Group #24920355 — subgroups

Subgroups — #24920355

Order 10624.

324 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 133 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	#19128429	C₂	1	2	yes	maximal
2	#19128106	C₂	1	1	yes	maximal, characteristic
3	#19128103	C₂	1	2	yes	maximal
4	#19128009	C₂	1	4	yes	maximal
5	#19127987	C₂	1	2	yes	maximal
6	C₄×Dic₃₃₂	C₂	1	2	yes	maximal
7	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
10	#14581792	C₂²	1	2	yes	—
11	#14581790	—	2	2	no	—
12	#14581784	C₂²	1	4	yes	—
13	#14581782	—	2	4	no	—
14	C₄×Dic₁₆₆	C₂²	1	2	yes	—
15	C₄×Dic₁₆₆	C₂²	1	4	yes	—
16	#14581757	—	2	2	no	—
17	#14581726	C₂²	1	1	yes	characteristic
18	#14581726	C₂²	1	2	yes	—
19	#14581726	C₂²	1	2	yes	—
20	#14581724	C₂²	1	4	yes	—
21	#14581717	C₂²	1	4	yes	—
22	#14581717	C₂²	1	2	yes	—
23	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
26	C₄×C₈×C₈₃	C₂²	1	2	yes	—
27	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₆₆	—	2	2	no	—
29	C₂×Dic₁₆₆	C₂³	1	2	yes	—
30	Dic₃₃₂	Q₈	1	4	yes	complemented
31	C₄⋊Dic₈₃	—	2	2	no	—
32	C₄⋊Dic₈₃	C₂³	1	2	yes	—
33	C₄⋊Dic₈₃	C₂³	1	4	yes	—
34	Dic₈₃⋊C₄	—	2	2	no	—
35	Dic₈₃⋊C₄	—	2	4	no	—
36	Dic₈₃⋊C₄	—	2	4	no	—
37	Dic₈₃⋊C₄	—	2	4	no	—
38	C₄×Dic₈₃	—	2	4	no	—
39	C₄×Dic₈₃	—	2	2	no	—
40	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₃	C₂³	1	2	yes	—
42	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₈₃	D₄	1	2	yes	—
45	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
46	C₄²×C₈₃	C₂³	1	2	yes	—
47	C₄²×C₈₃	C₂³	1	1	yes	characteristic
48	Q₈×C₈₃	—	2	2	no	—
49	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₈₃	—	2	2	no	—
55	C₂×Dic₈₃	—	2	2	no	—
56	C₂×Dic₈₃	—	2	4	no	—
57	Dic₁₆₆	—	2	2	no	—
58	Dic₁₆₆	—	4	4	no	—
59	Dic₁₆₆	C₂×Q₈	1	4	yes	—
60	C₈×C₈₃	—	2	2	no	—
61	C₈×C₈₃	—	2	1	no	—
62	C₈×C₈₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₈₃	—	2	2	no	—
65	C₄×C₈₃	—	2	2	no	—
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	#135	1	2	yes	—
68	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₈₃	—	2	4	no	—
71	Dic₈₃	—	4	4	no	—
72	Dic₈₃	—	4	2	no	—
73	C₂×C₈₃	D₄×Q₈	1	1	yes	characteristic
74	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
75	C₂×C₈₃	#578	1	1	yes	characteristic
76	#3383	—	83	1	no	maximal, Sylow
77	C₈₃	#3383	1	1	yes	characteristic, complemented, Sylow
78	#557	—	83	2	no	—
79	#500	—	83	1	no	—
80	#499	—	83	2	no	—
81	#479	—	83	4	no	—
82	#475	—	83	2	no	—
83	C₈×Q₈	D₈₃	1	1	yes	characteristic
84	C₄×Q₁₆	—	83	2	no	—
85	C₄×Q₁₆	—	83	1	no	—
86	C₂×Q₁₆	—	83	1	no	—
87	C₄⋊Q₈	—	83	2	no	—
88	C₄⋊Q₈	—	166	2	no	—
89	#125	—	83	4	no	—
90	#125	—	166	4	no	—
91	C₄×Q₈	—	83	2	no	—
92	C₄×Q₈	—	83	4	no	—
93	C₄×Q₈	—	166	2	no	—
94	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
95	#107	—	83	1	no	—
96	#107	—	83	2	no	—
97	#107	—	83	2	no	—
98	#106	—	83	4	no	—
99	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
101	#103	—	83	4	no	—
102	#103	—	83	2	no	—
103	C₄×C₈	C₂×D₈₃	1	2	yes	—
104	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
105	C₂×Q₈	—	83	2	no	—
106	C₂×Q₈	—	166	2	no	—
107	C₂×Q₈	D₃₃₂	1	1	yes	characteristic
108	Q₁₆	—	83	4	no	—
109	C₂×C₈	C₂²×D₈₃	1	2	yes	—
110	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
112	C₄⋊C₄	—	83	2	no	—
113	C₄⋊C₄	—	83	4	no	—
114	C₄⋊C₄	—	166	2	no	—
115	C₄⋊C₄	—	166	4	no	—
116	C₄⋊C₄	—	166	4	no	—
117	C₄⋊C₄	—	166	4	no	—
118	C₄⋊C₄	—	166	2	no	—
119	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
120	C₄⋊C₄	D₃₃₂	1	2	yes	—
121	C₄²	—	166	4	no	—
122	C₄²	—	166	2	no	—
123	C₄²	C₂²×D₈₃	1	2	yes	—
124	C₄²	C₂²×D₈₃	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	332	4	no	—
127	Q₈	—	83	4	no	—
128	Q₈	—	166	2	no	—
129	C₂×C₄	—	166	2	no	—
130	C₂×C₄	—	166	4	no	—
131	C₂×C₄	—	166	2	no	—
132	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
133	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
134	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
135	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	1	no	—
139	C₈	Q₈×D₈₃	1	2	yes	—
140	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	332	4	no	—
145	C₄	—	332	2	no	—
146	C₄	—	166	4	no	—
147	C₄	#14581814	1	2	yes	—
148	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
149	C₄	#14581839	1	1	yes	characteristic
150	C₂	Q₈×D₃₃₂	1	1	yes	characteristic
151	C₂	C₂×#14581814	1	1	yes	characteristic
152	C₂	#19128518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128429

