Group #22610023 — subgroups

Subgroups — #22610023

Order 9344.

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

normal subgroups only

This group has 300 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	C₂×#14556550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	#17988349	C₂	1	2	yes	maximal, complemented
4	#17988349	C₂	1	2	yes	maximal, complemented
5	#380×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₂²	1	2	yes	—
7	#14556550	—	2	2	no	—
8	#14556550	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₇₃	—	2	1	no	—
12	C₂×C₄×Dic₇₃	—	2	1	no	—
13	C₂²⋊#10596831	—	2	2	no	—
14	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
17	#95×C₇₃	—	2	1	no	—
18	C₂²×Dic₇₃	—	2	1	no	—
19	C₂²×Dic₇₃	—	2	1	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	—	2	4	no	—
22	C₄⋊Dic₇₃	—	2	1	no	—
23	C₄⋊Dic₇₃	—	2	1	no	—
24	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
25	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
26	C₂²⋊Dic₇₃	—	2	4	no	—
27	C₄×Dic₇₃	—	2	4	no	—
28	C₂⁴×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	—	2	4	no	—
30	C₂²×C₄×C₇₃	—	2	2	no	—
31	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₇₃	D₄	1	2	yes	—
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₃	—	2	4	no	—
36	C₂²⋊C₄×C₇₃	—	4	2	no	—
37	C₂³×C₇₃	—	2	2	no	—
38	C₂³×C₇₃	—	2	2	no	—
39	C₂³×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	—	2	1	no	—
41	C₂³×C₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	4	no	—
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	4	yes	—
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	—	4	2	no	—
56	C₂×Dic₇₃	—	4	1	no	—
57	C₂×Dic₇₃	—	4	1	no	—
58	#10596831	—	4	2	no	—
59	C₂²×C₇₃	—	2	2	no	—
60	C₂²×C₇₃	—	2	2	no	—
61	C₂²×C₇₃	—	2	2	no	—
62	C₂²×C₇₃	—	4	2	no	—
63	C₂²×C₇₃	—	4	2	no	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₇₃	#123	1	2	yes	—
66	C₂²×C₇₃	#135	1	2	yes	—
67	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₇₃	#137	1	1	yes	characteristic
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	8	2	no	—
71	C₄×C₇₃	#117	1	2	yes	—
72	C₄×C₇₃	#123	1	2	yes	—
73	Dic₇₃	—	4	2	no	—
74	Dic₇₃	—	4	2	no	—
75	C₂×C₇₃	—	4	2	no	—
76	C₂×C₇₃	#381	1	1	yes	characteristic
77	C₂×C₇₃	#416	1	1	yes	characteristic
78	C₂×C₇₃	#417	1	1	yes	characteristic
79	C₂×C₇₃	#482	1	2	yes	—
80	C₂×C₇₃	#485	1	2	yes	—
81	#1825	—	73	1	no	maximal, Sylow
82	C₇₃	#1825	1	1	yes	characteristic, complemented, Sylow
83	C₂×#117	—	73	1	no	—
84	C₂×C₂²⋊C₈	—	73	1	no	—
85	#380	D₇₃	1	1	yes	characteristic
86	#340	—	73	2	no	—
87	#340	—	73	2	no	—
88	C₂³×C₄	C₂×D₇₃	1	1	yes	characteristic
89	C₂²×C₈	—	73	2	no	—
90	#117	—	73	4	no	—
91	#117	—	146	2	no	—
92	C₂×C₄⋊C₄	—	73	1	no	—
93	C₂×C₄⋊C₄	—	73	1	no	—
94	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
95	C₂×C₂²⋊C₄	—	2	1	no	—
96	C₂×C₂²⋊C₄	—	146	1	no	—
97	C₂×C₄²	—	146	1	no	—
98	C₂²⋊C₈	—	146	2	no	—
99	#95	—	2	1	no	—
100	C₂⁴	C₂²⋊D₇₃	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	146	1	no	—
104	C₂²×C₄	—	146	1	no	—
105	C₂²×C₄	C₄×D₇₃	1	2	yes	—
106	C₂²×C₄	D₂₉₂	1	2	yes	—
107	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	—	146	4	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	73	2	no	—
113	C₄⋊C₄	—	73	2	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	146	4	no	—
118	C₄²	—	146	4	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	2	1	no	—
122	C₂³	—	2	1	no	—
123	C₂³	#11924005	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	292	2	no	—
132	C₂×C₄	—	292	1	no	—
133	C₂×C₄	—	292	1	no	—
134	C₂×C₄	—	146	2	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	#11924005	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
139	C₂×C₄	#11924027	1	4	yes	—
140	C₈	—	292	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	C₂×#11924005	1	1	yes	characteristic
147	C₂²	#14556572	1	2	yes	—
148	C₂²	#14556611	1	2	yes	—
149	C₂²	#14556617	1	1	yes	characteristic
150	C₂²	#14556622	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	292	2	no	—
154	C₄	—	292	2	no	—
155	C₄	#14556547	1	2	yes	—
156	C₄	#14556575	1	2	yes	—
157	C₂	—	4	2	no	—
158	C₂	#17988448	1	1	yes	characteristic
159	C₂	#17988581	1	1	yes	characteristic
160	C₂	#17988586	1	1	yes	characteristic
161	C₂	#17988908	1	2	yes	—
162	C₂	#17988925	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556550

