Group #22610456 — subgroups

Subgroups — #22610456

Order 9344.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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, complemented
2	#17989151	C₂	1	1	yes	maximal, characteristic
3	#17988591	C₂	1	2	yes	maximal, complemented
4	#17988343	C₂	1	2	yes	maximal, complemented
5	#410×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
7	#14556598	—	2	4	no	—
8	C₂×#11924012	—	2	2	no	—
9	C₂×#11924012	C₂²	1	2	yes	—
10	#14556587	C₄	1	2	yes	complemented
11	#14556580	C₄	1	2	yes	complemented
12	#14556565	—	2	2	no	—
13	#14556559	—	2	2	no	—
14	#14556550	C₂²	1	1	yes	characteristic
15	#14556511	—	2	4	no	—
16	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
17	#117×C₇₃	C₂²	1	2	yes	—
18	#99×C₇₃	—	2	2	no	—
19	C₂³×D₇₃	—	2	1	no	—
20	C₂×D₂₉₂	—	2	2	no	—
21	C₂×D₂₉₂	—	2	2	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
24	C₂×C₂²⋊D₇₃	—	2	4	no	—
25	#11924012	—	2	4	no	—
26	#11924012	—	2	2	no	—
27	#11924012	—	2	2	no	—
28	C₂×#10596831	—	2	4	no	—
29	C₂×#10596831	D₄	1	4	yes	complemented
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₂²⋊Dic₇₃	—	2	1	no	—
32	#11924005	—	2	4	no	—
33	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
34	#41×C₇₃	—	2	2	no	—
35	#41×C₇₃	—	2	2	no	—
36	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₇₃	D₄	1	2	yes	—
39	C₂²×C₄×C₇₃	D₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	—	2	2	no	—
42	C₂²⋊C₄×C₇₃	—	2	2	no	—
43	C₄²×C₇₃	—	2	2	no	—
44	C₂³×C₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₇₃	—	2	2	no	—
47	C₂²×D₇₃	—	4	2	no	—
48	C₂²×D₇₃	—	8	2	no	—
49	Q₈×C₇₃	—	2	2	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	—	4	2	no	—
55	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#41	1	2	yes	—
60	C₂²⋊D₇₃	—	8	4	no	—
61	C₂×Dic₇₃	—	2	2	no	—
62	D₂₉₂	—	2	4	no	—
63	D₂₉₂	—	4	1	no	—
64	D₂₉₂	—	4	1	no	—
65	#10596831	—	2	4	no	—
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	—	4	2	no	—
69	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₇₃	#121	1	2	yes	—
71	C₂×D₇₃	—	4	4	no	—
72	C₂×D₇₃	—	4	2	no	—
73	C₂×D₇₃	—	8	4	no	—
74	C₄×C₇₃	—	2	2	no	—
75	C₄×C₇₃	—	8	2	no	—
76	C₄×C₇₃	C₄×D₄	1	2	yes	—
77	C₄×C₇₃	#120	1	1	yes	characteristic
78	C₄×C₇₃	#123	1	1	yes	characteristic
79	Dic₇₃	—	4	2	no	—
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	2	no	—
82	C₂×C₇₃	—	4	2	no	—
83	C₂×C₇₃	#386	1	1	yes	characteristic
84	D₇₃	—	8	2	no	—
85	#1910	—	73	1	no	maximal, Sylow
86	C₇₃	#1910	1	1	yes	characteristic, complemented, Sylow
87	#568	—	73	1	no	—
88	#535	—	73	1	no	—
89	#418	—	73	2	no	—
90	#410	D₇₃	1	1	yes	characteristic, complemented
91	H(3,ℤ/4)	—	73	2	no	—
92	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
93	C₂²×D₄	—	73	1	no	—
94	#131	—	146	4	no	—
95	C₂×M₁₆	—	73	2	no	—
96	C₂×M₁₆	—	146	2	no	—
97	#127	—	73	2	no	—
98	#124	—	73	2	no	—
99	#121	—	146	2	no	—
100	#120	—	146	2	no	—
101	#117	—	73	1	no	—
102	#117	C₂×D₇₃	1	2	yes	—
103	#102	—	146	4	no	—
104	#99	—	2	2	no	—
105	C₂⁴	—	146	1	no	—
106	#41	—	2	2	no	—
107	#41	—	2	2	no	—
108	C₂×Q₈	C₄×D₇₃	1	1	yes	characteristic
109	C₂×D₄	—	73	2	no	—
110	C₂×D₄	—	146	4	no	—
111	C₂×D₄	—	146	2	no	—
112	C₂×D₄	—	146	1	no	—
113	C₂×D₄	—	146	2	no	—
114	C₂×D₄	C₄×D₇₃	1	1	yes	characteristic
115	C₂×D₄	C₂²⋊D₇₃	1	2	yes	—
116	C₂²×C₄	D₂₉₂	1	2	yes	—
117	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
118	M₁₆	—	146	4	no	—
119	M₁₆	—	146	2	no	—
120	M₁₆	—	146	2	no	—
121	C₂×C₈	—	73	4	no	—
122	C₂×C₈	—	146	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	146	1	no	—
127	C₂²⋊C₄	—	146	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	73	2	no	—
130	C₂³	—	292	2	no	—
131	C₂³	—	584	2	no	—
132	C₂³	—	146	2	no	—
133	C₂³	#11924005	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
135	Q₈	—	2	2	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	2	no	—
138	D₄	—	292	1	no	—
139	D₄	—	292	1	no	—
140	D₄	—	584	4	no	—
141	D₄	—	146	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	#11924005	1	2	yes	—
147	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₇₃	1	2	yes	—
149	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
150	C₂×C₄	#11924027	1	2	yes	—
151	C₈	—	146	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	292	4	no	—
156	C₂²	—	292	2	no	—
157	C₂²	—	584	4	no	—
158	C₂²	C₂×#11924005	1	1	yes	characteristic
159	C₂²	#14556563	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	292	2	no	—
163	C₄	#14556551	1	2	yes	—
164	C₄	#14556561	1	1	yes	characteristic
165	C₄	#14556575	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	—	2	1	no	—
168	C₂	—	4	2	no	—
169	C₂	—	584	2	no	—
170	C₂	#17988464	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989354

