Group #32973607 — subgroups

Subgroups — #32973607

Order 18688.

136 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 37 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	#22611506	C₂	1	1	yes	maximal, characteristic
2	#22607882	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1307	C₂	1	1	yes	maximal, characteristic
4	#17988636	C₄	1	1	yes	characteristic
5	#17988369	—	2	1	no	—
6	#17988359	—	2	1	no	—
7	#17988347	—	2	1	no	—
8	#17988344	C₄	1	1	yes	characteristic
9	#432×C₇₃	—	2	1	no	—
10	#423×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×#11924012	—	2	1	no	—
12	#14556550	—	2	1	no	—
13	#14556531	—	4	1	no	—
14	#14556527	—	4	1	no	—
15	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
18	C₄⋊C₈×C₇₃	—	4	1	no	—
19	C₄⋊C₈×C₇₃	—	4	1	no	—
20	C₂²⋊C₈×C₇₃	—	2	2	no	—
21	#97×C₇₃	—	4	1	no	—
22	#11924012	—	4	1	no	—
23	#11924012	—	4	1	no	—
24	C₂×#10596831	—	4	1	no	—
25	C₄⋊Dic₇₃	—	4	1	no	—
26	C₂²⋊Dic₇₃	—	4	1	no	—
27	C₄×Dic₇₃	—	4	1	no	—
28	#11924003	—	8	1	no	—
29	C₂²×C₄×C₇₃	—	2	1	no	—
30	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	M₁₆×C₇₃	—	4	1	no	—
32	M₁₆×C₇₃	—	4	1	no	—
33	C₂×C₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	4	2	no	—
36	C₄²×C₇₃	—	2	1	no	—
37	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
38	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
39	C₂³×C₇₃	#104	1	1	yes	characteristic
40	C₂×C₄×C₇₃	—	2	1	no	—
41	C₂×C₄×C₇₃	—	2	1	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	8	1	no	—
45	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
47	C₂×Dic₇₃	—	8	1	no	—
48	C₈×C₇₃	—	4	1	no	—
49	C₈×C₇₃	—	4	1	no	—
50	C₈×C₇₃	—	8	2	no	—
51	#10596831	—	8	1	no	—
52	C₂²×C₇₃	—	2	1	no	—
53	C₂²×C₇₃	—	4	1	no	—
54	C₂²×C₇₃	#328	1	1	yes	characteristic
55	C₄×C₇₃	—	2	1	no	—
56	C₄×C₇₃	—	4	1	no	—
57	C₄×C₇₃	—	4	1	no	—
58	C₄×C₇₃	#352	1	1	yes	characteristic
59	C₄×C₇₃	#355	1	1	yes	characteristic
60	Dic₇₃	—	16	1	no	—
61	#7441	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	2	1	no	—
63	C₂×C₇₃	—	4	1	no	—
64	C₂×C₇₃	#1353	1	1	yes	characteristic
65	#2102	—	73	1	no	—
66	#1327	—	73	1	no	—
67	#1307	D₇₃	1	1	yes	characteristic
68	C₇₃	#7441	1	1	yes	characteristic, complemented, Sylow
69	#432	—	2	1	no	—
70	#430	—	73	1	no	—
71	#423	C₂×D₇₃	1	1	yes	characteristic
72	#349	—	146	1	no	—
73	#344	—	146	1	no	—
74	#339	—	146	1	no	—
75	H(3,ℤ/4)	—	73	1	no	—
76	C₂×M₁₆	—	146	1	no	—
77	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
78	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
79	#117	—	146	1	no	—
80	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
81	M₃₂	—	292	1	no	—
82	#108	—	292	1	no	—
83	C₄⋊C₈	—	4	1	no	—
84	C₄⋊C₈	—	4	1	no	—
85	C₂²⋊C₈	—	2	2	no	—
86	#97	—	4	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	#11924005	1	1	yes	characteristic
89	M₁₆	—	4	1	no	—
90	M₁₆	—	4	1	no	—
91	M₁₆	—	292	1	no	—
92	M₁₆	—	292	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	292	1	no	—
97	C₄⋊C₄	—	292	1	no	—
98	C₂²⋊C₄	—	292	1	no	—
99	C₄²	—	2	1	no	—
100	C₄²	—	292	1	no	—
101	C₄²	#11924016	1	1	yes	characteristic
102	C₄²	#11924018	1	1	yes	characteristic
103	C₁₆	—	584	1	no	—
104	C₂³	#14556519	1	1	yes	characteristic
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	8	1	no	—
110	C₂×C₄	—	584	1	no	—
111	C₂×C₄	#14556505	1	1	yes	characteristic
112	C₂×C₄	#14556514	1	1	yes	characteristic
113	C₈	—	4	1	no	—
114	C₈	—	4	1	no	—
115	C₈	—	8	2	no	—
116	C₈	—	584	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	4	1	no	—
119	C₂²	#17988323	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	1	no	—
122	C₄	—	4	1	no	—
123	C₄	—	1168	1	no	—
124	C₄	#17988378	1	1	yes	characteristic
125	C₄	#17988383	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	1	no	—
128	C₂	#22607957	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611506