C₂

yes

maximal, characteristic

C₂×C₂²⋊#10596831

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, complemented

#17988349

C₂

yes

maximal, complemented

#380×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

#14556550

—

#14556550

C₄

yes

complemented

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₇₃

—

C₂×C₄×Dic₇₃

—

C₂²⋊#10596831

—

C₂³×C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₃

—

#95×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂⁴×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×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

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#123

yes

—

C₂²×C₇₃

#135

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#117

yes

—

C₄×C₇₃

#123

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#485

yes

—

#1825

—

maximal, Sylow

C₇₃

#1825

yes

characteristic, complemented, Sylow

C₂×#117

—

C₂×C₂²⋊C₈

—

#380

D₇₃

yes

characteristic

#340

—

#340

—

C₂³×C₄

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

146

—

C₂×C₄²

—

146

—

C₂²⋊C₈

—

146

—

#95

—

100

C₂⁴

C₂²⋊D₇₃

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

146

—

104

C₂²×C₄

—

146

—

105

C₂²×C₄

C₄×D₇₃

yes

—

106

C₂²×C₄

D₂₉₂

yes

—

107

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

108

C₂²×C₄

C₂²×D₇₃

yes

characteristic

109

C₂×C₈

—

146

—

110

C₂×C₈

—

146

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

—

146

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

146

—

118

C₄²

—

146

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

#11924005

yes

—

124

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

292

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

#11924005

yes

—

137

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

138

C₂×C₄

C₂×D₂₉₂

yes

characteristic

139

C₂×C₄

#11924027

yes

—

140

C₈

—

292

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×#11924005

yes

characteristic

147

C₂²

#14556572

yes

—

148

C₂²

#14556611

yes

—

149

C₂²

#14556617

yes

characteristic

150

C₂²

#14556622

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

292

—

154

C₄

—

292

—

155

C₄

#14556547

yes

—

156

C₄

#14556575

yes

—

157

C₂

—

158

C₂

#17988448

yes

characteristic

159

C₂

#17988581

yes

characteristic

160

C₂

#17988586

yes

characteristic

161

C₂

#17988908

yes

—

162

C₂

#17988925

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#380×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#380	D₇₃
hypocenter	C₇₃	#1825
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#380×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
3. #17988349	C₂	semidirect
4. #17988349	C₂	semidirect
8. #14556550	C₄	semidirect
82. C₇₃	#1825	semidirect

H ≅ G/N

type

3. #17988349

C₂

semidirect

4. #17988349

C₂

semidirect

8. #14556550

C₄

semidirect

82. C₇₃

#1825

semidirect

Subgroups — #22610023

Conjugacy classes of subgroups

Complements