Group #36353230 — subgroups

Subgroups — #36353230

Order 22144.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26011867	C₂	1	2	yes	maximal, complemented
2	#26011864	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011837	C₂	1	2	yes	maximal, complemented
4	#26011212	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583815	—	2	2	no	—
7	C₂×#14583811	—	2	4	no	—
8	#19149679	C₂²	1	1	yes	characteristic
9	#19149676	C₂²	1	1	yes	characteristic
10	#19149600	—	2	4	no	—
11	C₄×#13063436	C₂²	1	2	yes	—
12	C₂×D₈×C₁₇₃	—	2	2	no	—
13	#127×C₁₇₃	C₂²	1	2	yes	—
14	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₄₆	—	2	1	no	—
16	C₂×D₆₉₂	—	2	2	no	—
17	C₂×D₆₉₂	—	2	1	no	—
18	#14583815	—	4	4	no	—
19	#14583811	—	4	8	no	—
20	C₂×#13063436	D₄	1	4	yes	—
21	C₄⋊Dic₁₇₃	—	2	2	no	—
22	C₂×D₄×C₁₇₃	—	2	2	no	—
23	C₂×D₄×C₁₇₃	—	2	2	no	—
24	C₂×D₄×C₁₇₃	—	2	2	no	—
25	D₈×C₁₇₃	—	4	4	no	—
26	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
27	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
28	C₂²×D₁₇₃	—	4	1	no	—
29	Dic₃₄₆	—	4	2	no	—
30	D₆₉₂	—	4	2	no	—
31	D₆₉₂	—	4	4	no	—
32	C₂×Dic₁₇₃	—	4	1	no	—
33	#13063436	—	2	4	no	—
34	C₂³×C₁₇₃	—	4	2	no	—
35	D₄×C₁₇₃	—	4	4	no	—
36	D₄×C₁₇₃	—	4	4	no	—
37	D₄×C₁₇₃	—	4	4	no	—
38	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₇₃	D₈	1	4	yes	complemented
41	C₂²×C₁₇₃	—	8	2	no	—
42	C₂²×C₁₇₃	—	8	2	no	—
43	C₂²×C₁₇₃	—	8	2	no	—
44	C₂²×C₁₇₃	#127	1	1	yes	characteristic
45	C₂×D₁₇₃	—	8	1	no	—
46	C₂×D₁₇₃	—	8	2	no	—
47	C₄×C₁₇₃	C₂×D₈	1	2	yes	—
48	C₄×C₁₇₃	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₇₃	—	8	1	no	—
50	C₂×C₁₇₃	—	16	2	no	—
51	C₂×C₁₇₃	#493	1	1	yes	characteristic
52	C₂×C₁₇₃	#496	1	2	yes	—
53	D₁₇₃	—	16	1	no	—
54	C₁₇₃	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	173	1	no	maximal, Sylow
56	#493	—	173	2	no	—
57	#493	D₁₇₃	1	1	yes	characteristic, complemented
58	#492	—	173	1	no	—
59	#486	—	173	2	no	—
60	#322	—	173	1	no	—
61	C₂×SD₁₆	—	346	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	346	4	no	—
64	C₄⋊Q₈	—	173	1	no	—
65	#127	—	173	1	no	—
66	#127	C₂×D₁₇₃	1	2	yes	—
67	#102	—	346	4	no	—
68	C₄×C₈	—	173	2	no	—
69	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
70	C₂×Q₈	—	346	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	346	2	no	—
75	C₂×D₄	—	346	1	no	—
76	SD₁₆	—	692	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	692	8	no	—
79	C₂×C₈	—	173	4	no	—
80	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
81	C₄⋊C₄	—	346	2	no	—
82	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	692	1	no	—
85	Q₈	—	692	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	692	2	no	—
90	D₄	—	692	4	no	—
91	C₂×C₄	—	692	1	no	—
92	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
94	C₈	—	346	4	no	—
95	C₈	#14583811	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	1384	1	no	—
100	C₂²	—	1384	2	no	—
101	C₂²	#19149675	1	1	yes	characteristic
102	C₄	—	1384	1	no	—
103	C₄	C₂×#14583811	1	2	yes	—
104	C₄	#19149703	1	4	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	2768	1	no	—
107	C₂	#26011867	1	1	yes	characteristic
108	C₂	#26011879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26011867

