Group #22610478 — subgroups

Subgroups — #22610478

Order 9344.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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	#17989151	C₂	1	1	yes	maximal, characteristic
2	#17988571	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14556507	C₂	1	2	yes	maximal, complemented
4	#17988359	C₂	1	2	yes	maximal, complemented
5	#568×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
7	C₂×#11924012	C₂²	1	2	yes	—
8	#14556587	—	2	1	no	—
9	#14556580	—	2	1	no	—
10	#14556565	—	2	2	no	—
11	#14556559	C₄	1	4	yes	—
12	#14556550	C₂²	1	1	yes	characteristic
13	#14556510	—	2	1	no	—
14	#14556508	—	2	1	no	—
15	#14556507	—	2	4	no	—
16	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
17	#131×C₇₃	—	2	4	no	—
18	C₂×M₁₆×C₇₃	—	2	2	no	—
19	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
20	C₂³×D₇₃	C₂×C₄	1	2	yes	—
21	C₂×D₂₉₂	—	2	2	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	2	no	—
25	C₂×C₂²⋊D₇₃	—	2	4	no	—
26	#11924012	—	2	2	no	—
27	#11924012	—	2	2	no	—
28	C₂×#10596831	—	2	2	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	#11924005	—	2	4	no	—
32	C₄×Dic₇₃	—	2	1	no	—
33	#41×C₇₃	—	2	2	no	—
34	#41×C₇₃	—	2	2	no	—
35	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
36	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	D₄	1	2	yes	—
39	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₃	—	2	4	no	—
41	M₁₆×C₇₃	—	2	2	no	—
42	M₁₆×C₇₃	D₄	1	4	yes	complemented
43	C₂×C₈×C₇₃	—	2	4	no	—
44	C₂×C₈×C₇₃	—	2	2	no	—
45	C₂³×C₇₃	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₇₃	—	2	2	no	—
48	C₂²×D₇₃	—	2	4	no	—
49	C₂²×D₇₃	—	8	2	no	—
50	Q₈×C₇₃	—	2	2	no	—
51	D₄×C₇₃	—	2	4	no	—
52	D₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
63	D₂₉₂	—	4	2	no	—
64	D₂₉₂	—	4	1	no	—
65	C₈×C₇₃	—	2	4	no	—
66	#10596831	—	4	2	no	—
67	C₂²×C₇₃	—	2	2	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	—	4	2	no	—
70	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₇₃	C₄×D₄	1	2	yes	—
72	C₂×D₇₃	—	4	4	no	—
73	C₂×D₇₃	—	4	2	no	—
74	C₂×D₇₃	—	8	4	no	—
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	#120	1	1	yes	characteristic
77	C₄×C₇₃	#121	1	2	yes	—
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	#1914	—	73	1	no	maximal, Sylow
86	C₇₃	#1914	1	1	yes	characteristic, complemented, Sylow
87	#568	D₇₃	1	1	yes	characteristic, complemented
88	#535	—	73	1	no	—
89	#413	—	73	1	no	—
90	C₂×#100	—	73	2	no	—
91	#344	—	73	2	no	—
92	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
93	C₂²×D₄	—	73	1	no	—
94	#131	—	2	4	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	73	2	no	—
97	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
98	#127	—	146	1	no	—
99	#124	—	146	1	no	—
100	#121	—	146	2	no	—
101	#120	—	73	4	no	—
102	#117	—	73	1	no	—
103	#101	—	146	1	no	—
104	#100	—	146	1	no	—
105	#100	—	146	4	no	—
106	C₂⁴	—	73	2	no	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	D₂₉₂	1	1	yes	characteristic
110	C₂×D₄	—	146	4	no	—
111	C₂×D₄	—	146	2	no	—
112	C₂×D₄	—	146	1	no	—
113	C₂×D₄	—	146	1	no	—
114	C₂×D₄	—	146	2	no	—
115	C₂×D₄	C₄×D₇₃	1	2	yes	—
116	C₂×D₄	D₂₉₂	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
118	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
119	M₁₆	—	2	4	no	—
120	M₁₆	—	2	2	no	—
121	M₁₆	—	146	2	no	—
122	M₁₆	—	146	2	no	—
123	M₁₆	C₂²⋊D₇₃	1	4	yes	complemented
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₄⋊C₄	—	146	1	no	—
128	C₂²⋊C₄	—	73	2	no	—
129	C₂²⋊C₄	—	146	4	no	—
130	C₄²	—	146	1	no	—
131	C₂³	—	584	2	no	—
132	C₂³	—	146	2	no	—
133	C₂³	—	146	4	no	—
134	C₂³	#11924005	1	2	yes	—
135	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	292	1	no	—
140	D₄	—	292	2	no	—
141	D₄	—	292	1	no	—
142	D₄	—	584	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	#11924005	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
149	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
150	C₂×C₄	#11924028	1	2	yes	—
151	C₈	—	2	4	no	—
152	C₈	—	292	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	292	4	no	—
157	C₂²	—	292	2	no	—
158	C₂²	—	584	4	no	—
159	C₂²	C₂×#11924005	1	1	yes	characteristic
160	C₂²	C₄×C₂²⋊D₇₃	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	292	2	no	—
163	C₄	#14556560	1	1	yes	characteristic
164	C₄	#14556565	1	2	yes	—
165	C₄	#14556574	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	584	2	no	—
170	C₂	#17988466	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989151

