Group #33000702 — subgroups

Subgroups — #33000702

Order 18688.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22612187	C₂	1	4	yes	maximal, complemented
2	#22609549	C₂	1	1	yes	maximal, characteristic, complemented
3	#22609546	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556536	—	2	8	no	—
6	#17988992	—	2	2	no	—
7	#17988967	C₂²	1	2	yes	—
8	#17988964	—	2	2	no	—
9	#17988964	C₂²	1	2	yes	—
10	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
11	C₈²×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₉₂	—	2	4	no	—
13	C₂×D₅₈₄	—	2	4	no	—
14	C₂×D₅₈₄	—	4	4	no	—
15	#14556591	—	2	2	no	—
16	#14556588	—	2	2	no	—
17	#14556536	—	4	16	no	—
18	#14556525	—	4	4	no	—
19	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
20	C₄×C₈×C₇₃	D₄	1	2	yes	—
21	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₄₆	—	4	2	no	—
23	C₂×D₂₉₂	—	4	2	no	—
24	C₂×D₂₉₂	—	4	4	no	—
25	Dic₂₉₂	—	4	8	no	—
26	D₅₈₄	—	4	8	no	—
27	D₅₈₄	—	8	8	no	—
28	C₄⋊Dic₇₃	—	4	4	no	—
29	C₂×C₈×C₇₃	D₈	1	4	yes	—
30	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
31	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
32	C₁₆×C₇₃	D₈	1	8	yes	complemented
33	C₂²×D₇₃	—	8	2	no	—
34	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
36	C₂×Dic₇₃	—	8	2	no	—
37	D₂₉₂	—	8	4	no	—
38	D₂₉₂	—	8	8	no	—
39	Dic₁₄₆	—	8	4	no	—
40	C₈×C₇₃	SD₃₂	1	8	yes	complemented
41	C₈×C₇₃	C₂×D₈	1	4	yes	—
42	C₂²×C₇₃	#493	1	1	yes	characteristic
43	C₂×D₇₃	—	16	2	no	—
44	C₂×D₇₃	—	16	4	no	—
45	C₄×C₇₃	#493	1	2	yes	—
46	C₄×C₇₃	C₂×SD₃₂	1	4	yes	—
47	Dic₇₃	—	16	2	no	—
48	#12517	—	73	1	no	maximal, Sylow
49	C₂×C₇₃	#1712	1	1	yes	characteristic
50	C₂×C₇₃	#2248	1	2	yes	—
51	D₇₃	—	32	2	no	—
52	#2248	—	73	4	no	—
53	#1713	—	73	1	no	—
54	#1712	—	73	1	no	—
55	C₈×C₁₆	D₇₃	1	1	yes	characteristic, complemented
56	C₇₃	#12517	1	1	yes	characteristic, complemented, Sylow
57	C₂×SD₃₂	—	146	8	no	—
58	#500	—	146	2	no	—
59	#494	—	73	2	no	—
60	#493	—	73	2	no	—
61	#493	—	146	2	no	—
62	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
63	C₈²	C₂×D₇₃	1	1	yes	characteristic
64	C₂×Q₁₆	—	146	4	no	—
65	C₂×D₈	—	292	4	no	—
66	C₂×D₈	—	146	4	no	—
67	C₄⋊Q₈	—	146	2	no	—
68	#127	—	146	2	no	—
69	SD₃₂	—	292	16	no	—
70	C₂×C₁₆	D₂₉₂	1	4	yes	—
71	#107	—	292	4	no	—
72	C₄×C₈	D₂₉₂	1	2	yes	—
73	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
74	C₂×Q₈	—	292	2	no	—
75	C₂×D₄	—	292	2	no	—
76	C₂×D₄	—	292	4	no	—
77	Q₁₆	—	292	8	no	—
78	D₈	—	292	8	no	—
79	D₈	—	584	8	no	—
80	C₂×C₈	D₅₈₄	1	4	yes	—
81	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
82	C₄⋊C₄	—	292	4	no	—
83	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
84	C₁₆	D₅₈₄	1	8	yes	complemented
85	C₂³	—	584	2	no	—
86	Q₈	—	584	4	no	—
87	D₄	—	584	4	no	—
88	D₄	—	584	8	no	—
89	C₂×C₄	—	584	2	no	—
90	C₂×C₄	#14556588	1	1	yes	characteristic
91	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
92	C₈	#14556536	1	8	yes	complemented
93	C₈	C₂×D₅₈₄	1	4	yes	—
94	C₂²	—	1168	2	no	—
95	C₂²	—	1168	4	no	—
96	C₂²	#17988964	1	1	yes	characteristic
97	C₄	—	1168	2	no	—
98	C₄	#17988964	1	2	yes	—
99	C₄	C₂×#14556536	1	4	yes	—
100	C₂	—	2336	2	no	—
101	C₂	#22609546	1	1	yes	characteristic
102	C₂	#22612187	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612187

