Group #22611871 — subgroups

Subgroups — #22611871

Order 9344.

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

normal subgroups only

This group has 220 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	C₂×#14556537	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556535	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988828	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988391	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988388	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
9	#14556570	—	2	1	no	—
10	#14556537	—	2	2	no	—
11	#14556535	—	2	2	no	—
12	C₂×#11924003	C₂²	1	1	yes	characteristic
13	C₂×#11924003	C₂²	1	1	yes	characteristic
14	#14556525	C₂²	1	1	yes	characteristic
15	#14556516	—	2	1	no	—
16	C₂×#41×C₇₃	—	2	1	no	—
17	#135×C₇₃	—	2	2	no	—
18	#135×C₇₃	—	2	2	no	—
19	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₇₃	—	2	1	no	—
21	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	2	1	no	—
24	Dic₂₉₂	—	2	1	no	—
25	Dic₂₉₂	—	2	1	no	—
26	C₄⋊Dic₇₃	—	2	1	no	—
27	#11924007	—	4	1	no	—
28	C₂²⋊Dic₇₃	—	4	1	no	—
29	#11924003	—	2	1	no	—
30	#11924003	—	2	1	no	—
31	#41×C₇₃	—	2	4	no	—
32	#41×C₇₃	—	4	2	no	—
33	C₂×Q₈×C₇₃	—	2	1	no	—
34	C₂×D₄×C₇₃	—	2	1	no	—
35	C₂×D₄×C₇₃	—	4	1	no	—
36	C₂²×C₄×C₇₃	—	4	1	no	—
37	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
38	Q₁₆×C₇₃	—	2	1	no	—
39	Q₁₆×C₇₃	D₄	1	2	yes	—
40	SD₁₆×C₇₃	—	2	4	no	—
41	D₈×C₇₃	—	2	1	no	—
42	D₈×C₇₃	D₄	1	2	yes	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₂³×C₇₃	—	4	1	no	—
48	C₂³×C₇₃	D₈	1	1	yes	characteristic
49	Q₈×C₇₃	—	2	2	no	—
50	Q₈×C₇₃	—	4	1	no	—
51	D₄×C₇₃	—	2	2	no	—
52	D₄×C₇₃	—	4	4	no	—
53	D₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	4	4	no	—
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₇₃	—	4	1	no	—
61	C₂×Dic₇₃	—	4	1	no	—
62	Dic₁₄₆	—	4	1	no	—
63	Dic₁₄₆	—	4	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	4	2	no	—
71	C₂²×C₇₃	—	8	1	no	—
72	C₂²×C₇₃	—	8	1	no	—
73	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	4	2	no	—
76	C₄×C₇₃	#120	1	1	yes	characteristic
77	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	Dic₇₃	—	8	1	no	—
79	Dic₇₃	—	8	1	no	—
80	C₂×C₇₃	—	4	1	no	—
81	C₂×C₇₃	—	8	2	no	—
82	C₂×C₇₃	#447	1	1	yes	characteristic
83	C₂×C₇₃	#508	1	1	yes	characteristic
84	C₂×C₇₃	#510	1	1	yes	characteristic
85	#2189	—	73	1	no	maximal, Sylow
86	C₇₃	#2189	1	1	yes	characteristic, complemented, Sylow
87	C₂×#135	D₇₃	1	1	yes	characteristic
88	C₂×Q₃₂	—	73	1	no	—
89	C₂×SD₃₂	—	73	1	no	—
90	#467	—	73	1	no	—
91	#358	—	73	1	no	—
92	#357	—	73	1	no	—
93	C₂²⋊C₁₆	—	73	1	no	—
94	C₂×#41	—	2	1	no	—
95	#135	—	2	2	no	—
96	#135	—	2	2	no	—
97	C₂×Q₁₆	—	73	1	no	—
98	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
99	C₂×SD₁₆	—	2	1	no	—
100	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
102	C₂²⋊Q₈	—	146	1	no	—
103	Q₃₂	—	146	2	no	—
104	SD₃₂	—	146	2	no	—
105	C₂×C₁₆	—	73	1	no	—
106	C₂×C₁₆	—	73	1	no	—
107	#107	—	73	1	no	—
108	#103	—	146	1	no	—
109	#41	—	2	4	no	—
110	#41	—	4	2	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	146	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	4	1	no	—
115	C₂²×C₄	—	4	1	no	—
116	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
117	Q₁₆	—	2	1	no	—
118	Q₁₆	—	146	1	no	—
119	Q₁₆	—	146	1	no	—
120	Q₁₆	C₂²⋊D₇₃	1	2	yes	—
121	SD₁₆	—	2	4	no	—
122	D₈	—	2	1	no	—
123	D₈	C₂²⋊D₇₃	1	2	yes	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	C₄⋊C₄	—	292	1	no	—
129	C₄⋊C₄	—	146	1	no	—
130	C₂²⋊C₄	—	292	1	no	—
131	C₁₆	—	146	1	no	—
132	C₁₆	—	146	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	#11924013	1	1	yes	characteristic
135	Q₈	—	2	2	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	292	1	no	—
138	Q₈	—	292	1	no	—
139	D₄	—	2	2	no	—
140	D₄	—	4	4	no	—
141	D₄	—	4	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	292	1	no	—
147	C₂×C₄	—	292	1	no	—
148	C₂×C₄	#11924013	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
152	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	8	1	no	—
158	C₂²	—	8	1	no	—
159	C₂²	C₂×#11924013	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	4	2	no	—
162	C₄	—	584	1	no	—
163	C₄	—	584	1	no	—
164	C₄	#14556560	1	1	yes	characteristic
165	C₄	#14556617	1	1	yes	characteristic
166	C₂	—	4	1	no	—
167	C₂	—	8	2	no	—
168	C₂	#17988721	1	1	yes	characteristic
169	C₂	#17989017	1	1	yes	characteristic
170	C₂	#17989027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556537

