Group #22626162 — subgroups

Subgroups — #22626162

Order 9344.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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×#14556666	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556664	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989482	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11924031	C₂	1	2	yes	maximal, complemented
5	C₂×#14556599	C₂	1	1	yes	maximal, characteristic
6	#14556666	—	2	2	no	—
7	#14556664	—	2	2	no	—
8	C₂×#11924031	C₂²	1	2	yes	—
9	C₂×#11924031	C₂²	1	2	yes	—
10	C₂×#11924028	C₄	1	1	yes	characteristic
11	#14556599	—	2	2	no	—
12	#14556599	—	2	2	no	—
13	C₂×#11924011	C₄	1	1	yes	characteristic
14	C₂×#11924011	C₄	1	1	yes	characteristic
15	C₂²×#10596831	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₃	C₄	1	1	yes	characteristic
19	#11924031	—	2	2	no	—
20	#11924031	—	2	2	no	—
21	#11924028	—	2	2	no	—
22	#11924028	—	2	2	no	—
23	C₂×Dic₁₄₆	C₈	1	1	yes	characteristic
24	C₂×D₂₉₂	C₈	1	1	yes	characteristic
25	C₂×C₂²⋊D₇₃	C₈	1	1	yes	characteristic
26	C₂×C₂²⋊D₇₃	C₈	1	1	yes	characteristic
27	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924011	—	2	2	no	—
30	#11924011	—	2	2	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₂×#10596831	C₂³	1	1	yes	characteristic
35	C₈×D₇₃	—	2	1	no	—
36	C₈×D₇₃	—	2	1	no	—
37	C₈×D₇₃	D₄	1	2	yes	—
38	C₈×D₇₃	D₄	1	2	yes	—
39	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₇₃	—	2	2	no	—
41	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊D₇₃	—	2	2	no	—
51	C₂²⋊D₇₃	—	2	2	no	—
52	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
54	D₂₉₂	—	2	2	no	—
55	C₄×D₇₃	—	2	1	no	—
56	C₄×D₇₃	—	2	1	no	—
57	C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
58	C₄×D₇₃	M₁₆	1	2	yes	—
59	Dic₁₄₆	—	2	2	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
62	#10596831	—	2	1	no	—
63	#10596831	C₂×D₄	1	1	yes	characteristic
64	#10596831	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×D₇₃	—	2	2	no	—
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	—	2	2	no	—
71	C₂×D₇₃	C₂²⋊C₈	1	2	yes	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	Dic₇₃	—	2	1	no	—
76	Dic₇₃	C₂²⋊C₈	1	2	yes	—
77	C₂×C₇₃	—	4	1	no	—
78	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₇₃	#504	1	2	yes	—
80	D₇₃	—	2	2	no	—
81	D₇₃	—	4	1	no	—
82	#2113	—	73	1	no	maximal, Sylow
83	C₇₃	#2113	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	73	1	no	—
85	C₂×M₃₂	—	73	1	no	—
86	C₂²×C₁₆	—	73	1	no	—
87	C₂²⋊C₁₆	—	73	2	no	—
88	C₂²⋊C₁₆	—	73	2	no	—
89	C₂×#41	—	73	1	no	—
90	#131	—	146	2	no	—
91	#131	—	146	2	no	—
92	C₂×M₁₆	—	73	1	no	—
93	C₂×M₁₆	—	73	1	no	—
94	C₂×M₁₆	#64408	1	1	yes	characteristic
95	C₂²×C₈	—	73	1	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	M₃₂	—	146	2	no	—
99	C₂×C₁₆	—	73	2	no	—
100	C₂×C₁₆	—	73	2	no	—
101	C₂×C₁₆	—	146	2	no	—
102	#41	—	146	2	no	—
103	#41	—	146	2	no	—
104	C₂×Q₈	—	73	1	no	—
105	C₂×D₄	—	73	1	no	—
106	C₂×D₄	—	73	1	no	—
107	C₂×D₄	—	73	1	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	146	2	no	—
113	M₁₆	—	146	2	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	2	no	—
116	C₂×C₈	—	73	2	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	146	1	no	—
119	C₂×C₈	—	146	1	no	—
120	C₂×C₈	—	146	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₂×C₈	C₂×#64408	1	1	yes	characteristic
123	C₂×C₈	C₂×#64408	1	1	yes	characteristic
124	C₁₆	—	146	2	no	—
125	C₁₆	—	146	2	no	—
126	C₂³	—	73	1	no	—
127	C₂³	—	73	1	no	—
128	C₂³	#11924036	1	1	yes	characteristic
129	Q₈	—	146	2	no	—
130	D₄	—	146	2	no	—
131	D₄	—	146	2	no	—
132	D₄	—	146	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	73	2	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	146	1	no	—
140	C₂×C₄	—	146	1	no	—
141	C₂×C₄	#11924036	1	1	yes	characteristic
142	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	73	1	no	—
145	C₈	—	73	1	no	—
146	C₈	—	146	1	no	—
147	C₈	C₂²⋊#64408	1	2	yes	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	73	2	no	—
151	C₂²	—	146	2	no	—
152	C₂²	—	146	1	no	—
153	C₂²	—	146	2	no	—
154	C₂²	C₂×#11924036	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	73	2	no	—
157	C₄	—	146	1	no	—
158	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
159	C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
160	C₂	—	4	1	no	—
161	C₂	—	292	1	no	—
162	C₂	—	146	2	no	—
163	C₂	#17989555	1	1	yes	characteristic
164	C₂	#17989646	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556666

