Group #22611256 — subgroups

Subgroups — #22611256

Order 9344.

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×#14556524	C₂	1	2	yes	maximal
2	#17988485	C₂	1	1	yes	maximal, characteristic
3	#17988349	C₂	1	2	yes	maximal
4	#17988342	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂×#11924007	—	2	1	no	—
10	#14556524	—	2	4	no	—
11	#14556498	—	2	1	no	—
12	C₂²×Q₈×C₇₃	—	2	1	no	—
13	C₄⋊Q₈×C₇₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₇₃	—	2	1	no	—
18	C₂²×Dic₇₃	—	2	1	no	—
19	C₂×#10596831	—	2	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	Q₈	1	4	yes	complemented
22	C₄⋊Dic₇₃	—	2	1	no	—
23	C₄⋊Dic₇₃	—	2	2	no	—
24	C₄⋊Dic₇₃	Q₈	1	2	yes	complemented
25	#11924007	—	4	2	no	—
26	C₂×Q₈×C₇₃	—	2	4	no	—
27	C₂×Q₈×C₇₃	—	4	2	no	—
28	C₂×Q₈×C₇₃	—	4	2	no	—
29	C₂²×C₄×C₇₃	—	2	2	no	—
30	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	—	2	4	no	—
34	C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₄⋊C₄×C₇₃	D₄	1	4	yes	—
36	C₄²×C₇₃	—	2	2	no	—
37	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₇₃	—	4	8	no	—
39	C₂×C₄×C₇₃	—	2	1	no	—
40	C₂×C₄×C₇₃	—	2	1	no	—
41	C₂×C₄×C₇₃	—	2	1	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	4	no	—
44	C₂×C₄×C₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	D₈	1	2	yes	—
47	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	4	1	no	—
54	C₂×Dic₇₃	—	4	1	no	—
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	—	4	1	no	—
57	C₂×Dic₇₃	—	4	1	no	—
58	#10596831	—	2	4	no	—
59	C₂²×C₇₃	#120	1	1	yes	characteristic
60	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₇₃	#137	1	1	yes	characteristic
66	C₄×C₇₃	—	2	2	no	—
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	—	4	4	no	—
69	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
71	Dic₇₃	—	4	2	no	—
72	Dic₇₃	—	8	1	no	—
73	C₂×C₇₃	#395	1	1	yes	characteristic
74	C₂×C₇₃	#447	1	1	yes	characteristic
75	C₂×C₇₃	#451	1	1	yes	characteristic
76	C₂×C₇₃	#474	1	1	yes	characteristic
77	C₂×C₇₃	#477	1	1	yes	characteristic
78	C₂×C₇₃	#500	1	1	yes	characteristic
79	C₂×C₇₃	#501	1	1	yes	characteristic
80	#2057	—	73	1	no	maximal, Sylow
81	C₇₃	#2057	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
83	C₂×#107	—	73	2	no	—
84	#389	—	73	1	no	—
85	#340	—	73	2	no	—
86	#336	—	73	1	no	—
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	73	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	73	1	no	—
92	C₂×C₄⋊C₄	—	73	1	no	—
93	C₂×C₄⋊C₄	—	146	1	no	—
94	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
95	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
96	#107	—	146	4	no	—
97	#95	—	146	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	—	146	1	no	—
103	C₂²×C₄	—	146	1	no	—
104	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
107	C₂×C₈	—	73	4	no	—
108	C₂×C₈	—	146	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	292	2	no	—
113	C₄⋊C₄	—	73	2	no	—
114	C₄⋊C₄	—	146	2	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₄⋊C₄	C₂²⋊D₇₃	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₇₃	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	—	292	1	no	—
128	C₂×C₄	—	292	1	no	—
129	C₂×C₄	—	292	1	no	—
130	C₂×C₄	—	292	1	no	—
131	C₂×C₄	—	292	1	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	#11924013	1	2	yes	—
134	C₂×C₄	#11924018	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
136	C₂×C₄	Q₈×D₇₃	1	2	yes	—
137	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
138	C₂×C₄	#11924029	1	1	yes	characteristic
139	C₈	—	146	4	no	—
140	C₂²	#14556560	1	1	yes	characteristic
141	C₂²	#14556567	1	1	yes	characteristic
142	C₂²	#14556589	1	1	yes	characteristic
143	C₂²	C₂×#11924013	1	1	yes	characteristic
144	C₂²	C₂×#11924018	1	1	yes	characteristic
145	C₂²	#14556617	1	1	yes	characteristic
146	C₂²	#14556622	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	292	2	no	—
151	C₄	—	584	1	no	—
152	C₄	#14556567	1	2	yes	—
153	C₄	#14556589	1	2	yes	—
154	C₂	#17988506	1	1	yes	characteristic
155	C₂	#17988721	1	1	yes	characteristic
156	C₂	#17988743	1	1	yes	characteristic
157	C₂	#17988864	1	1	yes	characteristic
158	C₂	#17988879	1	1	yes	characteristic
159	C₂	#17988990	1	1	yes	characteristic
160	C₂	#17988993	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556524

C₂

yes

maximal

#17988485

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal

#17988342

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

C₂²×Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

Q₈

yes

complemented

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

Q₈

yes

complemented

#11924007

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

C₄⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

C₄⋊Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₃

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#107

—

146

—

#95

—

146

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

146

—

103

C₂²×C₄

—

146

—

104

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

106

C₂²×C₄

C₂²×D₇₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

292

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

—

146

—

116

C₄⋊C₄

C₂²⋊D₇₃

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₇₃

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

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

292

—

130

C₂×C₄

—

292

—

131

C₂×C₄

—

292

—

132

C₂×C₄

—

146

—

133

C₂×C₄

#11924013

yes

—

134

C₂×C₄

#11924018

yes

—

135

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

136

C₂×C₄

Q₈×D₇₃

yes

—

137

C₂×C₄

Q₈×D₇₃

yes

characteristic

138

C₂×C₄

#11924029

yes

characteristic

139

C₈

—

146

—

140

C₂²

#14556560

yes

characteristic

141

C₂²

#14556567

yes

characteristic

142

C₂²

#14556589

yes

characteristic

143

C₂²

C₂×#11924013

yes

characteristic

144

C₂²

C₂×#11924018

yes

characteristic

145

C₂²

#14556617

yes

characteristic

146

C₂²

#14556622

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

292

—

151

C₄

—

584

—

152

C₄

#14556567

yes

—

153

C₄

#14556589

yes

—

154

C₂

#17988506

yes

characteristic

155

C₂

#17988721

yes

characteristic

156

C₂

#17988743

yes

characteristic

157

C₂

#17988864

yes

characteristic

158

C₂

#17988879

yes

characteristic

159

C₂

#17988990

yes

characteristic

160

C₂

#17988993

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊Q₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#2057
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊Q₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₃

hypocenter

C₇₃

#2057

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. C₂×#10596831	Q₈	semidirect
24. C₄⋊Dic₇₃	Q₈	semidirect
81. C₇₃	#2057	semidirect

H ≅ G/N

type

21. C₂×#10596831

Q₈

semidirect

24. C₄⋊Dic₇₃

Q₈

semidirect

81. C₇₃

#2057

semidirect

Subgroups — #22611256

Conjugacy classes of subgroups

Complements