Group #24908374 — subgroups

Subgroups — #24908374

Order 10624.

316 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#14581728	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127706	C₂	1	2	yes	maximal, complemented
3	C₂×#14581709	C₂	1	2	yes	maximal, complemented
4	#19127656	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	C₂×#13050574	C₂²	1	1	yes	characteristic
8	#14581751	C₂²	1	1	yes	characteristic
9	#14581728	—	2	2	no	—
10	#14581713	—	2	4	no	—
11	#14581709	—	2	4	no	—
12	#14581702	—	2	1	no	—
13	C₈×Dic₈₃	—	2	1	no	—
14	C₂²×D₄×C₈₃	C₂²	1	2	yes	—
15	C₂×D₈×C₈₃	—	2	1	no	—
16	C₂×D₈×C₈₃	—	2	1	no	—
17	C₂×D₈×C₈₃	—	2	2	no	—
18	C₂×D₈×C₈₃	C₄	1	4	yes	complemented
19	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	#13050574	—	2	1	no	—
21	#13050574	—	2	2	no	—
22	#13050574	—	2	2	no	—
23	#13050574	D₄	1	2	yes	—
24	C₂×C₈₃⋊C₈	—	2	1	no	—
25	C₂×C₈₃⋊C₈	—	2	2	no	—
26	C₄⋊Dic₈₃	D₄	1	2	yes	—
27	C₂²⋊Dic₈₃	—	2	1	no	—
28	C₄×Dic₈₃	—	2	1	no	—
29	C₂⁴×C₈₃	—	2	2	no	—
30	C₂×D₄×C₈₃	—	2	2	no	—
31	C₂×D₄×C₈₃	—	2	2	no	—
32	C₂×D₄×C₈₃	—	2	2	no	—
33	C₂×D₄×C₈₃	—	2	2	no	—
34	C₂×D₄×C₈₃	—	2	2	no	—
35	C₂×D₄×C₈₃	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
37	D₈×C₈₃	—	2	4	no	—
38	D₈×C₈₃	—	2	4	no	—
39	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₃	D₄	1	2	yes	—
43	C₂³×C₈₃	—	2	4	no	—
44	C₂³×C₈₃	—	4	2	no	—
45	C₂³×C₈₃	—	4	2	no	—
46	C₂³×C₈₃	—	8	4	no	—
47	C₂³×C₈₃	#41	1	1	yes	characteristic
48	D₄×C₈₃	—	2	4	no	—
49	D₄×C₈₃	—	2	4	no	—
50	D₄×C₈₃	—	4	2	no	—
51	D₄×C₈₃	—	4	2	no	—
52	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
58	C₂×Dic₈₃	—	2	2	no	—
59	C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
60	C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
61	C₈₃⋊C₈	—	2	2	no	—
62	C₈₃⋊C₈	—	4	2	no	—
63	C₂²×C₈₃	—	2	2	no	—
64	C₂²×C₈₃	—	4	4	no	—
65	C₂²×C₈₃	—	8	4	no	—
66	C₂²×C₈₃	—	8	4	no	—
67	C₂²×C₈₃	—	8	4	no	—
68	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
69	C₂²×C₈₃	#121	1	1	yes	characteristic
70	C₂²×C₈₃	#124	1	1	yes	characteristic
71	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
73	C₄×C₈₃	#121	1	1	yes	characteristic
74	C₄×C₈₃	#127	1	1	yes	characteristic
75	Dic₈₃	—	4	2	no	—
76	C₂×C₈₃	—	2	1	no	—
77	C₂×C₈₃	—	2	1	no	—
78	C₂×C₈₃	—	8	4	no	—
79	C₂×C₈₃	#390	1	1	yes	characteristic
80	C₂×C₈₃	#472	1	2	yes	—
81	#1978	—	83	1	no	maximal, Sylow
82	C₈₃	#1978	1	1	yes	characteristic, complemented, Sylow
83	C₂²×D₈	D₈₃	1	1	yes	characteristic
84	C₂×#108	—	83	1	no	—
85	#418	—	83	2	no	—
86	C₂×#100	—	83	2	no	—
87	#405	—	83	1	no	—
88	C₂²×D₄	C₂×D₈₃	1	2	yes	—
89	C₂×D₈	—	2	1	no	—
90	C₂×D₈	—	2	1	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	Dic₈₃	1	4	yes	complemented
93	C₂×M₁₆	—	83	1	no	—
94	C₂×M₁₆	—	83	2	no	—
95	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
96	#117	—	83	1	no	—
97	#108	—	166	2	no	—
98	#102	—	166	4	no	—
99	#100	—	166	4	no	—
100	#97	—	166	1	no	—
101	C₄×C₈	—	166	1	no	—
102	C₂⁴	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	C₂×Dic₈₃	1	4	yes	—
109	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
110	D₈	—	2	4	no	—
111	D₈	—	2	4	no	—
112	M₁₆	—	83	2	no	—
113	M₁₆	—	166	2	no	—
114	M₁₆	—	166	2	no	—
115	M₁₆	—	166	1	no	—
116	C₂×C₈	—	166	2	no	—
117	C₂×C₈	—	166	1	no	—
118	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
119	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
122	C₄⋊C₄	—	83	2	no	—
123	C₂²⋊C₄	—	166	1	no	—
124	C₄²	—	166	1	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	8	4	no	—
129	C₂³	#13050589	1	1	yes	characteristic
130	D₄	—	2	4	no	—
131	D₄	—	2	4	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	166	2	no	—
135	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
136	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
137	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
140	C₂×C₄	#13050589	1	1	yes	characteristic
141	C₈	—	332	2	no	—
142	C₈	—	166	2	no	—
143	C₈	C₂²⋊Dic₈₃	1	2	yes	—
144	C₈	C₂²⋊Dic₈₃	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	4	no	—
147	C₂²	—	8	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	4	no	—
150	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
151	C₂²	#14581763	1	1	yes	characteristic
152	C₂²	#14581779	1	1	yes	characteristic
153	C₄	—	332	2	no	—
154	C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
155	C₄	D₄×Dic₈₃	1	1	yes	characteristic
156	C₄	#14581763	1	1	yes	characteristic
157	C₄	#14581788	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	8	4	no	—
161	C₂	#19127604	1	1	yes	characteristic
162	C₂	#19127968	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581728

