Group #24918202 — subgroups

Subgroups — #24918202

Order 10624.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19128252	C₂	1	2	yes	maximal, complemented
2	#19128078	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128049	C₂	1	2	yes	maximal, complemented
4	#19127955	C₂	1	4	yes	maximal, complemented
5	#19127812	C₂	1	4	yes	maximal, complemented
6	C₄×#13050573	C₂	1	1	yes	maximal, characteristic, complemented
7	#498×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13050591	—	2	2	no	—
9	C₂×D₆₆₄	C₂²	1	4	yes	—
10	C₂×#13050573	C₂²	1	2	yes	—
11	#14581789	C₂²	1	2	yes	—
12	#14581781	—	2	2	no	—
13	#14581767	—	2	2	no	—
14	#14581767	C₂²	1	4	yes	—
15	#14581754	—	2	2	no	—
16	#14581754	C₂²	1	4	yes	—
17	C₄²×D₈₃	C₂²	1	1	yes	characteristic
18	#14581712	C₂²	1	8	yes	—
19	#14581702	C₂²	1	2	yes	—
20	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
22	#106×C₈₃	C₂²	1	4	yes	—
23	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	#13050591	—	4	8	no	—
25	C₂×D₃₃₂	—	2	4	no	—
26	C₂×D₃₃₂	—	2	2	no	—
27	C₂×D₃₃₂	—	2	2	no	—
28	C₂×D₃₃₂	C₂³	1	4	yes	—
29	C₂×C₄×D₈₃	—	2	4	no	—
30	C₂×C₄×D₈₃	D₄	1	2	yes	—
31	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
32	D₆₆₄	—	2	8	no	—
33	#13050573	—	2	4	no	—
34	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
35	(C₂×D₈₃)⋊C₄	—	2	4	no	—
36	(C₂×D₈₃)⋊C₄	—	2	4	no	—
37	C₄×Dic₈₃	D₄	1	2	yes	—
38	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₈₃	—	2	2	no	—
40	C₂×C₈×C₈₃	C₂³	1	2	yes	—
41	C₄⋊C₄×C₈₃	—	2	2	no	—
42	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
43	C₄²×C₈₃	C₂³	1	1	yes	characteristic
44	C₂²×D₈₃	—	2	4	no	—
45	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₈₃	—	4	4	no	—
47	C₂×C₄×C₈₃	—	2	4	no	—
48	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
52	D₃₃₂	—	2	8	no	—
53	D₃₃₂	—	4	8	no	—
54	D₃₃₂	—	4	4	no	—
55	D₃₃₂	—	4	4	no	—
56	C₄×D₈₃	—	2	4	no	—
57	C₄×D₈₃	—	2	2	no	—
58	C₄×D₈₃	—	4	8	no	—
59	C₈×C₈₃	#41	1	4	yes	complemented
60	C₈₃⋊C₈	—	2	2	no	—
61	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₈₃	—	2	1	no	—
63	C₂×D₈₃	—	2	2	no	—
64	C₂×D₈₃	—	4	4	no	—
65	C₂×D₈₃	—	4	4	no	—
66	C₂×D₈₃	—	4	4	no	—
67	C₄×C₈₃	—	4	4	no	—
68	C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₈₃	C₂×#41	1	2	yes	—
70	Dic₈₃	—	2	1	no	—
71	Dic₈₃	—	2	2	no	—
72	C₂×C₈₃	#532	1	1	yes	characteristic
73	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	D₈₃	—	4	1	no	—
75	D₈₃	—	8	4	no	—
76	#3184	—	83	1	no	maximal, Sylow
77	C₈₃	#3184	1	1	yes	characteristic, complemented, Sylow
78	#532	—	83	2	no	—
79	#498	D₈₃	1	1	yes	characteristic, complemented
80	#493	—	83	1	no	—
81	#486	—	83	2	no	—
82	#469	—	83	4	no	—
83	#442	—	83	4	no	—
84	C₄×M₁₆	—	83	1	no	—
85	C₂×#41	—	166	2	no	—
86	C₂×D₈	—	83	4	no	—
87	C₂×M₁₆	—	83	2	no	—
88	C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
89	#127	—	83	2	no	—
90	#124	—	166	2	no	—
91	#121	—	83	4	no	—
92	#121	—	166	2	no	—
93	C₄×D₄	—	83	4	no	—
94	C₄×D₄	—	166	2	no	—
95	C₂×C₄²	—	83	1	no	—
96	#106	C₂×D₈₃	1	4	yes	—
97	#102	—	83	8	no	—
98	#97	—	83	2	no	—
99	C₄×C₈	—	83	1	no	—
100	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
101	#41	—	332	8	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×D₄	—	83	4	no	—
104	C₂×D₄	—	166	4	no	—
105	C₂×D₄	—	166	2	no	—
106	C₂×D₄	—	166	2	no	—
107	C₂²×C₄	—	83	2	no	—
108	C₂²×C₄	—	83	1	no	—
109	C₂²×C₄	—	166	4	no	—
110	D₈	—	166	8	no	—
111	M₁₆	—	166	4	no	—
112	C₂×C₈	—	83	2	no	—
113	C₂×C₈	C₂²×D₈₃	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
116	C₂²⋊C₄	—	166	4	no	—
117	C₂²⋊C₄	—	166	4	no	—
118	C₄²	—	83	2	no	—
119	C₄²	—	83	1	no	—
120	C₄²	C₂²×D₈₃	1	1	yes	characteristic
121	C₂³	—	83	1	no	—
122	C₂³	—	166	4	no	—
123	Q₈	—	4	4	no	—
124	D₄	—	332	8	no	—
125	D₄	—	332	4	no	—
126	D₄	—	332	4	no	—
127	D₄	—	166	8	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	332	8	no	—
130	C₂×C₄	—	83	1	no	—
131	C₂×C₄	—	83	2	no	—
132	C₂×C₄	—	166	4	no	—
133	C₂×C₄	—	166	2	no	—
134	C₂×C₄	D₄×D₈₃	1	2	yes	—
135	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
136	C₈	—	166	2	no	—
137	C₈	#13050591	1	4	yes	complemented
138	C₂²	—	332	4	no	—
139	C₂²	—	332	4	no	—
140	C₂²	—	332	4	no	—
141	C₂²	—	166	2	no	—
142	C₂²	—	166	1	no	—
143	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
144	C₄	—	4	4	no	—
145	C₄	—	166	2	no	—
146	C₄	—	166	1	no	—
147	C₄	#14581819	1	4	yes	—
148	C₄	C₂×#13050591	1	2	yes	—
149	C₂	—	332	1	no	—
150	C₂	—	664	4	no	—
151	C₂	#19128252	1	1	yes	characteristic
152	C₂	C₂×#14581819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128252

