Group #22609607 — subgroups

Subgroups — #22609607

Order 9344.

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

normal subgroups only

This group has 176 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
2	#17988983	C₂	1	1	yes	maximal, characteristic
3	#17988966	C₂	1	1	yes	maximal, characteristic
4	#17988938	C₂	1	1	yes	maximal, characteristic
5	#17988937	C₂	1	1	yes	maximal, characteristic
6	#17988309	C₂	1	1	yes	maximal, characteristic
7	#500×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924018	—	2	2	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556524	—	2	2	no	—
12	#14556522	—	2	2	no	—
13	#14556516	—	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	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	#107×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	#11924018	—	4	4	no	—
24	C₂×#10596831	D₄	1	2	yes	—
25	C₂×#10596831	D₄	1	2	yes	—
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	1	no	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₂×Q₈×C₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×C₈×C₇₃	D₄	1	2	yes	—
33	C₄⋊C₄×C₇₃	—	2	1	no	—
34	C₄⋊C₄×C₇₃	—	2	1	no	—
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄²×C₇₃	C₂³	1	1	yes	characteristic
38	Q₈×C₇₃	—	4	2	no	—
39	Q₈×C₇₃	—	4	2	no	—
40	C₂×C₄×C₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	—	4	1	no	—
42	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₇₃	—	4	1	no	—
46	C₂×Dic₇₃	—	4	1	no	—
47	Dic₁₄₆	—	4	2	no	—
48	Dic₁₄₆	—	4	2	no	—
49	C₈×C₇₃	—	2	2	no	—
50	#10596831	—	2	2	no	—
51	#10596831	Q₁₆	1	4	yes	complemented
52	C₂²×C₇₃	#127	1	1	yes	characteristic
53	C₄×C₇₃	—	8	1	no	—
54	C₄×C₇₃	—	8	1	no	—
55	C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
56	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₇₃	#137	1	2	yes	—
58	Dic₇₃	—	8	1	no	—
59	Dic₇₃	—	8	1	no	—
60	C₂×C₇₃	#494	1	1	yes	characteristic
61	C₂×C₇₃	#496	1	1	yes	characteristic
62	C₂×C₇₃	#497	1	1	yes	characteristic
63	#1723	—	73	1	no	maximal, Sylow
64	C₇₃	#1723	1	1	yes	characteristic, complemented, Sylow
65	#500	—	73	1	no	—
66	#500	D₇₃	1	1	yes	characteristic
67	#498	—	73	1	no	—
68	#494	—	73	1	no	—
69	#487	—	73	1	no	—
70	#487	—	73	1	no	—
71	#322	—	73	1	no	—
72	C₂×Q₁₆	—	146	2	no	—
73	C₄⋊Q₈	—	73	1	no	—
74	C₄⋊Q₈	—	73	1	no	—
75	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	146	2	no	—
79	#106	—	146	2	no	—
80	#103	—	146	2	no	—
81	#103	—	146	2	no	—
82	C₄×C₈	—	73	1	no	—
83	C₄×C₈	—	73	1	no	—
84	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	146	1	no	—
88	C₂×Q₈	—	146	1	no	—
89	Q₁₆	—	292	4	no	—
90	C₂×C₈	—	73	2	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	146	1	no	—
98	C₄⋊C₄	—	146	1	no	—
99	C₄⋊C₄	—	146	1	no	—
100	C₄⋊C₄	—	146	1	no	—
101	C₄²	C₂²×D₇₃	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	292	2	no	—
105	Q₈	—	292	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	292	1	no	—
109	C₂×C₄	—	292	1	no	—
110	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	73	4	no	—
115	C₈	—	146	2	no	—
116	C₂²	#14556587	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	584	1	no	—
120	C₄	—	584	1	no	—
121	C₄	Q₁₆×D₇₃	1	2	yes	—
122	C₄	#14556615	1	2	yes	—
123	C₄	#14556622	1	2	yes	—
124	C₂	#17988965	1	1	yes	characteristic
125	C₂	#17988975	1	1	yes	characteristic
126	C₂	#17988979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988990

C₂

yes

maximal, characteristic

#17988983

C₂

yes

maximal, characteristic

#17988966

C₂

yes

maximal, characteristic

#17988938

C₂

yes

maximal, characteristic

#17988937

C₂

yes

maximal, characteristic

#17988309

C₂

yes

maximal, characteristic

#500×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924018

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#107×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

#11924018

—

C₂×#10596831

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×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

Q₁₆

yes

complemented

C₂²×C₇₃

#127

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×Q₁₆

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

#137

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#500

—

#500

D₇₃

yes

—

—

—

—

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

#107

—

#107

—

146

—

#106

—

146

—

#103

—

146

—

#103

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

Q₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

100

C₄⋊C₄

—

146

—

101

C₄²

C₂²×D₇₃

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

292

—

105

Q₈

—

292

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

292

—

109

C₂×C₄

—

292

—

110

C₂×C₄

D₄×D₇₃

yes

characteristic

111

C₂×C₄

D₄×D₇₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

146

—

116

C₂²

#14556587

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

584

—

120

C₄

—

584

—

121

C₄

Q₁₆×D₇₃

yes

—

122

C₄

#14556615

yes

—

123

C₄

#14556622

yes

—

124

C₂

#17988965

yes

characteristic

125

C₂

#17988975

yes

characteristic

126

C₂

#17988979

yes

characteristic

	subgroup	G/H
center	C₂²	#14556587
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#500×C₇₃	C₂
socle	C₂²×C₇₃	#127
hypercenter	#500	D₇₃
hypocenter	C₇₃	#1723
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14556587

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#500×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
51. #10596831	Q₁₆	semidirect
64. C₇₃	#1723	semidirect

H ≅ G/N

type

51. #10596831

Q₁₆

semidirect

64. C₇₃

#1723

semidirect

Subgroups — #22609607

Conjugacy classes of subgroups

Complements