Group #22610822 — subgroups

Subgroups — #22610822

Order 9344.

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

normal subgroups only

This group has 292 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	C₂×#14556613	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556517	C₂	1	2	yes	maximal, complemented
3	#17988555	C₂	1	2	yes	maximal, complemented
4	C₂×#14556501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#108×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924028	C₂²	1	2	yes	—
7	#14556613	—	2	2	no	—
8	#14556613	—	2	2	no	—
9	C₂×Dic₂₉₂	C₄	1	1	yes	characteristic, complemented
10	C₂×#11924017	C₄	1	2	yes	complemented
11	C₂×D₅₈₄	C₄	1	1	yes	characteristic, complemented
12	C₂²×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	#14556517	—	2	4	no	—
15	#14556503	—	2	4	no	—
16	#14556501	—	2	2	no	—
17	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#108×C₇₃	—	2	2	no	—
20	#11924028	—	2	2	no	—
21	#11924028	—	2	2	no	—
22	#11924028	—	2	2	no	—
23	#11924028	—	2	2	no	—
24	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
25	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
26	C₂×C₂²⋊D₇₃	—	2	2	no	—
27	C₂²×Dic₇₃	—	2	1	no	—
28	C₂×C₄×D₇₃	—	2	2	no	—
29	Dic₂₉₂	—	2	2	no	—
30	#11924017	—	2	4	no	—
31	D₅₈₄	—	2	2	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	D₄	1	2	yes	—
35	C₄×Dic₇₃	—	2	1	no	—
36	C₄×Dic₇₃	D₄	1	2	yes	—
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	M₁₆×C₇₃	—	2	2	no	—
39	M₁₆×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	—	2	2	no	—
41	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	D₄	1	2	yes	—
44	C₂×C₈×C₇₃	D₄	1	2	yes	—
45	C₂³×C₇₃	#41	1	1	yes	characteristic
46	C₂²×D₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂²⋊D₇₃	—	4	2	no	—
54	C₂²⋊D₇₃	—	4	2	no	—
55	C₂×Dic₇₃	—	2	2	no	—
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	4	2	no	—
58	D₂₉₂	—	2	2	no	—
59	D₂₉₂	—	2	2	no	—
60	C₄×D₇₃	—	4	2	no	—
61	C₄×D₇₃	—	4	2	no	—
62	Dic₁₄₆	—	2	2	no	—
63	Dic₁₄₆	—	2	2	no	—
64	C₈×C₇₃	—	4	2	no	—
65	C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
66	C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
67	#10596831	—	2	2	no	—
68	C₂²×C₇₃	—	2	2	no	—
69	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂²×C₇₃	#121	1	1	yes	characteristic
71	C₂²×C₇₃	#124	1	1	yes	characteristic
72	C₂×D₇₃	—	4	2	no	—
73	C₂×D₇₃	—	4	4	no	—
74	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
76	C₄×C₇₃	#121	1	1	yes	characteristic
77	C₄×C₇₃	#127	1	1	yes	characteristic
78	Dic₇₃	—	4	2	no	—
79	Dic₇₃	—	4	2	no	—
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	#390	1	1	yes	characteristic
83	C₂×C₇₃	#444	1	2	yes	—
84	D₇₃	—	8	2	no	—
85	#1975	—	73	1	no	maximal, Sylow
86	C₇₃	#1975	1	1	yes	characteristic, complemented, Sylow
87	C₂×#135	—	73	1	no	—
88	C₂×#108	D₇₃	1	1	yes	characteristic, complemented
89	C₂×#104	—	73	2	no	—
90	#408	—	73	2	no	—
91	C₂×#97	—	73	1	no	—
92	C₂×#41	—	73	2	no	—
93	#135	—	146	2	no	—
94	#135	—	146	2	no	—
95	C₂×Q₁₆	—	73	1	no	—
96	C₂×SD₁₆	—	73	2	no	—
97	C₂×D₈	—	73	1	no	—
98	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
99	C₂²×C₈	—	73	1	no	—
100	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
101	C₂×C₄²	—	73	1	no	—
102	#108	—	2	2	no	—
103	#104	—	146	4	no	—
104	C₂²⋊C₈	—	146	4	no	—
105	#97	—	146	2	no	—
106	#41	—	146	2	no	—
107	#41	—	146	2	no	—
108	#41	—	146	2	no	—
109	#41	—	146	2	no	—
110	C₂×Q₈	—	73	2	no	—
111	C₂×D₄	—	73	2	no	—
112	C₂×D₄	—	146	2	no	—
113	C₂²×C₄	—	146	1	no	—
114	C₂²×C₄	—	146	2	no	—
115	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
116	Q₁₆	—	146	2	no	—
117	SD₁₆	—	146	4	no	—
118	D₈	—	146	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	73	2	no	—
123	C₂×C₈	—	146	1	no	—
124	C₂×C₈	—	146	1	no	—
125	C₂×C₈	C₄×D₇₃	1	1	yes	characteristic
126	C₂×C₈	C₄×D₇₃	1	1	yes	characteristic
127	C₂×C₈	D₂₉₂	1	2	yes	—
128	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
129	C₄²	—	73	2	no	—
130	C₄²	—	146	1	no	—
131	C₂³	—	146	2	no	—
132	C₂³	#11924029	1	1	yes	characteristic
133	Q₈	—	146	2	no	—
134	Q₈	—	146	2	no	—
135	D₄	—	292	2	no	—
136	D₄	—	292	2	no	—
137	D₄	—	146	2	no	—
138	D₄	—	146	2	no	—
139	C₂×C₄	—	292	2	no	—
140	C₂×C₄	—	292	2	no	—
141	C₂×C₄	—	292	2	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
145	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
146	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
148	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
149	C₂×C₄	#11924029	1	1	yes	characteristic
150	C₈	—	4	2	no	—
151	C₈	—	146	2	no	—
152	C₈	#11924005	1	2	yes	—
153	C₈	#11924005	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	292	4	no	—
156	C₂²	—	292	2	no	—
157	C₂²	#14556553	1	1	yes	characteristic
158	C₂²	#14556566	1	1	yes	characteristic
159	C₂²	#14556580	1	1	yes	characteristic
160	C₄	—	292	2	no	—
161	C₄	—	292	2	no	—
162	C₄	C₂×#11924005	1	1	yes	characteristic
163	C₄	#14556553	1	1	yes	characteristic
164	C₄	#14556566	1	1	yes	characteristic
165	C₄	#14556587	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	584	2	no	—
169	C₂	#17988488	1	1	yes	characteristic
170	C₂	#17988708	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556613