C₂

yes

maximal, characteristic, complemented

#17989151

C₂

yes

maximal, characteristic

#17988591

C₂

yes

maximal, complemented

#17988343

C₂

yes

maximal, complemented

#410×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

—

—

C₂²

yes

—

#14556587

C₄

yes

complemented

#14556580

C₄

yes

—

—

C₂²

yes

characteristic

#14556511

—

C₂×#41×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

—

#99×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₂²⋊D₇₃

—

—

—

—

—

D₄

yes

complemented

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

C₂×C₄

yes

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×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

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#386

yes

characteristic

D₇₃

—

#1910

—

maximal, Sylow

C₇₃

#1910

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic, complemented

H(3,ℤ/4)

—

C₂×#41

C₂×D₇₃

yes

characteristic

C₂²×D₄

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

#127

—

#124

—

#121

—

146

—

100

#120

—

146

—

101

#117

—

102

#117

C₂×D₇₃

yes

—

103

#102

—

146

—

104

#99

—

105

C₂⁴

—

146

—

106

#41

—

107

#41

—

108

C₂×Q₈

C₄×D₇₃

yes

characteristic

109

C₂×D₄

—

110

C₂×D₄

—

146

—

111

C₂×D₄

—

146

—

112

C₂×D₄

—

146

—

113

C₂×D₄

—

146

—

114

C₂×D₄

C₄×D₇₃

yes

characteristic

115

C₂×D₄

C₂²⋊D₇₃

yes

—

116

C₂²×C₄

D₂₉₂

yes

—

117

C₂²×C₄

C₂²×D₇₃

yes

characteristic

118

M₁₆

—

146

—

119

M₁₆

—

146

—

120

M₁₆

—

146

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

146

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

146

—

127

C₂²⋊C₄

—

146

—

128

C₄²

—

129

C₄²

—

130

C₂³

—

292

—

131

C₂³

—

584

—

132

C₂³

—

146

—

133

C₂³

#11924005

yes

—

134

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

292

—

139

D₄

—

292

—

140

D₄

—

584

—

141

D₄

—

146

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

—

146

C₂×C₄

#11924005

yes

—

147

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

148

C₂×C₄

D₄×D₇₃

yes

—

149

C₂×C₄

C₂×D₂₉₂

yes

characteristic

150

C₂×C₄

#11924027

yes

—

151

C₈

—

146

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

292

—

156

C₂²

—

292

—

157

C₂²

—

584

—

158

C₂²

C₂×#11924005

yes

characteristic

159

C₂²

#14556563

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

292

—

163

C₄

#14556551

yes

—

164

C₄

#14556561

yes

characteristic

165

C₄

#14556575

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

584

—

170

C₂

#17988464

yes

characteristic

	subgroup	G/H
center	C₂	#17988464
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#410×C₇₃	C₂
socle	C₂×C₇₃	#386
hypercenter	#410	D₇₃
hypocenter	C₇₃	#1910
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₂

#17988464

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#410×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
1. #17989354	C₂	semidirect
3. #17988591	C₂	semidirect
4. #17988343	C₂	semidirect
5. #410×C₇₃	C₂	semidirect
10. #14556587	C₄	semidirect
11. #14556580	C₄	semidirect
29. C₂×#10596831	D₄	semidirect
86. C₇₃	#1910	semidirect
90. #410	D₇₃	semidirect

H ≅ G/N

type

1. #17989354

C₂

semidirect

3. #17988591

C₂

semidirect

4. #17988343

C₂

semidirect

5. #410×C₇₃

C₂

semidirect

10. #14556587

C₄

semidirect

11. #14556580

C₄

semidirect

29. C₂×#10596831

D₄

semidirect

86. C₇₃

#1910

semidirect

90. #410

D₇₃

semidirect

Subgroups — #22610456

Conjugacy classes of subgroups

Complements