Group #22609477 — subgroups

Subgroups — #22609477

Order 9344.

160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17988937	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988864	C₂	1	1	yes	maximal, characteristic
3	#17988793	C₂	1	1	yes	maximal, characteristic
4	#17988664	C₂	1	1	yes	maximal, characteristic
5	#17988339	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988332	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924017	—	2	1	no	—
9	C₂×#11924011	—	2	1	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
12	#14556524	—	2	1	no	—
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	#14556516	—	2	1	no	—
15	#14556515	—	2	1	no	—
16	#14556515	—	2	1	no	—
17	#14556511	—	2	1	no	—
18	#14556503	—	2	1	no	—
19	C₄×#10596831	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×D₂₉₂	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₃	—	2	1	no	—
26	#11924017	—	4	2	no	—
27	#11924011	—	4	2	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₂×#10596831	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
34	#11924005	—	2	1	no	—
35	C₂×Q₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂²×D₇₃	—	2	1	no	—
42	Q₈×C₇₃	—	4	1	no	—
43	Q₈×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	—	4	1	no	—
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
48	C₂×Dic₇₃	—	2	1	no	—
49	C₂×Dic₇₃	—	4	1	no	—
50	D₂₉₂	—	2	1	no	—
51	D₂₉₂	—	2	1	no	—
52	C₄×D₇₃	—	4	1	no	—
53	C₄×D₇₃	—	4	1	no	—
54	Dic₁₄₆	—	4	1	no	—
55	Dic₁₄₆	—	4	1	no	—
56	C₈×C₇₃	—	4	1	no	—
57	C₈×C₇₃	—	4	1	no	—
58	#10596831	—	2	1	no	—
59	#10596831	—	2	1	no	—
60	#10596831	—	4	1	no	—
61	C₂²×C₇₃	#121	1	1	yes	characteristic
62	C₂×D₇₃	—	4	1	no	—
63	C₂×D₇₃	—	4	1	no	—
64	C₂×D₇₃	—	4	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	8	1	no	—
67	C₄×C₇₃	#135	1	1	yes	characteristic
68	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₇₃	#137	1	1	yes	characteristic
71	Dic₇₃	—	4	1	no	—
72	Dic₇₃	—	8	1	no	—
73	C₂×C₇₃	#463	1	1	yes	characteristic
74	C₂×C₇₃	#468	1	1	yes	characteristic
75	C₂×C₇₃	#473	1	1	yes	characteristic
76	D₇₃	—	8	1	no	—
77	#1699	—	73	1	no	maximal, Sylow
78	C₇₃	#1699	1	1	yes	characteristic, complemented, Sylow
79	#487	—	73	1	no	—
80	#474	—	73	1	no	—
81	#461	—	73	1	no	—
82	#436	—	73	1	no	—
83	#335	—	73	1	no	—
84	#332	—	73	1	no	—
85	#332	D₇₃	1	1	yes	characteristic, complemented
86	C₂×SD₁₆	—	146	1	no	—
87	C₂×M₁₆	—	146	1	no	—
88	C₄⋊Q₈	—	73	1	no	—
89	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
90	C₄×D₄	—	73	1	no	—
91	#107	—	146	1	no	—
92	C₄⋊C₈	—	73	1	no	—
93	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
95	#103	—	146	1	no	—
96	#103	—	146	1	no	—
97	#103	—	146	1	no	—
98	#102	—	146	1	no	—
99	C₂²⋊C₈	—	146	1	no	—
100	C₄×C₈	—	73	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	146	1	no	—
103	C₂×D₄	—	73	1	no	—
104	C₂²×C₄	—	146	1	no	—
105	SD₁₆	—	292	2	no	—
106	M₁₆	—	292	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	146	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	73	1	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₄⋊C₄	—	146	1	no	—
117	C₂²⋊C₄	—	146	1	no	—
118	C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	C₂³	—	146	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	292	1	no	—
123	Q₈	—	292	1	no	—
124	D₄	—	146	1	no	—
125	D₄	—	146	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	292	1	no	—
128	C₂×C₄	—	292	1	no	—
129	C₂×C₄	—	292	1	no	—
130	C₂×C₄	—	146	1	no	—
131	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
132	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
133	C₂×C₄	#11924028	1	1	yes	characteristic
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	292	1	no	—
137	C₈	—	146	1	no	—
138	C₈	—	146	1	no	—
139	C₂²	—	292	1	no	—
140	C₂²	—	292	1	no	—
141	C₂²	—	292	1	no	—
142	C₂²	#14556564	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	8	1	no	—
145	C₄	—	292	1	no	—
146	C₄	—	584	1	no	—
147	C₄	#14556612	1	1	yes	characteristic
148	C₄	#14556615	1	1	yes	characteristic
149	C₄	#14556618	1	1	yes	characteristic
150	C₄	#14556623	1	1	yes	characteristic
151	C₂	—	584	1	no	—
152	C₂	#17988801	1	1	yes	characteristic
153	C₂	#17988836	1	1	yes	characteristic
154	C₂	#17988861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988937

