Group #22609089 — subgroups

Subgroups — #22609089

Order 9344.

188 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17988939	C₂	1	1	yes	maximal, characteristic
2	#17988882	C₂	1	1	yes	maximal, characteristic
3	#17988882	C₂	1	1	yes	maximal, characteristic
4	#17988333	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#10596831	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊#10596831	C₂	1	1	yes	maximal, characteristic
7	#557×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556591	C₂²	1	1	yes	characteristic
9	#14556524	—	2	1	no	—
10	#14556524	—	2	1	no	—
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	#14556515	—	2	1	no	—
14	#14556515	—	2	1	no	—
15	#14556514	—	2	2	no	—
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	—	2	1	no	—
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	#125×C₇₃	—	2	2	no	—
20	#125×C₇₃	—	2	1	no	—
21	C₄×Q₈×C₇₃	—	2	1	no	—
22	C₄×Q₈×C₇₃	—	2	1	no	—
23	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₄⋊C₄×C₇₃	—	2	2	no	—
36	C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
43	C₄²×C₇₃	—	2	2	no	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	—	2	1	no	—
46	C₄²×C₇₃	C₂³	1	1	yes	characteristic
47	Q₈×C₇₃	—	2	1	no	—
48	Q₈×C₇₃	—	2	1	no	—
49	Q₈×C₇₃	—	2	1	no	—
50	Q₈×C₇₃	—	4	1	no	—
51	Q₈×C₇₃	—	4	1	no	—
52	Q₈×C₇₃	Q₁₆	1	2	yes	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₇₃	—	4	1	no	—
62	Dic₁₄₆	—	4	2	no	—
63	#10596831	—	4	1	no	—
64	#10596831	—	4	1	no	—
65	#10596831	—	4	1	no	—
66	C₂²×C₇₃	#120	1	1	yes	characteristic
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	4	1	no	—
70	C₄×C₇₃	—	4	2	no	—
71	C₄×C₇₃	—	4	1	no	—
72	C₄×C₇₃	—	4	1	no	—
73	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₇₃	#135	1	1	yes	characteristic
75	C₄×C₇₃	#137	1	1	yes	characteristic
76	C₄×C₇₃	#137	1	1	yes	characteristic
77	Dic₇₃	—	8	1	no	—
78	C₂×C₇₃	#451	1	1	yes	characteristic
79	C₂×C₇₃	#452	1	1	yes	characteristic
80	C₂×C₇₃	#454	1	1	yes	characteristic
81	#1636	—	73	1	no	maximal, Sylow
82	C₇₃	#1636	1	1	yes	characteristic, complemented, Sylow
83	#557	D₇₃	1	1	yes	characteristic
84	#487	—	73	1	no	—
85	#477	—	73	1	no	—
86	#477	—	73	1	no	—
87	#332	—	73	1	no	—
88	Q₈⋊C₈	—	73	1	no	—
89	Q₈⋊C₈	—	73	1	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	C₄⋊Q₈	—	73	1	no	—
92	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
93	#125	—	2	2	no	—
94	#125	—	2	1	no	—
95	C₄×Q₈	—	2	1	no	—
96	C₄×Q₈	—	2	1	no	—
97	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
98	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
99	#107	—	146	1	no	—
100	#107	—	146	1	no	—
101	C₄⋊C₈	—	73	1	no	—
102	C₄⋊C₈	—	73	1	no	—
103	#103	—	146	1	no	—
104	#103	—	146	1	no	—
105	#103	—	146	2	no	—
106	C₄×C₈	—	73	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	146	1	no	—
109	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
110	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
111	C₂×C₈	—	146	1	no	—
112	C₂×C₈	—	146	1	no	—
113	C₂×C₈	—	146	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	146	1	no	—
121	C₄⋊C₄	—	146	1	no	—
122	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
123	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
124	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₄²	—	2	2	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	C₂²×D₇₃	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	2	1	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	292	2	no	—
135	Q₈	#11924018	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
145	C₈	—	292	1	no	—
146	C₈	—	292	1	no	—
147	C₈	—	292	1	no	—
148	C₂²	#14556560	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	584	1	no	—
156	C₄	C₂×#11924018	1	1	yes	characteristic
157	C₄	#14556611	1	1	yes	characteristic
158	C₄	#14556621	1	1	yes	characteristic
159	C₄	#14556622	1	1	yes	characteristic
160	C₂	#17988743	1	1	yes	characteristic
161	C₂	#17988749	1	1	yes	characteristic
162	C₂	#17988760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988939

C₂

yes

maximal, characteristic

#17988882

C₂

yes

maximal, characteristic

#17988882

C₂

yes

maximal, characteristic

#17988333

C₂

yes

maximal, characteristic

Q₈⋊#10596831

C₂

yes

maximal, characteristic

Q₈⋊#10596831

C₂

yes

maximal, characteristic

#557×C₇₃

C₂

yes

maximal, characteristic

#14556591

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#125×C₇₃

—

#125×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#557

D₇₃

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₃

yes

characteristic

C₄×Q₈

C₂×D₇₃

yes

characteristic

#107

—

146

—

100

#107

—

146

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

#103

—

146

—

104

#103

—

146

—

105

#103

—

146

—

106

C₄×C₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

146

—

109

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

110

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

111

C₂×C₈

—

146

—

112

C₂×C₈

—

146

—

113

C₂×C₈

—

146

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

146

—

121

C₄⋊C₄

—

146

—

122

C₄⋊C₄

C₂²⋊D₇₃

yes

—

123

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

124

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×D₇₃

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

292

—

135

Q₈

#11924018

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

292

—

142

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

145

C₈

—

292

—

146

C₈

—

292

—

147

C₈

—

292

—

148

C₂²

#14556560

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

584

—

156

C₄

C₂×#11924018

yes

characteristic

157

C₄

#14556611

yes

characteristic

158

C₄

#14556621

yes

characteristic

159

C₄

#14556622

yes

characteristic

160

C₂

#17988743

yes

characteristic

161

C₂

#17988749

yes

characteristic

162

C₂

#17988760

yes

characteristic

	subgroup	G/H
center	C₂²	#14556560
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#557×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#557	D₇₃
hypocenter	C₇₃	#1636
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #557

subgroup

G/H

center

C₂²

#14556560

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#557×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #557

N	H ≅ G/N	type
82. C₇₃	#1636	semidirect

H ≅ G/N

type

82. C₇₃

#1636

semidirect

Subgroups — #22609089

Conjugacy classes of subgroups

Complements