C₂

yes

maximal, characteristic

C₂×#14556517

C₂

yes

maximal, complemented

#17988555

C₂

yes

maximal, complemented

C₂×#14556501

C₂

yes

maximal, characteristic, complemented

C₂×#108×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

—

#14556613

—

#14556613

—

C₂×Dic₂₉₂

C₄

yes

characteristic, complemented

C₂×#11924017

C₄

yes

complemented

C₂×D₅₈₄

C₄

yes

characteristic, complemented

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

—

—

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#108×C₇₃

—

—

—

—

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

#11924017

—

D₅₈₄

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

D₄

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

D₄

yes

—

C₂³×C₇₃

#41

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊C₄

yes

—

C₈×C₇₃

C₂²⋊C₄

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×D₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#390

yes

characteristic

C₂×C₇₃

#444

yes

—

D₇₃

—

#1975

—

maximal, Sylow

C₇₃

#1975

yes

characteristic, complemented, Sylow

C₂×#135

—

C₂×#108

D₇₃

yes

characteristic, complemented

C₂×#104

—

#408

—

C₂×#97

—

C₂×#41

—

#135

—

146

—

#135

—

146

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

C₂×D₇₃

yes

—

C₂²×C₈

—

100

C₂²×C₈

C₂×D₇₃

yes

characteristic

101

C₂×C₄²

—

102

#108

—

103

#104

—

146

—

104

C₂²⋊C₈

—

146

—

105

#97

—

146

—

106

#41

—

146

—

107

#41

—

146

—

108

#41

—

146

—

109

#41

—

146

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

146

—

113

C₂²×C₄

—

146

—

114

C₂²×C₄

—

146

—

115

C₂²×C₄

C₂²×D₇₃

yes

characteristic

116

Q₁₆

—

146

—

117

SD₁₆

—

146

—

118

D₈

—

146

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₂×C₈

C₄×D₇₃

yes

characteristic

126

C₂×C₈

C₄×D₇₃

yes

characteristic

127

C₂×C₈

D₂₉₂

yes

—

128

C₂×C₈

C₂²⋊D₇₃

yes

—

129

C₄²

—

130

C₄²

—

146

—

131

C₂³

—

146

—

132

C₂³

#11924029

yes

characteristic

133

Q₈

—

146

—

134

Q₈

—

146

—

135

D₄

—

292

—

136

D₄

—

292

—

137

D₄

—

146

—

138

D₄

—

146

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

145

C₂×C₄

D₄×D₇₃

yes

characteristic

146

C₂×C₄

D₄×D₇₃

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

148

C₂×C₄

C₂×D₂₉₂

yes

characteristic

149

C₂×C₄

#11924029

yes

characteristic

150

C₈

—

151

C₈

—

146

—

152

C₈

#11924005

yes

—

153

C₈

#11924005

yes

—

154

C₂²

—

155

C₂²

—

292

—

156

C₂²

—

292

—

157

C₂²

#14556553

yes

characteristic

158

C₂²

#14556566

yes

characteristic

159

C₂²

#14556580

yes

characteristic

160

C₄

—

292

—

161

C₄

—

292

—

162

C₄

C₂×#11924005

yes

characteristic

163

C₄

#14556553

yes

characteristic

164

C₄

#14556566

yes

characteristic

165

C₄

#14556587

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

584

—

169

C₂

#17988488

yes

characteristic

170

C₂

#17988708

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₄×D₄
hypercenter	C₂×#108	D₇₃
hypocenter	C₇₃	#1975
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#108×C₇₃

C₂

socle

C₂²×C₇₃

C₄×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#14556517	C₂	semidirect
3. #17988555	C₂	semidirect
4. C₂×#14556501	C₂	semidirect
5. C₂×#108×C₇₃	C₂	semidirect
9. C₂×Dic₂₉₂	C₄	semidirect
10. C₂×#11924017	C₄	semidirect
11. C₂×D₅₈₄	C₄	semidirect
86. C₇₃	#1975	semidirect
88. C₂×#108	D₇₃	semidirect

H ≅ G/N

type

2. C₂×#14556517

C₂

semidirect

3. #17988555

C₂

semidirect

4. C₂×#14556501

C₂

semidirect

5. C₂×#108×C₇₃

C₂

semidirect

9. C₂×Dic₂₉₂

C₄

semidirect

10. C₂×#11924017

C₄

semidirect

11. C₂×D₅₈₄

C₄

semidirect

86. C₇₃

#1975

semidirect

88. C₂×#108

D₇₃

semidirect

Subgroups — #22610822

Conjugacy classes of subgroups

Complements