C₂

yes

maximal, characteristic, complemented

#17988864

C₂

yes

maximal, characteristic

#17988793

C₂

yes

maximal, characteristic

#17988664

C₂

yes

maximal, characteristic

#17988339

C₂

yes

maximal, characteristic, complemented

#17988332

C₂

yes

maximal, characteristic, complemented

#332×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924017

—

C₂×#11924011

—

#14556591

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

characteristic

#14556524

—

C₄⋊#10596831

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924005

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

#121

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

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1699

—

maximal, Sylow

C₇₃

#1699

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂×SD₁₆

—

146

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄×D₄

—

#107

—

146

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

146

—

#103

—

146

—

#103

—

146

—

#102

—

146

—

C₂²⋊C₈

—

146

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

146

—

103

C₂×D₄

—

104

C₂²×C₄

—

146

—

105

SD₁₆

—

292

—

106

M₁₆

—

292

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

146

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

146

—

116

C₄⋊C₄

—

146

—

117

C₂²⋊C₄

—

146

—

118

C₄²

C₂²×D₇₃

yes

characteristic

119

C₂³

—

146

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

292

—

123

Q₈

—

292

—

124

D₄

—

146

—

125

D₄

—

146

—

126

C₂×C₄

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

292

—

130

C₂×C₄

—

146

—

131

C₂×C₄

D₄×D₇₃

yes

characteristic

132

C₂×C₄

D₄×D₇₃

yes

characteristic

133

C₂×C₄

#11924028

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

292

—

137

C₈

—

146

—

138

C₈

—

146

—

139

C₂²

—

292

—

140

C₂²

—

292

—

141

C₂²

—

292

—

142

C₂²

#14556564

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

292

—

146

C₄

—

584

—

147

C₄

#14556612

yes

characteristic

148

C₄

#14556615

yes

characteristic

149

C₄

#14556618

yes

characteristic

150

C₄

#14556623

yes

characteristic

151

C₂

—

584

—

152

C₂

#17988801

yes

characteristic

153

C₂

#17988836

yes

characteristic

154

C₂

#17988861

yes

characteristic

	subgroup	G/H
center	C₂²	#14556564
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#332×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#332	D₇₃
hypocenter	C₇₃	#1699
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14556564

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#332×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #17988937	C₂	semidirect
5. #17988339	C₂	semidirect
6. #17988332	C₂	semidirect
7. #332×C₇₃	C₂	semidirect
78. C₇₃	#1699	semidirect
85. #332	D₇₃	semidirect

H ≅ G/N

type

1. #17988937

C₂

semidirect

5. #17988339

C₂

semidirect

6. #17988332

C₂

semidirect

7. #332×C₇₃

C₂

semidirect

78. C₇₃

#1699

semidirect

85. #332

D₇₃

semidirect

Subgroups — #22609477

Conjugacy classes of subgroups

Complements