Group #32973621 — subgroups

Subgroups — #32973621

Order 18688.

132 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22612184	C₂	1	1	yes	maximal, characteristic
2	#22608003	C₂	1	1	yes	maximal, characteristic
3	C₇₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₈₄	—	2	1	no	—
5	C₂×Dic₅₈₄	C₄	1	2	yes	—
6	#17988967	—	2	1	no	—
7	#17988340	—	2	1	no	—
8	C₂×C₃₂×C₇₃	—	2	1	no	—
9	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₉₂	—	2	2	no	—
11	C₂×Dic₂₉₂	—	4	1	no	—
12	#14556591	—	4	1	no	—
13	Dic₅₈₄	—	4	2	no	—
14	Dic₅₈₄	—	4	1	no	—
15	Dic₅₈₄	—	4	1	no	—
16	C₄⋊#10596831	—	4	1	no	—
17	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
19	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
20	C₃₂×C₇₃	—	4	1	no	—
21	C₂×Dic₁₄₆	—	4	2	no	—
22	Dic₂₉₂	—	4	2	no	—
23	Dic₂₉₂	—	4	2	no	—
24	Dic₂₉₂	—	8	2	no	—
25	C₂×#10596831	—	8	1	no	—
26	C₄⋊Dic₇₃	—	8	1	no	—
27	C₄⋊Dic₇₃	—	8	1	no	—
28	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
30	C₄²×C₇₃	D₈	1	1	yes	characteristic
31	C₁₆×C₇₃	—	2	1	no	—
32	C₁₆×C₇₃	D₈	1	1	yes	characteristic
33	C₁₆×C₇₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
35	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
36	C₂×C₄×C₇₃	SD₃₂	1	1	yes	characteristic
37	C₂×Dic₇₃	—	8	2	no	—
38	Dic₁₄₆	—	8	2	no	—
39	Dic₁₄₆	—	8	2	no	—
40	C₈×C₇₃	—	2	1	no	—
41	C₈×C₇₃	#100	1	1	yes	characteristic
42	C₈×C₇₃	#102	1	1	yes	characteristic
43	#10596831	—	16	1	no	—
44	C₂²×C₇₃	#357	1	1	yes	characteristic
45	C₄×C₇₃	—	2	1	no	—
46	C₄×C₇₃	#331	1	1	yes	characteristic
47	C₄×C₇₃	#361	1	1	yes	characteristic
48	C₄×C₇₃	SD₆₄	1	1	yes	characteristic
49	C₄×C₇₃	Q₆₄	1	1	yes	characteristic
50	Dic₇₃	—	16	2	no	—
51	#7445	—	73	1	no	maximal, Sylow
52	C₂×C₇₃	#1361	1	1	yes	characteristic
53	C₂×C₇₃	#1416	1	1	yes	characteristic
54	C₂×C₇₃	#1420	1	1	yes	characteristic
55	#2247	—	73	1	no	—
56	C₄⋊C₃₂	D₇₃	1	1	yes	characteristic
57	#1371	—	73	1	no	—
58	C₇₃	#7445	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₃₂	—	73	2	no	—
60	C₂×Q₃₂	—	146	1	no	—
61	#494	—	146	1	no	—
62	C₂×C₃₂	—	2	1	no	—
63	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
64	#335	—	146	1	no	—
65	C₂×Q₁₆	—	292	1	no	—
66	C₂×Q₁₆	—	146	2	no	—
67	C₄⋊Q₈	—	292	1	no	—
68	Q₃₂	—	292	2	no	—
69	Q₃₂	—	292	1	no	—
70	Q₃₂	—	292	1	no	—
71	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
72	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
73	C₄⋊C₈	—	292	1	no	—
74	C₄×C₈	D₂₉₂	1	1	yes	characteristic
75	C₃₂	—	4	1	no	—
76	C₂×Q₈	—	292	2	no	—
77	Q₁₆	—	292	2	no	—
78	Q₁₆	—	292	2	no	—
79	Q₁₆	—	584	2	no	—
80	C₂×C₈	—	584	1	no	—
81	C₂×C₈	#11924005	1	1	yes	characteristic
82	C₂×C₈	#11924017	1	1	yes	characteristic
83	C₄⋊C₄	—	584	1	no	—
84	C₄⋊C₄	—	584	1	no	—
85	C₄²	D₅₈₄	1	1	yes	characteristic
86	C₁₆	—	2	1	no	—
87	C₁₆	#11924013	1	1	yes	characteristic
88	C₁₆	#11924015	1	1	yes	characteristic
89	Q₈	—	584	2	no	—
90	Q₈	—	584	2	no	—
91	C₂×C₄	—	584	2	no	—
92	C₂×C₄	#14556513	1	1	yes	characteristic
93	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
94	C₂×C₄	#14556536	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	1168	1	no	—
97	C₈	#14556507	1	1	yes	characteristic
98	C₈	#14556511	1	1	yes	characteristic
99	C₂²	#17988389	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	1168	2	no	—
102	C₄	#17988330	1	1	yes	characteristic
103	C₄	#17988401	1	1	yes	characteristic
104	C₄	#17988426	1	1	yes	characteristic
105	C₄	Dic₁₁₆₈	1	1	yes	characteristic
106	C₂	#22607978	1	1	yes	characteristic
107	C₂	#22608113	1	1	yes	characteristic
108	C₂	#22608125	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612184

