Group #22608638 — subgroups

Subgroups — #22608638

Order 9344.

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

normal subgroups only

This group has 198 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	#17989142	C₂	1	1	yes	maximal, characteristic
2	#17988643	C₂	1	1	yes	maximal, characteristic
3	#17988639	C₂	1	1	yes	maximal, characteristic
4	#17988335	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988335	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988333	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556591	C₂²	1	1	yes	characteristic
9	#14556582	—	2	1	no	—
10	#14556571	C₄	1	2	yes	—
11	#14556567	—	2	1	no	—
12	Q₈×Dic₇₃	—	2	1	no	—
13	#14556556	C₄	1	2	yes	—
14	#14556549	—	2	1	no	—
15	C₄²×D₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	#14556503	—	2	1	no	—
19	#14556503	—	2	1	no	—
20	#14556501	—	2	1	no	—
21	C₈×Dic₇₃	—	2	1	no	—
22	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	—	2	2	no	—
34	#11924007	—	2	2	no	—
35	#11924005	—	2	2	no	—
36	C₄×Dic₇₃	—	2	2	no	—
37	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₇₃	D₄	1	1	yes	characteristic
39	C₄×Dic₇₃	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
45	C₄²×C₇₃	C₂³	1	1	yes	characteristic
46	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₇₃	—	4	1	no	—
48	Q₈×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	2	1	no	—
61	C₄×D₇₃	—	4	1	no	—
62	Dic₁₄₆	—	4	2	no	—
63	C₈×C₇₃	—	4	1	no	—
64	C₈×C₇₃	—	4	1	no	—
65	#10596831	—	4	1	no	—
66	#10596831	—	4	1	no	—
67	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×D₇₃	—	2	1	no	—
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	4	2	no	—
73	C₄×C₇₃	#135	1	1	yes	characteristic
74	C₄×C₇₃	#135	1	1	yes	characteristic
75	C₄×C₇₃	#137	1	1	yes	characteristic
76	C₄×C₇₃	#137	1	1	yes	characteristic
77	Dic₇₃	—	2	1	no	—
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	4	2	no	—
81	C₂×C₇₃	#413	1	1	yes	characteristic
82	C₂×C₇₃	#416	1	1	yes	characteristic
83	C₂×C₇₃	#419	1	1	yes	characteristic
84	D₇₃	—	4	1	no	—
85	#1554	—	73	1	no	maximal, Sylow
86	C₇₃	#1554	1	1	yes	characteristic, complemented, Sylow
87	#533	—	73	1	no	—
88	#432	—	73	1	no	—
89	#431	—	73	1	no	—
90	#333	—	73	1	no	—
91	#333	—	73	1	no	—
92	#332	—	73	1	no	—
93	#332	D₇₃	1	1	yes	characteristic, complemented
94	C₄⋊Q₈	—	73	1	no	—
95	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
96	#125	—	146	1	no	—
97	C₂²⋊Q₈	—	73	2	no	—
98	C₂²⋊Q₈	—	146	1	no	—
99	C₄×Q₈	—	73	2	no	—
100	C₄×Q₈	—	146	1	no	—
101	#117	—	146	1	no	—
102	C₂×C₄²	—	73	1	no	—
103	C₄⋊C₈	—	73	1	no	—
104	C₄⋊C₈	—	73	1	no	—
105	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
106	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
107	C₂²⋊C₈	—	146	1	no	—
108	C₂²⋊C₈	—	146	1	no	—
109	#97	—	146	1	no	—
110	C₄×C₈	—	146	1	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	73	2	no	—
113	C₂²×C₄	—	73	1	no	—
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	146	1	no	—
119	C₂×C₈	—	146	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	146	2	no	—
122	C₄⋊C₄	—	146	2	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₄⋊C₄	C₄×D₇₃	1	2	yes	—
126	C₂²⋊C₄	—	146	2	no	—
127	C₄²	—	73	1	no	—
128	C₄²	—	73	1	no	—
129	C₄²	—	73	1	no	—
130	C₄²	—	146	2	no	—
131	C₄²	C₂²×D₇₃	1	1	yes	characteristic
132	C₂³	—	73	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	292	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	292	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	—	146	1	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
147	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	292	1	no	—
152	C₈	—	292	1	no	—
153	C₂²	—	146	1	no	—
154	C₂²	—	146	1	no	—
155	C₂²	—	146	1	no	—
156	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	292	2	no	—
160	C₄	—	146	1	no	—
161	C₄	—	146	1	no	—
162	C₄	—	146	1	no	—
163	C₄	#14556610	1	1	yes	characteristic
164	C₄	#14556612	1	1	yes	characteristic
165	C₄	#14556620	1	1	yes	characteristic
166	C₄	#14556623	1	1	yes	characteristic
167	C₂	—	292	1	no	—
168	C₂	#17988570	1	1	yes	characteristic
169	C₂	#17988580	1	1	yes	characteristic
170	C₂	#17988594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989142

C₂

yes

maximal, characteristic

#17988643

C₂

yes

maximal, characteristic

#17988639

C₂

yes

maximal, characteristic

#17988335

C₂

yes

maximal, characteristic, complemented

#17988335

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, characteristic, complemented

#332×C₇₃

C₂

yes

maximal, characteristic, complemented

#14556591

C₂²

yes

characteristic

#14556582

—

#14556571

C₄

yes

—

#14556567

—

Q₈×Dic₇₃

—

#14556556

C₄

yes

—

#14556549

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₈×Dic₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1554

—

maximal, Sylow

C₇₃

#1554

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

#125

—

146

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

100

C₄×Q₈

—

146

—

101

#117

—

146

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

C₂×D₇₃

yes

characteristic

106

C₄⋊C₈

C₂×D₇₃

yes

characteristic

107

C₂²⋊C₈

—

146

—

108

C₂²⋊C₈

—

146

—

109

#97

—

146

—

110

C₄×C₈

—

146

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

146

—

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

C₄×D₇₃

yes

—

126

C₂²⋊C₄

—

146

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

146

—

131

C₄²

C₂²×D₇₃

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

292

—

136

C₂×C₄

—

137

C₂×C₄

—

292

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

147

C₂×C₄

D₄×D₇₃

yes

characteristic

148

C₂×C₄

D₄×D₇₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

292

—

152

C₈

—

292

—

153

C₂²

—

146

—

154

C₂²

—

146

—

155

C₂²

—

146

—

156

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

292

—

160

C₄

—

146

—

161

C₄

—

146

—

162

C₄

—

146

—

163

C₄

#14556610

yes

characteristic

164

C₄

#14556612

yes

characteristic

165

C₄

#14556620

yes

characteristic

166

C₄

#14556623

yes

characteristic

167

C₂

—

292

—

168

C₂

#17988570

yes

characteristic

169

C₂

#17988580

yes

characteristic

170

C₂

#17988594

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#332×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#332	D₇₃
hypocenter	C₇₃	#1554
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#332×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #17988335	C₂	semidirect
5. #17988335	C₂	semidirect
6. #17988333	C₂	semidirect
7. #332×C₇₃	C₂	semidirect
86. C₇₃	#1554	semidirect
93. #332	D₇₃	semidirect

H ≅ G/N

type

4. #17988335

C₂

semidirect

5. #17988335

C₂

semidirect

6. #17988333

C₂

semidirect

7. #332×C₇₃

C₂

semidirect

86. C₇₃

#1554

semidirect

93. #332

D₇₃

semidirect

Subgroups — #22608638

Conjugacy classes of subgroups

Complements