Group #22611328 — subgroups

Subgroups — #22611328

Order 9344.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17988636	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988635	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14556507	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556507	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556507	C₂	1	1	yes	maximal, characteristic
6	#17988345	C₂	1	1	yes	maximal, characteristic, complemented
7	#430×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	1	yes	characteristic
11	C₂×#11924012	C₂²	1	1	yes	characteristic
12	#14556527	—	2	2	no	—
13	#14556526	—	2	2	no	—
14	#14556507	—	2	2	no	—
15	#14556507	—	2	2	no	—
16	#14556507	—	2	2	no	—
17	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	#108×C₇₃	—	2	2	no	—
21	C₂³×D₇₃	Q₈	1	2	yes	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	—	2	1	no	—
28	#11924012	—	2	1	no	—
29	#11924012	—	2	1	no	—
30	#11924012	—	2	1	no	—
31	#11924012	—	2	1	no	—
32	#11924012	—	2	1	no	—
33	#11924012	D₄	1	2	yes	—
34	C₂×#10596831	—	2	1	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	—	2	1	no	—
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	M₁₆×C₇₃	—	2	1	no	—
39	M₁₆×C₇₃	—	2	1	no	—
40	M₁₆×C₇₃	—	2	1	no	—
41	M₁₆×C₇₃	—	2	1	no	—
42	M₁₆×C₇₃	D₄	1	2	yes	—
43	M₁₆×C₇₃	D₄	1	2	yes	—
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂³×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×D₇₃	—	2	2	no	—
49	C₂²×D₇₃	—	2	2	no	—
50	C₂²×D₇₃	—	2	2	no	—
51	C₂²×D₇₃	—	8	2	no	—
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
58	D₂₉₂	—	4	1	no	—
59	D₂₉₂	—	4	1	no	—
60	D₂₉₂	—	4	1	no	—
61	D₂₉₂	—	4	1	no	—
62	C₈×C₇₃	—	2	2	no	—
63	C₈×C₇₃	—	2	2	no	—
64	C₈×C₇₃	—	4	1	no	—
65	#10596831	—	2	2	no	—
66	#10596831	—	4	1	no	—
67	#10596831	—	4	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×D₇₃	—	4	2	no	—
73	C₂×D₇₃	—	8	2	no	—
74	C₂×D₇₃	—	8	2	no	—
75	C₂×D₇₃	—	8	2	no	—
76	C₄×C₇₃	#121	1	1	yes	characteristic
77	C₄×C₇₃	#121	1	1	yes	characteristic
78	C₄×C₇₃	#121	1	1	yes	characteristic
79	C₄×C₇₃	#126	1	1	yes	characteristic
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	—	2	1	no	—
83	C₂×C₇₃	#396	1	1	yes	characteristic
84	D₇₃	—	8	2	no	—
85	#2069	—	73	1	no	maximal, Sylow
86	C₇₃	#2069	1	1	yes	characteristic, complemented, Sylow
87	#430	—	73	1	no	—
88	#430	—	73	1	no	—
89	#430	D₇₃	1	1	yes	characteristic, complemented
90	C₂×#100	—	73	1	no	—
91	C₂×#100	—	73	1	no	—
92	C₂×#100	—	73	1	no	—
93	#338	—	73	1	no	—
94	C₂²×D₄	—	73	1	no	—
95	C₂×M₁₆	—	73	1	no	—
96	C₂×M₁₆	—	73	1	no	—
97	C₂×M₁₆	—	73	1	no	—
98	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
99	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
101	#108	—	2	2	no	—
102	#108	—	146	2	no	—
103	#108	—	146	2	no	—
104	#100	—	146	2	no	—
105	#100	—	146	2	no	—
106	#100	—	146	2	no	—
107	C₂⁴	—	73	2	no	—
108	C₂×D₄	—	146	1	no	—
109	C₂×D₄	—	146	1	no	—
110	C₂×D₄	—	146	1	no	—
111	C₂×D₄	—	146	1	no	—
112	C₂×D₄	—	146	1	no	—
113	C₂×D₄	—	146	1	no	—
114	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	2	1	no	—
119	M₁₆	—	73	2	no	—
120	M₁₆	—	146	1	no	—
121	M₁₆	—	146	1	no	—
122	M₁₆	—	146	1	no	—
123	M₁₆	—	146	1	no	—
124	M₁₆	—	146	1	no	—
125	M₁₆	D₂₉₂	1	2	yes	—
126	M₁₆	C₂²⋊D₇₃	1	2	yes	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₂×C₈	—	146	1	no	—
132	C₂×C₈	—	146	1	no	—
133	C₂³	—	584	2	no	—
134	C₂³	—	146	2	no	—
135	C₂³	—	146	2	no	—
136	C₂³	—	146	2	no	—
137	C₂³	Q₈×D₇₃	1	1	yes	characteristic
138	D₄	—	292	1	no	—
139	D₄	—	292	1	no	—
140	D₄	—	292	1	no	—
141	D₄	—	292	1	no	—
142	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
144	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
145	C₂×C₄	#11924027	1	1	yes	characteristic
146	C₂×C₄	#11924028	1	1	yes	characteristic
147	C₂×C₄	#11924029	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	292	1	no	—
152	C₈	—	292	1	no	—
153	C₈	—	146	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	292	2	no	—
156	C₂²	—	584	2	no	—
157	C₂²	—	584	2	no	—
158	C₂²	—	584	2	no	—
159	C₂²	#14556567	1	1	yes	characteristic
160	C₂²	#14556569	1	1	yes	characteristic
161	C₂²	#14556571	1	1	yes	characteristic
162	C₄	#14556563	1	1	yes	characteristic
163	C₄	#14556565	1	1	yes	characteristic
164	C₄	#14556566	1	1	yes	characteristic
165	C₄	#14556586	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	584	2	no	—
170	C₂	#17988509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988636

