Group #22609402 — subgroups

Subgroups — #22609402

Order 9344.

168 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17988959	C₂	1	1	yes	maximal, characteristic
2	#17988873	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988784	C₂	1	1	yes	maximal, characteristic
4	#17988714	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988339	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988334	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924016	—	2	1	no	—
9	C₂×#11924016	—	2	1	no	—
10	C₂×#11924016	—	2	1	no	—
11	#14556591	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
13	#14556523	—	2	1	no	—
14	C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	#14556520	—	2	1	no	—
16	#14556516	—	2	1	no	—
17	#14556512	—	2	1	no	—
18	#14556501	—	2	1	no	—
19	C₄×#10596831	C₂²	1	1	yes	characteristic
20	#127×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×Dic₁₄₆	D₄	1	1	yes	characteristic
25	#11924016	—	4	2	no	—
26	#11924016	—	4	2	no	—
27	#11924016	—	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	#11924007	—	2	1	no	—
35	C₄×Dic₇₃	—	2	1	no	—
36	C₂×D₄×C₇₃	—	2	1	no	—
37	C₂×D₄×C₇₃	—	2	1	no	—
38	C₂×D₄×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	—	4	1	no	—
43	D₄×C₇₃	—	4	1	no	—
44	D₄×C₇₃	—	4	1	no	—
45	D₄×C₇₃	—	4	1	no	—
46	D₄×C₇₃	—	4	1	no	—
47	D₄×C₇₃	—	8	1	no	—
48	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	2	1	no	—
53	C₂×Dic₇₃	—	4	1	no	—
54	Dic₁₄₆	—	2	1	no	—
55	Dic₁₄₆	—	2	1	no	—
56	Dic₁₄₆	—	4	1	no	—
57	Dic₁₄₆	—	4	1	no	—
58	C₈×C₇₃	—	4	1	no	—
59	C₈×C₇₃	—	4	1	no	—
60	#10596831	—	2	1	no	—
61	#10596831	—	4	1	no	—
62	#10596831	SD₁₆	1	2	yes	complemented
63	C₂²×C₇₃	—	8	1	no	—
64	C₂²×C₇₃	—	8	1	no	—
65	C₂²×C₇₃	—	8	1	no	—
66	C₂²×C₇₃	#121	1	1	yes	characteristic
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₇₃	#137	1	1	yes	characteristic
72	Dic₇₃	—	4	1	no	—
73	Dic₇₃	—	4	1	no	—
74	Dic₇₃	—	8	1	no	—
75	C₂×C₇₃	—	16	1	no	—
76	C₂×C₇₃	#460	1	1	yes	characteristic
77	C₂×C₇₃	#468	1	1	yes	characteristic
78	C₂×C₇₃	#471	1	1	yes	characteristic
79	#1686	—	73	1	no	maximal, Sylow
80	C₇₃	#1686	1	1	yes	characteristic, complemented, Sylow
81	#492	—	73	1	no	—
82	#475	—	73	1	no	—
83	#460	—	73	1	no	—
84	#446	—	73	1	no	—
85	#335	—	73	1	no	—
86	#332	—	73	1	no	—
87	#331	D₇₃	1	1	yes	characteristic
88	C₂×SD₁₆	—	146	1	no	—
89	C₂×SD₁₆	—	146	1	no	—
90	C₂×SD₁₆	—	146	1	no	—
91	C₄⋊Q₈	—	73	1	no	—
92	#127	C₂×D₇₃	1	1	yes	characteristic
93	C₄×Q₈	—	73	1	no	—
94	#106	—	146	1	no	—
95	C₄⋊C₈	—	73	1	no	—
96	C₄⋊C₈	—	146	1	no	—
97	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
98	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
99	#103	—	146	1	no	—
100	#102	—	146	1	no	—
101	#97	—	146	1	no	—
102	C₄×C₈	—	73	1	no	—
103	C₂×Q₈	—	73	1	no	—
104	C₂×Q₈	—	146	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	SD₁₆	—	292	2	no	—
109	SD₁₆	—	292	2	no	—
110	SD₁₆	—	292	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	146	1	no	—
116	C₄⋊C₄	—	73	1	no	—
117	C₄⋊C₄	—	146	1	no	—
118	C₄⋊C₄	—	146	1	no	—
119	C₄⋊C₄	—	146	1	no	—
120	C₄²	—	146	1	no	—
121	C₄²	C₂²×D₇₃	1	1	yes	characteristic
122	C₂³	—	4	1	no	—
123	Q₈	—	292	1	no	—
124	Q₈	—	292	1	no	—
125	Q₈	—	146	1	no	—
126	Q₈	—	146	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	C₂×C₄	—	292	1	no	—
133	C₂×C₄	—	146	1	no	—
134	C₂×C₄	—	146	1	no	—
135	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
136	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
137	C₂×C₄	#11924028	1	1	yes	characteristic
138	C₈	—	4	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	292	1	no	—
141	C₈	—	73	2	no	—
142	C₈	—	146	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	#14556564	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	292	1	no	—
149	C₄	—	292	1	no	—
150	C₄	—	584	1	no	—
151	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
152	C₄	#14556615	1	1	yes	characteristic
153	C₄	#14556615	1	1	yes	characteristic
154	C₄	#14556620	1	1	yes	characteristic
155	C₂	—	16	1	no	—
156	C₂	#17988786	1	1	yes	characteristic
157	C₂	#17988832	1	1	yes	characteristic
158	C₂	#17988849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988959

