Group #22609103 — subgroups

Subgroups — #22609103

Order 9344.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988939	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988882	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988877	C₂	1	1	yes	maximal, characteristic
4	#17988333	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
7	#556×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556591	C₂²	1	1	yes	characteristic
9	#14556524	—	2	1	no	—
10	#14556522	—	2	1	no	—
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	#14556515	—	2	1	no	—
14	#14556514	—	2	2	no	—
15	#14556512	—	2	1	no	—
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
18	#126×C₇₃	—	2	2	no	—
19	#124×C₇₃	—	2	1	no	—
20	C₂²⋊Q₈×C₇₃	—	2	1	no	—
21	C₄×Q₈×C₇₃	—	2	1	no	—
22	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
24	#117×C₇₃	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	—	2	1	no	—
35	C₄⋊C₄×C₇₃	—	2	2	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
41	C₂²⋊C₄×C₇₃	—	2	2	no	—
42	C₂²⋊C₄×C₇₃	—	2	2	no	—
43	C₂²⋊C₄×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	—	2	2	no	—
45	C₄²×C₇₃	—	2	1	no	—
46	C₄²×C₇₃	C₂³	1	1	yes	characteristic
47	C₂³×C₇₃	—	2	1	no	—
48	Q₈×C₇₃	—	2	1	no	—
49	Q₈×C₇₃	—	2	1	no	—
50	Q₈×C₇₃	—	4	1	no	—
51	Q₈×C₇₃	—	4	1	no	—
52	D₄×C₇₃	—	2	1	no	—
53	D₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₇₃	—	4	1	no	—
64	Dic₁₄₆	—	4	2	no	—
65	#10596831	—	4	1	no	—
66	#10596831	—	4	1	no	—
67	#10596831	—	4	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	—	4	1	no	—
70	C₂²×C₇₃	—	4	1	no	—
71	C₂²×C₇₃	#120	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	—	4	1	no	—
75	C₄×C₇₃	—	4	2	no	—
76	C₄×C₇₃	—	4	1	no	—
77	C₄×C₇₃	#135	1	1	yes	characteristic
78	C₄×C₇₃	#135	1	1	yes	characteristic
79	C₄×C₇₃	#137	1	1	yes	characteristic
80	C₄×C₇₃	#137	1	1	yes	characteristic
81	Dic₇₃	—	8	1	no	—
82	C₂×C₇₃	—	8	1	no	—
83	C₂×C₇₃	#452	1	1	yes	characteristic
84	C₂×C₇₃	#452	1	1	yes	characteristic
85	C₂×C₇₃	#454	1	1	yes	characteristic
86	#1638	—	73	1	no	maximal, Sylow
87	C₇₃	#1638	1	1	yes	characteristic, complemented, Sylow
88	#556	D₇₃	1	1	yes	characteristic
89	#487	—	73	1	no	—
90	#477	—	73	1	no	—
91	#476	—	73	1	no	—
92	#332	—	73	1	no	—
93	Q₈⋊C₈	—	73	1	no	—
94	D₄⋊C₈	—	73	1	no	—
95	C₄⋊Q₈	—	73	1	no	—
96	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
97	#126	—	2	2	no	—
98	#124	—	2	1	no	—
99	C₂²⋊Q₈	—	2	1	no	—
100	C₄×Q₈	—	2	1	no	—
101	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
102	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
103	#117	—	2	1	no	—
104	#107	—	146	1	no	—
105	#106	—	146	1	no	—
106	C₄⋊C₈	—	73	1	no	—
107	C₄⋊C₈	—	73	1	no	—
108	#103	—	146	1	no	—
109	#103	—	146	2	no	—
110	#102	—	146	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	146	1	no	—
114	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂×D₄	C₂²⋊D₇₃	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂×C₈	—	146	1	no	—
118	C₂×C₈	—	146	1	no	—
119	C₂×C₈	—	146	1	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
126	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
128	C₂²⋊C₄	—	2	2	no	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₂²⋊C₄	—	2	1	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	1	no	—
133	C₄²	C₂²×D₇₃	1	1	yes	characteristic
134	C₂³	—	2	1	no	—
135	Q₈	—	2	1	no	—
136	Q₈	—	2	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	292	2	no	—
140	D₄	—	2	1	no	—
141	D₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	292	1	no	—
149	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
152	C₈	—	292	1	no	—
153	C₈	—	292	1	no	—
154	C₈	—	292	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	#14556560	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	1	no	—
164	C₄	—	584	1	no	—
165	C₄	#14556611	1	1	yes	characteristic
166	C₄	#14556611	1	1	yes	characteristic
167	C₄	#14556621	1	1	yes	characteristic
168	C₄	#14556621	1	1	yes	characteristic
169	C₂	—	8	1	no	—
170	C₂	#17988749	1	1	yes	characteristic
171	C₂	#17988749	1	1	yes	characteristic
172	C₂	#17988760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988939

C₂

yes

maximal, characteristic, complemented

#17988882

C₂

yes

maximal, characteristic, complemented

#17988877

C₂

yes

maximal, characteristic

#17988333

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596831

C₂

yes

maximal, characteristic, complemented

D₄⋊#10596831

C₂

yes

maximal, characteristic

#556×C₇₃

C₂

yes

maximal, characteristic

#14556591

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#126×C₇₃

—

#124×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

#117×C₇₃

—

C₂×Dic₁₄₆

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#120

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#556

D₇₃

yes

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

#126

—

#124

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

C₂×D₇₃

yes

characteristic

102

C₄×D₄

C₂×D₇₃

yes

characteristic

103

#117

—

104

#107

—

146

—

105

#106

—

146

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

#103

—

146

—

109

#103

—

146

—

110

#102

—

146

—

111

C₄×C₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

146

—

114

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

115

C₂×D₄

C₂²⋊D₇₃

yes

characteristic

116

C₂²×C₄

—

117

C₂×C₈

—

146

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

C₂²⋊D₇₃

yes

—

126

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

127

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²×D₇₃

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

292

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

292

—

149

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

151

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

152

C₈

—

292

—

153

C₈

—

292

—

154

C₈

—

292

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#14556560

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

584

—

165

C₄

#14556611

yes

characteristic

166

C₄

#14556611

yes

characteristic

167

C₄

#14556621

yes

characteristic

168

C₄

#14556621

yes

characteristic

169

C₂

—

170

C₂

#17988749

yes

characteristic

171

C₂

#17988749

yes

characteristic

172

C₂

#17988760

yes

characteristic

	subgroup	G/H
center	C₂²	#14556560
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#556×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#556	D₇₃
hypocenter	C₇₃	#1638
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #556

subgroup

G/H

center

C₂²

#14556560

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#556×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #556

N	H ≅ G/N	type
1. #17988939	C₂	semidirect
2. #17988882	C₂	semidirect
4. #17988333	C₂	semidirect
5. Q₈⋊#10596831	C₂	semidirect
87. C₇₃	#1638	semidirect

H ≅ G/N

type

1. #17988939

C₂

semidirect

2. #17988882

C₂

semidirect

4. #17988333

C₂

semidirect

5. Q₈⋊#10596831

C₂

semidirect

87. C₇₃

#1638

semidirect

Subgroups — #22609103

Conjugacy classes of subgroups

Complements