Group #22609692 — subgroups

Subgroups — #22609692

Order 9344.

412 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 137 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×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556503	C₂	1	2	yes	maximal
3	C₂×#14556501	C₂	1	2	yes	maximal, complemented
4	#17988342	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₇₃	C₄	1	2	yes	—
7	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
8	C₂²×#10596831	C₂²	1	2	yes	—
9	C₂×C₄×Dic₇₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₄²×D₇₃	—	2	4	no	—
12	#14556503	C₄	1	8	yes	complemented
13	#14556501	—	2	4	no	—
14	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂³×D₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×D₇₃	—	2	4	no	—
20	C₂×C₄×D₇₃	—	2	4	no	—
21	C₂×C₄×D₇₃	—	2	2	no	—
22	C₂×C₄×D₇₃	—	2	1	no	—
23	C₂×C₄×D₇₃	—	2	1	no	—
24	C₂×C₄×D₇₃	—	2	4	no	—
25	C₂×C₄×D₇₃	C₂×C₄	1	4	yes	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	C₂×C₄	1	4	yes	complemented
29	C₄×Dic₇₃	—	2	4	no	—
30	C₄×Dic₇₃	D₄	1	4	yes	—
31	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
36	C₄²×C₇₃	—	2	2	no	—
37	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×D₇₃	—	2	1	no	—
39	C₂²×D₇₃	—	2	1	no	—
40	C₂²×D₇₃	—	2	2	no	—
41	C₂²×D₇₃	C₄²	1	2	yes	—
42	C₂²×D₇₃	M₁₆	1	4	yes	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	M₁₆	1	4	yes	—
46	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	1	no	—
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	C₄²	1	2	yes	—
55	C₂×Dic₇₃	C₂²⋊C₄	1	4	yes	—
56	C₄×D₇₃	—	2	16	no	—
57	C₄×D₇₃	—	2	4	no	—
58	C₄×D₇₃	—	2	4	no	—
59	C₈×C₇₃	—	2	4	no	—
60	#10596831	—	2	4	no	—
61	C₂²×C₇₃	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₃	#117	1	1	yes	characteristic
64	C₂²×C₇₃	C₂×M₁₆	1	2	yes	—
65	C₂²×C₇₃	C₂×M₁₆	1	2	yes	—
66	C₂×D₇₃	—	2	2	no	—
67	C₂×D₇₃	—	2	2	no	—
68	C₂×D₇₃	—	2	4	no	—
69	C₂×D₇₃	—	2	4	no	—
70	C₂×D₇₃	#97	1	4	yes	complemented
71	C₄×C₇₃	—	2	4	no	—
72	C₄×C₇₃	C₄×D₄	1	2	yes	—
73	C₄×C₇₃	C₄×D₄	1	2	yes	—
74	Dic₇₃	—	2	2	no	—
75	Dic₇₃	—	2	4	no	—
76	C₂×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₇₃	C₂×#97	1	1	yes	characteristic
78	C₂×C₇₃	C₄×M₁₆	1	1	yes	characteristic
79	C₂×C₇₃	#407	1	2	yes	—
80	C₂×C₇₃	#432	1	2	yes	—
81	D₇₃	—	2	4	no	—
82	#1749	—	73	1	no	maximal, Sylow
83	C₇₃	#1749	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₄²	—	73	1	no	—
85	C₂×C₂²⋊C₈	—	73	2	no	—
86	C₂×#97	—	73	2	no	—
87	#336	—	73	1	no	—
88	#336	D₇₃	1	1	yes	characteristic, complemented
89	C₂³×C₄	—	73	1	no	—
90	C₂³×C₄	—	73	2	no	—
91	C₂²×C₈	—	73	2	no	—
92	C₂²×C₈	C₂×D₇₃	1	2	yes	—
93	C₂×C₄²	—	73	1	no	—
94	C₂×C₄²	—	73	2	no	—
95	C₂×C₄²	—	146	4	no	—
96	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
97	C₂²⋊C₈	—	73	8	no	—
98	#97	—	146	4	no	—
99	C₂⁴	—	73	1	no	—
100	C₂²×C₄	—	73	1	no	—
101	C₂²×C₄	—	73	2	no	—
102	C₂²×C₄	—	73	4	no	—
103	C₂²×C₄	—	146	4	no	—
104	C₂²×C₄	—	146	4	no	—
105	C₂²×C₄	—	146	4	no	—
106	C₂²×C₄	—	146	2	no	—
107	C₂²×C₄	—	146	1	no	—
108	C₂²×C₄	—	146	1	no	—
109	C₂²×C₄	C₄×D₇₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	73	4	no	—
114	C₂×C₈	—	146	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₂×C₈	C₄×D₇₃	1	4	yes	complemented
117	C₄²	—	2	2	no	—
118	C₄²	—	73	4	no	—
119	C₄²	—	146	4	no	—
120	C₂³	—	73	2	no	—
121	C₂³	—	73	4	no	—
122	C₂³	—	146	1	no	—
123	C₂³	—	146	1	no	—
124	C₂³	—	146	2	no	—
125	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	4	no	—
130	C₂×C₄	—	146	16	no	—
131	C₂×C₄	—	146	1	no	—
132	C₂×C₄	—	146	1	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	—	146	2	no	—
135	C₂×C₄	—	146	4	no	—
136	C₂×C₄	—	146	4	no	—
137	C₂×C₄	#11924011	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
139	C₂×C₄	D₄×D₇₃	1	2	yes	—
140	C₂×C₄	#11924027	1	1	yes	characteristic
141	C₂×C₄	#11924029	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	146	4	no	—
144	C₂²	—	73	4	no	—
145	C₂²	—	146	4	no	—
146	C₂²	—	146	2	no	—
147	C₂²	—	146	2	no	—
148	C₂²	—	146	4	no	—
149	C₂²	C₄²×D₇₃	1	1	yes	characteristic
150	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
151	C₂²	#14556549	1	1	yes	characteristic
152	C₂²	M₁₆×D₇₃	1	2	yes	—
153	C₂²	C₂×#11924011	1	2	yes	—
154	C₄	—	2	4	no	—
155	C₄	—	146	2	no	—
156	C₄	—	146	4	no	—
157	C₄	#14556551	1	2	yes	—
158	C₄	#14556553	1	2	yes	—
159	C₂	—	146	4	no	—
160	C₂	#17988434	1	1	yes	characteristic
161	C₂	#97×D₇₃	1	1	yes	characteristic
162	C₂	C₄×#11924011	1	1	yes	characteristic
163	C₂	#17988549	1	2	yes	—
164	C₂	#17988643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556503

