Group #32979740 — subgroups

Subgroups — #32979740

Order 18688.

402 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 211 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	#22610083	C₂	1	2	yes	maximal
2	C₂×#17988340	C₂	1	4	yes	maximal
3	C₇₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10596831	—	2	4	no	—
5	C₂×C₄⋊#10596831	C₂²	1	4	yes	—
6	#17988340	—	2	8	no	—
7	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
8	C₄³×C₇₃	C₂²	1	1	yes	characteristic
9	#336×C₇₃	C₄	1	4	yes	—
10	C₂²×#10596831	—	2	4	no	—
11	C₄⋊#10596831	—	2	8	no	—
12	C₄⋊#10596831	—	4	8	no	—
13	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₇₃	D₄	1	2	yes	—
16	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
17	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₇₃	D₄	1	4	yes	—
19	C₄×C₈×C₇₃	Q₈	1	4	yes	—
20	C₂×#10596831	—	4	4	no	—
21	C₂×#10596831	—	4	8	no	—
22	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
23	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈×C₇₃	—	2	8	no	—
27	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
28	C₄²×C₇₃	—	2	4	no	—
29	C₄²×C₇₃	—	4	2	no	—
30	C₄²×C₇₃	D₈	1	4	yes	—
31	C₄²×C₇₃	Q₁₆	1	4	yes	—
32	C₄²×C₇₃	C₂×D₄	1	2	yes	—
33	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
34	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₃	—	2	4	no	—
36	C₂×C₄×C₇₃	—	2	4	no	—
37	C₂×C₄×C₇₃	—	2	2	no	—
38	C₂×C₄×C₇₃	#107	1	8	yes	—
39	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#131	1	4	yes	—
44	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
46	C₈×C₇₃	—	2	8	no	—
47	#10596831	—	8	4	no	—
48	C₂²×C₇₃	#384	1	1	yes	characteristic
49	C₂²×C₇₃	#424	1	2	yes	—
50	C₂²×C₇₃	C₂×#107	1	2	yes	—
51	C₂²×C₇₃	#430	1	2	yes	—
52	C₄×C₇₃	—	2	8	no	—
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	#424	1	4	yes	—
55	C₄×C₇₃	#493	1	1	yes	characteristic
56	C₄×C₇₃	#494	1	1	yes	characteristic
57	C₄×C₇₃	#500	1	2	yes	—
58	#8810	—	73	1	no	maximal, Sylow
59	C₂×C₇₃	#1571	1	4	yes	—
60	C₂×C₇₃	#1843	1	1	yes	characteristic
61	C₂×C₇₃	#1845	1	1	yes	characteristic
62	C₂×C₇₃	#1849	1	1	yes	characteristic
63	#1842	—	73	2	no	—
64	#1744	D₇₃	1	1	yes	characteristic
65	C₂×#335	—	73	4	no	—
66	C₇₃	#8810	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	73	4	no	—
68	C₂×C₄⋊C₈	—	146	4	no	—
69	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
70	C₄³	C₂×D₇₃	1	1	yes	characteristic
71	#336	Dic₇₃	1	4	yes	—
72	#335	—	146	8	no	—
73	C₂²×C₈	—	146	4	no	—
74	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
75	C₂×C₄²	Dic₁₄₆	1	2	yes	—
76	C₂×C₄²	D₂₉₂	1	2	yes	—
77	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
78	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
79	C₄⋊C₈	—	292	8	no	—
80	C₄⋊C₈	—	146	8	no	—
81	C₄×C₈	Dic₁₄₆	1	4	yes	—
82	C₄×C₈	D₂₉₂	1	4	yes	—
83	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
84	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
86	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	292	4	no	—
89	C₂×C₈	—	292	8	no	—
90	C₂×C₈	C₄⋊Dic₇₃	1	8	yes	—
91	C₄²	—	2	4	no	—
92	C₄²	—	4	2	no	—
93	C₄²	D₅₈₄	1	4	yes	—
94	C₄²	Dic₂₉₂	1	4	yes	—
95	C₄²	C₂×D₂₉₂	1	2	yes	—
96	C₄²	C₂×Dic₁₄₆	1	2	yes	—
97	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	#14556525	1	8	yes	—
102	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
103	C₂×C₄	#14556588	1	1	yes	characteristic
104	C₂×C₄	#14556591	1	2	yes	—
105	C₂×C₄	#14556591	1	1	yes	characteristic
106	C₂×C₄	#14556598	1	4	yes	—
107	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
109	C₈	—	2	8	no	—
110	C₈	—	584	4	no	—
111	C₂²	#17988457	1	1	yes	characteristic
112	C₂²	#17988614	1	2	yes	—
113	C₂²	C₂×#14556525	1	2	yes	—
114	C₂²	#17988636	1	2	yes	—
115	C₄	—	2	8	no	—
116	C₄	—	2	2	no	—
117	C₄	#17988614	1	4	yes	—
118	C₄	#17988964	1	1	yes	characteristic
119	C₄	#17988967	1	1	yes	characteristic
120	C₄	#17988992	1	2	yes	—
121	C₂	#22608706	1	4	yes	—
122	C₂	#22610085	1	1	yes	characteristic
123	C₂	#22610091	1	1	yes	characteristic
124	C₂	#22610106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610083

