Group #22609152 — subgroups

Subgroups — #22609152

Order 9344.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17988937	C₂	1	1	yes	maximal, characteristic
2	#17988872	C₂	1	1	yes	maximal, characteristic
3	#17988797	C₂	1	1	yes	maximal, characteristic
4	#17988336	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	Q₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924018	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556522	—	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	2	no	—
15	#14556514	—	2	1	no	—
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	—	2	2	no	—
18	C₄⋊Q₈×C₇₃	—	2	1	no	—
19	C₄⋊Q₈×C₇₃	—	2	1	no	—
20	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
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	#11924018	—	4	2	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₂×Q₈×C₇₃	—	2	2	no	—
33	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₇₃	D₄	1	2	yes	—
36	C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₄⋊C₄×C₇₃	—	2	2	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
44	C₄²×C₇₃	—	2	2	no	—
45	C₄²×C₇₃	—	2	1	no	—
46	C₄²×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	Q₈×C₇₃	—	2	1	no	—
49	Q₈×C₇₃	—	2	1	no	—
50	Q₈×C₇₃	—	4	2	no	—
51	Q₈×C₇₃	—	8	2	no	—
52	Q₈×C₇₃	SD₁₆	1	2	yes	—
53	Q₈×C₇₃	Q₁₆	1	2	yes	complemented
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₇₃	—	4	1	no	—
63	Dic₁₄₆	—	4	1	no	—
64	Dic₁₄₆	—	4	1	no	—
65	#10596831	—	4	1	no	—
66	#10596831	—	4	1	no	—
67	#10596831	—	4	1	no	—
68	C₂²×C₇₃	#120	1	1	yes	characteristic
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	4	2	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	—	4	1	no	—
75	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₄×C₇₃	#137	1	1	yes	characteristic
78	C₄×C₇₃	#137	1	1	yes	characteristic
79	Dic₇₃	—	8	1	no	—
80	C₂×C₇₃	#448	1	1	yes	characteristic
81	C₂×C₇₃	#451	1	1	yes	characteristic
82	C₂×C₇₃	#455	1	1	yes	characteristic
83	#1645	—	73	1	no	maximal, Sylow
84	C₇₃	#1645	1	1	yes	characteristic, complemented, Sylow
85	Q₈²	D₇₃	1	1	yes	characteristic
86	#487	—	73	1	no	—
87	#475	—	73	1	no	—
88	#462	—	73	1	no	—
89	#333	—	73	1	no	—
90	Q₈⋊C₈	—	73	1	no	—
91	Q₈⋊C₈	—	73	1	no	—
92	C₂×Q₁₆	—	146	1	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	73	1	no	—
97	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
98	C₄×Q₈	—	2	1	no	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
101	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
102	#106	—	146	1	no	—
103	C₄⋊C₈	—	73	1	no	—
104	C₄⋊C₈	—	73	1	no	—
105	#103	—	146	1	no	—
106	#103	—	146	2	no	—
107	#103	—	146	1	no	—
108	C₄×C₈	—	73	1	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	146	1	no	—
111	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
112	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
113	C₂×Q₈	C₂²⋊D₇₃	1	2	yes	—
114	Q₁₆	—	292	2	no	—
115	C₂×C₈	—	146	1	no	—
116	C₂×C₈	—	146	1	no	—
117	C₂×C₈	—	146	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₄⋊C₄	—	146	1	no	—
126	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	C₂²×D₇₃	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	4	2	no	—
135	Q₈	—	8	2	no	—
136	Q₈	—	292	1	no	—
137	Q₈	—	292	1	no	—
138	Q₈	#11924015	1	2	yes	—
139	Q₈	#11924018	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	292	1	no	—
146	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
149	C₈	—	292	1	no	—
150	C₈	—	292	1	no	—
151	C₈	—	292	1	no	—
152	C₂²	#14556560	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	584	1	no	—
160	C₄	C₂×#11924015	1	1	yes	characteristic
161	C₄	C₂×#11924018	1	1	yes	characteristic
162	C₄	#14556622	1	1	yes	characteristic
163	C₄	#14556622	1	1	yes	characteristic
164	C₂	#17988726	1	1	yes	characteristic
165	C₂	#17988743	1	1	yes	characteristic
166	C₂	#17988765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988937

C₂

yes

maximal, characteristic

#17988872

C₂

yes

maximal, characteristic

#17988797

C₂

yes

maximal, characteristic

#17988336

C₂

yes

maximal, characteristic

Q₈⋊#10596831

C₂

yes

maximal, characteristic

Q₈⋊#10596831

C₂

yes

maximal, characteristic

Q₈²×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924018

—

#14556591

C₂²

yes

characteristic

#14556522

—

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

—

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

D₄

yes

—

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

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

SD₁₆

yes

—

Q₈×C₇₃

Q₁₆

yes

complemented

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

—

Dic₁₄₆

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

Q₈²

D₇₃

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

C₂×D₇₃

yes

characteristic

101

C₄×Q₈

C₂×D₇₃

yes

characteristic

102

#106

—

146

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

146

—

106

#103

—

146

—

107

#103

—

146

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

146

—

111

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

112

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

113

C₂×Q₈

C₂²⋊D₇₃

yes

—

114

Q₁₆

—

292

—

115

C₂×C₈

—

146

—

116

C₂×C₈

—

146

—

117

C₂×C₈

—

146

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

127

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₇₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

292

—

137

Q₈

—

292

—

138

Q₈

#11924015

yes

—

139

Q₈

#11924018

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

292

—

146

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

149

C₈

—

292

—

150

C₈

—

292

—

151

C₈

—

292

—

152

C₂²

#14556560

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

584

—

160

C₄

C₂×#11924015

yes

characteristic

161

C₄

C₂×#11924018

yes

characteristic

162

C₄

#14556622

yes

characteristic

163

C₄

#14556622

yes

characteristic

164

C₂

#17988726

yes

characteristic

165

C₂

#17988743

yes

characteristic

166

C₂

#17988765

yes

characteristic

	subgroup	G/H
center	C₂²	#14556560
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	Q₈²×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	Q₈²	D₇₃
hypocenter	C₇₃	#1645
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#14556560

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

Q₈²×C₇₃

C₂

socle

C₂²×C₇₃

#120

hypercenter

Q₈²

D₇₃

hypocenter

C₇₃

#1645

LCS

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
53. Q₈×C₇₃	Q₁₆	semidirect
84. C₇₃	#1645	semidirect
139. Q₈	#11924018	semidirect

H ≅ G/N

type

53. Q₈×C₇₃

Q₁₆

semidirect

84. C₇₃

#1645

semidirect

139. Q₈

#11924018

semidirect

Subgroups — #22609152

Conjugacy classes of subgroups

Complements