Group #33000680 — subgroups

Subgroups — #33000680

Order 18688.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22612160	C₂	1	2	yes	maximal, complemented
2	#22612156	C₂	1	2	yes	maximal
3	#22609537	C₂	1	1	yes	maximal, characteristic
4	C₈×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988967	C₂²	1	1	yes	characteristic
7	#17988964	C₂²	1	1	yes	characteristic
8	#17988934	—	2	2	no	—
9	#17988390	—	2	4	no	—
10	#17988387	—	2	4	no	—
11	C₄×#11924003	C₂²	1	2	yes	—
12	#500×C₇₃	C₂²	1	2	yes	—
13	#487×C₇₃	—	2	2	no	—
14	C₈²×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₉₂	—	2	2	no	—
16	C₂×D₅₈₄	—	2	2	no	—
17	#14556591	—	2	1	no	—
18	#14556588	—	2	1	no	—
19	C₂×#11924003	D₄	1	4	yes	—
20	#14556513	—	4	4	no	—
21	C₄⋊Q₈×C₇₃	—	2	2	no	—
22	#107×C₇₃	—	2	2	no	—
23	#107×C₇₃	—	2	2	no	—
24	#103×C₇₃	—	4	4	no	—
25	C₄×C₈×C₇₃	D₄	1	2	yes	—
26	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₄₆	—	4	1	no	—
28	C₂×D₂₉₂	—	4	1	no	—
29	C₂×D₂₉₂	—	4	2	no	—
30	Dic₂₉₂	—	4	4	no	—
31	D₅₈₄	—	4	4	no	—
32	C₄⋊Dic₇₃	—	4	2	no	—
33	#11924003	—	2	4	no	—
34	C₂×Q₈×C₇₃	—	4	2	no	—
35	C₂×C₈×C₇₃	D₈	1	4	yes	—
36	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
37	C₄⋊C₄×C₇₃	—	4	2	no	—
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₇₃	—	8	1	no	—
41	Q₈×C₇₃	—	8	4	no	—
42	C₂×C₄×C₇₃	—	8	2	no	—
43	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
45	C₂×Dic₇₃	—	8	1	no	—
46	D₂₉₂	—	8	2	no	—
47	D₂₉₂	—	8	4	no	—
48	Dic₁₄₆	—	8	2	no	—
49	C₈×C₇₃	—	2	4	no	—
50	C₈×C₇₃	#135	1	4	yes	—
51	C₂²×C₇₃	#493	1	1	yes	characteristic
52	C₂×D₇₃	—	16	1	no	—
53	C₂×D₇₃	—	16	2	no	—
54	C₄×C₇₃	—	16	2	no	—
55	C₄×C₇₃	#495	1	2	yes	—
56	C₄×C₇₃	#508	1	4	yes	—
57	Dic₇₃	—	16	1	no	—
58	#12511	—	73	1	no	maximal, Sylow
59	C₂×C₇₃	#1709	1	1	yes	characteristic
60	C₂×C₇₃	#2249	1	2	yes	—
61	D₇₃	—	32	1	no	—
62	#2241	—	73	2	no	—
63	#2240	—	73	2	no	—
64	#1710	D₇₃	1	1	yes	characteristic, complemented
65	#1709	—	73	1	no	—
66	C₈×C₁₆	—	73	1	no	—
67	C₇₃	#12511	1	1	yes	characteristic, complemented, Sylow
68	#500	C₂×D₇₃	1	2	yes	—
69	#494	—	73	1	no	—
70	#493	—	73	1	no	—
71	#487	—	2	2	no	—
72	#486	—	146	2	no	—
73	#358	—	146	4	no	—
74	#357	—	146	4	no	—
75	C₄×C₁₆	—	73	2	no	—
76	C₈²	C₂×D₇₃	1	1	yes	characteristic
77	C₂×Q₁₆	—	146	2	no	—
78	C₂×D₈	—	146	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	146	1	no	—
81	#127	—	146	1	no	—
82	C₂×C₁₆	—	73	4	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#103	—	4	4	no	—
86	#102	—	292	4	no	—
87	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
88	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	292	1	no	—
91	C₂×D₄	—	292	1	no	—
92	C₂×D₄	—	292	2	no	—
93	Q₁₆	—	292	4	no	—
94	D₈	—	292	4	no	—
95	C₂×C₈	#11924013	1	4	yes	—
96	C₂×C₈	D₄×D₇₃	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	292	2	no	—
100	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
101	C₁₆	—	146	4	no	—
102	C₂³	—	584	1	no	—
103	Q₈	—	8	4	no	—
104	Q₈	—	584	2	no	—
105	D₄	—	584	2	no	—
106	D₄	—	584	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	584	1	no	—
109	C₂×C₄	#14556587	1	1	yes	characteristic
110	C₂×C₄	C₂×#11924013	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14556614	1	4	yes	—
113	C₂²	—	1168	1	no	—
114	C₂²	—	1168	2	no	—
115	C₂²	#17988963	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	1168	1	no	—
118	C₄	#17988972	1	2	yes	—
119	C₄	#17989017	1	4	yes	—
120	C₂	—	2336	1	no	—
121	C₂	#22609538	1	1	yes	characteristic
122	C₂	#22612190	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612160

