Group #22608347 — subgroups

Subgroups — #22608347

Order 9344.

262 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 77 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	#17989137	C₂	1	1	yes	maximal, characteristic
2	C₄×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988314	C₂	1	4	yes	maximal, complemented
4	#431×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11924028	—	2	1	no	—
6	C₂×#11924012	—	2	1	no	—
7	#14556591	C₄	1	1	yes	characteristic
8	#14556588	C₄	1	1	yes	characteristic
9	#14556579	—	2	1	no	—
10	#14556565	—	2	1	no	—
11	#14556565	C₄	1	2	yes	—
12	C₄×D₂₉₂	—	2	1	no	—
13	C₄×D₂₉₂	C₂²	1	2	yes	—
14	#14556502	—	2	1	no	—
15	C₄×#10596831	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
18	C₂²⋊C₈×C₇₃	—	2	1	no	—
19	C₄×C₈×C₇₃	—	2	1	no	—
20	#11924028	—	4	4	no	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
25	C₂×C₂²⋊D₇₃	—	2	2	no	—
26	C₂×C₄×D₇₃	—	2	2	no	—
27	#11924012	—	2	4	no	—
28	C₂×#10596831	—	2	2	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	#11924005	—	2	2	no	—
32	#11924005	—	2	2	no	—
33	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×D₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₄	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	C₂²⋊D₇₃	—	4	4	no	—
53	C₂×Dic₇₃	—	2	2	no	—
54	D₂₉₂	—	2	4	no	—
55	D₂₉₂	—	4	2	no	—
56	D₂₉₂	—	4	2	no	—
57	C₄×D₇₃	—	4	4	no	—
58	Dic₁₄₆	—	4	2	no	—
59	C₈×C₇₃	—	4	2	no	—
60	#10596831	—	2	4	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₇₃	—	4	2	no	—
66	C₂×D₇₃	—	4	2	no	—
67	C₂×D₇₃	—	4	2	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	#104	1	2	yes	—
72	C₄×C₇₃	#131	1	2	yes	—
73	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₇₃	—	4	2	no	—
75	C₂×C₇₃	—	4	1	no	—
76	C₂×C₇₃	#408	1	1	yes	characteristic
77	C₂×C₇₃	#418	1	1	yes	characteristic
78	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
79	D₇₃	—	8	2	no	—
80	#1484	—	73	1	no	maximal, Sylow
81	C₇₃	#1484	1	1	yes	characteristic, complemented, Sylow
82	#532	—	73	1	no	—
83	#431	D₇₃	1	1	yes	characteristic, complemented
84	C₄×M₁₆	—	73	1	no	—
85	D₄⋊C₈	—	73	4	no	—
86	C₂×#41	—	146	1	no	—
87	C₂×M₁₆	—	146	1	no	—
88	C₄⋊Q₈	—	73	1	no	—
89	#127	—	73	1	no	—
90	#124	—	146	1	no	—
91	#121	—	73	2	no	—
92	#121	—	146	1	no	—
93	C₄×D₄	—	73	2	no	—
94	C₄×D₄	—	146	1	no	—
95	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
96	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
97	C₂²⋊C₈	—	2	1	no	—
98	#97	—	146	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	73	2	no	—
101	#41	—	292	4	no	—
102	C₂×Q₈	—	146	1	no	—
103	C₂×D₄	—	73	2	no	—
104	C₂×D₄	—	146	2	no	—
105	C₂×D₄	—	146	1	no	—
106	C₂×D₄	—	146	1	no	—
107	C₂²×C₄	—	146	2	no	—
108	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
109	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
111	M₁₆	—	146	4	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	146	2	no	—
114	C₄⋊C₄	—	73	2	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₂²⋊C₄	—	146	2	no	—
117	C₂²⋊C₄	—	146	2	no	—
118	C₄²	C₄×D₇₃	1	1	yes	characteristic
119	C₄²	D₂₉₂	1	1	yes	characteristic
120	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
121	C₄²	C₂²×D₇₃	1	1	yes	characteristic
122	C₂³	—	146	2	no	—
123	C₂³	#11924005	1	1	yes	characteristic
124	Q₈	—	292	2	no	—
125	D₄	—	292	2	no	—
126	D₄	—	292	4	no	—
127	D₄	—	292	2	no	—
128	D₄	—	146	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	292	4	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	#11924005	1	1	yes	characteristic
135	C₂×C₄	#11924005	1	1	yes	characteristic
136	C₂×C₄	#11924005	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
140	C₈	—	4	2	no	—
141	C₈	—	146	4	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	292	2	no	—
146	C₂²	—	292	2	no	—
147	C₂²	—	292	2	no	—
148	C₂²	C₂×#11924005	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	292	2	no	—
153	C₄	#14556517	1	2	yes	—
154	C₄	#14556597	1	2	yes	—
155	C₄	#14556616	1	2	yes	—
156	C₂	—	4	1	no	—
157	C₂	—	584	2	no	—
158	C₂	#17988553	1	1	yes	characteristic
159	C₂	#17988591	1	1	yes	characteristic
160	C₂	C₂×#14556517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989137

