Group #33000695 — subgroups

Subgroups — #33000695

Order 18688.

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

normal subgroups only

This group has 398 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	#22612180	C₂	1	4	yes	maximal, complemented
2	#22609546	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, complemented
5	C₂×#14556532	—	2	8	no	—
6	#17988964	—	2	2	no	—
7	#17988964	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×D₅₈₄	—	2	4	no	—
13	C₂×D₅₈₄	—	4	4	no	—
14	#14556588	—	2	2	no	—
15	#14556532	—	4	16	no	—
16	C₂×#11924003	D₄	1	4	yes	—
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×D₂₉₂	—	4	2	no	—
23	C₂×D₂₉₂	—	4	4	no	—
24	D₅₈₄	—	4	8	no	—
25	D₅₈₄	—	8	8	no	—
26	#11924003	D₈	1	8	yes	complemented
27	C₂×D₄×C₇₃	—	4	2	no	—
28	C₂×D₄×C₇₃	—	4	4	no	—
29	D₈×C₇₃	—	4	8	no	—
30	D₈×C₇₃	—	8	8	no	—
31	C₂×C₈×C₇₃	D₈	1	4	yes	—
32	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
33	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂³×C₇₃	—	8	2	no	—
35	C₂²×D₇₃	—	8	2	no	—
36	D₄×C₇₃	—	8	4	no	—
37	D₄×C₇₃	—	8	8	no	—
38	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
40	D₂₉₂	—	8	4	no	—
41	D₂₉₂	—	8	8	no	—
42	C₈×C₇₃	D₁₆	1	8	yes	complemented
43	C₈×C₇₃	C₂×D₈	1	4	yes	—
44	C₂²×C₇₃	—	16	2	no	—
45	C₂²×C₇₃	—	16	4	no	—
46	C₂²×C₇₃	#493	1	1	yes	characteristic
47	C₂×D₇₃	—	16	2	no	—
48	C₂×D₇₃	—	16	4	no	—
49	C₄×C₇₃	#493	1	2	yes	—
50	C₄×C₇₃	C₂×D₁₆	1	4	yes	—
51	#12515	—	73	1	no	maximal, Sylow
52	C₂×C₇₃	—	32	2	no	—
53	C₂×C₇₃	#1712	1	1	yes	characteristic
54	C₂×C₇₃	#2246	1	2	yes	—
55	D₇₃	—	32	2	no	—
56	#2246	—	73	4	no	—
57	#1712	—	73	1	no	—
58	#1712	D₇₃	1	1	yes	characteristic, complemented
59	C₈×C₁₆	—	73	1	no	—
60	C₇₃	#12515	1	1	yes	characteristic, complemented, Sylow
61	C₂×D₁₆	—	146	8	no	—
62	#493	—	2	2	no	—
63	#493	—	73	2	no	—
64	#493	—	146	2	no	—
65	#493	C₂×D₇₃	1	2	yes	—
66	C₄×C₁₆	—	73	2	no	—
67	C₈²	C₂×D₇₃	1	1	yes	characteristic
68	C₂×D₈	—	2	4	no	—
69	C₂×D₈	—	4	4	no	—
70	C₂×D₈	—	292	4	no	—
71	C₂×D₈	—	146	4	no	—
72	#127	—	2	2	no	—
73	#127	—	146	2	no	—
74	D₁₆	—	292	16	no	—
75	C₂×C₁₆	—	73	4	no	—
76	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
77	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	292	2	no	—
81	C₂×D₄	—	292	4	no	—
82	D₈	—	4	8	no	—
83	D₈	—	8	8	no	—
84	D₈	—	292	8	no	—
85	D₈	—	584	8	no	—
86	C₂×C₈	#11924013	1	4	yes	—
87	C₂×C₈	D₄×D₇₃	1	2	yes	—
88	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
89	C₁₆	—	73	8	no	—
90	C₂³	—	8	2	no	—
91	C₂³	—	584	2	no	—
92	D₄	—	8	4	no	—
93	D₄	—	8	8	no	—
94	D₄	—	584	4	no	—
95	D₄	—	584	8	no	—
96	C₂×C₄	#14556587	1	1	yes	characteristic
97	C₂×C₄	C₂×#11924013	1	2	yes	—
98	C₈	#14556532	1	8	yes	complemented
99	C₈	D₈×D₇₃	1	4	yes	—
100	C₂²	—	16	2	no	—
101	C₂²	—	16	4	no	—
102	C₂²	—	1168	2	no	—
103	C₂²	—	1168	4	no	—
104	C₂²	#17988963	1	1	yes	characteristic
105	C₄	#17988962	1	2	yes	—
106	C₄	C₂×#14556532	1	4	yes	—
107	C₂	—	32	2	no	—
108	C₂	—	2336	2	no	—
109	C₂	#22609545	1	1	yes	characteristic
110	C₂	#22612180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612180

