Group #22609504 — subgroups

Subgroups — #22609504

Order 9344.

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

normal subgroups only

This group has 154 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	#17988937	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988877	C₂	1	1	yes	maximal, characteristic
3	#17988665	C₂	1	1	yes	maximal, characteristic
4	#17988337	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	#474×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924012	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
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	#14556515	—	2	1	no	—
15	#14556512	—	2	1	no	—
16	C₂²⋊#10596831	—	2	1	no	—
17	C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	#107×C₇₃	—	2	1	no	—
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#102×C₇₃	—	2	1	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	#11924012	—	4	2	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	D₄	1	1	yes	characteristic
28	C₂×#10596831	D₄	1	1	yes	characteristic
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₇₃	—	2	1	no	—
39	C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₂³×C₇₃	—	2	1	no	—
41	Q₈×C₇₃	—	4	1	no	—
42	Q₈×C₇₃	—	4	1	no	—
43	D₄×C₇₃	—	2	1	no	—
44	D₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	4	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
52	C₂×Dic₇₃	—	4	1	no	—
53	Dic₁₄₆	—	8	1	no	—
54	C₈×C₇₃	—	4	1	no	—
55	#10596831	—	2	1	no	—
56	#10596831	—	2	1	no	—
57	#10596831	—	4	1	no	—
58	#10596831	—	4	1	no	—
59	C₂²×C₇₃	—	4	1	no	—
60	C₂²×C₇₃	—	4	1	no	—
61	C₂²×C₇₃	—	4	1	no	—
62	C₂²×C₇₃	#121	1	1	yes	characteristic
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	—	4	1	no	—
65	C₄×C₇₃	—	8	1	no	—
66	C₄×C₇₃	#135	1	1	yes	characteristic
67	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₄×C₇₃	#137	1	1	yes	characteristic
69	C₄×C₇₃	#137	1	1	yes	characteristic
70	Dic₇₃	—	8	1	no	—
71	C₂×C₇₃	—	8	1	no	—
72	C₂×C₇₃	#463	1	1	yes	characteristic
73	C₂×C₇₃	#470	1	1	yes	characteristic
74	C₂×C₇₃	#471	1	1	yes	characteristic
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	D₇₃	1	1	yes	characteristic
80	#436	—	73	1	no	—
81	#334	—	73	1	no	—
82	#333	—	73	1	no	—
83	#332	—	73	1	no	—
84	C₂×M₁₆	—	146	1	no	—
85	C₄⋊Q₈	—	73	1	no	—
86	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
87	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
88	#107	—	2	1	no	—
89	#106	—	146	1	no	—
90	C₄⋊C₈	—	73	1	no	—
91	C₄⋊C₈	—	73	1	no	—
92	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
93	#103	—	146	1	no	—
94	#103	—	146	1	no	—
95	#102	—	2	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	C₂²⋊D₇₃	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	M₁₆	—	292	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	146	1	no	—
108	C₂×C₈	—	146	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	146	1	no	—
112	C₄⋊C₄	—	146	1	no	—
113	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
114	C₂²⋊C₄	—	2	1	no	—
115	C₄²	C₂²×D₇₃	1	1	yes	characteristic
116	C₂³	—	2	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	584	1	no	—
120	D₄	—	2	1	no	—
121	D₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	292	1	no	—
127	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
129	C₂×C₄	#11924027	1	1	yes	characteristic
130	C₈	—	4	1	no	—
131	C₈	—	292	1	no	—
132	C₈	—	292	1	no	—
133	C₈	—	146	1	no	—
134	C₈	—	146	1	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#14556563	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	584	1	no	—
143	C₄	#14556610	1	1	yes	characteristic
144	C₄	#14556615	1	1	yes	characteristic
145	C₄	#14556621	1	1	yes	characteristic
146	C₄	#14556622	1	1	yes	characteristic
147	C₂	—	8	1	no	—
148	C₂	#17988800	1	1	yes	characteristic
149	C₂	#17988843	1	1	yes	characteristic
150	C₂	#17988852	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988937

C₂

yes

maximal, characteristic, complemented

#17988877

C₂

yes

maximal, characteristic

#17988665

C₂

yes

maximal, characteristic

#17988337

C₂

yes

maximal, characteristic, complemented

#17988335

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, characteristic, complemented

#474×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

—

#14556591

C₂²

yes

characteristic

#14556522

—

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

#107×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂×Dic₁₄₆

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×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×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

—

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

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

#487

—

#476

—

#474

D₇₃

yes

—

—

—

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄×D₄

C₂×D₇₃

yes

characteristic

#107

—

#106

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

146

—

#103

—

146

—

#102

—

#102

—

146

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

146

—

101

C₂×D₄

C₂²⋊D₇₃

yes

characteristic

102

C₂²×C₄

—

103

M₁₆

—

292

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

146

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

114

C₂²⋊C₄

—

115

C₄²

C₂²×D₇₃

yes

characteristic

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

584

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

292

—

127

C₂×C₄

D₄×D₇₃

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

129

C₂×C₄

#11924027

yes

characteristic

130

C₈

—

131

C₈

—

292

—

132

C₈

—

292

—

133

C₈

—

146

—

134

C₈

—

146

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14556563

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

584

—

143

C₄

#14556610

yes

characteristic

144

C₄

#14556615

yes

characteristic

145

C₄

#14556621

yes

characteristic

146

C₄

#14556622

yes

characteristic

147

C₂

—

148

C₂

#17988800

yes

characteristic

149

C₂

#17988843

yes

characteristic

150

C₂

#17988852

yes

characteristic

	subgroup	G/H
center	C₂²	#14556563
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#474×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#474	D₇₃
hypocenter	C₇₃	#1703
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

subgroup

G/H

center

C₂²

#14556563

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#474×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #474

N	H ≅ G/N	type
1. #17988937	C₂	semidirect
4. #17988337	C₂	semidirect
5. #17988335	C₂	semidirect
6. #17988333	C₂	semidirect
76. C₇₃	#1703	semidirect

H ≅ G/N

type

1. #17988937

C₂

semidirect

4. #17988337

C₂

semidirect

5. #17988335

C₂

semidirect

6. #17988333

C₂

semidirect

76. C₇₃

#1703

semidirect

Subgroups — #22609504

Conjugacy classes of subgroups

Complements