Group #22611040 — subgroups

Subgroups — #22611040

Order 9344.

382 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14556616	C₂	1	2	yes	maximal, complemented
2	C₂²×D₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14556507	C₂	1	2	yes	maximal, complemented
4	#17988352	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₉₂	C₂²	1	2	yes	—
7	#14556616	—	2	4	no	—
8	#14556616	—	2	4	no	—
9	C₂×#11924015	—	2	2	no	—
10	C₂×D₅₈₄	—	2	4	no	—
11	C₂×D₅₈₄	—	2	2	no	—
12	C₂×#11924013	—	2	2	no	—
13	C₂×#11924012	C₂²	1	2	yes	—
14	#14556507	—	2	4	no	—
15	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₇₃	—	2	2	no	—
19	C₂³×D₇₃	—	2	2	no	—
20	C₂×D₂₉₂	—	2	2	no	—
21	C₂×D₂₉₂	—	2	2	no	—
22	C₂×D₂₉₂	—	2	2	no	—
23	C₂×D₂₉₂	—	2	2	no	—
24	C₂×D₂₉₂	—	2	2	no	—
25	C₂×D₂₉₂	D₄	1	4	yes	—
26	#11924015	—	2	8	no	—
27	D₅₈₄	—	2	8	no	—
28	#11924013	—	2	8	no	—
29	#11924012	—	2	2	no	—
30	#11924012	D₄	1	4	yes	—
31	C₂×#10596831	—	2	2	no	—
32	#41×C₇₃	—	2	2	no	—
33	#41×C₇₃	—	2	2	no	—
34	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
35	C₂×D₄×C₇₃	—	2	1	no	—
36	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	—	2	1	no	—
38	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	M₁₆×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	—	2	2	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	D₄	1	2	yes	—
43	C₂³×C₇₃	—	2	1	no	—
44	C₂³×C₇₃	#41	1	1	yes	characteristic
45	C₂²×D₇₃	—	2	4	no	—
46	C₂²×D₇₃	—	4	2	no	—
47	C₂²×D₇₃	—	4	2	no	—
48	C₂²×D₇₃	—	8	4	no	—
49	Q₈×C₇₃	—	2	2	no	—
50	D₄×C₇₃	—	2	2	no	—
51	D₄×C₇₃	—	4	2	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	D₂₉₂	—	2	4	no	—
60	D₂₉₂	—	2	4	no	—
61	D₂₉₂	—	4	2	no	—
62	D₂₉₂	—	4	2	no	—
63	C₈×C₇₃	—	2	2	no	—
64	C₈×C₇₃	—	4	1	no	—
65	#10596831	—	2	4	no	—
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	—	4	2	no	—
69	C₂²×C₇₃	#121	1	2	yes	—
70	C₂²×C₇₃	#121	1	1	yes	characteristic
71	C₂×D₇₃	—	4	4	no	—
72	C₂×D₇₃	—	8	4	no	—
73	C₂×D₇₃	—	8	4	no	—
74	C₂×D₇₃	—	8	4	no	—
75	C₄×C₇₃	—	4	1	no	—
76	C₄×C₇₃	#120	1	1	yes	characteristic
77	C₄×C₇₃	#120	1	2	yes	—
78	C₄×C₇₃	#127	1	1	yes	characteristic
79	C₂×C₇₃	—	2	2	no	—
80	C₂×C₇₃	—	8	1	no	—
81	C₂×C₇₃	#392	1	1	yes	characteristic
82	C₂×C₇₃	#472	1	2	yes	—
83	D₇₃	—	8	4	no	—
84	#2018	—	73	1	no	maximal, Sylow
85	C₇₃	#2018	1	1	yes	characteristic, complemented, Sylow
86	C₂×AGL(1,ℤ/8)	—	73	2	no	—
87	C₂²×D₈	—	73	1	no	—
88	C₂×#100	—	73	2	no	—
89	#408	D₇₃	1	1	yes	characteristic, complemented
90	#341	—	73	1	no	—
91	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
92	C₂²×D₄	—	73	2	no	—
93	AGL(1,ℤ/8)	—	146	4	no	—
94	AGL(1,ℤ/8)	—	146	4	no	—
95	C₂×SD₁₆	—	146	2	no	—
96	C₂×D₈	—	146	4	no	—
97	C₂×D₈	—	146	2	no	—
98	C₂×D₈	—	146	2	no	—
99	C₂×M₁₆	—	73	2	no	—
100	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
102	#100	—	146	4	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂⁴	—	146	2	no	—
105	#41	—	2	2	no	—
106	#41	—	2	2	no	—
107	C₂×Q₈	D₂₉₂	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	73	4	no	—
110	C₂×D₄	—	146	2	no	—
111	C₂×D₄	—	146	2	no	—
112	C₂×D₄	—	146	2	no	—
113	C₂×D₄	—	146	2	no	—
114	C₂×D₄	—	146	2	no	—
115	C₂×D₄	D₂₉₂	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
118	SD₁₆	—	146	8	no	—
119	D₈	—	146	8	no	—
120	D₈	—	146	8	no	—
121	M₁₆	—	2	2	no	—
122	M₁₆	—	73	4	no	—
123	M₁₆	—	146	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
128	C₂³	—	2	1	no	—
129	C₂³	—	292	2	no	—
130	C₂³	—	292	2	no	—
131	C₂³	—	584	4	no	—
132	C₂³	—	146	4	no	—
133	C₂³	#11924028	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	292	2	no	—
138	D₄	—	292	2	no	—
139	D₄	—	146	4	no	—
140	D₄	—	146	4	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	D₄×D₇₃	1	2	yes	—
145	C₂×C₄	D₄×D₇₃	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
147	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	146	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	292	4	no	—
155	C₂²	—	584	4	no	—
156	C₂²	—	584	4	no	—
157	C₂²	—	584	4	no	—
158	C₂²	#14556564	1	2	yes	—
159	C₂²	#14556565	1	1	yes	characteristic
160	C₄	—	4	1	no	—
161	C₄	#14556559	1	1	yes	characteristic
162	C₄	#14556561	1	2	yes	—
163	C₄	#14556587	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	1	no	—
166	C₂	—	584	4	no	—
167	C₂	#17988496	1	1	yes	characteristic
168	C₂	#17988857	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556616