C₂

yes

maximal, complemented

#22609549

C₂

yes

maximal, characteristic, complemented

#22609546

C₂

yes

maximal, characteristic

C₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14556536

—

#17988992

—

#17988967

C₂²

yes

—

#17988964

—

#17988964

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

—

—

—

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

Dic₂₉₂

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

D₈

yes

complemented

C₂²×D₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

SD₃₂

yes

complemented

C₈×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

#493

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₇₃

yes

characteristic, complemented

C₇₃

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

146

—

#500

—

146

—

#494

—

#493

—

#493

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×D₈

—

292

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

146

—

#127

—

146

—

SD₃₂

—

292

—

C₂×C₁₆

D₂₉₂

yes

—

#107

—

292

—

C₄×C₈

D₂₉₂

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

Q₁₆

—

292

—

D₈

—

292

—

D₈

—

584

—

C₂×C₈

D₅₈₄

yes

—

C₂×C₈

C₂×D₂₉₂

yes

—

C₄⋊C₄

—

292

—

C₄²

C₂×D₂₉₂

yes

characteristic

C₁₆

D₅₈₄

yes

complemented

C₂³

—

584

—

Q₈

—

584

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

—

584

—

C₂×C₄

#14556588

yes

characteristic

C₂×C₄

C₂×D₅₈₄

yes

—

C₈

#14556536

yes

complemented

C₈

C₂×D₅₈₄

yes

—

C₂²

—

1168

—

C₂²

—

1168

—

C₂²

#17988964

yes

characteristic

C₄

—

1168

—

C₄

#17988964

yes

—

C₄

C₂×#14556536

yes

—

100

C₂

—

2336

—

101

C₂

#22609546

yes

characteristic

102

C₂

#22612187

yes

—

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₈×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#493
prime radical	#22609546	C₂
hypercenter	C₈×C₁₆	D₇₃
hypocenter	C₇₃	#12517
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₈×C₁₆×C₇₃

C₂

C₂²×C₇₃

C₂

C₈×C₁₆

D₇₃

hypocenter

C₇₃

#12517

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #22612187	C₂	semidirect
2. #22609549	C₂	semidirect
4. C₈×C₁₆×C₇₃	C₂	semidirect
32. C₁₆×C₇₃	D₈	semidirect
40. C₈×C₇₃	SD₃₂	semidirect
55. C₈×C₁₆	D₇₃	semidirect
56. C₇₃	#12517	semidirect
84. C₁₆	D₅₈₄	semidirect
92. C₈	#14556536	semidirect

H ≅ G/N

type

1. #22612187

C₂

semidirect

2. #22609549

C₂

semidirect

4. C₈×C₁₆×C₇₃

C₂

semidirect

32. C₁₆×C₇₃

D₈

semidirect

40. C₈×C₇₃

SD₃₂

semidirect

55. C₈×C₁₆

D₇₃

semidirect

56. C₇₃

#12517

semidirect

84. C₁₆

D₅₈₄

semidirect

92. C₈

#14556536

semidirect

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

subgroup	index
#22612187	2
#22609549	2
#22609546	2
C₈×C₁₆×C₇₃	2
#12517	73

subgroup

index

C₈×C₁₆×C₇₃

Subgroups — #33000702

Conjugacy classes of subgroups

Complements

Maximal subgroups