Group #24908016 — subgroups

Subgroups — #24908016

Order 10624.

308 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19128468	C₂	1	1	yes	maximal, characteristic
2	#19127685	C₂	1	1	yes	maximal, characteristic
3	C₂×#14581711	C₂	1	2	yes	maximal, complemented
4	#19127471	C₂	1	2	yes	maximal, complemented
5	#536×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#14581799	—	2	4	no	—
7	C₂×#13050574	—	2	2	no	—
8	C₂×#13050574	C₂²	1	2	yes	—
9	C₂×#13050574	C₂²	1	2	yes	—
10	#14581711	—	2	4	no	—
11	#14581711	—	2	1	no	—
12	#14581709	—	2	1	no	—
13	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₈₃	—	2	1	no	—
16	#124×C₈₃	—	2	1	no	—
17	#123×C₈₃	C₄	1	4	yes	—
18	C₂²⋊Q₈×C₈₃	—	2	2	no	—
19	#117×C₈₃	C₂²	1	1	yes	characteristic
20	#13050574	—	2	4	no	—
21	#13050574	—	2	2	no	—
22	#13050574	—	2	2	no	—
23	#13050574	—	2	2	no	—
24	#13050574	D₄	1	4	yes	—
25	C₂×C₈₃⋊C₈	—	2	4	no	—
26	C₂×C₈₃⋊C₈	—	2	2	no	—
27	C₂×C₈₃⋊C₈	—	2	2	no	—
28	#41×C₈₃	—	2	2	no	—
29	#41×C₈₃	—	2	2	no	—
30	C₂×Q₈×C₈₃	—	2	1	no	—
31	C₂×Q₈×C₈₃	—	2	1	no	—
32	C₂×Q₈×C₈₃	—	2	2	no	—
33	C₂×Q₈×C₈₃	—	2	2	no	—
34	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
35	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₈₃	D₄	1	2	yes	—
39	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₃	—	2	4	no	—
41	C₄⋊C₄×C₈₃	—	2	1	no	—
42	C₂²⋊C₄×C₈₃	—	2	4	no	—
43	C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
44	C₄²×C₈₃	—	2	1	no	—
45	C₂³×C₈₃	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₈₃	—	2	2	no	—
48	Q₈×C₈₃	—	4	1	no	—
49	Q₈×C₈₃	—	4	2	no	—
50	Q₈×C₈₃	—	4	1	no	—
51	D₄×C₈₃	—	2	4	no	—
52	D₄×C₈₃	—	2	2	no	—
53	C₂×C₄×C₈₃	—	2	2	no	—
54	C₂×C₄×C₈₃	—	2	2	no	—
55	C₂×C₄×C₈₃	—	2	2	no	—
56	C₂×C₄×C₈₃	—	2	2	no	—
57	C₂×C₄×C₈₃	—	2	4	no	—
58	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#41	1	2	yes	—
63	C₈₃⋊C₈	—	2	4	no	—
64	C₈₃⋊C₈	—	4	2	no	—
65	C₂²×C₈₃	—	2	2	no	—
66	C₂²×C₈₃	—	4	1	no	—
67	C₂²×C₈₃	—	4	2	no	—
68	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈₃	C₄×D₄	1	2	yes	—
70	C₄×C₈₃	—	2	2	no	—
71	C₄×C₈₃	—	4	2	no	—
72	C₄×C₈₃	—	4	2	no	—
73	C₄×C₈₃	#120	1	1	yes	characteristic
74	C₄×C₈₃	#121	1	2	yes	—
75	C₄×C₈₃	#123	1	1	yes	characteristic
76	C₂×C₈₃	—	2	1	no	—
77	C₂×C₈₃	—	2	2	no	—
78	C₂×C₈₃	—	4	2	no	—
79	C₂×C₈₃	#386	1	1	yes	characteristic
80	#1915	—	83	1	no	maximal, Sylow
81	C₈₃	#1915	1	1	yes	characteristic, complemented, Sylow
82	#568	—	83	1	no	—
83	#536	D₈₃	1	1	yes	characteristic
84	#413	—	83	1	no	—
85	C₂×#101	—	83	2	no	—
86	#344	—	83	2	no	—
87	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
88	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
89	#131	—	166	4	no	—
90	C₂×M₁₆	—	83	2	no	—
91	C₂×M₁₆	—	83	2	no	—
92	C₂×M₁₆	—	166	2	no	—
93	C₄⋊Q₈	—	2	1	no	—
94	#124	—	2	1	no	—
95	#123	Dic₈₃	1	4	yes	—
96	C₂²⋊Q₈	—	2	2	no	—
97	#117	C₂×D₈₃	1	1	yes	characteristic
98	#101	—	166	4	no	—
99	#101	—	166	1	no	—
100	#100	—	166	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
108	C₂×D₄	C₂×Dic₈₃	1	2	yes	—
109	C₂×D₄	C₂²⋊D₈₃	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
111	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
113	M₁₆	—	83	4	no	—
114	M₁₆	—	166	2	no	—
115	M₁₆	—	166	2	no	—
116	M₁₆	—	166	4	no	—
117	M₁₆	—	166	2	no	—
118	C₂×C₈	—	166	2	no	—
119	C₂×C₈	—	166	4	no	—
120	C₂×C₈	—	166	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	C₂×Dic₈₃	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₂³	C₂²⋊Dic₈₃	1	2	yes	—
127	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂²⋊Dic₈₃	1	2	yes	—
140	C₂×C₄	D₄×D₈₃	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
143	C₂×C₄	#13050589	1	2	yes	—
144	C₈	—	332	2	no	—
145	C₈	—	166	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
150	C₂²	D₄×Dic₈₃	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#14581760	1	1	yes	characteristic
155	C₄	#14581764	1	2	yes	—
156	C₄	#14581773	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#19127581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128468

