes

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×D₅

yes

—

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

#683

yes

—

101

C₂×C₈

#687

yes

—

102

C₂×C₈

#688

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

C₂×Dic₁₀

yes

characteristic

108

C₄²

C₂×C₄×D₅

yes

characteristic

109

C₄²

Q₈×D₅

yes

—

110

C₂×C₅

#1705

yes

—

111

C₂×C₅

#1705

yes

—

112

C₂×C₅

#1844

yes

—

113

C₂×C₅

#1844

yes

characteristic

114

C₂³

#4086

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#4034

yes

—

120

C₂×C₄

D₅×C₄⋊C₄

yes

—

121

C₂×C₄

D₅×SD₁₆

yes

—

122

C₂×C₄

C₂×#683

yes

characteristic

123

C₂×C₄

C₂×#687

yes

—

124

C₂×C₄

C₂×#688

yes

—

125

C₂×C₄

#4090

yes

characteristic

126

C₈

—

127

C₈

#3940

yes

complemented

128

C₅

#8813

yes

characteristic, complemented, Sylow

129

C₂²

#65265

yes

characteristic

130

C₂²

#65365

yes

characteristic

131

C₂²

#65374

yes

characteristic

132

C₂²

#65464

yes

—

133

C₄

—

134

C₄

—

135

C₄

#65144

yes

—

136

C₄

D₅×#106

yes

—

137

C₄

C₂×#3940

yes

—

138

C₄

#65368

yes

—

139

C₂

#10600369

yes

characteristic

140

C₂

#10601347

yes

—

141

C₂

#10602047

yes

—

142

C₂

#10602125

yes

—

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1844	C₂
socle, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1844	D₅
hypocenter	C₅	#8813
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#4086

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
12. C₈×#168	C₄	semidirect
68. C₅×C₈	#106	semidirect
69. #168	#106	semidirect
127. C₈	#3940	semidirect
128. C₅	#8813	semidirect

H ≅ G/N

type

12. C₈×#168

C₄

semidirect

68. C₅×C₈

#106

semidirect

69. #168

#106

semidirect

127. C₈

#3940

semidirect

128. C₅

#8813

semidirect

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

subgroup	index
C₅×#1844	2
#10600557	2
C₂×C₈×#168	2
#10600369	2
#8813	5

subgroup

index

C₂×C₈×#168

Subgroups — #11993066

Conjugacy classes of subgroups

Complements

Maximal subgroups