Group #32991059 — subgroups

Subgroups — #32991059

Order 18688.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22610737	C₂	1	3	yes	maximal
2	#22610059	C₂	1	3	yes	maximal, complemented
3	C₇₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556588	C₂²	1	1	yes	characteristic
5	C₂×#14556513	—	2	6	no	—
6	#17988457	C₂²	1	3	yes	—
7	#17988350	—	2	6	no	—
8	C₂×C₄×C₈×C₇₃	C₂²	1	3	yes	—
9	C₂²×D₂₉₂	—	2	3	no	—
10	#14556588	—	2	3	no	—
11	#14556588	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₇₃	—	2	3	no	—
13	C₂×C₄⋊Dic₇₃	—	2	3	no	—
14	C₂×C₄⋊Dic₇₃	—	2	3	no	—
15	#14556513	—	4	12	no	—
16	C₂²×C₈×C₇₃	D₄	1	6	yes	—
17	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₇₃	—	2	6	no	—
19	C₂³×D₇₃	—	4	1	no	—
20	C₂×D₂₉₂	—	2	6	no	—
21	C₂×D₂₉₂	—	4	6	no	—
22	C₂×D₂₉₂	—	4	6	no	—
23	C₂×D₂₉₂	—	4	3	no	—
24	C₂²×Dic₇₃	—	4	3	no	—
25	C₄⋊Dic₇₃	—	4	6	no	—
26	C₄⋊Dic₇₃	—	4	6	no	—
27	C₄⋊Dic₇₃	—	4	6	no	—
28	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
29	C₂×C₈×C₇₃	—	2	6	no	—
30	C₂×C₈×C₇₃	—	2	6	no	—
31	C₂×C₈×C₇₃	—	2	6	no	—
32	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₇₃	#41	1	3	yes	—
34	C₂³×C₇₃	#127	1	1	yes	characteristic
35	C₂²×D₇₃	—	4	2	no	—
36	C₂²×D₇₃	—	8	3	no	—
37	C₂²×D₇₃	—	8	3	no	—
38	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₇₃	#123	1	3	yes	—
40	C₂×C₄×C₇₃	#135	1	6	yes	—
41	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₇₃	—	8	3	no	—
43	C₂×Dic₇₃	—	8	3	no	—
44	C₂×Dic₇₃	—	8	3	no	—
45	D₂₉₂	—	4	12	no	—
46	D₂₉₂	—	8	6	no	—
47	C₈×C₇₃	—	4	6	no	—
48	C₂²×C₇₃	#399	1	1	yes	characteristic
49	C₂²×C₇₃	#495	1	3	yes	—
50	C₂²×C₇₃	#496	1	3	yes	—
51	C₂×D₇₃	—	8	6	no	—
52	C₂×D₇₃	—	16	1	no	—
53	C₂×D₇₃	—	16	3	no	—
54	C₄×C₇₃	#478	1	6	yes	—
55	C₄×C₇₃	#482	1	6	yes	—
56	Dic₇₃	—	16	3	no	—
57	#10678	—	73	1	no	maximal, Sylow
58	C₂×C₇₃	#1711	1	1	yes	characteristic
59	C₂×C₇₃	#1717	1	3	yes	—
60	C₂×C₇₃	#2092	1	3	yes	—
61	D₇₃	—	16	2	no	—
62	#1957	—	73	3	no	—
63	#1834	—	73	3	no	—
64	#1273	D₇₃	1	1	yes	characteristic, complemented
65	C₇₃	#10678	1	1	yes	characteristic, complemented, Sylow
66	C₂×#127	—	73	1	no	—
67	C₂×#102	—	146	6	no	—
68	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	—
69	#384	—	73	3	no	—
70	#340	—	146	6	no	—
71	C₂²×D₄	—	146	3	no	—
72	C₂²×C₈	D₂₉₂	1	6	yes	—
73	#127	—	73	2	no	—
74	#127	—	146	3	no	—
75	C₂×C₄⋊C₄	—	146	3	no	—
76	C₂×C₄⋊C₄	—	146	3	no	—
77	C₂×C₄⋊C₄	—	146	3	no	—
78	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
79	#102	—	292	12	no	—
80	C₄×C₈	—	2	6	no	—
81	C₂⁴	—	292	1	no	—
82	C₂×D₄	—	292	3	no	—
83	C₂×D₄	—	292	6	no	—
84	C₂×D₄	—	292	6	no	—
85	C₂×D₄	—	146	6	no	—
86	C₂²×C₄	—	292	3	no	—
87	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	292	6	no	—
92	C₄⋊C₄	—	292	6	no	—
93	C₄⋊C₄	—	292	6	no	—
94	C₄²	Q₈×D₇₃	1	1	yes	characteristic
95	C₄²	#11924027	1	3	yes	—
96	C₂³	—	292	2	no	—
97	C₂³	—	584	3	no	—
98	C₂³	—	584	3	no	—
99	C₂³	#14556588	1	1	yes	characteristic
100	D₄	—	292	12	no	—
101	D₄	—	584	6	no	—
102	C₂×C₄	—	584	3	no	—
103	C₂×C₄	—	584	3	no	—
104	C₂×C₄	—	584	3	no	—
105	C₂×C₄	#14556571	1	3	yes	—
106	C₂×C₄	#14556575	1	3	yes	—
107	C₂×C₄	#14556613	1	6	yes	—
108	C₂×C₄	#14556619	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	584	6	no	—
111	C₂²	—	1168	1	no	—
112	C₂²	—	1168	3	no	—
113	C₂²	#17988524	1	1	yes	characteristic
114	C₂²	#17988971	1	3	yes	—
115	C₂²	#17988976	1	3	yes	—
116	C₄	—	1168	3	no	—
117	C₄	#17988888	1	6	yes	—
118	C₄	#17988912	1	6	yes	—
119	C₂	—	1168	2	no	—
120	C₂	#22609543	1	1	yes	characteristic
121	C₂	#22609570	1	3	yes	—
122	C₂	#22611451	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610737