C₂

yes

maximal, complemented

#22609546

C₂

yes

maximal, characteristic, complemented

C₈×#11924003

C₂

yes

maximal, characteristic, complemented

C₇₃×#1712

C₂

yes

maximal, characteristic, complemented

C₂×#14556532

—

#17988964

—

#17988964

C₂²

yes

—

C₄×#11924003

C₂²

yes

—

#493×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

—

#14556532

—

C₂×#11924003

D₄

yes

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

#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

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

D₁₆

yes

complemented

C₈×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

#493

yes

—

C₄×C₇₃

C₂×D₁₆

yes

—

#12515

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1712

yes

characteristic

C₂×C₇₃

#2246

yes

—

D₇₃

—

#2246

—

#1712

—

#1712

D₇₃

yes

characteristic, complemented

C₈×C₁₆

—

C₇₃

#12515

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

146

—

#493

—

#493

—

#493

—

146

—

#493

C₂×D₇₃

yes

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

292

—

C₂×D₈

—

146

—

#127

—

#127

—

146

—

D₁₆

—

292

—

C₂×C₁₆

—

C₄×C₈

C₂²⋊D₇₃

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

D₈

—

D₈

—

D₈

—

292

—

D₈

—

584

—

C₂×C₈

#11924013

yes

—

C₂×C₈

D₄×D₇₃

yes

—

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

584

—

D₄

—

D₄

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

#14556587

yes

characteristic

C₂×C₄

C₂×#11924013

yes

—

C₈

#14556532

yes

complemented

C₈

D₈×D₇₃

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

1168

—

103

C₂²

—

1168

—

104

C₂²

#17988963

yes

characteristic

105

C₄

#17988962

yes

—

106

C₄

C₂×#14556532

yes

—

107

C₂

—

108

C₂

—

2336

—

109

C₂

#22609545

yes

characteristic

110

C₂

#22612180

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1712	C₂
socle	C₂²×C₇₃	#493
hypercenter	#1712	D₇₃
hypocenter	C₇₃	#12515
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

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. #22612180	C₂	semidirect
2. #22609546	C₂	semidirect
3. C₈×#11924003	C₂	semidirect
4. C₇₃×#1712	C₂	semidirect
26. #11924003	D₈	semidirect
42. C₈×C₇₃	D₁₆	semidirect
58. #1712	D₇₃	semidirect
60. C₇₃	#12515	semidirect
98. C₈	#14556532	semidirect

H ≅ G/N

type

1. #22612180

C₂

semidirect

2. #22609546

C₂

semidirect

3. C₈×#11924003

C₂

semidirect

4. C₇₃×#1712

C₂

semidirect

26. #11924003

D₈

semidirect

42. C₈×C₇₃

D₁₆

semidirect

58. #1712

D₇₃

semidirect

60. C₇₃

#12515

semidirect

98. C₈

#14556532

semidirect

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

subgroup	index
#22612180	2
#22609546	2
C₈×#11924003	2
C₇₃×#1712	2
#12515	73

subgroup

index

C₇₃×#1712

Subgroups — #33000695

Conjugacy classes of subgroups

Complements

Maximal subgroups