C₂

yes

maximal, characteristic, complemented

C₂×#14556535

C₂

yes

maximal, characteristic, complemented

#17988828

C₂

yes

maximal, characteristic, complemented

#17988391

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11924003

C₂

yes

maximal, characteristic, complemented

C₂×#135×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₂₉₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556516

—

C₂×#41×C₇₃

—

#135×C₇₃

—

#135×C₇₃

—

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924003

—

#11924003

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

D₄

yes

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

D₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#135

D₇₃

yes

characteristic

C₂×Q₃₂

—

C₂×SD₃₂

—

#467

—

#358

—

#357

—

C₂²⋊C₁₆

—

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

C₂×SD₁₆

—

100

C₂×D₈

C₂×D₇₃

yes

characteristic

101

C₂²×C₈

C₂×D₇₃

yes

characteristic

102

C₂²⋊Q₈

—

146

—

103

Q₃₂

—

146

—

104

SD₃₂

—

146

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

#107

—

108

#103

—

146

—

109

#41

—

110

#41

—

111

C₂×Q₈

—

112

C₂×Q₈

—

146

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

117

Q₁₆

—

118

Q₁₆

—

146

—

119

Q₁₆

—

146

—

120

Q₁₆

C₂²⋊D₇₃

yes

—

121

SD₁₆

—

122

D₈

—

123

D₈

C₂²⋊D₇₃

yes

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

127

C₂×C₈

C₂²×D₇₃

yes

characteristic

128

C₄⋊C₄

—

292

—

129

C₄⋊C₄

—

146

—

130

C₂²⋊C₄

—

292

—

131

C₁₆

—

146

—

132

C₁₆

—

146

—

133

C₂³

—

134

C₂³

#11924013

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

292

—

138

Q₈

—

292

—

139

D₄

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

292

—

147

C₂×C₄

—

292

—

148

C₂×C₄

#11924013

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

150

C₈

—

151

C₈

C₂×C₂²⋊D₇₃

yes

characteristic

152

C₈

C₂×C₂²⋊D₇₃

yes

characteristic

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

C₂×#11924013

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

584

—

163

C₄

—

584

—

164

C₄

#14556560

yes

characteristic

165

C₄

#14556617

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#17988721

yes

characteristic

169

C₂

#17989017

yes

characteristic

170

C₂

#17989027

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂×#135×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₂×#135	D₇₃
hypocenter	C₇₃	#2189
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₂×#135×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#14556537	C₂	semidirect
2. C₂×#14556535	C₂	semidirect
3. #17988828	C₂	semidirect
4. #17988391	C₂	semidirect
5. #17988388	C₂	semidirect
6. C₂²⋊#11924003	C₂	semidirect
86. C₇₃	#2189	semidirect

H ≅ G/N

type

1. C₂×#14556537

C₂

semidirect

2. C₂×#14556535

C₂

semidirect

3. #17988828

C₂

semidirect

4. #17988391

C₂

semidirect

5. #17988388

C₂

semidirect

6. C₂²⋊#11924003

C₂

semidirect

86. C₇₃

#2189

semidirect

Subgroups — #22611871

Conjugacy classes of subgroups

Complements