Group #24906164 — subgroups

Subgroups — #24906164

Order 10624.

208 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19128258	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19127448	C₂	1	2	yes	maximal, complemented
4	#19127446	C₂	1	2	yes	maximal, complemented
5	#432×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	—	2	1	no	—
7	#14581792	C₂²	1	1	yes	characteristic
8	#14581792	C₂²	1	1	yes	characteristic
9	#14581784	C₄	1	2	yes	—
10	#14581771	—	2	1	no	—
11	#14581771	—	2	1	no	—
12	C₄×Dic₁₆₆	—	2	1	no	—
13	C₄×Dic₁₆₆	—	2	1	no	—
14	#14581751	C₄	1	2	yes	—
15	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
16	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₈₃	—	2	1	no	—
20	#97×C₈₃	—	2	1	no	—
21	C₂×Dic₁₆₆	—	2	1	no	—
22	C₂×Dic₁₆₆	—	2	1	no	—
23	C₂×C₈₃⋊C₈	—	2	2	no	—
24	C₂×C₈₃⋊C₈	—	4	2	no	—
25	C₄⋊Dic₈₃	—	2	1	no	—
26	C₄⋊Dic₈₃	—	2	1	no	—
27	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
28	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
29	Dic₈₃⋊C₄	—	2	2	no	—
30	Dic₈₃⋊C₄	—	2	2	no	—
31	C₂²⋊Dic₈₃	—	2	2	no	—
32	C₄×Dic₈₃	—	2	2	no	—
33	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₈₃	—	2	2	no	—
37	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₄²×C₈₃	D₄	1	1	yes	characteristic
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	—	2	2	no	—
43	C₂×C₄×C₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	—	2	1	no	—
45	C₂×C₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₈₃	—	2	2	no	—
53	C₂×Dic₈₃	—	2	2	no	—
54	Dic₁₆₆	—	4	2	no	—
55	Dic₁₆₆	—	4	1	no	—
56	Dic₁₆₆	—	4	1	no	—
57	C₈×C₈₃	—	4	2	no	—
58	C₈₃⋊C₈	—	4	2	no	—
59	C₂²×C₈₃	—	2	1	no	—
60	C₂²×C₈₃	—	2	1	no	—
61	C₂²×C₈₃	#101	1	2	yes	—
62	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₈₃	—	2	1	no	—
64	C₄×C₈₃	—	2	1	no	—
65	C₄×C₈₃	—	2	1	no	—
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	#135	1	2	yes	—
68	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₈₃	#137	1	1	yes	characteristic
70	Dic₈₃	—	4	2	no	—
71	Dic₈₃	—	4	2	no	—
72	C₂×C₈₃	—	2	2	no	—
73	C₂×C₈₃	C₂×#101	1	1	yes	characteristic
74	C₂×C₈₃	#416	1	1	yes	characteristic
75	C₂×C₈₃	#417	1	1	yes	characteristic
76	#1552	—	83	1	no	maximal, Sylow
77	C₈₃	#1552	1	1	yes	characteristic, complemented, Sylow
78	#533	—	83	1	no	—
79	#432	D₈₃	1	1	yes	characteristic
80	C₂×C₄⋊C₈	—	83	1	no	—
81	#332	—	83	2	no	—
82	#332	—	83	2	no	—
83	C₂²×C₈	—	166	1	no	—
84	C₄⋊Q₈	—	83	1	no	—
85	C₄⋊Q₈	—	83	1	no	—
86	#125	—	83	2	no	—
87	C₂²⋊Q₈	—	166	1	no	—
88	C₂²⋊Q₈	—	166	1	no	—
89	C₄×Q₈	—	166	1	no	—
90	C₄×Q₈	—	166	1	no	—
91	#117	—	83	2	no	—
92	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
93	C₄⋊C₈	—	83	2	no	—
94	C₄⋊C₈	—	166	1	no	—
95	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
96	C₂²⋊C₈	—	2	1	no	—
97	#97	—	2	1	no	—
98	C₂×Q₈	—	166	1	no	—
99	C₂×Q₈	—	166	1	no	—
100	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
101	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	332	2	no	—
105	C₂×C₈	—	166	2	no	—
106	C₄⋊C₄	—	83	2	no	—
107	C₄⋊C₄	—	83	2	no	—
108	C₄⋊C₄	—	166	2	no	—
109	C₄⋊C₄	—	166	2	no	—
110	C₄⋊C₄	—	166	1	no	—
111	C₄⋊C₄	—	166	1	no	—
112	C₂²⋊C₄	—	166	2	no	—
113	C₄²	—	166	2	no	—
114	C₄²	C₄×D₈₃	1	1	yes	characteristic
115	C₄²	D₃₃₂	1	1	yes	characteristic
116	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
117	C₄²	C₂²×D₈₃	1	1	yes	characteristic
118	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
119	Q₈	—	332	2	no	—
120	Q₈	—	332	1	no	—
121	Q₈	—	332	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	166	2	no	—
127	C₂×C₄	—	166	2	no	—
128	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	332	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#14581710	1	2	yes	—
139	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	332	2	no	—
145	C₄	—	332	2	no	—
146	C₄	#14581812	1	2	yes	—
147	C₄	#14581820	1	1	yes	characteristic
148	C₄	#14581825	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	C₂×#14581710	1	1	yes	characteristic
151	C₂	#19127695	1	1	yes	characteristic
152	C₂	#19127703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127448