C₂

yes

maximal, characteristic, complemented

#17988635

C₂

yes

maximal, characteristic, complemented

C₂×#14556507

C₂

yes

maximal, characteristic

C₂×#14556507

C₂

yes

maximal, characteristic

C₂×#14556507

C₂

yes

maximal, characteristic

#17988345

C₂

yes

maximal, characteristic, complemented

#430×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#108×C₇₃

—

C₂³×D₇₃

Q₈

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

—

—

—

—

—

D₄

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

D₄

yes

—

M₁₆×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₂×Q₈

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#396

yes

characteristic

D₇₃

—

#2069

—

maximal, Sylow

C₇₃

#2069

yes

characteristic, complemented, Sylow

#430

—

#430

—

#430

D₇₃

yes

characteristic, complemented

—

—

—

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂×M₁₆

C₂×D₇₃

yes

characteristic

100

C₂×M₁₆

C₂×D₇₃

yes

characteristic

101

#108

—

102

#108

—

146

—

103

#108

—

146

—

104

#100

—

146

—

105

#100

—

146

—

106

#100

—

146

—

107

C₂⁴

—

108

C₂×D₄

—

146

—

109

C₂×D₄

—

146

—

110

C₂×D₄

—

146

—

111

C₂×D₄

—

146

—

112

C₂×D₄

—

146

—

113

C₂×D₄

—

146

—

114

C₂²×C₄

C₂²×D₇₃

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

146

—

121

M₁₆

—

146

—

122

M₁₆

—

146

—

123

M₁₆

—

146

—

124

M₁₆

—

146

—

125

M₁₆

D₂₉₂

yes

—

126

M₁₆

C₂²⋊D₇₃

yes

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

146

—

131

C₂×C₈

—

146

—

132

C₂×C₈

—

146

—

133

C₂³

—

584

—

134

C₂³

—

146

—

135

C₂³

—

146

—

136

C₂³

—

146

—

137

C₂³

Q₈×D₇₃

yes

characteristic

138

D₄

—

292

—

139

D₄

—

292

—

140

D₄

—

292

—

141

D₄

—

292

—

142

C₂×C₄

D₄×D₇₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

144

C₂×C₄

C₂×D₂₉₂

yes

characteristic

145

C₂×C₄

#11924027

yes

characteristic

146

C₂×C₄

#11924028

yes

characteristic

147

C₂×C₄

#11924029

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

292

—

152

C₈

—

292

—

153

C₈

—

146

—

154

C₂²

—

155

C₂²

—

292

—

156

C₂²

—

584

—

157

C₂²

—

584

—

158

C₂²

—

584

—

159

C₂²

#14556567

yes

characteristic

160

C₂²

#14556569

yes

characteristic

161

C₂²

#14556571

yes

characteristic

162

C₄

#14556563

yes

characteristic

163

C₄

#14556565

yes

characteristic

164

C₄

#14556566

yes

characteristic

165

C₄

#14556586

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

584

—

170

C₂

#17988509

yes

characteristic

	subgroup	G/H
center	C₂	#17988509
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#430×C₇₃	C₂
socle	C₂×C₇₃	#396
hypercenter	#430	D₇₃
hypocenter	C₇₃	#2069
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₂

#17988509

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#430×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #17988636	C₂	semidirect
2. #17988635	C₂	semidirect
6. #17988345	C₂	semidirect
7. #430×C₇₃	C₂	semidirect
86. C₇₃	#2069	semidirect
89. #430	D₇₃	semidirect

H ≅ G/N

type

1. #17988636

C₂

semidirect

2. #17988635

C₂

semidirect

6. #17988345

C₂

semidirect

7. #430×C₇₃

C₂

semidirect

86. C₇₃

#2069

semidirect

89. #430

D₇₃

semidirect

Subgroups — #22611328

Conjugacy classes of subgroups

Complements