Group #22610454 — subgroups

Subgroups — #22610454

Order 9344.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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	#17989354	C₂	1	1	yes	maximal, characteristic
2	#17988596	C₂	1	2	yes	maximal, complemented
3	#17988561	C₂	1	1	yes	maximal, characteristic
4	#17988343	C₂	1	2	yes	maximal, complemented
5	#536×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556598	—	2	4	no	—
7	C₂×#11924012	—	2	2	no	—
8	C₂×#11924012	C₂²	1	2	yes	—
9	#14556550	C₂²	1	2	yes	—
10	#14556515	—	2	4	no	—
11	#14556506	—	2	2	no	—
12	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₇₃	C₄	1	2	yes	complemented
15	#124×C₇₃	C₄	1	2	yes	complemented
16	#123×C₇₃	—	2	2	no	—
17	C₂²⋊Q₈×C₇₃	—	2	2	no	—
18	#117×C₇₃	C₂²	1	1	yes	characteristic
19	#11924012	—	2	4	no	—
20	#11924012	—	2	2	no	—
21	#11924012	—	2	2	no	—
22	C₂×#10596831	—	2	4	no	—
23	C₂×#10596831	D₄	1	4	yes	—
24	C₄⋊Dic₇₃	—	2	2	no	—
25	C₂²⋊Dic₇₃	—	2	2	no	—
26	C₄×Dic₇₃	—	2	2	no	—
27	#41×C₇₃	—	2	2	no	—
28	#41×C₇₃	—	2	2	no	—
29	C₂×Q₈×C₇₃	—	2	1	no	—
30	C₂×Q₈×C₇₃	—	2	2	no	—
31	C₂×Q₈×C₇₃	—	2	2	no	—
32	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₇₃	D₄	1	2	yes	—
36	C₂²×C₄×C₇₃	—	2	1	no	—
37	C₂²×C₄×C₇₃	D₄	1	2	yes	—
38	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	4	no	—
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₂²⋊C₄×C₇₃	—	2	4	no	—
42	C₂²⋊C₄×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂³×C₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₇₃	—	2	2	no	—
47	Q₈×C₇₃	—	2	4	no	—
48	Q₈×C₇₃	—	4	1	no	—
49	Q₈×C₇₃	—	4	1	no	—
50	D₄×C₇₃	—	2	4	no	—
51	D₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	4	2	no	—
57	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#41	1	2	yes	—
62	C₂×Dic₇₃	—	4	2	no	—
63	#10596831	—	2	4	no	—
64	C₂²×C₇₃	—	2	2	no	—
65	C₂²×C₇₃	—	4	1	no	—
66	C₂²×C₇₃	—	4	2	no	—
67	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₇₃	#121	1	2	yes	—
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	4	2	no	—
71	C₄×C₇₃	—	4	2	no	—
72	C₄×C₇₃	C₄×D₄	1	2	yes	—
73	C₄×C₇₃	#120	1	1	yes	characteristic
74	C₄×C₇₃	#123	1	1	yes	characteristic
75	Dic₇₃	—	8	2	no	—
76	C₂×C₇₃	—	2	2	no	—
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	—	4	2	no	—
79	C₂×C₇₃	#386	1	1	yes	characteristic
80	#1909	—	73	1	no	maximal, Sylow
81	C₇₃	#1909	1	1	yes	characteristic, complemented, Sylow
82	#568	—	73	1	no	—
83	#536	D₇₃	1	1	yes	characteristic
84	#419	—	73	2	no	—
85	#410	—	73	1	no	—
86	H(3,ℤ/4)	—	73	2	no	—
87	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
88	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
89	#131	—	146	4	no	—
90	C₂×M₁₆	—	73	2	no	—
91	C₂×M₁₆	—	146	2	no	—
92	C₄⋊Q₈	Dic₇₃	1	2	yes	complemented
93	#124	Dic₇₃	1	2	yes	complemented
94	#123	—	2	2	no	—
95	C₂²⋊Q₈	—	2	2	no	—
96	#117	—	73	2	no	—
97	#117	C₂×D₇₃	1	1	yes	characteristic
98	#103	—	146	4	no	—
99	#99	—	146	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	C₂×Dic₇₃	1	2	yes	—
106	C₂×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
107	C₂×D₄	C₂×Dic₇₃	1	1	yes	characteristic
108	C₂×D₄	C₂²⋊D₇₃	1	2	yes	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
112	M₁₆	—	146	4	no	—
113	M₁₆	—	146	2	no	—
114	M₁₆	—	146	2	no	—
115	C₂×C₈	—	73	4	no	—
116	C₂×C₈	—	146	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	146	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	146	2	no	—
123	C₄²	—	146	2	no	—
124	C₄²	C₂×Dic₇₃	1	2	yes	—
125	C₂³	C₂²⋊Dic₇₃	1	2	yes	—
126	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	4	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	292	2	no	—
139	C₂×C₄	C₂²⋊Dic₇₃	1	2	yes	—
140	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
141	C₂×C₄	D₄×D₇₃	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂×C₄	#11924027	1	2	yes	—
144	C₈	—	146	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
149	C₂²	#14556563	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	584	2	no	—
154	C₄	D₄×Dic₇₃	1	2	yes	—
155	C₄	#14556559	1	1	yes	characteristic
156	C₄	#14556572	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#17988467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989354

