Group #32990669 — subgroups

Subgroups — #32990669

Order 18688.

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

normal subgroups only

This group has 328 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	#22608722	C₂	1	3	yes	maximal, complemented
2	#22608600	C₂	1	3	yes	maximal
3	C₇₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556588	C₂²	1	1	yes	characteristic
5	#17988640	C₂²	1	3	yes	—
6	#17988339	—	2	3	no	—
7	#17988337	—	2	3	no	—
8	#17988330	—	2	3	no	—
9	#17988330	—	2	3	no	—
10	#423×C₇₃	C₂²	1	3	yes	—
11	C₂²×D₂₉₂	—	2	3	no	—
12	#14556588	—	2	3	no	—
13	#14556588	Q₈	1	2	yes	—
14	C₄⋊#10596831	D₄	1	3	yes	—
15	C₄⋊#10596831	D₄	1	3	yes	—
16	C₂²⋊#10596831	—	2	3	no	—
17	C₄×#10596831	—	2	3	no	—
18	C₂×M₁₆×C₇₃	—	2	3	no	—
19	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₇₃	—	2	3	no	—
21	C₄⋊C₈×C₇₃	—	2	3	no	—
22	C₂³×D₇₃	—	4	1	no	—
23	C₂×D₂₉₂	—	2	6	no	—
24	C₂×D₂₉₂	—	4	3	no	—
25	C₂×D₂₉₂	—	8	3	no	—
26	C₂×D₂₉₂	—	8	3	no	—
27	C₂×#10596831	—	2	3	no	—
28	C₂×#10596831	—	2	3	no	—
29	C₂²×C₄×C₇₃	#41	1	3	yes	—
30	M₁₆×C₇₃	—	8	3	no	—
31	C₂×C₈×C₇₃	—	4	3	no	—
32	C₄²×C₇₃	C₂×D₄	1	3	yes	—
33	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
34	C₂³×C₇₃	#126	1	1	yes	characteristic
35	C₂²×D₇₃	—	4	2	no	—
36	C₂²×D₇₃	—	8	6	no	—
37	C₂×C₄×C₇₃	—	2	3	no	—
38	C₂×C₄×C₇₃	—	2	3	no	—
39	C₂×C₄×C₇₃	#121	1	3	yes	—
40	C₂×C₄×C₇₃	C₂²⋊Q₈	1	3	yes	—
41	D₂₉₂	—	8	6	no	—
42	D₂₉₂	—	8	3	no	—
43	D₂₉₂	—	8	3	no	—
44	C₈×C₇₃	—	8	3	no	—
45	#10596831	—	4	3	no	—
46	#10596831	—	4	3	no	—
47	C₂²×C₇₃	—	2	3	no	—
48	C₂²×C₇₃	#396	1	1	yes	characteristic
49	C₂×D₇₃	—	8	6	no	—
50	C₂×D₇₃	—	16	4	no	—
51	C₄×C₇₃	—	2	3	no	—
52	C₄×C₇₃	#471	1	3	yes	—
53	C₄×C₇₃	#472	1	3	yes	—
54	#10612	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	—	4	1	no	—
56	C₂×C₇₃	#2063	1	3	yes	—
57	D₇₃	—	16	2	no	—
58	#1574	—	73	3	no	—
59	#1545	—	73	3	no	—
60	#1309	D₇₃	1	1	yes	characteristic, complemented
61	C₇₃	#10612	1	1	yes	characteristic, complemented, Sylow
62	C₂×#127	—	73	1	no	—
63	#431	—	73	3	no	—
64	#423	C₂×D₇₃	1	3	yes	—
65	#335	—	146	3	no	—
66	#334	—	146	3	no	—
67	#331	—	146	3	no	—
68	#331	—	146	3	no	—
69	C₂²×D₄	—	146	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	73	2	no	—
72	#127	—	146	3	no	—
73	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	73	3	no	—
77	C₄⋊C₈	—	73	3	no	—
78	C₂²⋊C₈	—	146	3	no	—
79	C₄×C₈	—	146	3	no	—
80	C₂⁴	—	292	1	no	—
81	C₂×D₄	—	292	3	no	—
82	C₂×D₄	—	584	3	no	—
83	C₂×D₄	—	584	3	no	—
84	C₂×D₄	—	146	6	no	—
85	C₂²×C₄	#11924028	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	146	3	no	—
89	C₂×C₈	—	146	3	no	—
90	C₄²	D₄×D₇₃	1	3	yes	—
91	C₄²	Q₈×D₇₃	1	1	yes	characteristic
92	C₂³	—	292	2	no	—
93	C₂³	—	584	6	no	—
94	C₂³	#14556585	1	1	yes	characteristic
95	D₄	—	584	6	no	—
96	D₄	—	584	3	no	—
97	D₄	—	584	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#14556564	1	3	yes	—
101	C₂×C₄	#14556569	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	292	3	no	—
104	C₈	—	292	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	584	6	no	—
107	C₂²	—	1168	4	no	—
108	C₂²	#17988511	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#17988853	1	3	yes	—
111	C₄	#17988856	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	1168	2	no	—
114	C₂	#22611292	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22608722