C₂

yes

maximal, characteristic

#22607882

C₂

yes

maximal, characteristic

C₇₃×#1307

C₂

yes

maximal, characteristic

#17988636

C₄

yes

—

—

—

C₄

yes

characteristic

#432×C₇₃

—

#423×C₇₃

C₂²

yes

—

—

—

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

characteristic

C₄²×C₇₃

Q₁₆

yes

characteristic

C₂³×C₇₃

#104

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#328

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#7441

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

C₇₃

#7441

yes

characteristic, complemented, Sylow

#432

—

#430

—

#423

C₂×D₇₃

yes

characteristic

#349

—

146

—

#344

—

146

—

#339

—

146

—

H(3,ℤ/4)

—

C₂×M₁₆

—

146

—

C₂×M₁₆

C₄×D₇₃

yes

characteristic

C₂×M₁₆

D₂₉₂

yes

characteristic

#117

—

146

—

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

M₃₂

—

292

—

#108

—

292

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂²×C₄

—

C₂²×C₄

#11924005

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

292

—

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₄⋊C₄

—

292

—

C₂²⋊C₄

—

292

—

C₄²

—

100

C₄²

—

292

—

101

C₄²

#11924016

yes

characteristic

102

C₄²

#11924018

yes

characteristic

103

C₁₆

—

584

—

104

C₂³

#14556519

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

584

—

111

C₂×C₄

#14556505

yes

characteristic

112

C₂×C₄

#14556514

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

584

—

117

C₂²

—

118

C₂²

—

119

C₂²

#17988323

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

1168

—

124

C₄

#17988378

yes

characteristic

125

C₄

#17988383

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#22607957

yes

characteristic

	subgroup	G/H
center	C₂	#22607957
commutator	C₂×M₁₆×C₇₃	C₂×C₄
Frattini	#423	C₂×D₇₃
Fitting	C₇₃×#1307	C₂
socle	C₂×C₇₃	#1353
prime radical	C₂³×C₇₃	#104
cosocle	#423×C₇₃	C₂²
hypercenter	#1307	D₇₃
hypocenter	C₇₃	#7441
LCS	G ▷ C₂×M₁₆×C₇₃ ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1307

subgroup

G/H

center

C₂

#22607957

commutator

C₂×M₁₆×C₇₃

C₂×C₄

Frattini

#423

C₂×D₇₃

Fitting

C₇₃×#1307

C₂

C₂×C₇₃

C₂³×C₇₃

#423×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×M₁₆×C₇₃ ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1307

N	H ≅ G/N	type
68. C₇₃	#7441	semidirect

H ≅ G/N

type

68. C₇₃

#7441

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22611506	2
#22607882	2
C₇₃×#1307	2
#7441	73

subgroup

index

#22611506

#22607882

C₇₃×#1307

#7441

Subgroups — #32973607

Conjugacy classes of subgroups

Complements

Maximal subgroups