C₂

yes

maximal, characteristic

#17988596

C₂

yes

maximal, complemented

#17988561

C₂

yes

maximal, characteristic

#17988343

C₂

yes

maximal, complemented

#536×C₇₃

C₂

yes

maximal, characteristic

#14556598

—

C₂×#11924012

—

C₂×#11924012

C₂²

yes

—

#14556550

C₂²

yes

—

#14556515

—

#14556506

—

C₂×#41×C₇₃

C₂²

yes

characteristic

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

complemented

#124×C₇₃

C₄

yes

complemented

#123×C₇₃

—

C₂²⋊Q₈×C₇₃

—

#117×C₇₃

C₂²

yes

—

—

—

—

D₄

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂²⋊C₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

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

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#568

—

#536

D₇₃

yes

—

—

—

C₂×#41

C₂×D₇₃

yes

characteristic

C₂²×Q₈

C₂×D₇₃

yes

characteristic

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

Dic₇₃

yes

complemented

#124

Dic₇₃

yes

complemented

#123

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₇₃

yes

characteristic

#103

—

146

—

#99

—

146

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×Dic₇₃

yes

—

106

C₂×Q₈

C₂×Dic₇₃

yes

characteristic

107

C₂×D₄

C₂×Dic₇₃

yes

characteristic

108

C₂×D₄

C₂²⋊D₇₃

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₇₃

yes

—

111

C₂²×C₄

C₂²×D₇₃

yes

characteristic

112

M₁₆

—

146

—

113

M₁₆

—

146

—

114

M₁₆

—

146

—

115

C₂×C₈

—

116

C₂×C₈

—

146

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

146

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

146

—

123

C₄²

—

146

—

124

C₄²

C₂×Dic₇₃

yes

—

125

C₂³

C₂²⋊Dic₇₃

yes

—

126

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

292

—

139

C₂×C₄

C₂²⋊Dic₇₃

yes

—

140

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

141

C₂×C₄

D₄×D₇₃

yes

—

142

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

143

C₂×C₄

#11924027

yes

—

144

C₈

—

146

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

149

C₂²

#14556563

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

584

—

154

C₄

D₄×Dic₇₃

yes

—

155

C₄

#14556559

yes

characteristic

156

C₄

#14556572

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#17988467

yes

characteristic

	subgroup	G/H
center	C₂	#17988467
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#536×C₇₃	C₂
socle	C₂×C₇₃	#386
hypercenter	#536	D₇₃
hypocenter	C₇₃	#1909
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#17988467

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#536×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
2. #17988596	C₂	semidirect
4. #17988343	C₂	semidirect
14. C₄⋊Q₈×C₇₃	C₄	semidirect
15. #124×C₇₃	C₄	semidirect
81. C₇₃	#1909	semidirect
92. C₄⋊Q₈	Dic₇₃	semidirect
93. #124	Dic₇₃	semidirect

H ≅ G/N

type

2. #17988596

C₂

semidirect

4. #17988343

C₂

semidirect

14. C₄⋊Q₈×C₇₃

C₄

semidirect

15. #124×C₇₃

C₄

semidirect

81. C₇₃

#1909

semidirect

92. C₄⋊Q₈

Dic₇₃

semidirect

93. #124

Dic₇₃

semidirect

Subgroups — #22610454

Conjugacy classes of subgroups

Complements