Group #24912671 — subgroups

Subgroups — #24912671

Order 10624.

500 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 205 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×#14581792	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#19127622	C₂	1	4	yes	maximal
4	#19127607	C₂	1	2	yes	maximal
5	#19127600	C₂	1	4	yes	maximal
6	#19127571	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
10	#14581792	—	2	4	no	—
11	C₄×Dic₁₆₆	—	2	4	no	—
12	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₈₃	C₂²	1	8	yes	—
16	C₂×Dic₈₃⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
18	C₈₃⋊#95	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
22	#95×C₈₃	C₂²	1	2	yes	—
23	C₂×Dic₁₆₆	—	2	2	no	—
24	C₂×Dic₁₆₆	—	2	2	no	—
25	C₂×Dic₁₆₆	—	2	4	no	—
26	C₂×Dic₁₆₆	—	4	4	no	—
27	C₂×Dic₁₆₆	Q₈	1	4	yes	complemented
28	C₂²×Dic₈₃	C₂³	1	4	yes	—
29	C₂²×Dic₈₃	C₂³	1	2	yes	—
30	C₂²×Dic₈₃	C₂³	1	2	yes	—
31	C₄⋊Dic₈₃	—	2	4	no	—
32	C₄⋊Dic₈₃	—	2	4	no	—
33	C₄⋊Dic₈₃	—	2	2	no	—
34	C₄⋊Dic₈₃	—	4	8	no	—
35	Dic₈₃⋊C₄	—	2	4	no	—
36	C₄×Dic₈₃	—	2	4	no	—
37	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₈₃	Q₈	1	8	yes	complemented
42	C₄²×C₈₃	—	2	4	no	—
43	C₄²×C₈₃	D₄	1	4	yes	—
44	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₈₃	—	2	4	no	—
46	C₂×C₄×C₈₃	—	2	4	no	—
47	C₂×C₄×C₈₃	—	2	2	no	—
48	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₈₃	C₂×Q₈	1	8	yes	—
53	C₂×C₄×C₈₃	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₈₃	#41	1	2	yes	—
55	C₂×Dic₈₃	—	2	2	no	—
56	C₂×Dic₈₃	—	2	2	no	—
57	C₂×Dic₈₃	—	2	2	no	—
58	C₂×Dic₈₃	—	2	4	no	—
59	C₂×Dic₈₃	—	2	4	no	—
60	C₂×Dic₈₃	—	2	8	no	—
61	C₂×Dic₈₃	C₂×Q₈	1	4	yes	—
62	Dic₁₆₆	—	2	8	no	—
63	Dic₁₆₆	—	4	8	no	—
64	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₈₃	C₂²×Q₈	1	2	yes	—
67	C₂²×C₈₃	C₂×#41	1	1	yes	characteristic
68	C₂²×C₈₃	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₈₃	—	2	2	no	—
70	C₄×C₈₃	—	2	8	no	—
71	C₄×C₈₃	C₂²⋊Q₈	1	4	yes	—
72	C₄×C₈₃	C₄⋊Q₈	1	4	yes	—
73	Dic₈₃	—	2	4	no	—
74	Dic₈₃	—	4	4	no	—
75	Dic₈₃	—	4	2	no	—
76	C₂×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₈₃	#535	1	1	yes	characteristic
79	C₂×C₈₃	#554	1	2	yes	—
80	C₂×C₈₃	Q₈²	1	2	yes	—
81	#2567	—	83	1	no	maximal, Sylow
82	C₈₃	#2567	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊Q₈	—	83	1	no	—