C₂

yes

maximal, characteristic

#19127685

C₂

yes

maximal, characteristic

C₂×#14581711

C₂

yes

maximal, complemented

#19127471

C₂

yes

maximal, complemented

#536×C₈₃

C₂

yes

maximal, characteristic

#14581799

—

C₂×#13050574

—

C₂×#13050574

C₂²

yes

—

C₂×#13050574

C₂²

yes

—

#14581711

—

#14581711

—

#14581709

—

C₂×#41×C₈₃

C₂²

yes

characteristic

C₂²×Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

—

#124×C₈₃

—

#123×C₈₃

C₄

yes

—

C₂²⋊Q₈×C₈₃

—

#117×C₈₃

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

D₄

yes

characteristic

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂²⋊C₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

#121

yes

—

C₄×C₈₃

#123

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#568

—

#536

D₈₃

yes

—

—

—

C₂×#41

C₂×D₈₃

yes

characteristic

C₂²×Q₈

C₂×D₈₃

yes

characteristic

#131

—

166

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₄⋊Q₈

—

#124

—

#123

Dic₈₃

yes

—

C₂²⋊Q₈

—

#117

C₂×D₈₃

yes

characteristic

#101

—

166

—

#101

—

166

—

100

#100

—

166

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₈₃

yes

characteristic

108

C₂×D₄

C₂×Dic₈₃

yes

—

109

C₂×D₄

C₂²⋊D₈₃

yes

characteristic

110

C₂²×C₄

C₂×Dic₈₃

yes

—

111

C₂²×C₄

C₂²⋊D₈₃

yes

—

112

C₂²×C₄

C₂²×D₈₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

166

—

115

M₁₆

—

166

—

116

M₁₆

—

166

—

117

M₁₆

—

166

—

118

C₂×C₈

—

166

—

119

C₂×C₈

—

166

—

120

C₂×C₈

—

166

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

C₂×Dic₈₃

yes

—

125

C₄²

—

126

C₂³

C₂²⋊Dic₈₃

yes

—

127

C₂³

C₂²×Dic₈₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₈₃

yes

—

140

C₂×C₄

D₄×D₈₃

yes

—

141

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

143

C₂×C₄

#13050589

yes

—

144

C₈

—

332

—

145

C₈

—

166

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₈₃

yes

characteristic

150

C₂²

D₄×Dic₈₃

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#14581760

yes

characteristic

155

C₄

#14581764

yes

—

156

C₄

#14581773

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#19127581

yes

characteristic

	subgroup	G/H
center	C₂	#19127581
commutator	C₂³×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#536×C₈₃	C₂
socle	C₂×C₈₃	#386
prime radical	C₂×#41×C₈₃	C₂²
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#536	D₈₃
hypocenter	C₈₃	#1915
LCS	G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#19127581

commutator

C₂³×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#536×C₈₃

C₂

socle

C₂×C₈₃

#386

prime radical

C₂×#41×C₈₃

C₂²

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
3. C₂×#14581711	C₂	semidirect
4. #19127471	C₂	semidirect
81. C₈₃	#1915	semidirect

H ≅ G/N

type

3. C₂×#14581711

C₂

semidirect

4. #19127471

C₂

semidirect

81. C₈₃

#1915

semidirect

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

subgroup	index
#19128468	2
#19127685	2
C₂×#14581711	2
#19127471	2
#536×C₈₃	2
#1915	83

subgroup

index

#536×C₈₃

Subgroups — #24908016

Conjugacy classes of subgroups

Complements

Maximal subgroups