Group #22611390 — subgroups

Subgroups — #22611390

Order 9344.

204 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 59 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	#17988636	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988593	C₂	1	1	yes	maximal, characteristic
3	#17988591	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556507	C₂	1	1	yes	maximal, characteristic
5	#17988358	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988343	C₂	1	1	yes	maximal, characteristic, complemented
7	#344×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	1	yes	characteristic
11	#14556550	C₂²	1	1	yes	characteristic
12	#14556527	—	2	2	no	—
13	#14556513	—	2	2	no	—
14	#14556511	—	2	2	no	—
15	#14556507	—	2	2	no	—
16	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	#117×C₇₃	C₂²	1	1	yes	characteristic
19	C₂³×D₇₃	—	2	1	no	—
20	C₂×D₂₉₂	—	2	1	no	—
21	C₂×D₂₉₂	—	2	1	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	D₄	1	2	yes	—
26	#11924012	—	2	1	no	—
27	#11924012	—	2	1	no	—
28	#11924012	—	2	1	no	—
29	#11924012	—	2	1	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₂²⋊Dic₇₃	—	2	1	no	—
34	C₄×Dic₇₃	—	2	1	no	—
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₃	—	2	1	no	—
37	M₁₆×C₇₃	—	2	1	no	—
38	M₁₆×C₇₃	—	2	1	no	—
39	M₁₆×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	D₄	1	2	yes	—
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₂²⋊C₄×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₂³×C₇₃	#41	1	1	yes	characteristic
46	C₂²×D₇₃	—	2	2	no	—
47	C₂²×D₇₃	—	4	1	no	—
48	C₂²×D₇₃	—	4	1	no	—
49	C₂²×D₇₃	—	8	2	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×Dic₇₃	—	4	1	no	—
58	D₂₉₂	—	2	2	no	—
59	D₂₉₂	—	2	2	no	—
60	D₂₉₂	—	4	1	no	—
61	D₂₉₂	—	4	1	no	—
62	C₈×C₇₃	—	2	2	no	—
63	C₈×C₇₃	—	4	1	no	—
64	#10596831	—	4	1	no	—
65	#10596831	—	4	1	no	—
66	C₂²×C₇₃	—	4	1	no	—
67	C₂²×C₇₃	#123	1	1	yes	characteristic
68	C₂²×C₇₃	#124	1	1	yes	characteristic
69	C₂²×C₇₃	#126	1	1	yes	characteristic
70	C₂×D₇₃	—	4	2	no	—
71	C₂×D₇₃	—	8	2	no	—
72	C₂×D₇₃	—	8	2	no	—
73	C₂×D₇₃	—	8	2	no	—
74	C₄×C₇₃	—	8	1	no	—
75	C₄×C₇₃	#121	1	1	yes	characteristic
76	C₄×C₇₃	#123	1	1	yes	characteristic
77	C₄×C₇₃	#123	1	1	yes	characteristic
78	C₄×C₇₃	#126	1	1	yes	characteristic
79	Dic₇₃	—	8	1	no	—
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	—	2	1	no	—
83	C₂×C₇₃	#397	1	1	yes	characteristic
84	D₇₃	—	8	2	no	—
85	#2081	—	73	1	no	maximal, Sylow
86	C₇₃	#2081	1	1	yes	characteristic, complemented, Sylow
87	#430	—	73	1	no	—
88	#418	—	73	1	no	—
89	#418	—	73	1	no	—
90	C₂×#100	—	73	1	no	—
91	#344	—	73	1	no	—
92	#344	D₇₃	1	1	yes	characteristic, complemented
93	H(3,ℤ/4)	—	73	1	no	—
94	C₂²×D₄	—	73	1	no	—
95	C₂×M₁₆	—	73	1	no	—
96	C₂×M₁₆	—	73	1	no	—
97	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
98	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
99	#117	—	73	1	no	—
100	#117	C₂×D₇₃	1	1	yes	characteristic
101	#108	—	146	2	no	—
102	#102	—	146	2	no	—
103	#102	—	146	2	no	—
104	#100	—	146	2	no	—
105	C₂⁴	—	146	1	no	—
106	C₂×D₄	—	73	2	no	—
107	C₂×D₄	—	146	1	no	—
108	C₂×D₄	—	146	1	no	—
109	C₂×D₄	—	146	1	no	—
110	C₂×D₄	—	146	1	no	—
111	C₂×D₄	—	146	1	no	—
112	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	146	1	no	—
118	M₁₆	—	146	1	no	—
119	M₁₆	—	146	1	no	—
120	M₁₆	—	146	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₂×C₈	—	146	1	no	—
124	C₂×C₈	D₂₉₂	1	2	yes	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	146	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	146	1	no	—
131	C₂³	—	292	1	no	—
132	C₂³	—	292	1	no	—
133	C₂³	—	584	2	no	—
134	C₂³	—	146	2	no	—
135	C₂³	#11924028	1	1	yes	characteristic
136	D₄	—	292	1	no	—
137	D₄	—	292	1	no	—
138	D₄	—	146	2	no	—
139	D₄	—	146	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
144	C₂×C₄	#11924027	1	1	yes	characteristic
145	C₂×C₄	#11924027	1	1	yes	characteristic
146	C₂×C₄	#11924028	1	1	yes	characteristic
147	C₂×C₄	#11924029	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	292	1	no	—
151	C₈	—	292	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	292	2	no	—
154	C₂²	—	584	2	no	—
155	C₂²	—	584	2	no	—
156	C₂²	—	584	2	no	—
157	C₂²	#14556573	1	1	yes	characteristic
158	C₂²	#14556579	1	1	yes	characteristic
159	C₂²	#14556586	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	584	1	no	—
162	C₄	#14556566	1	1	yes	characteristic
163	C₄	#14556573	1	1	yes	characteristic
164	C₄	#14556575	1	1	yes	characteristic
165	C₄	#14556586	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	584	2	no	—
170	C₂	#17988514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988636

