Group #24908793 — subgroups

Subgroups — #24908793

Order 10624.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14581723	C₂	1	2	yes	maximal
2	#19127599	C₂	1	1	yes	maximal, characteristic
3	#19127463	C₂	1	2	yes	maximal
4	#19127456	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₈₃⋊C₄	—	2	1	no	—
10	#14581723	—	2	4	no	—
11	C₈₃⋊#95	—	2	1	no	—
12	C₂²×Q₈×C₈₃	—	2	1	no	—
13	C₄⋊Q₈×C₈₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₃	—	2	1	no	—
18	C₂²×Dic₈₃	—	2	1	no	—
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	Q₈	1	4	yes	complemented
22	C₄⋊Dic₈₃	—	2	1	no	—
23	C₄⋊Dic₈₃	—	2	2	no	—
24	C₄⋊Dic₈₃	Q₈	1	2	yes	complemented
25	Dic₈₃⋊C₄	—	4	2	no	—
26	C₂×Q₈×C₈₃	—	2	4	no	—
27	C₂×Q₈×C₈₃	—	4	2	no	—
28	C₂×Q₈×C₈₃	—	4	2	no	—
29	C₂²×C₄×C₈₃	—	2	2	no	—
30	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₃	—	2	4	no	—
34	C₄⋊C₄×C₈₃	—	2	2	no	—
35	C₄⋊C₄×C₈₃	D₄	1	4	yes	—
36	C₄²×C₈₃	—	2	2	no	—
37	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₈₃	—	4	8	no	—
39	C₂×C₄×C₈₃	—	2	1	no	—
40	C₂×C₄×C₈₃	—	2	1	no	—
41	C₂×C₄×C₈₃	—	2	1	no	—
42	C₂×C₄×C₈₃	—	2	1	no	—
43	C₂×C₄×C₈₃	—	2	4	no	—
44	C₂×C₄×C₈₃	—	4	2	no	—
45	C₂×C₄×C₈₃	—	4	2	no	—
46	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
48	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
52	C₂×Dic₈₃	—	2	2	no	—
53	C₂×Dic₈₃	—	4	1	no	—
54	C₂×Dic₈₃	—	4	1	no	—
55	C₂×Dic₈₃	—	4	1	no	—
56	C₂×Dic₈₃	—	4	1	no	—
57	C₂×Dic₈₃	—	4	1	no	—
58	C₈₃⋊C₈	—	2	4	no	—
59	C₂²×C₈₃	#120	1	1	yes	characteristic
60	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₈₃	#137	1	1	yes	characteristic
66	C₄×C₈₃	—	2	2	no	—
67	C₄×C₈₃	—	2	2	no	—
68	C₄×C₈₃	—	4	4	no	—
69	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
71	Dic₈₃	—	4	2	no	—
72	Dic₈₃	—	8	1	no	—
73	C₂×C₈₃	#395	1	1	yes	characteristic
74	C₂×C₈₃	#448	1	1	yes	characteristic
75	C₂×C₈₃	#450	1	1	yes	characteristic
76	C₂×C₈₃	#475	1	1	yes	characteristic
77	C₂×C₈₃	#476	1	1	yes	characteristic
78	C₂×C₈₃	#498	1	1	yes	characteristic
79	C₂×C₈₃	#501	1	1	yes	characteristic
80	#2058	—	83	1	no	maximal, Sylow
81	C₈₃	#2058	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	D₈₃	1	1	yes	characteristic
83	C₂×#106	—	83	2	no	—
84	#389	—	83	1	no	—
85	#340	—	83	2	no	—
86	#336	—	83	1	no	—
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	83	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	83	1	no	—
92	C₂×C₄⋊C₄	—	83	1	no	—
93	C₂×C₄⋊C₄	—	166	1	no	—
94	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
95	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
96	#106	—	166	4	no	—
97	#95	—	166	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	166	1	no	—
103	C₂²×C₄	—	166	1	no	—
104	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
107	C₂×C₈	—	83	4	no	—
108	C₂×C₈	—	166	2	no	—
109	C₂×C₈	—	166	2	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	332	2	no	—
113	C₄⋊C₄	—	83	2	no	—
114	C₄⋊C₄	—	166	2	no	—
115	C₄⋊C₄	—	166	1	no	—
116	C₄⋊C₄	C₂²⋊D₈₃	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	332	1	no	—
128	C₂×C₄	—	332	1	no	—
129	C₂×C₄	—	332	1	no	—
130	C₂×C₄	—	332	1	no	—
131	C₂×C₄	—	332	1	no	—
132	C₂×C₄	—	166	2	no	—
133	C₂×C₄	#13050577	1	2	yes	—
134	C₂×C₄	#13050578	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
136	C₂×C₄	Q₈×D₈₃	1	2	yes	—
137	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
138	C₂×C₄	#13050591	1	1	yes	characteristic
139	C₈	—	166	4	no	—
140	C₂²	#14581761	1	1	yes	characteristic
141	C₂²	#14581768	1	1	yes	characteristic
142	C₂²	#14581790	1	1	yes	characteristic
143	C₂²	C₂×#13050577	1	1	yes	characteristic
144	C₂²	C₂×#13050578	1	1	yes	characteristic
145	C₂²	#14581818	1	1	yes	characteristic
146	C₂²	#14581823	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	332	2	no	—
151	C₄	—	664	1	no	—
152	C₄	#14581768	1	2	yes	—
153	C₄	#14581790	1	2	yes	—
154	C₂	#19127620	1	1	yes	characteristic
155	C₂	#19127840	1	1	yes	characteristic
156	C₂	#19127852	1	1	yes	characteristic
157	C₂	#19127983	1	1	yes	characteristic
158	C₂	#19127988	1	1	yes	characteristic
159	C₂	#19128097	1	1	yes	characteristic
160	C₂	#19128107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581723