C₂

yes

maximal, characteristic

#17988873

C₂

yes

maximal, characteristic, complemented

#17988784

C₂

yes

maximal, characteristic

#17988714

C₂

yes

maximal, characteristic, complemented

#17988339

C₂

yes

maximal, characteristic, complemented

#17988334

C₂

yes

maximal, characteristic, complemented

#331×C₇₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₄₆

C₂²

yes

characteristic

#14556523

—

C₄⋊#10596831

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×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

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

SD₁₆

yes

complemented

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₃

yes

characteristic

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₄⋊Q₈

—

#127

C₂×D₇₃

yes

characteristic

C₄×Q₈

—

#106

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

146

—

100

#102

—

146

—

101

#97

—

146

—

102

C₄×C₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

146

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

292

—

109

SD₁₆

—

292

—

110

SD₁₆

—

292

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

—

146

—

119

C₄⋊C₄

—

146

—

120

C₄²

—

146

—

121

C₄²

C₂²×D₇₃

yes

characteristic

122

C₂³

—

123

Q₈

—

292

—

124

Q₈

—

292

—

125

Q₈

—

146

—

126

Q₈

—

146

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

D₄×D₇₃

yes

characteristic

136

C₂×C₄

D₄×D₇₃

yes

characteristic

137

C₂×C₄

#11924028

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

292

—

141

C₈

—

142

C₈

—

146

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#14556564

yes

characteristic

147

C₄

—

148

C₄

—

292

—

149

C₄

—

292

—

150

C₄

—

584

—

151

C₄

SD₁₆×D₇₃

yes

characteristic

152

C₄

#14556615

yes

characteristic

153

C₄

#14556615

yes

characteristic

154

C₄

#14556620

yes

characteristic

155

C₂

—

156

C₂

#17988786

yes

characteristic

157

C₂

#17988832

yes

characteristic

158

C₂

#17988849

yes

characteristic

	subgroup	G/H
center	C₂²	#14556564
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#331×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#331	D₇₃
hypocenter	C₇₃	#1686
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14556564

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#331×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #17988873	C₂	semidirect
4. #17988714	C₂	semidirect
5. #17988339	C₂	semidirect
6. #17988334	C₂	semidirect
62. #10596831	SD₁₆	semidirect
80. C₇₃	#1686	semidirect

H ≅ G/N

type

2. #17988873

C₂

semidirect

4. #17988714

C₂

semidirect

5. #17988339

C₂

semidirect

6. #17988334

C₂

semidirect

62. #10596831

SD₁₆

semidirect

80. C₇₃

#1686

semidirect

Subgroups — #22609402

Conjugacy classes of subgroups

Complements