C₂

yes

maximal, complemented

#22608600

C₂

yes

maximal

C₇₃×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#14556588

C₂²

yes

characteristic

#17988640

C₂²

yes

—

—

—

—

—

#423×C₇₃

C₂²

yes

—

C₂²×D₂₉₂

—

#14556588

—

#14556588

Q₈

yes

—

C₄⋊#10596831

D₄

yes

—

C₄⋊#10596831

D₄

yes

—

C₂²⋊#10596831

—

C₄×#10596831

—

C₂×M₁₆×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×#10596831

—

C₂×#10596831

—

C₂²×C₄×C₇₃

#41

yes

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

#126

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#396

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#471

yes

—

C₄×C₇₃

#472

yes

—

#10612

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2063

yes

—

D₇₃

—

#1574

—

#1545

—

#1309

D₇₃

yes

characteristic, complemented

C₇₃

#10612

yes

characteristic, complemented, Sylow

C₂×#127

—

#431

—

#423

C₂×D₇₃

yes

—

#335

—

146

—

#334

—

146

—

#331

—

146

—

#331

—

146

—

C₂²×D₄

—

146

—

C₂×M₁₆

—

#127

—

#127

—

146

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

146

—

C₂⁴

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

584

—

C₂×D₄

—

584

—

C₂×D₄

—

146

—

C₂²×C₄

#11924028

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄²

D₄×D₇₃

yes

—

C₄²

Q₈×D₇₃

yes

characteristic

C₂³

—

292

—

C₂³

—

584

—

C₂³

#14556585

yes

characteristic

D₄

—

584

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#14556564

yes

—

101

C₂×C₄

#14556569

yes

—

102

C₈

—

103

C₈

—

292

—

104

C₈

—

292

—

105

C₂²

—

106

C₂²

—

584

—

107

C₂²

—

1168

—

108

C₂²

#17988511

yes

characteristic

109

C₄

—

110

C₄

#17988853

yes

—

111

C₄

#17988856

yes

—

112

C₂

—

113

C₂

—

1168

—

114

C₂

#22611292

yes

—

	subgroup	G/H
center	C₂²	#17988511
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1309	C₂
socle	C₂²×C₇₃	#396
prime radical	C₂×#14556588	C₂²
hypercenter	#1309	D₇₃
hypocenter	C₇₃	#10612
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#17988511

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1309

C₂

C₂²×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #22608722	C₂	semidirect
3. C₇₃×#1309	C₂	semidirect
60. #1309	D₇₃	semidirect
61. C₇₃	#10612	semidirect

H ≅ G/N

type

1. #22608722

C₂

semidirect

3. C₇₃×#1309

C₂

semidirect

60. #1309

D₇₃

semidirect

61. C₇₃

#10612

semidirect

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

subgroup	index
#22608722	2
#22608600	2
C₇₃×#1309	2
#10612	73

subgroup

index

#22608722

#22608600

C₇₃×#1309

#10612

Subgroups — #32990669

Conjugacy classes of subgroups

Complements

Maximal subgroups