C₂

yes

maximal

#19127599

C₂

yes

maximal, characteristic

#19127463

C₂

yes

maximal

#19127456

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×Dic₈₃⋊C₄

—

#14581723

—

C₈₃⋊#95

—

C₂²×Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

Q₈

yes

complemented

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

Q₈

yes

complemented

Dic₈₃⋊C₄

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

D₄

yes

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

#120

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₃

C₄⋊Q₈

yes

characteristic

C₂²×C₈₃

C₂×SD₁₆

yes

characteristic

C₂²×C₈₃

C₂×SD₁₆

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

C₄⋊Q₈

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₈₃

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#106

—

166

—

#95

—

166

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

166

—

103

C₂²×C₄

—

166

—

104

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

106

C₂²×C₄

C₂²×D₈₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

166

—

109

C₂×C₈

—

166

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

332

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

166

—

115

C₄⋊C₄

—

166

—

116

C₄⋊C₄

C₂²⋊D₈₃

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

332

—

128

C₂×C₄

—

332

—

129

C₂×C₄

—

332

—

130

C₂×C₄

—

332

—

131

C₂×C₄

—

332

—

132

C₂×C₄

—

166

—

133

C₂×C₄

#13050577

yes

—

134

C₂×C₄

#13050578

yes

—

135

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

136

C₂×C₄

Q₈×D₈₃

yes

—

137

C₂×C₄

Q₈×D₈₃

yes

characteristic

138

C₂×C₄

#13050591

yes

characteristic

139

C₈

—

166

—

140

C₂²

#14581761

yes

characteristic

141

C₂²

#14581768

yes

characteristic

142

C₂²

#14581790

yes

characteristic

143

C₂²

C₂×#13050577

yes

characteristic

144

C₂²

C₂×#13050578

yes

characteristic

145

C₂²

#14581818

yes

characteristic

146

C₂²

#14581823

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

332

—

151

C₄

—

664

—

152

C₄

#14581768

yes

—

153

C₄

#14581790

yes

—

154

C₂

#19127620

yes

characteristic

155

C₂

#19127840

yes

characteristic

156

C₂

#19127852

yes

characteristic

157

C₂

#19127983

yes

characteristic

158

C₂

#19127988

yes

characteristic

159

C₂

#19128097

yes

characteristic

160

C₂

#19128107

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×C₄⋊Q₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₈₃
hypocenter	C₈₃	#2058
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×C₄⋊Q₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₈₃

hypocenter

C₈₃

#2058

LCS

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. C₂×C₈₃⋊C₈	Q₈	semidirect
24. C₄⋊Dic₈₃	Q₈	semidirect
81. C₈₃	#2058	semidirect

H ≅ G/N

type

21. C₂×C₈₃⋊C₈

Q₈

semidirect

24. C₄⋊Dic₈₃

Q₈

semidirect

81. C₈₃

#2058

semidirect

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

subgroup	index
C₂×#14581723	2
#19127599	2
#19127463	2
#19127456	2
C₂×C₄⋊Q₈×C₈₃	2
#2058	83

subgroup

index

C₂×C₄⋊Q₈×C₈₃

Subgroups — #24908793

Conjugacy classes of subgroups

Complements

Maximal subgroups