C₂

yes

maximal, characteristic

C₄×#11924012

C₂

yes

maximal, characteristic, complemented

#17988314

C₂

yes

maximal, complemented

#431×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

—

C₂×#11924012

—

#14556591

C₄

yes

characteristic

#14556588

C₄

yes

—

—

C₄

yes

—

C₄×D₂₉₂

—

C₄×D₂₉₂

C₂²

yes

—

#14556502

—

C₄×#10596831

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924005

—

#11924005

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#104

yes

—

C₄×C₇₃

#131

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1484

—

maximal, Sylow

C₇₃

#1484

yes

characteristic, complemented, Sylow

#532

—

#431

D₇₃

yes

characteristic, complemented

C₄×M₁₆

—

D₄⋊C₈

—

C₂×#41

—

146

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

#127

—

#124

—

146

—

#121

—

#121

—

146

—

C₄×D₄

—

C₄×D₄

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

C₂×D₇₃

yes

—

C₂²⋊C₈

—

#97

—

146

—

C₄×C₈

—

100

C₄×C₈

—

101

#41

—

292

—

102

C₂×Q₈

—

146

—

103

C₂×D₄

—

104

C₂×D₄

—

146

—

105

C₂×D₄

—

146

—

106

C₂×D₄

—

146

—

107

C₂²×C₄

—

146

—

108

C₂²×C₄

C₄×D₇₃

yes

characteristic

109

C₂²×C₄

D₂₉₂

yes

characteristic

110

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

111

M₁₆

—

146

—

112

C₂×C₈

—

113

C₂×C₈

—

146

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

146

—

116

C₂²⋊C₄

—

146

—

117

C₂²⋊C₄

—

146

—

118

C₄²

C₄×D₇₃

yes

characteristic

119

C₄²

D₂₉₂

yes

characteristic

120

C₄²

C₂²⋊D₇₃

yes

characteristic

121

C₄²

C₂²×D₇₃

yes

characteristic

122

C₂³

—

146

—

123

C₂³

#11924005

yes

characteristic

124

Q₈

—

292

—

125

D₄

—

292

—

126

D₄

—

292

—

127

D₄

—

292

—

128

D₄

—

146

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

146

—

134

C₂×C₄

#11924005

yes

characteristic

135

C₂×C₄

#11924005

yes

characteristic

136

C₂×C₄

#11924005

yes

characteristic

137

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

139

C₂×C₄

C₂×D₂₉₂

yes

characteristic

140

C₈

—

141

C₈

—

146

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

292

—

146

C₂²

—

292

—

147

C₂²

—

292

—

148

C₂²

C₂×#11924005

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

292

—

153

C₄

#14556517

yes

—

154

C₄

#14556597

yes

—

155

C₄

#14556616

yes

—

156

C₂

—

157

C₂

—

584

—

158

C₂

#17988553

yes

characteristic

159

C₂

#17988591

yes

characteristic

160

C₂

C₂×#14556517

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#431×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#431	D₇₃
hypocenter	C₇₃	#1484
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#431×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. C₄×#11924012	C₂	semidirect
3. #17988314	C₂	semidirect
4. #431×C₇₃	C₂	semidirect
81. C₇₃	#1484	semidirect
83. #431	D₇₃	semidirect

H ≅ G/N

type

2. C₄×#11924012

C₂

semidirect

3. #17988314

C₂

semidirect

4. #431×C₇₃

C₂

semidirect

81. C₇₃

#1484

semidirect

83. #431

D₇₃

semidirect

Subgroups — #22608347

Conjugacy classes of subgroups

Complements