C₂

yes

maximal, complemented

#19128078

C₂

yes

maximal, characteristic, complemented

#19128049

C₂

yes

maximal, complemented

#19127955

C₂

yes

maximal, complemented

#19127812

C₂

yes

maximal, complemented

C₄×#13050573

C₂

yes

maximal, characteristic, complemented

#498×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050591

—

C₂×D₆₆₄

C₂²

yes

—

C₂×#13050573

C₂²

yes

—

#14581789

C₂²

yes

—

#14581781

—

#14581767

—

#14581767

C₂²

yes

—

#14581754

—

#14581754

C₂²

yes

—

C₄²×D₈₃

C₂²

yes

characteristic

#14581712

C₂²

yes

—

#14581702

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

C₂²

yes

—

#106×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

#13050591

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

C₂³

yes

—

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

D₄

yes

—

C₂×C₄×D₈₃

C₂³

yes

characteristic

D₆₆₄

—

#13050573

—

C₂×C₈₃⋊C₈

C₂³

yes

—

(C₂×D₈₃)⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

C₂³

yes

characteristic

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

C₂×D₄

yes

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

#41

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₄×C₈₃

C₂×#41

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#532

yes

characteristic

C₂×C₈₃

C₂×AGL(1,ℤ/8)

yes

—

D₈₃

—

D₈₃

—

#3184

—

maximal, Sylow

C₈₃

#3184

yes

characteristic, complemented, Sylow

#532

—

#498

D₈₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×#41

—

166

—

C₂×D₈

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₈₃

yes

—

#127

—

#124

—

166

—

#121

—

#121

—

166

—

C₄×D₄

—

C₄×D₄

—

166

—

C₂×C₄²

—

#106

C₂×D₈₃

yes

—

#102

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₈₃

yes

characteristic

101

#41

—

332

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

166

—

105

C₂×D₄

—

166

—

106

C₂×D₄

—

166

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

166

—

110

D₈

—

166

—

111

M₁₆

—

166

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₈₃

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²×D₈₃

yes

—

116

C₂²⋊C₄

—

166

—

117

C₂²⋊C₄

—

166

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²×D₈₃

yes

characteristic

121

C₂³

—

122

C₂³

—

166

—

123

Q₈

—

124

D₄

—

332

—

125

D₄

—

332

—

126

D₄

—

332

—

127

D₄

—

166

—

128

C₂×C₄

—

129

C₂×C₄

—

332

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

166

—

133

C₂×C₄

—

166

—

134

C₂×C₄

D₄×D₈₃

yes

—

135

C₂×C₄

C₂³×D₈₃

yes

characteristic

136

C₈

—

166

—

137

C₈

#13050591

yes

complemented

138

C₂²

—

332

—

139

C₂²

—

332

—

140

C₂²

—

332

—

141

C₂²

—

166

—

142

C₂²

—

166

—

143

C₂²

C₂×D₄×D₈₃

yes

characteristic

144

C₄

—

145

C₄

—

166

—

146

C₄

—

166

—

147

C₄

#14581819

yes

—

148

C₄

C₂×#13050591

yes

—

149

C₂

—

332

—

150

C₂

—

664

—

151

C₂

#19128252

yes

characteristic

152

C₂

C₂×#14581819

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#498×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#498	D₈₃
hypocenter	C₈₃	#3184
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#498×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. #19128252	C₂	semidirect
2. #19128078	C₂	semidirect
3. #19128049	C₂	semidirect
4. #19127955	C₂	semidirect
5. #19127812	C₂	semidirect
6. C₄×#13050573	C₂	semidirect
7. #498×C₈₃	C₂	semidirect
59. C₈×C₈₃	#41	semidirect
77. C₈₃	#3184	semidirect
79. #498	D₈₃	semidirect
137. C₈	#13050591	semidirect

H ≅ G/N

type

1. #19128252

C₂

semidirect

2. #19128078

C₂

semidirect

3. #19128049

C₂

semidirect

4. #19127955

C₂

semidirect

5. #19127812

C₂

semidirect

6. C₄×#13050573

C₂

semidirect

7. #498×C₈₃

C₂

semidirect

59. C₈×C₈₃

#41

semidirect

77. C₈₃

#3184

semidirect

79. #498

D₈₃

semidirect

137. C₈

#13050591

semidirect

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

subgroup	index
#19128252	2
#19128078	2
#19128049	2
#19127955	2
#19127812	2
C₄×#13050573	2
#498×C₈₃	2
#3184	83

subgroup

index

#498×C₈₃

Subgroups — #24918202

Conjugacy classes of subgroups

Complements

Maximal subgroups