Group #22609552 — subgroups

Subgroups — #22609552

Order 9344.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17988990	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988985	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988959	C₂	1	1	yes	maximal, characteristic
4	#17988959	C₂	1	1	yes	maximal, characteristic
5	#17988937	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988309	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924016	—	2	2	no	—
9	C₂×#11924016	—	2	2	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556591	C₂²	1	1	yes	characteristic
12	#14556524	—	2	2	no	—
13	#14556523	—	2	2	no	—
14	#14556515	—	2	2	no	—
15	C₄×#10596831	C₂²	1	1	yes	characteristic
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
18	#127×C₇₃	C₂²	1	1	yes	characteristic
19	#102×C₇₃	—	2	2	no	—
20	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×Dic₁₄₆	—	2	1	no	—
23	#11924016	—	4	4	no	—
24	#11924016	—	4	4	no	—
25	C₂×#10596831	D₄	1	2	yes	—
26	C₂×#10596831	D₄	1	2	yes	—
27	C₄⋊Dic₇₃	—	2	1	no	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×D₄×C₇₃	—	2	1	no	—
33	C₂×D₄×C₇₃	—	2	1	no	—
34	C₂×D₄×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	D₄	1	2	yes	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	C₂³×C₇₃	—	4	1	no	—
40	Q₈×C₇₃	—	4	2	no	—
41	D₄×C₇₃	—	4	2	no	—
42	D₄×C₇₃	—	4	2	no	—
43	D₄×C₇₃	—	4	2	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	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₇₃	—	4	1	no	—
49	C₂×Dic₇₃	—	4	1	no	—
50	Dic₁₄₆	—	4	2	no	—
51	Dic₁₄₆	—	4	2	no	—
52	C₈×C₇₃	—	2	2	no	—
53	#10596831	—	2	2	no	—
54	#10596831	SD₁₆	1	4	yes	complemented
55	C₂²×C₇₃	—	8	1	no	—
56	C₂²×C₇₃	—	8	1	no	—
57	C₂²×C₇₃	—	8	1	no	—
58	C₂²×C₇₃	#127	1	1	yes	characteristic
59	C₄×C₇₃	—	8	1	no	—
60	C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
61	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₇₃	#137	1	2	yes	—
63	Dic₇₃	—	8	1	no	—
64	Dic₇₃	—	8	1	no	—
65	C₂×C₇₃	—	16	1	no	—
66	C₂×C₇₃	#492	1	1	yes	characteristic
67	C₂×C₇₃	#496	1	1	yes	characteristic
68	C₂×C₇₃	#497	1	1	yes	characteristic
69	#1714	—	73	1	no	maximal, Sylow
70	C₇₃	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	—	73	1	no	—
72	#498	—	73	1	no	—
73	#492	—	73	1	no	—
74	#492	—	73	1	no	—
75	#487	—	73	1	no	—
76	#486	D₇₃	1	1	yes	characteristic
77	#322	—	73	1	no	—
78	C₂×SD₁₆	—	146	2	no	—
79	C₂×SD₁₆	—	146	2	no	—
80	C₄⋊Q₈	—	73	1	no	—
81	C₄⋊Q₈	—	73	1	no	—
82	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
83	#127	C₂×D₇₃	1	1	yes	characteristic
84	#107	—	146	2	no	—
85	#106	—	146	2	no	—
86	#103	—	146	2	no	—
87	#102	—	2	2	no	—
88	C₄×C₈	—	73	1	no	—
89	C₄×C₈	—	73	1	no	—
90	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	146	1	no	—
93	C₂×Q₈	—	146	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	292	4	no	—
98	SD₁₆	—	292	4	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	146	1	no	—
105	C₄⋊C₄	—	146	1	no	—
106	C₄⋊C₄	—	146	1	no	—
107	C₄⋊C₄	—	146	1	no	—
108	C₄²	C₂²×D₇₃	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	292	2	no	—
112	Q₈	—	292	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	292	1	no	—
118	C₂×C₄	—	292	1	no	—
119	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
120	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	73	4	no	—
124	C₈	—	146	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#14556587	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	584	1	no	—
131	C₄	—	584	1	no	—
132	C₄	SD₁₆×D₇₃	1	2	yes	—
133	C₄	#14556615	1	2	yes	—
134	C₄	#14556621	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#17988958	1	1	yes	characteristic
137	C₂	#17988975	1	1	yes	characteristic
138	C₂	#17988982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988990

C₂

yes

maximal, characteristic, complemented

#17988985

C₂

yes

maximal, characteristic, complemented

#17988959

C₂

yes

maximal, characteristic

#17988959

C₂

yes

maximal, characteristic

#17988937

C₂

yes

maximal, characteristic, complemented

#17988309

C₂

yes

maximal, characteristic, complemented

#486×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924016

—

C₂×#11924016

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

#11924016

—

#11924016

—

C₂×#10596831

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×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

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

SD₁₆

yes

complemented

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#127

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

#137

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic

#322

—

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

#127

C₂×D₇₃

yes

characteristic

#107

—

146

—

#106

—

146

—

#103

—

146

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

292

—

SD₁₆

—

292

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₇₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

146

—

105

C₄⋊C₄

—

146

—

106

C₄⋊C₄

—

146

—

107

C₄⋊C₄

—

146

—

108

C₄²

C₂²×D₇₃

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

292

—

112

Q₈

—

292

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

292

—

118

C₂×C₄

—

292

—

119

C₂×C₄

D₄×D₇₃

yes

characteristic

120

C₂×C₄

D₄×D₇₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

146

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#14556587

yes

characteristic

129

C₄

—

130

C₄

—

584

—

131

C₄

—

584

—

132

C₄

SD₁₆×D₇₃

yes

—

133

C₄

#14556615

yes

—

134

C₄

#14556621

yes

—

135

C₂

—

136

C₂

#17988958

yes

characteristic

137

C₂

#17988975

yes

characteristic

138

C₂

#17988982

yes

characteristic

	subgroup	G/H
center	C₂²	#14556587
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#486×C₇₃	C₂
socle	C₂²×C₇₃	#127
hypercenter	#486	D₇₃
hypocenter	C₇₃	#1714
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14556587

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#486×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #17988990	C₂	semidirect
2. #17988985	C₂	semidirect
5. #17988937	C₂	semidirect
6. #17988309	C₂	semidirect
54. #10596831	SD₁₆	semidirect
70. C₇₃	#1714	semidirect

H ≅ G/N

type

1. #17988990

C₂

semidirect

2. #17988985

C₂

semidirect

5. #17988937

C₂

semidirect

6. #17988309

C₂

semidirect

54. #10596831

SD₁₆

semidirect

70. C₇₃

#1714

semidirect

Subgroups — #22609552

Conjugacy classes of subgroups

Complements