Group #33000701 — subgroups

Subgroups — #33000701

Order 18688.

354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22612186	C₂	1	4	yes	maximal, complemented
2	#22609549	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1712	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556535	—	2	8	no	—
6	#17988992	—	2	2	no	—
7	#17988967	C₂²	1	2	yes	—
8	C₄×#11924003	C₂²	1	2	yes	—
9	#493×C₇₃	—	2	2	no	—
10	#493×C₇₃	C₂²	1	2	yes	—
11	C₈²×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₉₂	—	2	4	no	—
13	#14556591	—	2	2	no	—
14	#14556535	—	4	16	no	—
15	C₂×#11924003	D₄	1	4	yes	—
16	#14556525	—	4	4	no	—
17	C₂×D₈×C₇₃	—	2	4	no	—
18	C₂×D₈×C₇₃	—	4	4	no	—
19	#127×C₇₃	—	2	2	no	—
20	C₄×C₈×C₇₃	D₄	1	2	yes	—
21	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₄₆	—	4	2	no	—
23	Dic₂₉₂	—	4	8	no	—
24	C₄⋊Dic₇₃	—	4	4	no	—
25	#11924003	D₈	1	8	yes	complemented
26	C₂×D₄×C₇₃	—	4	2	no	—
27	C₂×D₄×C₇₃	—	4	4	no	—
28	D₈×C₇₃	—	4	8	no	—
29	D₈×C₇₃	—	8	8	no	—
30	C₂×C₈×C₇₃	D₈	1	4	yes	—
31	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
32	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂³×C₇₃	—	8	2	no	—
34	D₄×C₇₃	—	8	4	no	—
35	D₄×C₇₃	—	8	8	no	—
36	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
38	C₂×Dic₇₃	—	8	2	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	—	16	2	no	—
43	C₂²×C₇₃	—	16	4	no	—
44	C₂²×C₇₃	#493	1	1	yes	characteristic
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	—	32	2	no	—
50	C₂×C₇₃	#1712	1	1	yes	characteristic
51	C₂×C₇₃	#2248	1	2	yes	—
52	#2248	—	73	4	no	—
53	#1713	—	73	1	no	—
54	#1712	D₇₃	1	1	yes	characteristic
55	C₈×C₁₆	—	73	1	no	—
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	—	2	2	no	—
61	#493	C₂×D₇₃	1	2	yes	—
62	C₄×C₁₆	—	73	2	no	—
63	C₈²	C₂×D₇₃	1	1	yes	characteristic
64	C₂×Q₁₆	—	146	4	no	—
65	C₂×D₈	—	2	4	no	—
66	C₂×D₈	—	4	4	no	—
67	C₄⋊Q₈	—	146	2	no	—
68	#127	—	2	2	no	—
69	SD₃₂	—	292	16	no	—
70	C₂×C₁₆	—	73	4	no	—
71	#107	—	292	4	no	—
72	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
73	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
74	C₂×Q₈	—	292	2	no	—
75	C₂×D₄	—	4	2	no	—
76	C₂×D₄	—	4	4	no	—
77	Q₁₆	—	292	8	no	—
78	D₈	—	4	8	no	—
79	D₈	—	8	8	no	—
80	C₂×C₈	#11924013	1	4	yes	—
81	C₂×C₈	D₄×D₇₃	1	2	yes	—
82	C₄⋊C₄	—	292	4	no	—
83	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
84	C₁₆	—	73	8	no	—
85	C₂³	—	8	2	no	—
86	Q₈	—	584	4	no	—
87	D₄	—	8	4	no	—
88	D₄	—	8	8	no	—
89	C₂×C₄	—	584	2	no	—
90	C₂×C₄	#14556587	1	1	yes	characteristic
91	C₂×C₄	C₂×#11924013	1	2	yes	—
92	C₈	#14556535	1	8	yes	complemented
93	C₈	D₈×D₇₃	1	4	yes	—
94	C₂²	—	16	2	no	—
95	C₂²	—	16	4	no	—
96	C₂²	#17988963	1	1	yes	characteristic
97	C₄	—	1168	2	no	—
98	C₄	#17988962	1	2	yes	—
99	C₄	C₂×#14556535	1	4	yes	—
100	C₂	—	32	2	no	—
101	C₂	#22609545	1	1	yes	characteristic
102	C₂	#22612186	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612186

C₂

yes

maximal, complemented

#22609549

C₂

yes

maximal, characteristic, complemented

C₈×#11924003

C₂

yes

maximal, characteristic, complemented

C₇₃×#1712

C₂

yes

maximal, characteristic

C₂×#14556535

—

#17988992

—

#17988967

C₂²

yes

—

C₄×#11924003

C₂²

yes

—

#493×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

#14556591

—

#14556535

—

C₂×#11924003

D₄

yes

—

#14556525

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

#11924003

D₈

yes

complemented

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

SD₃₂

yes

complemented

C₈×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

#493

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

Dic₇₃

—

#12517

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1712

yes

characteristic

C₂×C₇₃

#2248

yes

—

#2248

—

#1713

—

#1712

D₇₃

yes

characteristic

C₈×C₁₆

—

C₇₃

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

146

—

#500

—

146

—

#494

—

#493

—

#493

C₂×D₇₃

yes

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

146

—

#127

—

SD₃₂

—

292

—

C₂×C₁₆

—

#107

—

292

—

C₄×C₈

C₂²⋊D₇₃

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

292

—

D₈

—

D₈

—

C₂×C₈

#11924013

yes

—

C₂×C₈

D₄×D₇₃

yes

—

C₄⋊C₄

—

292

—

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

584

—

D₄

—

D₄

—

C₂×C₄

—

584

—

C₂×C₄

#14556587

yes

characteristic

C₂×C₄

C₂×#11924013

yes

—

C₈

#14556535

yes

complemented

C₈

D₈×D₇₃

yes

—

C₂²

—

C₂²

—

C₂²

#17988963

yes

characteristic

C₄

—

1168

—

C₄

#17988962

yes

—

C₄

C₂×#14556535

yes

—

100

C₂

—

101

C₂

#22609545

yes

characteristic

102

C₂

#22612186

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting, prime radical	C₇₃×#1712	C₂
socle	C₂²×C₇₃	#493
hypercenter	#1712	D₇₃
hypocenter	C₇₃	#12517
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting, prime radical

C₇₃×#1712

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #22612186	C₂	semidirect
2. #22609549	C₂	semidirect
3. C₈×#11924003	C₂	semidirect
25. #11924003	D₈	semidirect
40. C₈×C₇₃	SD₃₂	semidirect
56. C₇₃	#12517	semidirect
92. C₈	#14556535	semidirect

H ≅ G/N

type

1. #22612186

C₂

semidirect

2. #22609549

C₂

semidirect

3. C₈×#11924003

C₂

semidirect

25. #11924003

D₈

semidirect

40. C₈×C₇₃

SD₃₂

semidirect

56. C₇₃

#12517

semidirect

92. C₈

#14556535

semidirect

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

subgroup	index
#22612186	2
#22609549	2
C₈×#11924003	2
C₇₃×#1712	2
#12517	73

subgroup

index

C₇₃×#1712

Subgroups — #33000701

Conjugacy classes of subgroups

Complements

Maximal subgroups