Group #32973549 — subgroups

Subgroups — #32973549

Order 18688.

140 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 45 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	#22612022	C₂	1	1	yes	maximal, characteristic
2	#22608025	C₂	1	1	yes	maximal, characteristic
3	C₇₃×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₈₄	C₄	1	1	yes	characteristic
5	#17988828	—	2	1	no	—
6	C₂×#14556527	—	2	1	no	—
7	#17988413	C₄	1	1	yes	characteristic
8	#17988392	—	2	1	no	—
9	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₃₂×C₇₃	—	2	1	no	—
11	C₂×Dic₂₉₂	—	2	1	no	—
12	C₂×#11924012	—	4	1	no	—
13	#14556570	—	4	1	no	—
14	Dic₅₈₄	—	4	1	no	—
15	#14556527	—	2	2	no	—
16	#14556527	—	4	1	no	—
17	#14556525	—	2	1	no	—
18	#14556516	—	4	1	no	—
19	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₇₃	—	4	1	no	—
21	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
23	C₃₂×C₇₃	—	4	1	no	—
24	C₂×Dic₁₄₆	—	4	1	no	—
25	Dic₂₉₂	—	4	1	no	—
26	Dic₂₉₂	—	4	1	no	—
27	#11924012	—	4	2	no	—
28	#11924012	—	8	1	no	—
29	C₂×#10596831	—	8	1	no	—
30	C₄⋊Dic₇₃	—	4	1	no	—
31	#11924007	—	8	1	no	—
32	C₂²⋊Dic₇₃	—	8	1	no	—
33	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
38	C₁₆×C₇₃	—	2	1	no	—
39	C₁₆×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	SD₃₂	1	1	yes	characteristic
41	C₂×C₄×C₇₃	—	2	1	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
44	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
45	C₂×Dic₇₃	—	8	1	no	—
46	C₂×Dic₇₃	—	8	1	no	—
47	Dic₁₄₆	—	8	1	no	—
48	Dic₁₄₆	—	8	1	no	—
49	C₈×C₇₃	—	2	1	no	—
50	C₈×C₇₃	#99	1	1	yes	characteristic
51	C₈×C₇₃	#104	1	1	yes	characteristic
52	#10596831	—	8	2	no	—
53	C₂²×C₇₃	—	2	1	no	—
54	C₂²×C₇₃	—	2	1	no	—
55	C₂²×C₇₃	—	2	1	no	—
56	C₂²×C₇₃	#357	1	1	yes	characteristic
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	#327	1	1	yes	characteristic
59	C₄×C₇₃	#360	1	1	yes	characteristic
60	Dic₇₃	—	16	1	no	—
61	Dic₇₃	—	16	1	no	—
62	#7421	—	73	1	no	maximal, Sylow
63	C₂×C₇₃	—	4	1	no	—
64	C₂×C₇₃	#1347	1	1	yes	characteristic
65	C₂×C₇₃	#1417	1	1	yes	characteristic
66	C₂×C₇₃	#1420	1	1	yes	characteristic
67	#2216	—	73	1	no	—
68	C₂²⋊C₃₂	D₇₃	1	1	yes	characteristic
69	#1381	—	73	1	no	—
70	C₇₃	#7421	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	73	1	no	—
72	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
73	#467	—	146	1	no	—
74	C₂×#108	—	146	1	no	—
75	C₂×C₃₂	—	2	1	no	—
76	#366	—	73	1	no	—
77	#358	—	146	1	no	—
78	C₂×Q₁₆	—	146	1	no	—
79	C₂×M₁₆	—	292	1	no	—
80	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
81	C₂²⋊Q₈	—	292	1	no	—
82	Q₃₂	—	292	1	no	—
83	C₂×C₁₆	—	4	1	no	—
84	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
85	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
86	#108	—	292	1	no	—
87	#108	—	146	2	no	—
88	#107	—	146	1	no	—
89	#103	—	292	1	no	—
90	C₃₂	—	4	1	no	—
91	C₂×Q₈	—	292	1	no	—
92	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
93	Q₁₆	—	292	1	no	—
94	Q₁₆	—	292	1	no	—
95	M₁₆	—	292	2	no	—
96	M₁₆	—	584	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	584	1	no	—
100	C₂×C₈	#11924005	1	1	yes	characteristic
101	C₂×C₈	#11924017	1	1	yes	characteristic
102	C₄⋊C₄	—	292	1	no	—
103	C₄⋊C₄	—	584	1	no	—
104	C₂²⋊C₄	—	584	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	2	1	no	—
107	C₂³	#14556536	1	1	yes	characteristic
108	Q₈	—	584	1	no	—
109	Q₈	—	584	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	584	1	no	—
113	C₂×C₄	—	584	1	no	—
114	C₂×C₄	#14556513	1	1	yes	characteristic
115	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	584	2	no	—
118	C₈	#14556505	1	1	yes	characteristic
119	C₈	#14556517	1	1	yes	characteristic
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#17988389	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	1168	1	no	—
126	C₄	—	1168	1	no	—
127	C₄	#17988322	1	1	yes	characteristic
128	C₄	#17988398	1	1	yes	characteristic
129	C₂	—	4	1	no	—
130	C₂	#22607941	1	1	yes	characteristic
131	C₂	#22608116	1	1	yes	characteristic
132	C₂	#22608125	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612022

