Group #22618296 — subgroups

Subgroups — #22618296

Order 9344.

516 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556585	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988487	C₂	1	3	yes	maximal
3	#17988482	C₂	1	3	yes	maximal, complemented
4	#17988481	C₂	1	3	yes	maximal, complemented
5	#17988477	C₂	1	3	yes	maximal, complemented
6	C₄²×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#126×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₇₃	C₂²	1	1	yes	characteristic
9	#14556585	—	2	4	no	—
10	C₂×#11924007	C₂²	1	3	yes	—
11	C₂×#11924007	C₂²	1	3	yes	—
12	C₂×C₂²⋊Dic₇₃	C₂²	1	3	yes	—
13	C₂×#11924005	C₂²	1	3	yes	—
14	C₂×#11924005	C₂²	1	3	yes	—
15	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
16	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
17	#14556498	C₂²	1	6	yes	—
18	#126×C₇₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₇₃	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄×C₇₃	C₂²	1	3	yes	—
21	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
22	C₂³×D₇₃	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₇₃	—	2	6	no	—
24	C₂²×Dic₇₃	C₂³	1	3	yes	—
25	C₂²×Dic₇₃	C₂³	1	3	yes	—
26	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
27	#11924007	—	2	6	no	—
28	#11924007	—	2	6	no	—
29	C₂²⋊Dic₇₃	—	2	6	no	—
30	#11924005	—	2	6	no	—
31	#11924005	—	2	6	no	—
32	C₄×Dic₇₃	—	2	6	no	—
33	C₄×Dic₇₃	—	2	6	no	—
34	C₂⁴×C₇₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
36	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
37	C₄⋊C₄×C₇₃	—	2	6	no	—
38	C₂²⋊C₄×C₇₃	—	2	6	no	—
39	C₄²×C₇₃	D₄	1	4	yes	complemented
40	C₂³×C₇₃	—	2	1	no	—
41	C₂³×C₇₃	—	2	3	no	—
42	C₂³×C₇₃	—	2	3	no	—
43	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
44	C₂²×D₇₃	—	2	1	no	—
45	C₂²×D₇₃	—	2	3	no	—
46	C₂²×D₇₃	—	2	3	no	—
47	C₂×C₄×C₇₃	—	2	3	no	—
48	C₂×C₄×C₇₃	—	2	3	no	—
49	C₂×C₄×C₇₃	—	2	3	no	—
50	C₂×C₄×C₇₃	—	2	3	no	—
51	C₂×C₄×C₇₃	—	2	3	no	—
52	C₂×C₄×C₇₃	C₂×D₄	1	6	yes	—
53	C₂²⋊D₇₃	—	4	4	no	—
54	C₂×Dic₇₃	—	2	3	no	—
55	C₂×Dic₇₃	—	2	3	no	—
56	C₂×Dic₇₃	—	2	3	no	—
57	C₂×Dic₇₃	—	2	3	no	—
58	C₂×Dic₇₃	—	2	3	no	—
59	C₂×Dic₇₃	#41	1	6	yes	—
60	C₂×Dic₇₃	#41	1	6	yes	—
61	C₂²×C₇₃	—	4	1	no	—
62	C₂²×C₇₃	—	4	3	no	—
63	C₂²×C₇₃	—	4	3	no	—
64	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₇₃	C₂×#41	1	3	yes	—
66	C₂²×C₇₃	C₂×#41	1	3	yes	—
67	C₂×D₇₃	—	4	1	no	—
68	C₂×D₇₃	—	4	3	no	—
69	C₂×D₇₃	—	4	3	no	—
70	C₄×C₇₃	—	2	6	no	—
71	C₄×C₇₃	—	4	3	no	—
72	Dic₇₃	—	2	6	no	—
73	Dic₇₃	—	4	3	no	—
74	Dic₇₃	#126	1	4	yes	complemented
75	C₂×C₇₃	—	8	1	no	—
76	C₂×C₇₃	C₂×#126	1	1	yes	characteristic
77	C₂×C₇₃	#529	1	3	yes	—
78	C₂×C₇₃	#532	1	3	yes	—
79	D₇₃	—	8	1	no	—
80	#2860	—	73	1	no	maximal, Sylow
81	C₇₃	#2860	1	1	yes	characteristic, complemented, Sylow
82	C₂×#126	—	73	1	no	—
83	C₂×#126	D₇₃	1	1	yes	characteristic, complemented
84	#390	—	73	3	no	—
85	#389	—	73	3	no	—
86	#388	—	73	3	no	—
87	#388	—	73	3	no	—
88	C₄³	—	73	1	no	—
89	C₂²×D₄	—	73	1	no	—
90	#126	—	2	4	no	—
91	#126	—	146	4	no	—
92	C₂×C₄⋊C₄	—	73	3	no	—
93	C₂×C₄⋊C₄	—	73	3	no	—
94	C₂×C₄⋊C₄	C₂×D₇₃	1	3	yes	—
95	C₂×C₂²⋊C₄	—	73	3	no	—
96	C₂×C₂²⋊C₄	—	73	3	no	—
97	C₂×C₂²⋊C₄	—	73	3	no	—
98	C₂×C₂²⋊C₄	C₂×D₇₃	1	3	yes	—
99	C₂×C₄²	—	73	3	no	—
100	C₂×C₄²	—	73	3	no	—
101	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
102	#95	—	73	6	no	—
103	C₂⁴	—	73	1	no	—
104	C₂⁴	C₂²×D₇₃	1	1	yes	characteristic
105	C₂×D₄	—	146	6	no	—
106	C₂²×C₄	—	73	1	no	—
107	C₂²×C₄	—	73	3	no	—
108	C₂²×C₄	—	73	3	no	—
109	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
110	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
111	C₄⋊C₄	—	2	6	no	—
112	C₄⋊C₄	—	146	6	no	—
113	C₄⋊C₄	—	146	6	no	—
114	C₂²⋊C₄	—	2	6	no	—
115	C₂²⋊C₄	—	146	6	no	—
116	C₂²⋊C₄	—	146	6	no	—
117	C₂²⋊C₄	—	146	6	no	—
118	C₄²	—	146	6	no	—
119	C₄²	—	146	6	no	—
120	C₄²	C₂²⋊D₇₃	1	4	yes	complemented
121	C₂³	—	2	1	no	—
122	C₂³	—	2	3	no	—
123	C₂³	—	2	3	no	—
124	C₂³	—	146	1	no	—
125	C₂³	—	146	3	no	—
126	C₂³	—	146	3	no	—
127	C₂³	C₂³×D₇₃	1	1	yes	characteristic
128	D₄	—	292	4	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	2	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	73	6	no	—
135	C₂×C₄	—	73	6	no	—
136	C₂×C₄	—	146	3	no	—
137	C₂×C₄	—	146	3	no	—
138	C₂×C₄	—	146	3	no	—
139	C₂×C₄	—	146	3	no	—
140	C₂×C₄	—	146	3	no	—
141	C₂×C₄	C₂×C₂²⋊D₇₃	1	6	yes	—
142	C₂²	—	4	3	no	—
143	C₂²	—	4	3	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	292	3	no	—
146	C₂²	—	292	3	no	—
147	C₂²	—	292	1	no	—
148	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
149	C₂²	#41×D₇₃	1	3	yes	—
150	C₂²	#41×D₇₃	1	3	yes	—
151	C₄	—	2	6	no	—
152	C₄	—	4	3	no	—
153	C₄	—	292	3	no	—
154	C₄	—	73	4	no	—
155	C₄	—	146	6	no	—
156	C₂	—	8	1	no	—
157	C₂	—	584	1	no	—
158	C₂	#126×D₇₃	1	1	yes	characteristic
159	C₂	#17989120	1	3	yes	—
160	C₂	#17989133	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556585