C₂

yes

maximal

C₂×#14556501

C₂

yes

maximal, complemented

#17988342

C₂

yes

maximal, characteristic, complemented

#336×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇₃

C₄

yes

—

C₂²×C₄×D₇₃

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄×Dic₇₃

C₄

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₄²×D₇₃

—

#14556503

C₄

yes

complemented

#14556501

—

C₂²×C₈×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

complemented

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

complemented

C₄²×C₇₃

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₄²

yes

—

C₂²×D₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄²

yes

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄²

yes

characteristic

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#117

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

—

C₂²×C₇₃

C₂×M₁₆

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#97

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

C₄×D₄

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₇₃

C₂×#97

yes

characteristic

C₂×C₇₃

C₄×M₁₆

yes

characteristic

C₂×C₇₃

#407

yes

—

C₂×C₇₃

#432

yes

—

D₇₃

—

#1749

—

maximal, Sylow

C₇₃

#1749

yes

characteristic, complemented, Sylow

C₂²×C₄²

—

C₂×C₂²⋊C₈

—

C₂×#97

—

#336

—

#336

D₇₃

yes

characteristic, complemented

C₂³×C₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₂²⋊C₈

—

#97

—

146

—

C₂⁴

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

146

—

104

C₂²×C₄

—

146

—

105

C₂²×C₄

—

146

—

106

C₂²×C₄

—

146

—

107

C₂²×C₄

—

146

—

108

C₂²×C₄

—

146

—

109

C₂²×C₄

C₄×D₇₃

yes

—

110

C₂²×C₄

C₂²×D₇₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₂×C₈

C₄×D₇₃

yes

complemented

117

C₄²

—

118

C₄²

—

119

C₄²

—

146

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

146

—

123

C₂³

—

146

—

124

C₂³

—

146

—

125

C₂³

C₂×C₄×D₇₃

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

#11924011

yes

—

138

C₂×C₄

C₂×C₄×D₇₃

yes

—

139

C₂×C₄

D₄×D₇₃

yes

—

140

C₂×C₄

#11924027

yes

characteristic

141

C₂×C₄

#11924029

yes

characteristic

142

C₈

—

143

C₈

—

146

—

144

C₂²

—

145

C₂²

—

146

—

146

C₂²

—

146

—

147

C₂²

—

146

—

148

C₂²

—

146

—

149

C₂²

C₄²×D₇₃

yes

characteristic

150

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

151

C₂²

#14556549

yes

characteristic

152

C₂²

M₁₆×D₇₃

yes

—

153

C₂²

C₂×#11924011

yes

—

154

C₄

—

155

C₄

—

146

—

156

C₄

—

146

—

157

C₄

#14556551

yes

—

158

C₄

#14556553

yes

—

159

C₂

—

146

—

160

C₂

#17988434

yes

characteristic

161

C₂

#97×D₇₃

yes

characteristic

162

C₂

C₄×#11924011

yes

characteristic

163

C₂

#17988549

yes

—

164

C₂

#17988643

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂×C₄²
Fitting	#336×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₄
hypercenter	#336	D₇₃
hypocenter	C₇₃	#1749
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₃

commutator

C₂²×C₇₃

C₂×C₄²

Fitting

#336×C₇₃

C₂

socle

C₂³×C₇₃

C₂²×C₄

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
3. C₂×#14556501	C₂	semidirect
4. #17988342	C₂	semidirect
5. #336×C₇₃	C₂	semidirect
12. #14556503	C₄	semidirect
28. C₂×#10596831	C₂×C₄	semidirect
35. C₂×C₈×C₇₃	C₂×C₄	semidirect
70. C₂×D₇₃	#97	semidirect
83. C₇₃	#1749	semidirect
88. #336	D₇₃	semidirect
116. C₂×C₈	C₄×D₇₃	semidirect

H ≅ G/N

type

3. C₂×#14556501

C₂

semidirect

4. #17988342

C₂

semidirect

5. #336×C₇₃

C₂

semidirect

12. #14556503

C₄

semidirect

28. C₂×#10596831

C₂×C₄

semidirect

35. C₂×C₈×C₇₃

C₂×C₄

semidirect

70. C₂×D₇₃

#97

semidirect

83. C₇₃

#1749

semidirect

88. #336

D₇₃

semidirect

116. C₂×C₈

C₄×D₇₃

semidirect

Subgroups — #22609692

Conjugacy classes of subgroups

Complements