C₂

yes

maximal, complemented

#22612156

C₂

yes

maximal

#22609537

C₂

yes

maximal, characteristic

C₈×#11924003

C₂

yes

maximal, characteristic, complemented

C₇₃×#1710

C₂

yes

maximal, characteristic, complemented

#17988967

C₂²

yes

characteristic

#17988964

C₂²

yes

—

—

—

C₂²

yes

—

#500×C₇₃

C₂²

yes

—

#487×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×D₅₈₄

—

#14556591

—

#14556588

—

C₂×#11924003

D₄

yes

—

#14556513

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

Dic₂₉₂

—

D₅₈₄

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#495

yes

—

C₄×C₇₃

#508

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2241

—

#2240

—

#1710

D₇₃

yes

characteristic, complemented

#1709

—

C₈×C₁₆

—

C₇₃

#12511

yes

characteristic, complemented, Sylow

#500

C₂×D₇₃

yes

—

—

—

—

—

146

—

#358

—

146

—

#357

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

#127

—

146

—

C₂×C₁₆

—

—

—

—

—

292

—

C₄×C₈

C₂²⋊D₇₃

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

Q₁₆

—

292

—

D₈

—

292

—

C₂×C₈

#11924013

yes

—

C₂×C₈

D₄×D₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

100

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

101

C₁₆

—

146

—

102

C₂³

—

584

—

103

Q₈

—

104

Q₈

—

584

—

105

D₄

—

584

—

106

D₄

—

584

—

107

C₂×C₄

—

108

C₂×C₄

—

584

—

109

C₂×C₄

#14556587

yes

characteristic

110

C₂×C₄

C₂×#11924013

yes

—

111

C₈

—

112

C₈

#14556614

yes

—

113

C₂²

—

1168

—

114

C₂²

—

1168

—

115

C₂²

#17988963

yes

characteristic

116

C₄

—

117

C₄

—

1168

—

118

C₄

#17988972

yes

—

119

C₄

#17989017

yes

—

120

C₂

—

2336

—

121

C₂

#22609538

yes

characteristic

122

C₂

#22612190

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1710	C₂
socle	C₂²×C₇₃	#493
prime radical	#17988964	C₂²
hypercenter	#1710	D₇₃
hypocenter	C₇₃	#12511
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1710

C₂

C₂²×C₇₃

C₂²

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #22612160	C₂	semidirect
4. C₈×#11924003	C₂	semidirect
5. C₇₃×#1710	C₂	semidirect
64. #1710	D₇₃	semidirect
67. C₇₃	#12511	semidirect

H ≅ G/N

type

1. #22612160

C₂

semidirect

4. C₈×#11924003

C₂

semidirect

5. C₇₃×#1710

C₂

semidirect

64. #1710

D₇₃

semidirect

67. C₇₃

#12511

semidirect

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

subgroup	index
#22612160	2
#22612156	2
#22609537	2
C₈×#11924003	2
C₇₃×#1710	2
#12511	73

subgroup

index

C₇₃×#1710

Subgroups — #33000680

Conjugacy classes of subgroups

Complements

Maximal subgroups