Group #32988750 — subgroups

Subgroups — #32988750

Order 18688.

478 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22621325	C₂	1	3	yes	maximal
2	#22610907	C₂	1	3	yes	maximal
3	C₇₃×#1309	C₂	1	1	yes	maximal, characteristic
4	#17989144	C₂²	1	3	yes	—
5	#17988873	—	2	6	no	—
6	#17988845	—	2	6	no	—
7	#17988798	—	2	6	no	—
8	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
9	#17988348	—	2	6	no	—
10	#423×C₇₃	C₂²	1	3	yes	—
11	C₂×Dic₂₉₂	—	2	6	no	—
12	#14556591	D₄	1	6	yes	—
13	#14556583	—	2	3	no	—
14	#14556570	—	2	6	no	—
15	C₄×Dic₁₄₆	—	2	6	no	—
16	#14556550	—	2	3	no	—
17	C₂×C₄×Dic₇₃	—	2	3	no	—
18	#14556523	—	2	6	no	—
19	#14556516	—	2	12	no	—
20	C₂×M₁₆×C₇₃	D₄	1	6	yes	—
21	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₇₃	—	2	6	no	—
23	C₂×Dic₁₄₆	—	2	6	no	—
24	C₂²×Dic₇₃	—	4	1	no	—
25	Dic₂₉₂	—	8	6	no	—
26	C₄⋊Dic₇₃	—	2	6	no	—
27	C₄⋊Dic₇₃	—	2	6	no	—
28	#11924007	—	4	6	no	—
29	C₂²⋊Dic₇₃	—	4	3	no	—
30	C₄×Dic₇₃	—	2	6	no	—
31	C₄×Dic₇₃	—	4	3	no	—
32	C₄×Dic₇₃	—	4	3	no	—
33	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
34	M₁₆×C₇₃	—	4	6	no	—
35	C₂×C₈×C₇₃	—	2	6	no	—
36	C₄²×C₇₃	C₂×D₄	1	3	yes	—
37	C₄²×C₇₃	#41	1	1	yes	characteristic
38	C₂³×C₇₃	#127	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	2	3	no	—
40	C₂×C₄×C₇₃	—	2	3	no	—
41	C₂×C₄×C₇₃	#120	1	3	yes	—
42	C₂×C₄×C₇₃	#121	1	3	yes	—
43	C₂×Dic₇₃	—	4	6	no	—
44	C₂×Dic₇₃	—	4	6	no	—
45	Dic₁₄₆	—	4	6	no	—
46	Dic₁₄₆	—	4	6	no	—
47	C₈×C₇₃	—	4	6	no	—
48	C₂²×C₇₃	—	2	3	no	—
49	C₂²×C₇₃	#392	1	1	yes	characteristic
50	C₄×C₇₃	—	2	3	no	—
51	C₄×C₇₃	#455	1	3	yes	—
52	C₄×C₇₃	#456	1	3	yes	—
53	Dic₇₃	—	4	4	no	—
54	Dic₇₃	—	8	6	no	—
55	#10303	—	73	1	no	maximal, Sylow
56	C₂×C₇₃	—	4	1	no	—
57	C₂×C₇₃	#2007	1	3	yes	—
58	#3255	—	73	3	no	—
59	#1994	—	73	3	no	—
60	#1309	D₇₃	1	1	yes	characteristic
61	C₇₃	#10303	1	1	yes	characteristic, complemented, Sylow
62	#533	—	73	3	no	—
63	#475	—	146	6	no	—
64	#470	—	146	6	no	—
65	#462	—	146	6	no	—
66	#423	C₂×D₇₃	1	3	yes	—
67	C₄³	—	73	1	no	—
68	#339	—	146	6	no	—
69	C₂×Q₁₆	—	146	6	no	—
70	C₂×M₁₆	D₂₉₂	1	6	yes	—
71	C₄⋊Q₈	—	73	6	no	—
72	#125	—	146	3	no	—
73	C₂²⋊Q₈	—	146	6	no	—
74	C₄×Q₈	—	146	6	no	—
75	#117	—	146	3	no	—
76	C₂×C₄²	—	146	3	no	—
77	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
78	#106	—	146	6	no	—
79	C₄⋊C₈	—	2	6	no	—
80	#103	—	146	12	no	—
81	C₂×Q₈	—	146	6	no	—
82	C₂²×C₄	—	292	1	no	—
83	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
84	Q₁₆	—	584	6	no	—
85	M₁₆	—	4	6	no	—
86	C₂×C₈	—	2	6	no	—
87	C₄⋊C₄	—	292	6	no	—
88	C₄⋊C₄	—	146	6	no	—
89	C₄⋊C₄	—	146	6	no	—
90	C₂²⋊C₄	—	292	3	no	—
91	C₄²	—	292	3	no	—
92	C₄²	—	292	3	no	—
93	C₄²	—	146	6	no	—
94	C₄²	D₄×D₇₃	1	3	yes	—
95	C₄²	#11924029	1	1	yes	characteristic
96	C₂³	#14556588	1	1	yes	characteristic
97	Q₈	—	292	6	no	—
98	Q₈	—	292	6	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	292	6	no	—
102	C₂×C₄	—	292	6	no	—
103	C₂×C₄	#14556561	1	3	yes	—
104	C₂×C₄	#14556566	1	3	yes	—
105	C₈	—	4	6	no	—
106	C₂²	—	2	3	no	—
107	C₂²	#17988497	1	1	yes	characteristic
108	C₄	—	2	3	no	—
109	C₄	—	292	4	no	—
110	C₄	—	584	6	no	—
111	C₄	#17988767	1	3	yes	—
112	C₄	#17988772	1	3	yes	—
113	C₂	—	4	1	no	—
114	C₂	#22610980	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22621325

