Group #24906307 — subgroups

Subgroups — #24906307

Order 10624.

166 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 73 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	Q₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127779	C₂	1	1	yes	maximal, characteristic
3	#19127452	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#19127423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050574	—	2	1	no	—
9	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
10	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
13	C₄×C₈₃⋊C₈	—	2	1	no	—
14	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₈₃	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
19	#106×C₈₃	C₄	1	1	yes	characteristic
20	#103×C₈₃	C₄	1	1	yes	characteristic
21	#102×C₈₃	C₄	1	1	yes	characteristic
22	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	#13050574	—	4	2	no	—
24	C₂×C₈₃⋊C₈	—	2	1	no	—
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₂×C₈₃⋊C₈	—	2	1	no	—
27	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	—	2	1	no	—
32	SD₁₆×C₈₃	—	2	2	no	—
33	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₈₃	—	2	1	no	—
36	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₈₃	—	2	1	no	—
39	C₄²×C₈₃	—	2	1	no	—
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₂³×C₈₃	—	2	1	no	—
42	Q₈×C₈₃	—	2	1	no	—
43	Q₈×C₈₃	M₁₆	1	2	yes	complemented
44	D₄×C₈₃	—	2	1	no	—
45	D₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	—	2	1	no	—
47	C₂×C₄×C₈₃	—	2	1	no	—
48	C₂×C₄×C₈₃	—	2	1	no	—
49	C₂×C₄×C₈₃	—	4	1	no	—
50	C₂×C₄×C₈₃	—	4	1	no	—
51	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
54	C₈×C₈₃	—	2	1	no	—
55	C₈×C₈₃	—	2	1	no	—
56	C₈₃⋊C₈	—	2	1	no	—
57	C₈₃⋊C₈	—	4	1	no	—
58	C₈₃⋊C₈	—	4	1	no	—
59	C₈₃⋊C₈	—	4	1	no	—
60	C₈₃⋊C₈	SD₁₆	1	2	yes	complemented
61	C₂²×C₈₃	—	4	1	no	—
62	C₂²×C₈₃	—	4	1	no	—
63	C₂²×C₈₃	—	4	1	no	—
64	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
65	C₄×C₈₃	—	2	2	no	—
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	—	4	1	no	—
68	C₄×C₈₃	—	4	1	no	—
69	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
70	C₄×C₈₃	#131	1	1	yes	characteristic
71	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₈₃	#135	1	1	yes	characteristic
73	C₂×C₈₃	—	8	1	no	—
74	C₂×C₈₃	#435	1	1	yes	characteristic
75	C₂×C₈₃	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₈₃	#444	1	1	yes	characteristic
77	#1583	—	83	1	no	maximal, Sylow
78	C₈₃	#1583	1	1	yes	characteristic, complemented, Sylow
79	C₈×Q₈	—	83	1	no	—
80	C₄×SD₁₆	D₈₃	1	1	yes	characteristic
81	#436	—	83	1	no	—
82	#334	—	83	1	no	—
83	Q₈⋊C₈	—	83	1	no	—
84	D₄⋊C₈	—	83	1	no	—
85	#322	—	83	1	no	—
86	C₂×SD₁₆	Dic₈₃	1	1	yes	characteristic
87	C₂×M₁₆	—	166	1	no	—
88	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
89	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
90	#106	Dic₈₃	1	1	yes	characteristic
91	C₄⋊C₈	—	83	1	no	—
92	C₄⋊C₈	—	83	1	no	—
93	C₄⋊C₈	—	166	1	no	—
94	#103	Dic₈₃	1	1	yes	characteristic
95	#102	Dic₈₃	1	1	yes	characteristic
96	C₂²⋊C₈	—	166	1	no	—
97	C₄×C₈	—	83	1	no	—
98	C₄×C₈	—	83	1	no	—
99	C₄×C₈	—	166	1	no	—
100	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
101	C₂×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
102	C₂×D₄	C₂×Dic₈₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	2	2	no	—
105	M₁₆	—	332	2	no	—
106	C₂×C₈	—	83	1	no	—
107	C₂×C₈	—	83	1	no	—
108	C₂×C₈	—	166	1	no	—
109	C₂×C₈	—	166	1	no	—
110	C₂×C₈	—	166	1	no	—
111	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
115	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₈₃	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#13050574	1	2	yes	complemented
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
130	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
131	C₂×C₄	#13050589	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	332	1	no	—
135	C₈	—	332	1	no	—
136	C₈	—	332	1	no	—
137	C₈	—	83	2	no	—
138	C₈	—	166	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	C₂×#13050574	1	1	yes	characteristic
148	C₄	#14581799	1	1	yes	characteristic
149	C₄	SD₁₆×D₈₃	1	1	yes	characteristic
150	C₄	#14581813	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	#19127773	1	1	yes	characteristic
153	C₂	SD₁₆×Dic₈₃	1	1	yes	characteristic
154	C₂	#19127823	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#19127779