C₂

yes

maximal

#22610059

C₂

yes

maximal, complemented

C₇₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14556588

C₂²

yes

characteristic

C₂×#14556513

—

#17988457

C₂²

yes

—

#17988350

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₂²×D₂₉₂

—

#14556588

—

#14556588

Q₈

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556513

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#127

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₂²×C₇₃

#399

yes

characteristic

C₂²×C₇₃

#495

yes

—

C₂²×C₇₃

#496

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

#478

yes

—

C₄×C₇₃

#482

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2092

yes

—

D₇₃

—

#1957

—

#1834

—

#1273

D₇₃

yes

characteristic, complemented

C₇₃

#10678

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

—

#384

—

#340

—

146

—

C₂²×D₄

—

146

—

C₂²×C₈

D₂₉₂

yes

—

#127

—

#127

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

#102

—

292

—

C₄×C₈

—

C₂⁴

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

C₂²×C₄

—

292

—

C₂²×C₄

C₂×D₂₉₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄²

Q₈×D₇₃

yes

characteristic

C₄²

#11924027

yes

—

C₂³

—

292

—

C₂³

—

584

—

C₂³

—

584

—

C₂³

#14556588

yes

characteristic

100

D₄

—

292

—

101

D₄

—

584

—

102

C₂×C₄

—

584

—

103

C₂×C₄

—

584

—

104

C₂×C₄

—

584

—

105

C₂×C₄

#14556571

yes

—

106

C₂×C₄

#14556575

yes

—

107

C₂×C₄

#14556613

yes

—

108

C₂×C₄

#14556619

yes

—

109

C₈

—

110

C₂²

—

584

—

111

C₂²

—

1168

—

112

C₂²

—

1168

—

113

C₂²

#17988524

yes

characteristic

114

C₂²

#17988971

yes

—

115

C₂²

#17988976

yes

—

116

C₄

—

1168

—

117

C₄

#17988888

yes

—

118

C₄

#17988912

yes

—

119

C₂

—

1168

—

120

C₂

#22609543

yes

characteristic

121

C₂

#22609570

yes

—

122

C₂

#22611451

yes

—

	subgroup	G/H
center	C₂³	#14556588
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1273	C₂
socle	C₂³×C₇₃	#127
prime radical	C₂×#14556588	C₂²
hypercenter	#1273	D₇₃
hypocenter	C₇₃	#10678
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14556588

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1273

C₂

C₂³×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #22610059	C₂	semidirect
3. C₇₃×#1273	C₂	semidirect
64. #1273	D₇₃	semidirect
65. C₇₃	#10678	semidirect

H ≅ G/N

type

2. #22610059

C₂

semidirect

3. C₇₃×#1273

C₂

semidirect

64. #1273

D₇₃

semidirect

65. C₇₃

#10678

semidirect

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

subgroup	index
#22610737	2
#22610059	2
C₇₃×#1273	2
#10678	73

subgroup

index

#22610737

#22610059

C₇₃×#1273

#10678

Subgroups — #32991059

Conjugacy classes of subgroups

Complements

Maximal subgroups