C₂

yes

maximal, complemented

#19127446

C₂

yes

maximal, complemented

#432×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

—

#14581792

C₂²

yes

characteristic

#14581792

C₂²

yes

characteristic

#14581784

C₄

yes

—

#14581771

—

#14581771

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

—

#14581751

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#101

yes

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#135

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#533

—

#432

D₈₃

yes

characteristic

C₂×C₄⋊C₈

—

#332

—

#332

—

C₂²×C₈

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

166

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

#117

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

—

C₂²⋊C₈

—

#97

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

100

C₂²×C₄

C₄×D₈₃

yes

characteristic

101

C₂²×C₄

D₃₃₂

yes

characteristic

102

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

332

—

105

C₂×C₈

—

166

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

166

—

109

C₄⋊C₄

—

166

—

110

C₄⋊C₄

—

166

—

111

C₄⋊C₄

—

166

—

112

C₂²⋊C₄

—

166

—

113

C₄²

—

166

—

114

C₄²

C₄×D₈₃

yes

characteristic

115

C₄²

D₃₃₂

yes

characteristic

116

C₄²

C₂²⋊D₈₃

yes

characteristic

117

C₄²

C₂²×D₈₃

yes

characteristic

118

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

119

Q₈

—

332

—

120

Q₈

—

332

—

121

Q₈

—

332

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

166

—

127

C₂×C₄

—

166

—

128

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

129

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

130

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

133

C₂×C₄

C₂×D₃₃₂

yes

characteristic

134

C₈

—

135

C₈

—

332

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14581710

yes

—

139

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

332

—

145

C₄

—

332

—

146

C₄

#14581812

yes

—

147

C₄

#14581820

yes

characteristic

148

C₄

#14581825

yes

characteristic

149

C₂

—

150

C₂

C₂×#14581710

yes

characteristic

151

C₂

#19127695

yes

characteristic

152

C₂

#19127703

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#432×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#432	D₈₃
hypocenter	C₈₃	#1552
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#432×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #19127448	C₂	semidirect
4. #19127446	C₂	semidirect
77. C₈₃	#1552	semidirect

H ≅ G/N

type

3. #19127448

C₂

semidirect

4. #19127446

C₂

semidirect

77. C₈₃

#1552

semidirect

Subgroups — #24906164

Conjugacy classes of subgroups

Complements