C₂

yes

maximal, characteristic

#22608025

C₂

yes

maximal, characteristic

C₇₃×C₂²⋊C₃₂

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

C₄

yes

—

—

C₄

yes

characteristic

#17988392

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

—

C₂×Dic₂₉₂

—

C₂×#11924012

—

#14556570

—

Dic₅₈₄

—

—

—

—

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

SD₃₂

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#357

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7421

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂²⋊C₃₂

D₇₃

yes

characteristic

#1381

—

C₇₃

#7421

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

#467

—

146

—

C₂×#108

—

146

—

C₂×C₃₂

—

#366

—

#358

—

146

—

C₂×Q₁₆

—

146

—

C₂×M₁₆

—

292

—

C₂²×C₈

D₂₉₂

yes

characteristic

C₂²⋊Q₈

—

292

—

Q₃₂

—

292

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

#108

—

292

—

#108

—

146

—

#107

—

146

—

#103

—

292

—

C₃₂

—

C₂×Q₈

—

292

—

C₂²×C₄

D₅₈₄

yes

characteristic

Q₁₆

—

292

—

Q₁₆

—

292

—

M₁₆

—

292

—

M₁₆

—

584

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

584

—

100

C₂×C₈

#11924005

yes

characteristic

101

C₂×C₈

#11924017

yes

characteristic

102

C₄⋊C₄

—

292

—

103

C₄⋊C₄

—

584

—

104

C₂²⋊C₄

—

584

—

105

C₁₆

—

106

C₁₆

—

107

C₂³

#14556536

yes

characteristic

108

Q₈

—

584

—

109

Q₈

—

584

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

584

—

113

C₂×C₄

—

584

—

114

C₂×C₄

#14556513

yes

characteristic

115

C₂×C₄

D₁₁₆₈

yes

characteristic

116

C₈

—

117

C₈

—

584

—

118

C₈

#14556505

yes

characteristic

119

C₈

#14556517

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#17988389

yes

characteristic

124

C₄

—

125

C₄

—

1168

—

126

C₄

—

1168

—

127

C₄

#17988322

yes

characteristic

128

C₄

#17988398

yes

characteristic

129

C₂

—

130

C₂

#22607941

yes

characteristic

131

C₂

#22608116

yes

characteristic

132

C₂

#22608125

yes

characteristic

	subgroup	G/H
center	C₂²	#17988389
commutator	C₂×C₁₆×C₇₃	C₂×C₄
Frattini	C₂²×C₁₆	C₂×D₇₃
Fitting	C₇₃×C₂²⋊C₃₂	C₂
socle	C₂²×C₇₃	#357
prime radical	C₂³×C₇₃	SD₃₂
cosocle	C₂²×C₁₆×C₇₃	C₂²
hypercenter	C₂²⋊C₃₂	D₇₃
hypocenter	C₇₃	#7421
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²⋊C₃₂

subgroup

G/H

center

C₂²

#17988389

commutator

C₂×C₁₆×C₇₃

C₂×C₄

Frattini

C₂²×C₁₆

C₂×D₇₃

Fitting

C₇₃×C₂²⋊C₃₂

C₂

socle

C₂²×C₇₃

#357

prime radical

C₂³×C₇₃

SD₃₂

cosocle

C₂²×C₁₆×C₇₃

C₂²

hypercenter

C₂²⋊C₃₂

D₇₃

hypocenter

C₇₃

#7421

LCS

G ▷ C₂×C₁₆×C₇₃ ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²⋊C₃₂

N	H ≅ G/N	type
70. C₇₃	#7421	semidirect

H ≅ G/N

type

70. C₇₃

#7421

semidirect

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

subgroup	index
#22612022	2
#22608025	2
C₇₃×C₂²⋊C₃₂	2
#7421	73

subgroup

index

#22612022

#22608025

C₇₃×C₂²⋊C₃₂

#7421

Subgroups — #32973549

Conjugacy classes of subgroups

Complements

Maximal subgroups