Group #33007009 — subgroups

Subgroups — #33007009

Order 18688.

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

normal subgroups only

This group has 158 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	#22612206	C₂	1	1	yes	maximal, characteristic
2	#22608132	C₂	1	4	yes	maximal
3	C₄×#14556497	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2252	C₂	1	1	yes	maximal, characteristic
5	#17988992	—	2	1	no	—
6	C₂×#14556497	C₂²	1	2	yes	—
7	#17988413	—	2	1	no	—
8	#17988413	C₂²	1	2	yes	—
9	#500×C₇₃	—	2	1	no	—
10	#366×C₇₃	—	2	1	no	—
11	#366×C₇₃	C₂²	1	2	yes	—
12	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	#14556591	—	4	1	no	—
14	#14556525	—	2	2	no	—
15	#14556525	—	4	1	no	—
16	#14556497	Q₈	1	4	yes	complemented
17	C₄⋊Q₈×C₇₃	—	4	1	no	—
18	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
20	#107×C₇₃	—	2	2	no	—
21	#107×C₇₃	—	4	1	no	—
22	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	8	1	no	—
24	C₄⋊Dic₇₃	—	4	2	no	—
25	C₄⋊Dic₇₃	—	8	1	no	—
26	C₂×Q₈×C₇₃	—	8	1	no	—
27	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
28	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	4	2	no	—
30	C₄⋊C₄×C₇₃	—	8	1	no	—
31	C₄²×C₇₃	D₈	1	1	yes	characteristic
32	C₁₆×C₇₃	Q₁₆	1	2	yes	—
33	C₁₆×C₇₃	C₂×Q₈	1	2	yes	—
34	Q₈×C₇₃	—	16	2	no	—
35	C₂×C₄×C₇₃	—	8	2	no	—
36	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
37	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
39	C₂×Dic₇₃	—	8	2	no	—
40	Dic₁₄₆	—	16	2	no	—
41	C₈×C₇₃	Q₃₂	1	2	yes	—
42	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
43	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
44	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
45	C₄×C₇₃	—	16	2	no	—
46	C₄×C₇₃	SD₆₄	1	2	yes	—
47	C₄×C₇₃	SD₆₄	1	2	yes	—
48	C₄×C₇₃	#500	1	1	yes	characteristic
49	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
50	Dic₇₃	—	16	2	no	—
51	#13733	—	73	1	no	maximal, Sylow
52	C₂×C₇₃	#2252	1	1	yes	characteristic
53	C₂×C₇₃	C₂×SD₆₄	1	1	yes	characteristic
54	C₂×C₇₃	C₂×SD₆₄	1	1	yes	characteristic
55	#2252	—	73	1	no	—
56	#2252	D₇₃	1	1	yes	characteristic
57	#1424	—	73	4	no	—
58	C₄×C₃₂	—	73	1	no	—
59	C₇₃	#13733	1	1	yes	characteristic, complemented, Sylow
60	#500	—	2	1	no	—
61	#500	—	146	1	no	—
62	C₂×C₃₂	—	73	2	no	—
63	#366	—	2	1	no	—
64	#366	—	73	2	no	—
65	#366	—	146	1	no	—
66	#366	C₂×D₇₃	1	2	yes	—
67	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
68	C₄⋊Q₈	—	4	1	no	—
69	C₄⋊Q₈	—	292	1	no	—
70	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
71	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
72	#107	—	2	2	no	—
73	#107	—	4	1	no	—
74	#107	—	292	1	no	—
75	#107	—	146	2	no	—
76	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
77	C₃₂	—	73	4	no	—
78	C₂×Q₈	—	8	1	no	—
79	C₂×Q₈	—	584	1	no	—
80	C₂×C₈	#11924013	1	1	yes	characteristic
81	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	8	1	no	—
84	C₄⋊C₄	—	292	2	no	—
85	C₄⋊C₄	—	584	1	no	—
86	C₄²	#11924013	1	1	yes	characteristic
87	C₁₆	#11924018	1	2	yes	—
88	C₁₆	Q₈×D₇₃	1	2	yes	—
89	Q₈	—	16	2	no	—
90	Q₈	—	1168	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	—	584	2	no	—
93	C₂×C₄	#14556532	1	1	yes	characteristic
94	C₂×C₄	#14556532	1	1	yes	characteristic
95	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
96	C₈	#14556537	1	2	yes	—
97	C₈	#14556589	1	1	yes	characteristic
98	C₈	C₂×#11924018	1	1	yes	characteristic
99	C₂²	C₂×#14556532	1	1	yes	characteristic
100	C₄	—	16	2	no	—
101	C₄	—	1168	2	no	—
102	C₄	#17988424	1	2	yes	—
103	C₄	#17988425	1	2	yes	—
104	C₄	#17988990	1	1	yes	characteristic
105	C₄	C₂×#14556537	1	1	yes	characteristic
106	C₂	#22612204	1	1	yes	characteristic
107	C₂	C₂×#17988424	1	1	yes	characteristic
108	C₂	C₂×#17988425	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22608132