C₂

yes

maximal, complemented

#26011864

C₂

yes

maximal, characteristic, complemented

#26011837

C₂

yes

maximal, complemented

#26011212

C₂

yes

maximal, characteristic, complemented

#493×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14583815

—

C₂×#14583811

—

#19149679

C₂²

yes

characteristic

#19149676

C₂²

yes

characteristic

#19149600

—

C₄×#13063436

C₂²

yes

—

C₂×D₈×C₁₇₃

—

#127×C₁₇₃

C₂²

yes

—

C₄×C₈×C₁₇₃

C₂²

yes

characteristic

C₂×Dic₃₄₆

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

#14583815

—

#14583811

—

C₂×#13063436

D₄

yes

—

C₄⋊Dic₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

D₈×C₁₇₃

—

C₂×C₈×C₁₇₃

D₄

yes

—

C₄²×C₁₇₃

C₂³

yes

characteristic

C₂²×D₁₇₃

—

Dic₃₄₆

—

D₆₉₂

—

D₆₉₂

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₈×C₁₇₃

D₈

yes

complemented

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#127

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

C₂×D₈

yes

—

C₄×C₁₇₃

AGL(1,ℤ/8)

yes

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#493

yes

characteristic

C₂×C₁₇₃

#496

yes

—

D₁₇₃

—

C₁₇₃

#1721

yes

characteristic, complemented, Sylow

#1721

—

173

maximal, Sylow

#493

—

173

—

#493

D₁₇₃

yes

characteristic, complemented

#492

—

173

—

#486

—

173

—

#322

—

173

—

C₂×SD₁₆

—

346

—

C₂×D₈

—

C₂×D₈

—

346

—

C₄⋊Q₈

—

173

—

#127

—

173

—

#127

C₂×D₁₇₃

yes

—

#102

—

346

—

C₄×C₈

—

173

—

C₄×C₈

C₂×D₁₇₃

yes

characteristic

C₂×Q₈

—

346

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

346

—

C₂×D₄

—

346

—

SD₁₆

—

692

—

D₈

—

D₈

—

692

—

C₂×C₈

—

173

—

C₂×C₈

C₂²⋊D₁₇₃

yes

—

C₄⋊C₄

—

346

—

C₄²

C₂²×D₁₇₃

yes

characteristic

C₂³

—

C₂³

—

692

—

Q₈

—

692

—

D₄

—

D₄

—

D₄

—

D₄

—

692

—

D₄

—

692

—

C₂×C₄

—

692

—

C₂×C₄

D₄×D₁₇₃

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

C₈

—

346

—

C₈

#14583811

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

1384

—

100

C₂²

—

1384

—

101

C₂²

#19149675

yes

characteristic

102

C₄

—

1384

—

103

C₄

C₂×#14583811

yes

—

104

C₄

#19149703

yes

—

105

C₂

—

106

C₂

—

2768

—

107

C₂

#26011867

yes

characteristic

108

C₂

#26011879

yes

—

	subgroup	G/H
center	C₂²	#19149675
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#493×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#127
hypercenter	#493	D₁₇₃
hypocenter	C₁₇₃	#1721
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#19149675

commutator, cosocle

C₄²×C₁₇₃

C₂³

Frattini

C₄²

C₂²×D₁₇₃

Fitting

#493×C₁₇₃

C₂

C₂²×C₁₇₃

D₁₇₃

C₁₇₃

G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #26011867	C₂	semidirect
2. #26011864	C₂	semidirect
3. #26011837	C₂	semidirect
4. #26011212	C₂	semidirect
5. #493×C₁₇₃	C₂	semidirect
40. C₈×C₁₇₃	D₈	semidirect
54. C₁₇₃	#1721	semidirect
57. #493	D₁₇₃	semidirect
95. C₈	#14583811	semidirect

H ≅ G/N

type

1. #26011867

C₂

semidirect

2. #26011864

C₂

semidirect

3. #26011837

C₂

semidirect

4. #26011212

C₂

semidirect

5. #493×C₁₇₃

C₂

semidirect

40. C₈×C₁₇₃

D₈

semidirect

54. C₁₇₃

#1721

semidirect

57. #493

D₁₇₃

semidirect

95. C₈

#14583811

semidirect

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

subgroup	index
#26011867	2
#26011864	2
#26011837	2
#26011212	2
#493×C₁₇₃	2
#1721	173

subgroup

index

#493×C₁₇₃

173

Subgroups — #36353230

Conjugacy classes of subgroups

Complements

Maximal subgroups