C₂

yes

maximal

#19128106

C₂

yes

maximal, characteristic

#19128103

C₂

yes

maximal

#19128009

C₂

yes

maximal

#19127987

C₂

yes

maximal

C₄×Dic₃₃₂

C₂

yes

maximal

C₄×Dic₃₃₂

C₂

yes

maximal, characteristic

C₈×Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

C₂²

yes

characteristic

#14581792

C₂²

yes

—

#14581790

—

#14581784

C₂²

yes

—

#14581782

—

C₄×Dic₁₆₆

C₂²

yes

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581757

—

#14581726

C₂²

yes

characteristic

#14581726

C₂²

yes

—

#14581726

C₂²

yes

—

#14581724

C₂²

yes

—

#14581717

C₂²

yes

—

#14581717

C₂²

yes

—

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

—

Dic₃₃₂

Q₈

yes

complemented

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

C₂×Q₈

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×Q₈

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#135

yes

—

C₄×C₈₃

C₂²×Q₈

yes

characteristic

C₄×C₈₃

2₋¹⁺⁴

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

D₄×Q₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

D₈₃

yes

characteristic

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

#125

—

#125

—

166

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

166

—

C₄×Q₈

C₂×D₈₃

yes

—

—

—

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

100

C₄⋊C₈

C₂×D₈₃

yes

—

101

#103

—

102

#103

—

103

C₄×C₈

C₂×D₈₃

yes

—

104

C₄×C₈

C₂×D₈₃

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

166

—

107

C₂×Q₈

D₃₃₂

yes

characteristic

108

Q₁₆

—

109

C₂×C₈

C₂²×D₈₃

yes

—

110

C₂×C₈

C₂²×D₈₃

yes

characteristic

111

C₂×C₈

C₂²×D₈₃

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

166

—

115

C₄⋊C₄

—

166

—

116

C₄⋊C₄

—

166

—

117

C₄⋊C₄

—

166

—

118

C₄⋊C₄

—

166

—

119

C₄⋊C₄

D₃₃₂

yes

characteristic

120

C₄⋊C₄

D₃₃₂

yes

—

121

C₄²

—

166

—

122

C₄²

—

166

—

123

C₄²

C₂²×D₈₃

yes

—

124

C₄²

C₂²×D₈₃

yes

characteristic

125

Q₈

—

126

Q₈

—

332

—

127

Q₈

—

128

Q₈

—

166

—

129

C₂×C₄

—

166

—

130

C₂×C₄

—

166

—

131

C₂×C₄

—

166

—

132

C₂×C₄

C₂×D₃₃₂

yes

—

133

C₂×C₄

C₂×D₃₃₂

yes

characteristic

134

C₂×C₄

C₂×D₃₃₂

yes

—

135

C₂×C₄

C₂×D₃₃₂

yes

characteristic

136

C₂×C₄

C₂³×D₈₃

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

Q₈×D₈₃

yes

—

140

C₂²

C₂²×D₃₃₂

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

332

—

145

C₄

—

332

—

146

C₄

—

166

—

147

C₄

#14581814

yes

—

148

C₄

C₂×Q₈×D₈₃

yes

characteristic

149

C₄

#14581839

yes

characteristic

150

C₂

Q₈×D₃₃₂

yes

characteristic

151

C₂

C₂×#14581814

yes

characteristic

152

C₂

#19128518

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₈×Q₈×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂²×D₄
hypercenter	C₈×Q₈	D₈₃
hypocenter	C₈₃	#3383
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₃₂

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

C₈×Q₈×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂²×D₄

hypercenter

C₈×Q₈

D₈₃

hypocenter

C₈₃

#3383

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
30. Dic₃₃₂	Q₈	semidirect
77. C₈₃	#3383	semidirect

H ≅ G/N

type

30. Dic₃₃₂

Q₈

semidirect

77. C₈₃

#3383

semidirect

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

subgroup	index
#19128429	2
#19128106	2
#19128103	2
#19128009	2
#19127987	2
C₄×Dic₃₃₂	2
C₈×Q₈×C₈₃	2
#3383	83

subgroup

index

C₄×Dic₃₃₂

C₈×Q₈×C₈₃

#3383

Subgroups — #24920355

Conjugacy classes of subgroups

Complements

Maximal subgroups