C₂

yes

maximal, characteristic

#17988571

C₂

yes

maximal, characteristic, complemented

C₂×#14556507

C₂

yes

maximal, complemented

#17988359

C₂

yes

maximal, complemented

#568×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

—

—

—

—

C₄

yes

—

#14556550

C₂²

yes

—

—

—

C₂×#41×C₇₃

C₂²

yes

characteristic

#131×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂³×D₇₃

C₂×C₄

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

C₂×C₄

yes

—

#11924005

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

D₄

yes

complemented

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

characteristic

C₄×C₇₃

#121

yes

—

C₄×C₇₃

#123

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#386

yes

characteristic

D₇₃

—

#1914

—

maximal, Sylow

C₇₃

#1914

yes

characteristic, complemented, Sylow

#568

D₇₃

yes

characteristic, complemented

—

—

—

—

C₂×#41

C₂×D₇₃

yes

characteristic

C₂²×D₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

—

#127

—

146

—

#124

—

146

—

100

#121

—

146

—

101

#120

—

102

#117

—

103

#101

—

146

—

104

#100

—

146

—

105

#100

—

146

—

106

C₂⁴

—

107

#41

—

108

#41

—

109

C₂×Q₈

D₂₉₂

yes

characteristic

110

C₂×D₄

—

146

—

111

C₂×D₄

—

146

—

112

C₂×D₄

—

146

—

113

C₂×D₄

—

146

—

114

C₂×D₄

—

146

—

115

C₂×D₄

C₄×D₇₃

yes

—

116

C₂×D₄

D₂₉₂

yes

characteristic

117

C₂²×C₄

C₂²⋊D₇₃

yes

—

118

C₂²×C₄

C₂²×D₇₃

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

146

—

122

M₁₆

—

146

—

123

M₁₆

C₂²⋊D₇₃

yes

complemented

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₄⋊C₄

—

146

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

146

—

130

C₄²

—

146

—

131

C₂³

—

584

—

132

C₂³

—

146

—

133

C₂³

—

146

—

134

C₂³

#11924005

yes

—

135

C₂³

C₂×C₄×D₇₃

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

292

—

140

D₄

—

292

—

141

D₄

—

292

—

142

D₄

—

584

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

—

146

C₂×C₄

#11924005

yes

—

147

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

148

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

149

C₂×C₄

C₂×D₂₉₂

yes

characteristic

150

C₂×C₄

#11924028

yes

—

151

C₈

—

152

C₈

—

292

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

292

—

157

C₂²

—

292

—

158

C₂²

—

584

—

159

C₂²

C₂×#11924005

yes

characteristic

160

C₂²

C₄×C₂²⋊D₇₃

yes

—

161

C₄

—

162

C₄

—

292

—

163

C₄

#14556560

yes

characteristic

164

C₄

#14556565

yes

—

165

C₄

#14556574

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

584

—

170

C₂

#17988466

yes

characteristic

	subgroup	G/H
center	C₂	#17988466
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#568×C₇₃	C₂
socle	C₂×C₇₃	#386
hypercenter	#568	D₇₃
hypocenter	C₇₃	#1914
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#17988466

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#568×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #17988571	C₂	semidirect
3. C₂×#14556507	C₂	semidirect
4. #17988359	C₂	semidirect
5. #568×C₇₃	C₂	semidirect
42. M₁₆×C₇₃	D₄	semidirect
86. C₇₃	#1914	semidirect
87. #568	D₇₃	semidirect
123. M₁₆	C₂²⋊D₇₃	semidirect

H ≅ G/N

type

2. #17988571

C₂

semidirect

3. C₂×#14556507

C₂

semidirect

4. #17988359

C₂

semidirect

5. #568×C₇₃

C₂

semidirect

42. M₁₆×C₇₃

D₄

semidirect

86. C₇₃

#1914

semidirect

87. #568

D₇₃

semidirect

123. M₁₆

C₂²⋊D₇₃

semidirect

Subgroups — #22610478

Conjugacy classes of subgroups

Complements