Group #22609201 — subgroups

Subgroups — #22609201

Order 9344.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988983	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988867	C₂	1	1	yes	maximal, characteristic
3	#17988797	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988333	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
7	#556×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924018	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556524	—	2	1	no	—
11	#14556522	—	2	2	no	—
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	#14556514	—	2	1	no	—
15	#14556512	—	2	1	no	—
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
18	#126×C₇₃	—	2	2	no	—
19	#124×C₇₃	—	2	1	no	—
20	C₂²⋊Q₈×C₇₃	—	2	1	no	—
21	C₄×Q₈×C₇₃	—	2	1	no	—
22	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
24	#117×C₇₃	—	2	1	no	—
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	1	no	—
33	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	—	2	1	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	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
42	C₂²⋊C₄×C₇₃	—	2	2	no	—
43	C₂²⋊C₄×C₇₃	—	2	2	no	—
44	C₂²⋊C₄×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	—	2	2	no	—
46	C₄²×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂³×C₇₃	—	2	1	no	—
49	Q₈×C₇₃	—	2	1	no	—
50	Q₈×C₇₃	—	2	1	no	—
51	Q₈×C₇₃	—	4	1	no	—
52	Q₈×C₇₃	—	4	1	no	—
53	D₄×C₇₃	—	2	1	no	—
54	D₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	4	1	no	—
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₇₃	—	4	1	no	—
65	Dic₁₄₆	—	4	1	no	—
66	Dic₁₄₆	—	4	1	no	—
67	#10596831	—	2	2	no	—
68	#10596831	—	4	1	no	—
69	#10596831	—	4	1	no	—
70	C₂²×C₇₃	—	4	1	no	—
71	C₂²×C₇₃	—	4	1	no	—
72	C₂²×C₇₃	—	4	1	no	—
73	C₂²×C₇₃	#120	1	1	yes	characteristic
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	4	1	no	—
76	C₄×C₇₃	—	4	1	no	—
77	C₄×C₇₃	—	4	2	no	—
78	C₄×C₇₃	—	4	1	no	—
79	C₄×C₇₃	#135	1	1	yes	characteristic
80	C₄×C₇₃	#135	1	1	yes	characteristic
81	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₄×C₇₃	#137	1	1	yes	characteristic
83	Dic₇₃	—	8	1	no	—
84	C₂×C₇₃	—	8	1	no	—
85	C₂×C₇₃	#449	1	1	yes	characteristic
86	C₂×C₇₃	#452	1	1	yes	characteristic
87	C₂×C₇₃	#456	1	1	yes	characteristic
88	#1652	—	73	1	no	maximal, Sylow
89	C₇₃	#1652	1	1	yes	characteristic, complemented, Sylow
90	#556	D₇₃	1	1	yes	characteristic
91	#498	—	73	1	no	—
92	#474	—	73	1	no	—
93	#462	—	73	1	no	—
94	#332	—	73	1	no	—
95	Q₈⋊C₈	—	73	1	no	—
96	D₄⋊C₈	—	73	1	no	—
97	C₂×Q₁₆	—	146	1	no	—
98	C₄⋊Q₈	—	73	1	no	—
99	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
100	#126	—	2	2	no	—
101	#124	—	2	1	no	—
102	C₂²⋊Q₈	—	2	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
105	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
106	#117	—	2	1	no	—
107	#107	—	146	1	no	—
108	#106	—	146	2	no	—
109	C₄⋊C₈	—	73	1	no	—
110	C₄⋊C₈	—	73	1	no	—
111	#103	—	146	1	no	—
112	#102	—	146	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	146	1	no	—
116	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
117	C₂×D₄	C₂²⋊D₇₃	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	Q₁₆	—	292	2	no	—
120	C₂×C₈	—	146	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₄⋊C₄	—	146	1	no	—
128	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
129	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
130	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	C₂²×D₇₃	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	292	1	no	—
143	Q₈	—	292	1	no	—
144	D₄	—	2	1	no	—
145	D₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	292	1	no	—
153	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
156	C₈	—	292	1	no	—
157	C₈	—	292	1	no	—
158	C₈	—	146	2	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	#14556560	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	2	no	—
167	C₄	—	4	1	no	—
168	C₄	—	584	1	no	—
169	C₄	#14556611	1	1	yes	characteristic
170	C₄	#14556611	1	1	yes	characteristic
171	C₄	#14556616	1	1	yes	characteristic
172	C₄	#14556621	1	1	yes	characteristic
173	C₂	—	8	1	no	—
174	C₂	#17988732	1	1	yes	characteristic
175	C₂	#17988749	1	1	yes	characteristic
176	C₂	#17988770	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988983

C₂

yes

maximal, characteristic, complemented

#17988867

C₂

yes

maximal, characteristic

#17988797

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596831

C₂

yes

maximal, characteristic, complemented

D₄⋊#10596831

C₂

yes

maximal, characteristic

#556×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924018

—

#14556591

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#126×C₇₃

—

#124×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

#117×C₇₃

—

C₂×Dic₁₄₆

—

—

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×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

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

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

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#120

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#556

D₇₃

yes

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

100

#126

—

101

#124

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

C₄×Q₈

C₂×D₇₃

yes

characteristic

105

C₄×D₄

C₂×D₇₃

yes

characteristic

106

#117

—

107

#107

—

146

—

108

#106

—

146

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

146

—

112

#102

—

146

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

146

—

116

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

117

C₂×D₄

C₂²⋊D₇₃

yes

characteristic

118

C₂²×C₄

—

119

Q₁₆

—

292

—

120

C₂×C₈

—

146

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

—

146

—

128

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

129

C₄⋊C₄

C₂²⋊D₇₃

yes

—

130

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₇₃

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

292

—

143

Q₈

—

292

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

292

—

153

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

154

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

155

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

156

C₈

—

292

—

157

C₈

—

292

—

158

C₈

—

146

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#14556560

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

584

—

169

C₄

#14556611

yes

characteristic

170

C₄

#14556611

yes

characteristic

171

C₄

#14556616

yes

characteristic

172

C₄

#14556621

yes

characteristic

173

C₂

—

174

C₂

#17988732

yes

characteristic

175

C₂

#17988749

yes

characteristic

176

C₂

#17988770

yes

characteristic

	subgroup	G/H
center	C₂²	#14556560
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#556×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#556	D₇₃
hypocenter	C₇₃	#1652
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #556

subgroup

G/H

center

C₂²

#14556560

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#556×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #556

N	H ≅ G/N	type
1. #17988983	C₂	semidirect
3. #17988797	C₂	semidirect
4. #17988333	C₂	semidirect
5. Q₈⋊#10596831	C₂	semidirect
89. C₇₃	#1652	semidirect

H ≅ G/N

type

1. #17988983

C₂

semidirect

3. #17988797

C₂

semidirect

4. #17988333

C₂

semidirect

5. Q₈⋊#10596831

C₂

semidirect

89. C₇₃

#1652

semidirect

Subgroups — #22609201

Conjugacy classes of subgroups

Complements