Group #32991090 — subgroups

Subgroups — #32991090

Order 18688.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#22610088	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×#14556523	—	2	6	no	—
6	C₂×#14556513	—	2	6	no	—
7	#17988457	C₂²	1	3	yes	—
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	6	no	—
13	C₂×C₄⋊Dic₇₃	—	2	3	no	—
14	#14556523	—	4	12	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	12	no	—
27	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
28	C₂×C₈×C₇₃	—	2	6	no	—
29	C₂×C₈×C₇₃	—	2	6	no	—
30	C₂×C₈×C₇₃	SD₁₆	1	12	yes	—
31	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₇₃	#41	1	3	yes	—
33	C₂³×C₇₃	#127	1	1	yes	characteristic
34	C₂²×D₇₃	—	4	2	no	—
35	C₂²×D₇₃	—	8	3	no	—
36	C₂²×D₇₃	—	8	3	no	—
37	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₇₃	#123	1	3	yes	—
39	C₂×C₄×C₇₃	C₂×SD₁₆	1	6	yes	—
40	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	6	yes	—
41	C₂×Dic₇₃	—	8	3	no	—
42	C₂×Dic₇₃	—	8	6	no	—
43	D₂₉₂	—	4	12	no	—
44	D₂₉₂	—	8	6	no	—
45	C₈×C₇₃	—	2	12	no	—
46	C₂²×C₇₃	#399	1	1	yes	characteristic
47	C₂²×C₇₃	#492	1	3	yes	—
48	C₂²×C₇₃	#496	1	3	yes	—
49	C₂×D₇₃	—	8	6	no	—
50	C₂×D₇₃	—	16	1	no	—
51	C₂×D₇₃	—	16	3	no	—
52	C₄×C₇₃	#476	1	6	yes	—
53	C₄×C₇₃	#481	1	6	yes	—
54	Dic₇₃	—	16	3	no	—
55	#10683	—	73	1	no	maximal, Sylow
56	C₂×C₇₃	#1705	1	1	yes	characteristic
57	C₂×C₇₃	#1715	1	3	yes	—
58	C₂×C₇₃	#2094	1	3	yes	—
59	D₇₃	—	16	2	no	—
60	#1957	—	73	3	no	—
61	#1844	—	73	3	no	—
62	#1273	D₇₃	1	1	yes	characteristic, complemented
63	C₇₃	#10683	1	1	yes	characteristic, complemented, Sylow
64	C₂×#127	—	73	1	no	—
65	C₂×#106	—	146	6	no	—
66	C₂×#102	—	146	6	no	—
67	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	—
68	#384	—	73	3	no	—
69	C₂²×D₄	—	146	3	no	—
70	C₂²×C₈	D₂₉₂	1	6	yes	—
71	#127	—	73	2	no	—
72	#127	—	146	3	no	—
73	C₂×C₄⋊C₄	—	146	3	no	—
74	C₂×C₄⋊C₄	—	146	6	no	—
75	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
76	#106	—	292	12	no	—
77	#102	—	292	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	292	1	no	—
80	C₂×D₄	—	292	3	no	—
81	C₂×D₄	—	292	6	no	—
82	C₂×D₄	—	292	6	no	—
83	C₂×D₄	—	146	6	no	—
84	C₂²×C₄	—	292	3	no	—
85	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	#11924017	1	12	yes	—
89	C₄⋊C₄	—	292	6	no	—
90	C₄⋊C₄	—	292	12	no	—
91	C₄²	Q₈×D₇₃	1	1	yes	characteristic
92	C₄²	#11924027	1	3	yes	—
93	C₂³	—	292	2	no	—
94	C₂³	—	584	3	no	—
95	C₂³	—	584	3	no	—
96	C₂³	#14556588	1	1	yes	characteristic
97	D₄	—	292	12	no	—
98	D₄	—	584	6	no	—
99	C₂×C₄	—	584	3	no	—
100	C₂×C₄	—	584	6	no	—
101	C₂×C₄	#14556571	1	3	yes	—
102	C₂×C₄	#14556575	1	3	yes	—
103	C₂×C₄	C₂×#11924017	1	6	yes	—
104	C₂×C₄	#14556619	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	584	6	no	—
107	C₂²	—	1168	1	no	—
108	C₂²	—	1168	3	no	—
109	C₂²	#17988524	1	1	yes	characteristic
110	C₂²	#17988960	1	3	yes	—
111	C₂²	#17988976	1	3	yes	—
112	C₄	—	1168	3	no	—
113	C₄	#17988878	1	6	yes	—
114	C₄	#17988906	1	6	yes	—
115	C₂	—	1168	2	no	—
116	C₂	#22609518	1	1	yes	characteristic
117	C₂	#22609559	1	3	yes	—
118	C₂	#22611463	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610737

C₂

yes

maximal

#22610088

C₂

yes

maximal, complemented

C₇₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14556588

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₂²×D₂₉₂

—

#14556588

—

#14556588

Q₈

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556523

—

#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

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

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

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₂²×C₇₃

#399

yes

characteristic

C₂²×C₇₃

#492

yes

—

C₂²×C₇₃

#496

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

#476

yes

—

C₄×C₇₃

#481

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2094

yes

—

D₇₃

—

#1957

—

#1844

—

#1273

D₇₃

yes

characteristic, complemented

C₇₃

#10683

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#106

—

146

—

C₂×#102

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

—

#384

—

C₂²×D₄

—

146

—

C₂²×C₈

D₂₉₂

yes

—

#127

—

#127

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

#106

—

292

—

#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

#11924017

yes

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄²

Q₈×D₇₃

yes

characteristic

C₄²

#11924027

yes

—

C₂³

—

292

—

C₂³

—

584

—

C₂³

—

584

—

C₂³

#14556588

yes

characteristic

D₄

—

292

—

D₄

—

584

—

C₂×C₄

—

584

—

100

C₂×C₄

—

584

—

101

C₂×C₄

#14556571

yes

—

102

C₂×C₄

#14556575

yes

—

103

C₂×C₄

C₂×#11924017

yes

—

104

C₂×C₄

#14556619

yes

—

105

C₈

—

106

C₂²

—

584

—

107

C₂²

—

1168

—

108

C₂²

—

1168

—

109

C₂²

#17988524

yes

characteristic

110

C₂²

#17988960

yes

—

111

C₂²

#17988976

yes

—

112

C₄

—

1168

—

113

C₄

#17988878

yes

—

114

C₄

#17988906

yes

—

115

C₂

—

1168

—

116

C₂

#22609518

yes

characteristic

117

C₂

#22609559

yes

—

118

C₂

#22611463

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	#10683
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. #22610088	C₂	semidirect
3. C₇₃×#1273	C₂	semidirect
62. #1273	D₇₃	semidirect
63. C₇₃	#10683	semidirect

H ≅ G/N

type

2. #22610088

C₂

semidirect

3. C₇₃×#1273

C₂

semidirect

62. #1273

D₇₃

semidirect

63. C₇₃

#10683

semidirect

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

subgroup	index
#22610737	2
#22610088	2
C₇₃×#1273	2
#10683	73

subgroup

index

#22610737

#22610088

C₇₃×#1273

#10683

Subgroups — #32991090

Conjugacy classes of subgroups

Complements

Maximal subgroups