C₂

yes

maximal, characteristic, complemented

#19127706

C₂

yes

maximal, complemented

C₂×#14581709

C₂

yes

maximal, complemented

#19127656

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₂×#13050574

C₂²

yes

characteristic

#14581751

C₂²

yes

—

—

—

—

C₈×Dic₈₃

—

C₂²×D₄×C₈₃

C₂²

yes

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

C₄

yes

complemented

C₂²×C₈×C₈₃

C₂²

yes

—

—

—

D₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

D₄

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂⁴×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

D₈×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₈×C₈₃

C₂²⋊C₄

yes

—

C₈×C₈₃

C₂²⋊C₄

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#390

yes

characteristic

C₂×C₈₃

#472

yes

—

#1978

—

maximal, Sylow

C₈₃

#1978

yes

characteristic, complemented, Sylow

C₂²×D₈

D₈₃

yes

—

—

—

—

C₂²×D₄

C₂×D₈₃

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₈₃

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

#108

—

166

—

#102

—

166

—

#100

—

166

—

100

#97

—

166

—

101

C₄×C₈

—

166

—

102

C₂⁴

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

C₂×Dic₈₃

yes

—

109

C₂²×C₄

C₂²×D₈₃

yes

characteristic

110

D₈

—

111

D₈

—

112

M₁₆

—

113

M₁₆

—

166

—

114

M₁₆

—

166

—

115

M₁₆

—

166

—

116

C₂×C₈

—

166

—

117

C₂×C₈

—

166

—

118

C₂×C₈

C₂×Dic₈₃

yes

—

119

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

120

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

121

C₂×C₈

C₂²⋊D₈₃

yes

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

166

—

124

C₄²

—

166

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

#13050589

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

166

—

135

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

136

C₂×C₄

D₄×D₈₃

yes

characteristic

137

C₂×C₄

D₄×D₈₃

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

140

C₂×C₄

#13050589

yes

characteristic

141

C₈

—

332

—

142

C₈

—

166

—

143

C₈

C₂²⋊Dic₈₃

yes

—

144

C₈

C₂²⋊Dic₈₃

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

D₄×Dic₈₃

yes

characteristic

151

C₂²

#14581763

yes

characteristic

152

C₂²

#14581779

yes

characteristic

153

C₄

—

332

—

154

C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

155

C₄

D₄×Dic₈₃

yes

characteristic

156

C₄

#14581763

yes

characteristic

157

C₄

#14581788

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

#19127604

yes

characteristic

162

C₂

#19127968

yes

—

	subgroup	G/H
center	C₂²	#14581763
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting, prime radical	C₂²×D₈×C₈₃	C₂
socle	C₂²×C₈₃	#121
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂²×D₈	D₈₃
hypocenter	C₈₃	#1978
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

#14581763

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting, prime radical

C₂²×D₈×C₈₃

C₂

socle

C₂²×C₈₃

#121

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂²×D₈

D₈₃

hypocenter

C₈₃

#1978

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#14581728	C₂	semidirect
2. #19127706	C₂	semidirect
3. C₂×#14581709	C₂	semidirect
4. #19127656	C₂	semidirect
18. C₂×D₈×C₈₃	C₄	semidirect
82. C₈₃	#1978	semidirect
92. C₂×D₈	Dic₈₃	semidirect

H ≅ G/N

type

1. C₂×#14581728

C₂

semidirect

2. #19127706

C₂

semidirect

3. C₂×#14581709

C₂

semidirect

4. #19127656

C₂

semidirect

18. C₂×D₈×C₈₃

C₄

semidirect

82. C₈₃

#1978

semidirect

92. C₂×D₈

Dic₈₃

semidirect

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

subgroup	index
C₂×#14581728	2
#19127706	2
C₂×#14581709	2
#19127656	2
C₂²×D₈×C₈₃	2
#1978	83

subgroup

index

C₂²×D₈×C₈₃

Subgroups — #24908374

Conjugacy classes of subgroups

Complements

Maximal subgroups