C₂

yes

maximal, complemented

C₂²×D₅₈₄

C₂

yes

maximal, characteristic, complemented

C₂×#14556507

C₂

yes

maximal, complemented

#17988352

C₂

yes

maximal, characteristic, complemented

#408×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

—

#14556616

—

#14556616

—

C₂×#11924015

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×#11924013

—

C₂×#11924012

C₂²

yes

—

#14556507

—

C₂×#41×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊C₈×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

—

#11924015

—

D₅₈₄

—

#11924013

—

#11924012

—

#11924012

D₄

yes

—

C₂×#10596831

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

characteristic

C₄×C₇₃

#120

yes

—

C₄×C₇₃

#127

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#392

yes

characteristic

C₂×C₇₃

#472

yes

—

D₇₃

—

#2018

—

maximal, Sylow

C₇₃

#2018

yes

characteristic, complemented, Sylow

C₂×AGL(1,ℤ/8)

—

C₂²×D₈

—

C₂×#100

—

#408

D₇₃

yes

characteristic, complemented

#341

—

C₂×#41

C₂×D₇₃

yes

characteristic

C₂²×D₄

—

AGL(1,ℤ/8)

—

146

—

AGL(1,ℤ/8)

—

146

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₂×M₁₆

—

100

C₂×M₁₆

C₂×D₇₃

yes

characteristic

101

C₂²×C₈

C₂×D₇₃

yes

characteristic

102

#100

—

146

—

103

C₂²⋊C₈

—

104

C₂⁴

—

146

—

105

#41

—

106

#41

—

107

C₂×Q₈

D₂₉₂

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

146

—

111

C₂×D₄

—

146

—

112

C₂×D₄

—

146

—

113

C₂×D₄

—

146

—

114

C₂×D₄

—

146

—

115

C₂×D₄

D₂₉₂

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₇₃

yes

characteristic

118

SD₁₆

—

146

—

119

D₈

—

146

—

120

D₈

—

146

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

146

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₂×C₈

C₂²⋊D₇₃

yes

—

128

C₂³

—

129

C₂³

—

292

—

130

C₂³

—

292

—

131

C₂³

—

584

—

132

C₂³

—

146

—

133

C₂³

#11924028

yes

characteristic

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

292

—

138

D₄

—

292

—

139

D₄

—

146

—

140

D₄

—

146

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

D₄×D₇₃

yes

—

145

C₂×C₄

D₄×D₇₃

yes

—

146

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

147

C₂×C₄

C₂×D₂₉₂

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

146

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

292

—

155

C₂²

—

584

—

156

C₂²

—

584

—

157

C₂²

—

584

—

158

C₂²

#14556564

yes

—

159

C₂²

#14556565

yes

characteristic

160

C₄

—

161

C₄

#14556559

yes

characteristic

162

C₄

#14556561

yes

—

163

C₄

#14556587

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

584

—

167

C₂

#17988496

yes

characteristic

168

C₂

#17988857

yes

—

	subgroup	G/H
center	C₂²	#14556565
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#408×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#408	D₇₃
hypocenter	C₇₃	#2018
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#14556565

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#408×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. C₂×#14556616	C₂	semidirect
2. C₂²×D₅₈₄	C₂	semidirect
3. C₂×#14556507	C₂	semidirect
4. #17988352	C₂	semidirect
5. #408×C₇₃	C₂	semidirect
85. C₇₃	#2018	semidirect
89. #408	D₇₃	semidirect

H ≅ G/N

type

1. C₂×#14556616

C₂

semidirect

2. C₂²×D₅₈₄

C₂

semidirect

3. C₂×#14556507

C₂

semidirect

4. #17988352

C₂

semidirect

5. #408×C₇₃

C₂

semidirect

85. C₇₃

#2018

semidirect

89. #408

D₇₃

semidirect

Subgroups — #22611040

Conjugacy classes of subgroups

Complements