C₂

yes

maximal, characteristic, complemented

#17988593

C₂

yes

maximal, characteristic

#17988591

C₂

yes

maximal, characteristic

C₂×#14556507

C₂

yes

maximal, characteristic

#17988358

C₂

yes

maximal, characteristic, complemented

#17988343

C₂

yes

maximal, characteristic, complemented

#344×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

#14556550

C₂²

yes

—

—

—

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

—

—

—

—

—

—

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#397

yes

characteristic

D₇₃

—

#2081

—

maximal, Sylow

C₇₃

#2081

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic, complemented

H(3,ℤ/4)

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂×M₁₆

C₂×D₇₃

yes

characteristic

#117

—

100

#117

C₂×D₇₃

yes

characteristic

101

#108

—

146

—

102

#102

—

146

—

103

#102

—

146

—

104

#100

—

146

—

105

C₂⁴

—

146

—

106

C₂×D₄

—

107

C₂×D₄

—

146

—

108

C₂×D₄

—

146

—

109

C₂×D₄

—

146

—

110

C₂×D₄

—

146

—

111

C₂×D₄

—

146

—

112

C₂²×C₄

C₂²×D₇₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

146

—

118

M₁₆

—

146

—

119

M₁₆

—

146

—

120

M₁₆

—

146

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

D₂₉₂

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

146

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

146

—

129

C₄²

—

130

C₄²

—

146

—

131

C₂³

—

292

—

132

C₂³

—

292

—

133

C₂³

—

584

—

134

C₂³

—

146

—

135

C₂³

#11924028

yes

characteristic

136

D₄

—

292

—

137

D₄

—

292

—

138

D₄

—

146

—

139

D₄

—

146

—

140

C₂×C₄

—

141

C₂×C₄

—

292

—

142

C₂×C₄

D₄×D₇₃

yes

characteristic

143

C₂×C₄

C₂×D₂₉₂

yes

characteristic

144

C₂×C₄

#11924027

yes

characteristic

145

C₂×C₄

#11924027

yes

characteristic

146

C₂×C₄

#11924028

yes

characteristic

147

C₂×C₄

#11924029

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

292

—

151

C₈

—

292

—

152

C₂²

—

153

C₂²

—

292

—

154

C₂²

—

584

—

155

C₂²

—

584

—

156

C₂²

—

584

—

157

C₂²

#14556573

yes

characteristic

158

C₂²

#14556579

yes

characteristic

159

C₂²

#14556586

yes

characteristic

160

C₄

—

161

C₄

—

584

—

162

C₄

#14556566

yes

characteristic

163

C₄

#14556573

yes

characteristic

164

C₄

#14556575

yes

characteristic

165

C₄

#14556586

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

584

—

170

C₂

#17988514

yes

characteristic

	subgroup	G/H
center	C₂	#17988514
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#344×C₇₃	C₂
socle	C₂×C₇₃	#397
hypercenter	#344	D₇₃
hypocenter	C₇₃	#2081
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#17988514

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#344×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
1. #17988636	C₂	semidirect
5. #17988358	C₂	semidirect
6. #17988343	C₂	semidirect
7. #344×C₇₃	C₂	semidirect
86. C₇₃	#2081	semidirect
92. #344	D₇₃	semidirect

H ≅ G/N

type

1. #17988636

C₂

semidirect

5. #17988358

C₂

semidirect

6. #17988343

C₂

semidirect

7. #344×C₇₃

C₂

semidirect

86. C₇₃

#2081

semidirect

92. #344

D₇₃

semidirect

Subgroups — #22611390

Conjugacy classes of subgroups

Complements