Group #22610138 — subgroups

Subgroups — #22610138

Order 9344.

278 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂²×#11924012	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556527	C₂	1	2	yes	maximal, complemented
3	#17988551	C₂	1	1	yes	maximal, characteristic
4	#17988352	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	—	2	4	no	—
7	C₂×#11924012	—	2	1	no	—
8	C₂×#11924012	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	2	yes	—
11	C₂²×#10596831	—	2	1	no	—
12	#14556527	—	2	4	no	—
13	C₂²⋊#10596831	—	2	1	no	—
14	C₂²⋊#10596831	—	2	1	no	—
15	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₇₃	C₄	1	2	yes	—
18	C₂²⋊C₈×C₇₃	C₄	1	2	yes	—
19	#11924012	—	2	4	no	—
20	#11924012	—	2	2	no	—
21	#11924012	—	2	2	no	—
22	#11924012	—	2	2	no	—
23	#11924012	D₄	1	4	yes	—
24	C₂×#10596831	—	2	4	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂⁴×C₇₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	—	2	2	no	—
30	C₂²×C₄×C₇₃	—	2	2	no	—
31	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	C₂×C₄	1	4	yes	—
40	C₂³×C₇₃	—	2	2	no	—
41	C₂³×C₇₃	—	2	2	no	—
42	C₂³×C₇₃	—	4	1	no	—
43	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#41	1	2	yes	—
59	C₈×C₇₃	—	2	4	no	—
60	#10596831	—	2	4	no	—
61	#10596831	—	4	1	no	—
62	#10596831	—	4	1	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	2	1	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	4	4	no	—
71	C₂²×C₇₃	#108	1	2	yes	complemented
72	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₇₃	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	C₄×D₄	1	2	yes	—
77	C₄×C₇₃	#120	1	1	yes	characteristic
78	C₄×C₇₃	#123	1	1	yes	characteristic
79	C₂×C₇₃	—	2	2	no	—
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	—	4	1	no	—
82	C₂×C₇₃	#385	1	1	yes	characteristic
83	C₂×C₇₃	C₂×#108	1	1	yes	characteristic
84	C₂×C₇₃	#430	1	1	yes	characteristic
85	#1855	—	73	1	no	maximal, Sylow
86	C₇₃	#1855	1	1	yes	characteristic, complemented, Sylow
87	C₂²×M₁₆	—	73	1	no	—
88	C₂×#108	—	73	2	no	—
89	#407	—	73	1	no	—
90	C₂×C₂²⋊C₈	D₇₃	1	1	yes	characteristic
91	#341	—	73	2	no	—
92	C₂³×C₄	C₂×D₇₃	1	1	yes	characteristic
93	C₂×M₁₆	—	73	2	no	—
94	C₂×M₁₆	—	73	1	no	—
95	C₂×M₁₆	—	73	1	no	—
96	C₂×M₁₆	—	146	4	no	—
97	C₂×M₁₆	—	146	1	no	—
98	C₂²×C₈	—	146	1	no	—
99	C₂²×C₈	C₂×D₇₃	1	2	yes	—
100	#108	—	146	4	no	—
101	C₂²⋊C₈	—	146	1	no	—
102	C₂²⋊C₈	—	146	1	no	—
103	C₂²⋊C₈	Dic₇₃	1	2	yes	—
104	C₂²⋊C₈	Dic₇₃	1	2	yes	—
105	C₂⁴	Dic₁₄₆	1	1	yes	characteristic
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	Dic₁₄₆	1	1	yes	characteristic
109	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
110	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
114	M₁₆	—	73	4	no	—
115	M₁₆	—	146	2	no	—
116	M₁₆	—	146	2	no	—
117	M₁₆	—	146	4	no	—
118	M₁₆	—	146	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	146	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₂×C₈	—	146	4	no	—
124	C₂×C₈	—	146	2	no	—
125	C₂×C₈	C₂×Dic₇₃	1	4	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	4	1	no	—
129	C₂³	C₄⋊Dic₇₃	1	1	yes	characteristic
130	C₂³	C₄⋊Dic₇₃	1	1	yes	characteristic
131	C₂³	C₂×Dic₁₄₆	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
140	C₂×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
141	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
142	C₂×C₄	D₄×D₇₃	1	2	yes	—
143	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
144	C₂×C₄	#11924027	1	2	yes	—
145	C₈	—	2	4	no	—
146	C₈	—	292	1	no	—
147	C₈	—	292	1	no	—
148	C₈	—	146	4	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	4	4	no	—
157	C₂²	#14556527	1	2	yes	complemented
158	C₂²	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
159	C₂²	#14556568	1	2	yes	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	D₄×Dic₇₃	1	2	yes	—
163	C₄	#14556561	1	1	yes	characteristic
164	C₄	#14556575	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	—	2	2	no	—
167	C₂	—	4	1	no	—
168	C₂	#17988462	1	1	yes	characteristic
169	C₂	C₂×#14556527	1	1	yes	characteristic
170	C₂	#17988636	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924012

