Group #22609505 — subgroups

Subgroups — #22609505

Order 9344.

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

normal subgroups only

This group has 154 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	#17988878	C₂	1	1	yes	maximal, characteristic
3	#17988864	C₂	1	1	yes	maximal, characteristic
4	#17988664	C₂	1	1	yes	maximal, characteristic
5	#17988337	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988335	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924011	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	#14556524	—	2	1	no	—
12	#14556523	—	2	1	no	—
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	#14556515	—	2	1	no	—
15	#14556515	—	2	1	no	—
16	#14556513	—	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	#11924011	—	4	2	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	D₄	1	1	yes	characteristic
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
33	#11924005	—	2	1	no	—
34	C₂×Q₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₂²×D₇₃	—	2	1	no	—
41	Q₈×C₇₃	—	4	1	no	—
42	Q₈×C₇₃	—	4	1	no	—
43	C₂×C₄×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
47	C₂×Dic₇₃	—	2	1	no	—
48	C₂×Dic₇₃	—	4	1	no	—
49	D₂₉₂	—	2	1	no	—
50	D₂₉₂	—	2	1	no	—
51	C₄×D₇₃	—	4	1	no	—
52	C₄×D₇₃	—	4	1	no	—
53	Dic₁₄₆	—	8	1	no	—
54	C₈×C₇₃	—	4	1	no	—
55	C₈×C₇₃	—	4	1	no	—
56	#10596831	—	2	1	no	—
57	#10596831	—	2	1	no	—
58	#10596831	—	4	1	no	—
59	C₂²×C₇₃	#121	1	1	yes	characteristic
60	C₂×D₇₃	—	4	1	no	—
61	C₂×D₇₃	—	4	1	no	—
62	C₂×D₇₃	—	4	1	no	—
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	—	8	1	no	—
65	C₄×C₇₃	#135	1	1	yes	characteristic
66	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₄×C₇₃	#137	1	1	yes	characteristic
68	C₄×C₇₃	#137	1	1	yes	characteristic
69	Dic₇₃	—	4	1	no	—
70	Dic₇₃	—	8	1	no	—
71	C₂×C₇₃	#463	1	1	yes	characteristic
72	C₂×C₇₃	#470	1	1	yes	characteristic
73	C₂×C₇₃	#471	1	1	yes	characteristic
74	D₇₃	—	8	1	no	—
75	#1703	—	73	1	no	maximal, Sylow
76	C₇₃	#1703	1	1	yes	characteristic, complemented, Sylow
77	#487	—	73	1	no	—
78	#476	—	73	1	no	—
79	#474	—	73	1	no	—
80	#436	—	73	1	no	—
81	#334	—	73	1	no	—
82	#333	—	73	1	no	—
83	#332	D₇₃	1	1	yes	characteristic, complemented
84	C₂×M₁₆	—	146	1	no	—
85	C₄⋊Q₈	—	73	1	no	—
86	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
87	C₄×D₄	—	73	1	no	—
88	#107	—	146	1	no	—
89	#106	—	146	1	no	—
90	C₄⋊C₈	—	73	1	no	—
91	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
92	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
93	#103	—	146	1	no	—
94	#103	—	146	1	no	—
95	#102	—	146	1	no	—
96	#102	—	146	1	no	—
97	C₂²⋊C₈	—	146	1	no	—
98	C₄×C₈	—	73	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	146	1	no	—
101	C₂×D₄	—	73	1	no	—
102	C₂²×C₄	—	146	1	no	—
103	M₁₆	—	292	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	146	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	73	1	no	—
112	C₄⋊C₄	—	146	1	no	—
113	C₄⋊C₄	—	146	1	no	—
114	C₂²⋊C₄	—	146	1	no	—
115	C₄²	C₂²×D₇₃	1	1	yes	characteristic
116	C₂³	—	146	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	584	1	no	—
120	D₄	—	146	1	no	—
121	D₄	—	146	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	292	1	no	—
124	C₂×C₄	—	292	1	no	—
125	C₂×C₄	—	292	1	no	—
126	C₂×C₄	—	146	1	no	—
127	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
128	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
129	C₂×C₄	#11924028	1	1	yes	characteristic
130	C₈	—	4	1	no	—
131	C₈	—	4	1	no	—
132	C₈	—	292	1	no	—
133	C₈	—	146	1	no	—
134	C₈	—	146	1	no	—
135	C₂²	—	292	1	no	—
136	C₂²	—	292	1	no	—
137	C₂²	—	292	1	no	—
138	C₂²	#14556564	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	1	no	—
141	C₄	—	292	1	no	—
142	C₄	—	584	1	no	—
143	C₄	#14556612	1	1	yes	characteristic
144	C₄	#14556615	1	1	yes	characteristic
145	C₄	#14556620	1	1	yes	characteristic
146	C₄	#14556623	1	1	yes	characteristic
147	C₂	—	584	1	no	—
148	C₂	#17988801	1	1	yes	characteristic
149	C₂	#17988844	1	1	yes	characteristic
150	C₂	#17988853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988937

C₂

yes

maximal, characteristic, complemented

#17988878

C₂

yes

maximal, characteristic

#17988864

C₂

yes

maximal, characteristic

#17988664

C₂

yes

maximal, characteristic

#17988337

C₂

yes

maximal, characteristic, complemented

#17988335

C₂

yes

maximal, characteristic, complemented

#332×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924011

—

#14556591

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

—

—

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

—

#11924011

—

C₂×#10596831

—

C₂×#10596831

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

—

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

—

#1703

—

maximal, Sylow

C₇₃

#1703

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄×D₄

—

#107

—

146

—

#106

—

146

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

146

—

#103

—

146

—

#102

—

146

—

#102

—

146

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

146

—

101

C₂×D₄

—

102

C₂²×C₄

—

146

—

103

M₁₆

—

292

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₂²⋊C₄

—

146

—

115

C₄²

C₂²×D₇₃

yes

characteristic

116

C₂³

—

146

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

584

—

120

D₄

—

146

—

121

D₄

—

146

—

122

C₂×C₄

—

123

C₂×C₄

—

292

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

146

—

127

C₂×C₄

D₄×D₇₃

yes

characteristic

128

C₂×C₄

D₄×D₇₃

yes

characteristic

129

C₂×C₄

#11924028

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

292

—

133

C₈

—

146

—

134

C₈

—

146

—

135

C₂²

—

292

—

136

C₂²

—

292

—

137

C₂²

—

292

—

138

C₂²

#14556564

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

292

—

142

C₄

—

584

—

143

C₄

#14556612

yes

characteristic

144

C₄

#14556615

yes

characteristic

145

C₄

#14556620

yes

characteristic

146

C₄

#14556623

yes

characteristic

147

C₂

—

584

—

148

C₂

#17988801

yes

characteristic

149

C₂

#17988844

yes

characteristic

150

C₂

#17988853

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	#1703
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. #17988337	C₂	semidirect
6. #17988335	C₂	semidirect
7. #332×C₇₃	C₂	semidirect
76. C₇₃	#1703	semidirect
83. #332	D₇₃	semidirect

H ≅ G/N

type

1. #17988937

C₂

semidirect

5. #17988337

C₂

semidirect

6. #17988335

C₂

semidirect

7. #332×C₇₃

C₂

semidirect

76. C₇₃

#1703

semidirect

83. #332

D₇₃

semidirect

Subgroups — #22609505

Conjugacy classes of subgroups

Complements