C₂

yes

maximal

C₄×#14556497

C₂

yes

maximal, characteristic

C₇₃×#2252

C₂

yes

maximal, characteristic

#17988992

—

C₂×#14556497

C₂²

yes

—

#17988413

—

#17988413

C₂²

yes

—

#500×C₇₃

—

#366×C₇₃

—

#366×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

Q₈

yes

complemented

C₄⋊Q₈×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

Q₁₆

yes

—

C₁₆×C₇₃

C₂×Q₈

yes

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

Q₃₂

yes

—

C₈×C₇₃

C₄⋊Q₈

yes

characteristic

C₈×C₇₃

C₂×Q₁₆

yes

characteristic

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

SD₆₄

yes

—

C₄×C₇₃

SD₆₄

yes

—

C₄×C₇₃

#500

yes

characteristic

C₄×C₇₃

C₂×Q₃₂

yes

characteristic

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×SD₆₄

yes

characteristic

C₂×C₇₃

C₂×SD₆₄

yes

characteristic

#2252

—

#2252

D₇₃

yes

characteristic

#1424

—

C₄×C₃₂

—

C₇₃

#13733

yes

characteristic, complemented, Sylow

#500

—

#500

—

146

—

C₂×C₃₂

—

#366

—

#366

—

#366

—

146

—

#366

C₂×D₇₃

yes

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

292

—

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

—

—

—

292

—

#107

—

146

—

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

584

—

C₂×C₈

#11924013

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

584

—

C₄²

#11924013

yes

characteristic

C₁₆

#11924018

yes

—

C₁₆

Q₈×D₇₃

yes

—

Q₈

—

Q₈

—

1168

—

C₂×C₄

—

C₂×C₄

—

584

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

yes

—

C₈

yes

C₈

yes

C₂²

yes

100

C₄

—

101

C₄

—

1168

—

102

C₄

#17988424

yes

—

103

C₄

#17988425

yes

—

104

C₄

#17988990

yes

characteristic

105

C₄

C₂×#14556537

yes

characteristic

106

C₂

#22612204

yes

characteristic

107

C₂

C₂×#17988424

yes

characteristic

108

C₂

C₂×#17988425

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556532
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#2252	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
hypercenter	#2252	D₇₃
hypocenter	C₇₃	#13733
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

subgroup

G/H

center

C₂²

C₂×#14556532

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#2252

C₂

socle, prime radical

C₂²×C₇₃

C₂×D₁₆

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

N	H ≅ G/N	type
16. #14556497	Q₈	semidirect
59. C₇₃	#13733	semidirect

H ≅ G/N

type

16. #14556497

Q₈

semidirect

59. C₇₃

#13733

semidirect

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

subgroup	index
#22612206	2
#22608132	2
C₄×#14556497	2
C₇₃×#2252	2
#13733	73

subgroup

index

C₇₃×#2252

Subgroups — #33007009

Conjugacy classes of subgroups

Complements

Maximal subgroups