84	C₂×C₄×Q₈	—	83	1	no	—
85	#395	—	83	4	no	—
86	#391	—	83	2	no	—
87	#389	—	83	4	no	—
88	#384	—	83	2	no	—
89	C₄×C₄⋊C₄	D₈₃	1	1	yes	characteristic
90	C₂²×Q₈	—	83	2	no	—
91	C₂²×Q₈	—	83	1	no	—
92	C₄⋊Q₈	—	166	4	no	—
93	C₄×Q₈	—	166	4	no	—
94	C₂×C₄⋊C₄	—	83	8	no	—
95	C₂×C₄⋊C₄	—	83	1	no	—
96	C₂×C₄⋊C₄	—	83	2	no	—
97	C₂×C₄⋊C₄	—	83	2	no	—
98	C₂×C₄⋊C₄	—	83	2	no	—
99	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
100	C₂×C₄²	—	83	2	no	—
101	C₂×C₄²	C₂×D₈₃	1	2	yes	—
102	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
103	#95	—	83	8	no	—
104	#95	C₂×D₈₃	1	2	yes	—
105	C₂×Q₈	—	332	4	no	—
106	C₂×Q₈	—	83	4	no	—
107	C₂×Q₈	—	166	4	no	—
108	C₂×Q₈	—	166	2	no	—
109	C₂×Q₈	—	166	2	no	—
110	C₂²×C₄	—	83	2	no	—
111	C₂²×C₄	—	83	4	no	—
112	C₂²×C₄	—	83	2	no	—
113	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
115	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
117	C₄⋊C₄	—	332	8	no	—
118	C₄⋊C₄	—	166	4	no	—
119	C₄⋊C₄	—	166	2	no	—
120	C₄⋊C₄	—	166	4	no	—
121	C₄⋊C₄	—	166	4	no	—
122	C₄⋊C₄	Dic₁₆₆	1	8	yes	complemented
123	C₄²	—	2	4	no	—
124	C₄²	—	166	4	no	—
125	C₄²	D₃₃₂	1	4	yes	—
126	C₂³	C₂³×D₈₃	1	1	yes	characteristic
127	Q₈	—	332	8	no	—
128	Q₈	—	166	8	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	83	4	no	—
133	C₂×C₄	—	166	2	no	—
134	C₂×C₄	—	166	4	no	—
135	C₂×C₄	—	166	4	no	—
136	C₂×C₄	—	166	8	no	—
137	C₂×C₄	—	166	2	no	—
138	C₂×C₄	—	166	2	no	—
139	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
140	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
141	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
142	C₂×C₄	Q₈×D₈₃	1	2	yes	—
143	C₂×C₄	C₂×Dic₁₆₆	1	8	yes	—
144	C₂×C₄	C₂×Dic₁₆₆	1	4	yes	—
145	C₂×C₄	#13050589	1	2	yes	—
146	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
147	C₂²	C₂×Q₈×D₈₃	1	1	yes	characteristic
148	C₂²	C₂²×Dic₁₆₆	1	2	yes	—
149	C₂²	C₂×#13050589	1	1	yes	characteristic
150	C₂²	#14581838	1	2	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	2	2	no	—
153	C₄	—	332	4	no	—
154	C₄	—	332	2	no	—
155	C₄	—	166	4	no	—
156	C₄	#14581772	1	4	yes	—
157	C₄	#14581792	1	4	yes	—
158	C₂	C₂×#14581772	1	1	yes	characteristic
159	C₂	C₂×#14581792	1	1	yes	characteristic
160	C₂	#19128270	1	1	yes	characteristic
161	C₂	#19128408	1	2	yes	—
162	C₂	Q₈×Dic₁₆₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₆₆