C₂

yes

maximal, characteristic

#22608003

C₂

yes

maximal, characteristic

C₇₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

C₄

yes

—

#17988967

—

#17988340

—

C₂×C₃₂×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

#14556591

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

—

C₄⋊#10596831

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

characteristic

C₂×C₄×C₇₃

#102

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

SD₃₂

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

#357

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

SD₆₄

yes

characteristic

C₄×C₇₃

Q₆₄

yes

characteristic

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₄⋊C₃₂

D₇₃

yes

characteristic

#1371

—

C₇₃

#7445

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

146

—

#494

—

146

—

C₂×C₃₂

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

146

—

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

292

—

Q₃₂

—

292

—

Q₃₂

—

292

—

Q₃₂

—

292

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₄⋊C₈

—

292

—

C₄×C₈

D₂₉₂

yes

characteristic

C₃₂

—

C₂×Q₈

—

292

—

Q₁₆

—

292

—

Q₁₆

—

292

—

Q₁₆

—

584

—

C₂×C₈

—

584

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

584

—

C₄⋊C₄

—

584

—

C₄²

D₅₈₄

yes

characteristic

C₁₆

—

C₁₆

yes

C₁₆

yes

Q₈

—

584

—

Q₈

—

584

—

C₂×C₄

—

584

—

C₂×C₄

#14556513

yes

characteristic

C₂×C₄

D₁₁₆₈

yes

characteristic

C₂×C₄

#14556536

yes

characteristic

C₈

—

C₈

—

1168

—

C₈

yes

C₈

yes

C₂²

yes

100

C₄

—

101

C₄

—

1168

—

102

C₄

#17988330

yes

characteristic

103

C₄

#17988401

yes

characteristic

104

C₄

#17988426

yes

characteristic

105

C₄

Dic₁₁₆₈

yes

characteristic

106

C₂

#22607978

yes

characteristic

107

C₂

#22608113

yes

characteristic

108

C₂

#22608125

yes

characteristic

	subgroup	G/H
center	C₂²	#17988389
commutator	C₂×C₁₆×C₇₃	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₃
Fitting	C₇₃×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₇₃	#357
cosocle	C₄×C₁₆×C₇₃	C₂²
hypercenter	C₄⋊C₃₂	D₇₃
hypocenter	C₇₃	#7445
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center

C₂²

#17988389

commutator

C₂×C₁₆×C₇₃

C₂×C₄

Frattini

C₄×C₁₆

C₂×D₇₃

Fitting

C₇₃×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₇₃

#357

cosocle

C₄×C₁₆×C₇₃

C₂²

hypercenter

C₄⋊C₃₂

D₇₃

hypocenter

C₇₃

#7445

LCS

G ▷ C₂×C₁₆×C₇₃ ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
58. C₇₃	#7445	semidirect

H ≅ G/N

type

58. C₇₃

#7445

semidirect

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

subgroup	index
#22612184	2
#22608003	2
C₇₃×C₄⋊C₃₂	2
#7445	73

subgroup

index

#22612184

#22608003

C₇₃×C₄⋊C₃₂

#7445

Subgroups — #32973621

Conjugacy classes of subgroups

Complements

Maximal subgroups