C₂

yes

maximal, characteristic

#19127452

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127423

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₈₃

C₄

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄×D₄×C₈₃

C₂²

yes

characteristic

#106×C₈₃

C₄

yes

characteristic

#103×C₈₃

C₄

yes

characteristic

#102×C₈₃

C₄

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×Q₈×C₈₃

C₂×C₄

yes

characteristic

C₂×D₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

—

SD₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

M₁₆

yes

complemented

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

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

SD₁₆

yes

complemented

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₄×C₈₃

#131

yes

characteristic

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₄×C₈₃

#135

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

#435

yes

characteristic

C₂×C₈₃

C₄×SD₁₆

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×SD₁₆

D₈₃

yes

characteristic

#436

—

#334

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×SD₁₆

Dic₈₃

yes

characteristic

C₂×M₁₆

—

166

—

C₄×Q₈

C₂×D₈₃

yes

characteristic

C₄×D₄

C₂×D₈₃

yes

characteristic

#106

Dic₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

#103

Dic₈₃

yes

characteristic

#102

Dic₈₃

yes

characteristic

C₂²⋊C₈

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

166

—

100

C₄×C₈

C₂×D₈₃

yes

characteristic

101

C₂×Q₈

C₂×Dic₈₃

yes

characteristic

102

C₂×D₄

C₂×Dic₈₃

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

105

M₁₆

—

332

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

166

—

109

C₂×C₈

—

166

—

110

C₂×C₈

—

166

—

111

C₂×C₈

C₂×Dic₈₃

yes

characteristic

112

C₂×C₈

C₂×Dic₈₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×Dic₈₃

yes

characteristic

115

C₄⋊C₄

C₂×Dic₈₃

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₈₃

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

#13050574

yes

complemented

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

130

C₂×C₄

D₄×D₈₃

yes

characteristic

131

C₂×C₄

#13050589

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

332

—

135

C₈

—

332

—

136

C₈

—

332

—

137

C₈

—

138

C₈

—

166

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×Dic₈₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×#13050574

yes

characteristic

148

C₄

#14581799

yes

characteristic

149

C₄

SD₁₆×D₈₃

yes

characteristic

150

C₄

#14581813

yes

characteristic

151

C₂

—

152

C₂

#19127773

yes

characteristic

153

C₂

SD₁₆×Dic₈₃

yes

characteristic

154

C₂

#19127823

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₄×SD₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×D₄×C₈₃	C₂×C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₄×SD₁₆	D₈₃
hypocenter	C₈₃	#1583
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₄×SD₁₆×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂×D₄×C₈₃

C₂×C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₄×SD₁₆

D₈₃

hypocenter

C₈₃

#1583

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

N	H ≅ G/N	type
1. Q₈×C₈₃⋊C₈	C₂	semidirect
3. #19127452	C₂	semidirect
4. Q₈⋊(C₈₃⋊C₈)	C₂	semidirect
6. #19127423	C₂	semidirect
43. Q₈×C₈₃	M₁₆	semidirect
60. C₈₃⋊C₈	SD₁₆	semidirect
78. C₈₃	#1583	semidirect
121. Q₈	#13050574	semidirect

H ≅ G/N

type

1. Q₈×C₈₃⋊C₈

C₂

semidirect

3. #19127452

C₂

semidirect

4. Q₈⋊(C₈₃⋊C₈)

C₂

semidirect

6. #19127423

C₂

semidirect

43. Q₈×C₈₃

M₁₆

semidirect

60. C₈₃⋊C₈

SD₁₆

semidirect

78. C₈₃

#1583

semidirect

121. Q₈

#13050574

semidirect

Subgroups — #24906307

Conjugacy classes of subgroups

Complements