C₂

yes

maximal, characteristic, complemented

C₂×#14556664

C₂

yes

maximal, characteristic, complemented

#17989482

C₂

yes

maximal, complemented

C₂²⋊#11924031

C₂

yes

maximal, complemented

C₂×#14556599

C₂

yes

maximal, characteristic

#14556666

—

#14556664

—

C₂×#11924031

C₂²

yes

—

C₂×#11924031

C₂²

yes

—

C₂×#11924028

C₄

yes

—

—

C₄

yes

characteristic

C₂×#11924011

C₄

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₄

yes

—

—

—

—

C₂×Dic₁₄₆

C₈

yes

characteristic

C₂×D₂₉₂

C₈

yes

characteristic

C₂×C₂²⋊D₇₃

C₈

yes

characteristic

C₂×C₂²⋊D₇₃

C₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#10596831

C₂³

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

D₄

yes

—

C₈×D₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

characteristic

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²⋊C₄

yes

—

C₄×D₇₃

M₁₆

yes

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊C₄

yes

—

#10596831

—

#10596831

C₂×D₄

yes

characteristic

#10596831

C₂×D₄

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

C₂²⋊C₈

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇₃

#504

yes

—

D₇₃

—

D₇₃

—

#2113

—

maximal, Sylow

C₇₃

#2113

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₂×#41

—

#131

—

146

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

146

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

146

—

102

#41

—

146

—

103

#41

—

146

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×#64408

yes

characteristic

111

M₁₆

—

112

M₁₆

—

146

—

113

M₁₆

—

146

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₂×C₈

—

146

—

122

C₂×C₈

C₂×#64408

yes

characteristic

123

C₂×C₈

C₂×#64408

yes

characteristic

124

C₁₆

—

146

—

125

C₁₆

—

146

—

126

C₂³

—

127

C₂³

—

128

C₂³

#11924036

yes

characteristic

129

Q₈

—

146

—

130

D₄

—

146

—

131

D₄

—

146

—

132

D₄

—

146

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

#11924036

yes

characteristic

142

C₂×C₄

C₂²×#64408

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

146

—

147

C₈

C₂²⋊#64408

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

146

—

152

C₂²

—

146

—

153

C₂²

—

146

—

154

C₂²

C₂×#11924036

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

146

—

158

C₄

C₂×C₂²⋊#64408

yes

characteristic

159

C₄

D₇₃⋊M₁₆

yes

characteristic

160

C₂

—

161

C₂

—

292

—

162

C₂

—

146

—

163

C₂

#17989555

yes

characteristic

164

C₂

#17989646

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₂×M₁₆×C₇₃	C₄
hypercenter	C₂×M₁₆	#64408
hypocenter	C₇₃	#2113
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂²×C₈

Fitting

C₂×M₁₆×C₇₃

C₄

C₂×M₁₆

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#14556666	C₂	semidirect
2. C₂×#14556664	C₂	semidirect
3. #17989482	C₂	semidirect
4. C₂²⋊#11924031	C₂	semidirect
83. C₇₃	#2113	semidirect

H ≅ G/N

type

1. C₂×#14556666

C₂

semidirect

2. C₂×#14556664

C₂

semidirect

3. #17989482

C₂

semidirect

4. C₂²⋊#11924031

C₂

semidirect

83. C₇₃

#2113

semidirect

Subgroups — #22626162

Conjugacy classes of subgroups

Complements