C₂

yes

maximal, characteristic

#19127622

C₂

yes

maximal

#19127607

C₂

yes

maximal

#19127600

C₂

yes

maximal

#19127571

C₂

yes

maximal

C₄×C₄⋊C₄×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

#14581792

—

C₄×Dic₁₆₆

—

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#95×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

Q₈

yes

complemented

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

Q₈

yes

complemented

C₄²×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

C₂⁴

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×Q₈

yes

—

Dic₁₆₆

—

Dic₁₆₆

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂²×C₈₃

C₂²×Q₈

yes

characteristic

C₂²×C₈₃

C₂²×Q₈

yes

—

C₂²×C₈₃

C₂×#41

yes

characteristic

C₂²×C₈₃

H(5,𝔽₂)

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

C₄⋊Q₈

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₈₃

C₂×C₄⋊Q₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

Q₈²

yes

—

#2567

—

maximal, Sylow

C₈₃

#2567

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

D₈₃

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₄⋊Q₈

—

166

—

C₄×Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₈₃

yes

—

102

C₂×C₄²

C₂×D₈₃

yes

characteristic

103

#95

—

104

#95

C₂×D₈₃

yes

—

105

C₂×Q₈

—

332

—

106

C₂×Q₈

—

107

C₂×Q₈

—

166

—

108

C₂×Q₈

—

166

—

109

C₂×Q₈

—

166

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₈₃

yes

characteristic

114

C₂²×C₄

C₂²×D₈₃

yes

—

115

C₂²×C₄

C₂²×D₈₃

yes

characteristic

116

C₂²×C₄

C₂²×D₈₃

yes

characteristic

117

C₄⋊C₄

—

332

—

118

C₄⋊C₄

—

166

—

119

C₄⋊C₄

—

166

—

120

C₄⋊C₄

—

166

—

121

C₄⋊C₄

—

166

—

122

C₄⋊C₄

Dic₁₆₆

yes

complemented

123

C₄²

—

124

C₄²

—

166

—

125

C₄²

D₃₃₂

yes

—

126

C₂³

C₂³×D₈₃

yes

characteristic

127

Q₈

—

332

—

128

Q₈

—

166

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

166

—

134

C₂×C₄

—

166

—

135

C₂×C₄

—

166

—

136

C₂×C₄

—

166

—

137

C₂×C₄

—

166

—

138

C₂×C₄

—

166

—

139

C₂×C₄

C₂×D₃₃₂

yes

—

140

C₂×C₄

C₂×D₃₃₂

yes

—

141

C₂×C₄

C₂×D₃₃₂

yes

—

142

C₂×C₄

Q₈×D₈₃

yes

—

143

C₂×C₄

C₂×Dic₁₆₆

yes

—

144

C₂×C₄

C₂×Dic₁₆₆

yes

—

145

C₂×C₄

#13050589

yes

—

146

C₂²

C₂²×D₃₃₂

yes

characteristic

147

C₂²

C₂×Q₈×D₈₃

yes

characteristic

148

C₂²

C₂²×Dic₁₆₆

yes

—

149

C₂²

C₂×#13050589

yes

characteristic

150

C₂²

#14581838

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

332

—

154

C₄

—

332

—

155

C₄

—

166

—

156

C₄

#14581772

yes

—

157

C₄

#14581792

yes

—

158

C₂

C₂×#14581772

yes

characteristic

159

C₂

C₂×#14581792

yes

characteristic

160

C₂

#19128270

yes

characteristic

161

C₂

#19128408

yes

—

162

C₂

Q₈×Dic₁₆₆

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, prime radical, cosocle	C₂³×C₈₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₈₃	C₂
hypercenter	C₄×C₄⋊C₄	D₈₃
hypocenter	C₈₃	#2567
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, prime radical, cosocle

C₂³×C₈₃

C₂⁴

Fitting

C₄×C₄⋊C₄×C₈₃

C₂

hypercenter

C₄×C₄⋊C₄

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
27. C₂×Dic₁₆₆	Q₈	semidirect
41. C₄⋊C₄×C₈₃	Q₈	semidirect
82. C₈₃	#2567	semidirect
122. C₄⋊C₄	Dic₁₆₆	semidirect

H ≅ G/N

type

27. C₂×Dic₁₆₆

Q₈

semidirect

41. C₄⋊C₄×C₈₃

Q₈

semidirect

82. C₈₃

#2567

semidirect

122. C₄⋊C₄

Dic₁₆₆

semidirect

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

subgroup	index
C₂×#14581792	2
C₂×C₄×Dic₁₆₆	2
#19127622	2
#19127607	2
#19127600	2
#19127571	2
C₄×C₄⋊C₄×C₈₃	2
#2567	83

subgroup

index

C₂×C₄×Dic₁₆₆

C₄×C₄⋊C₄×C₈₃

Subgroups — #24912671

Conjugacy classes of subgroups

Complements

Maximal subgroups