C₂

yes

maximal, characteristic

C₂×#14556527

C₂

yes

maximal, complemented

#17988551

C₂

yes

maximal, characteristic

#17988352

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

—

C₂×#11924012

—

C₂×#11924012

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

—

—

—

—

—

C₂³×C₄×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

—

C₂²⋊C₈×C₇₃

C₄

yes

—

C₂²⋊C₈×C₇₃

C₄

yes

—

—

—

—

—

D₄

yes

—

—

—

—

—

C₂⁴×C₇₃

Q₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂³×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#108

yes

complemented

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×#108

—

#407

—

C₂×C₂²⋊C₈

D₇₃

yes

characteristic

#341

—

C₂³×C₄

C₂×D₇₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

146

—

C₂²×C₈

C₂×D₇₃

yes

—

100

#108

—

146

—

101

C₂²⋊C₈

—

146

—

102

C₂²⋊C₈

—

146

—

103

C₂²⋊C₈

Dic₇₃

yes

—

104

C₂²⋊C₈

Dic₇₃

yes

—

105

C₂⁴

Dic₁₄₆

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₁₄₆

yes

characteristic

109

C₂²×C₄

D₂₉₂

yes

characteristic

110

C₂²×C₄

D₂₉₂

yes

characteristic

111

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

112

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

113

C₂²×C₄

C₂²×D₇₃

yes

characteristic

114

M₁₆

—

115

M₁₆

—

146

—

116

M₁₆

—

146

—

117

M₁₆

—

146

—

118

M₁₆

—

146

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₂×C₈

C₂×Dic₇₃

yes

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₄⋊Dic₇₃

yes

characteristic

130

C₂³

C₄⋊Dic₇₃

yes

characteristic

131

C₂³

C₂×Dic₁₄₆

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₄⋊Dic₇₃

yes

characteristic

140

C₂×C₄

C₄⋊Dic₇₃

yes

characteristic

141

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

142

C₂×C₄

D₄×D₇₃

yes

—

143

C₂×C₄

C₂×D₂₉₂

yes

characteristic

144

C₂×C₄

#11924027

yes

—

145

C₈

—

146

C₈

—

292

—

147

C₈

—

292

—

148

C₈

—

146

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#14556527

yes

complemented

158

C₂²

C₂×C₄⋊Dic₇₃

yes

characteristic

159

C₂²

#14556568

yes

—

160

C₄

—

161

C₄

—

162

C₄

D₄×Dic₇₃

yes

—

163

C₄

#14556561

yes

characteristic

164

C₄

#14556575

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#17988462

yes

characteristic

169

C₂

C₂×#14556527

yes

characteristic

170

C₂

#17988636

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₂²⋊C₈×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄⋊C₄
hypercenter	C₂×C₂²⋊C₈	D₇₃
hypocenter	C₇₃	#1855
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₂²⋊C₈×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₄⋊C₄

hypercenter

C₂×C₂²⋊C₈

D₇₃

hypocenter

C₇₃

#1855

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#14556527	C₂	semidirect
4. #17988352	C₂	semidirect
71. C₂²×C₇₃	#108	semidirect
86. C₇₃	#1855	semidirect
157. C₂²	#14556527	semidirect

H ≅ G/N

type

2. C₂×#14556527

C₂

semidirect

4. #17988352

C₂

semidirect

71. C₂²×C₇₃

#108

semidirect

86. C₇₃

#1855

semidirect

157. C₂²

#14556527

semidirect

Subgroups — #22610138

Conjugacy classes of subgroups

Complements