C₂

yes

maximal

#22610907

C₂

yes

maximal

C₇₃×#1309

C₂

yes

maximal, characteristic

#17989144

C₂²

yes

—

#17988873

—

#17988845

—

#17988798

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988348

—

#423×C₇₃

C₂²

yes

—

C₂×Dic₂₉₂

—

#14556591

D₄

yes

—

#14556583

—

#14556570

—

C₄×Dic₁₄₆

—

#14556550

—

C₂×C₄×Dic₇₃

—

#14556523

—

#14556516

—

C₂×M₁₆×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#120

yes

—

C₂×C₄×C₇₃

#121

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#392

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#455

yes

—

C₄×C₇₃

#456

yes

—

Dic₇₃

—

Dic₇₃

—

#10303

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2007

yes

—

#3255

—

#1994

—

#1309

D₇₃

yes

characteristic

C₇₃

#10303

yes

characteristic, complemented, Sylow

#533

—

#475

—

146

—

#470

—

146

—

#462

—

146

—

#423

C₂×D₇₃

yes

—

C₄³

—

#339

—

146

—

C₂×Q₁₆

—

146

—

C₂×M₁₆

D₂₉₂

yes

—

C₄⋊Q₈

—

#125

—

146

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

146

—

#117

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

#106

—

146

—

C₄⋊C₈

—

#103

—

146

—

C₂×Q₈

—

146

—

C₂²×C₄

—

292

—

C₂²×C₄

C₂×D₂₉₂

yes

—

Q₁₆

—

584

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₄²

—

292

—

C₄²

—

292

—

C₄²

—

146

—

C₄²

D₄×D₇₃

yes

—

C₄²

yes

C₂³

yes

Q₈

—

292

—

Q₈

—

292

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

292

—

102

C₂×C₄

—

292

—

103

C₂×C₄

#14556561

yes

—

104

C₂×C₄

#14556566

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

#17988497

yes

characteristic

108

C₄

—

109

C₄

—

292

—

110

C₄

—

584

—

111

C₄

#17988767

yes

—

112

C₄

#17988772

yes

—

113

C₂

—

114

C₂

#22610980

yes

—

	subgroup	G/H
center	C₂²	#17988497
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1309	C₂
socle	C₂²×C₇₃	#392
prime radical	C₂³×C₇₃	#127
hypercenter	#1309	D₇₃
hypocenter	C₇₃	#10303
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#17988497

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1309

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
61. C₇₃	#10303	semidirect

H ≅ G/N

type

61. C₇₃

#10303

semidirect

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

subgroup	index
#22621325	2
#22610907	2
C₇₃×#1309	2
#10303	73

subgroup

index

#22621325

#22610907

C₇₃×#1309

#10303

Subgroups — #32988750

Conjugacy classes of subgroups

Complements

Maximal subgroups