C₂

yes

maximal, characteristic, complemented

#17988487

C₂

yes

maximal

#17988482

C₂

yes

maximal, complemented

#17988481

C₂

yes

maximal, complemented

#17988477

C₂

yes

maximal, complemented

C₄²×Dic₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₇₃

C₂²

yes

characteristic

#14556585

—

C₂×#11924007

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×C₂²⋊Dic₇₃

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

#14556498

C₂²

yes

—

#126×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

C₂³

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

characteristic

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂⁴×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

complemented

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#41

yes

—

C₂×Dic₇₃

#41

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

C₂×#41

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#126

yes

complemented

C₂×C₇₃

—

C₂×C₇₃

C₂×#126

yes

characteristic

C₂×C₇₃

#529

yes

—

C₂×C₇₃

#532

yes

—

D₇₃

—

#2860

—

maximal, Sylow

C₇₃

#2860

yes

characteristic, complemented, Sylow

C₂×#126

—

C₂×#126

D₇₃

yes

characteristic, complemented

—

—

—

—

C₄³

—

C₂²×D₄

—

#126

—

#126

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₇₃

yes

characteristic

102

#95

—

103

C₂⁴

—

104

C₂⁴

C₂²×D₇₃

yes

characteristic

105

C₂×D₄

—

146

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₇₃

yes

—

110

C₂²×C₄

C₂²×D₇₃

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

146

—

116

C₂²⋊C₄

—

146

—

117

C₂²⋊C₄

—

146

—

118

C₄²

—

146

—

119

C₄²

—

146

—

120

C₄²

C₂²⋊D₇₃

yes

complemented

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

146

—

125

C₂³

—

146

—

126

C₂³

—

146

—

127

C₂³

C₂³×D₇₃

yes

characteristic

128

D₄

—

292

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

292

—

146

C₂²

—

292

—

147

C₂²

—

292

—

148

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

149

C₂²

#41×D₇₃

yes

—

150

C₂²

#41×D₇₃

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

292

—

154

C₄

—

155

C₄

—

146

—

156

C₂

—

157

C₂

—

584

—

158

C₂

#126×D₇₃

yes

characteristic

159

C₂

#17989120

yes

—

160

C₂

#17989133

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	C₂×#126×C₇₃	C₂
hypercenter	C₂×#126	D₇₃
hypocenter	C₇₃	#2860
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

C₂×#126×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×#14556585	C₂	semidirect
3. #17988482	C₂	semidirect
4. #17988481	C₂	semidirect
5. #17988477	C₂	semidirect
6. C₄²×Dic₇₃	C₂	semidirect
7. C₂×#126×C₇₃	C₂	semidirect
39. C₄²×C₇₃	D₄	semidirect
74. Dic₇₃	#126	semidirect
81. C₇₃	#2860	semidirect
83. C₂×#126	D₇₃	semidirect
120. C₄²	C₂²⋊D₇₃	semidirect

H ≅ G/N

type

1. C₂×#14556585

C₂

semidirect

3. #17988482

C₂

semidirect

4. #17988481

C₂

semidirect

5. #17988477

C₂

semidirect

6. C₄²×Dic₇₃

C₂

semidirect

7. C₂×#126×C₇₃

C₂

semidirect

39. C₄²×C₇₃

D₄

semidirect

74. Dic₇₃

#126

semidirect

81. C₇₃

#2860

semidirect

83. C₂×#126

D₇₃

semidirect

120. C₄²

C₂²⋊D₇₃

semidirect

Subgroups — #22618296

Conjugacy classes of subgroups

Complements