C₂

yes

maximal

C₂×#17988340

C₂

yes

maximal

C₇₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

C₂²

yes

—

#17988340

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

characteristic

#336×C₇₃

C₄

yes

—

C₂²×#10596831

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

Q₈

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₈

yes

—

C₄²×C₇₃

Q₁₆

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#384

yes

characteristic

C₂²×C₇₃

#424

yes

—

C₂²×C₇₃

C₂×#107

yes

—

C₂²×C₇₃

#430

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#424

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#8810

—

maximal, Sylow

C₂×C₇₃

#1571

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

characteristic

C₂×#335

—

C₇₃

#8810

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

—

C₄³

C₂×D₇₃

yes

characteristic

#336

Dic₇₃

yes

—

#335

—

146

—

C₂²×C₈

—

146

—

C₂²×C₈

C₂×Dic₇₃

yes

—

C₂×C₄²

Dic₁₄₆

yes

—

C₂×C₄²

D₂₉₂

yes

—

C₂×C₄²

C₂×Dic₇₃

yes

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

C₄×C₈

Dic₁₄₆

yes

—

C₄×C₈

D₂₉₂

yes

—

C₂²×C₄

C₄⋊Dic₇₃

yes

—

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₄₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

292

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₄²

—

C₄²

—

C₄²

D₅₈₄

yes

—

C₄²

Dic₂₉₂

yes

—

C₄²

C₂×D₂₉₂

yes

—

C₄²

C₂×Dic₁₄₆

yes

—

C₂³

C₂×C₄⋊Dic₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#14556525

yes

—

102

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

—

103

C₂×C₄

#14556588

yes

characteristic

104

C₂×C₄

#14556591

yes

—

105

C₂×C₄

#14556591

yes

characteristic

106

C₂×C₄

#14556598

yes

—

107

C₂×C₄

C₂×D₅₈₄

yes

—

108

C₂×C₄

C₂×Dic₂₉₂

yes

—

109

C₈

—

110

C₈

—

584

—

111

C₂²

#17988457

yes

characteristic

112

C₂²

#17988614

yes

—

113

C₂²

C₂×#14556525

yes

—

114

C₂²

#17988636

yes

—

115

C₄

—

116

C₄

—

117

C₄

#17988614

yes

—

118

C₄

#17988964

yes

characteristic

119

C₄

#17988967

yes

characteristic

120

C₄

#17988992

yes

—

121

C₂

#22608706

yes

—

122

C₂

#22610085

yes

characteristic

123

C₂

#22610091

yes

characteristic

124

C₂

#22610106

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1744	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1744	D₇₃
hypocenter	C₇₃	#8810
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₉₂

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1744

C₂

socle, prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
66. C₇₃	#8810	semidirect

H ≅ G/N

type

66. C₇₃

#8810

semidirect

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

subgroup	index
#22610083	2
C₂×#17988340	2
C₇₃×#1744	2
#8810	73

subgroup

index

#22610083

C₂×#17988340

C₇₃×#1744

#8810

Subgroups — #32979740

Conjugacy classes of subgroups

Complements

Maximal subgroups