Group #46812902 — subgroups

Subgroups — #46812902

Order 34432.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#31770777	C₂	1	3	yes	maximal, complemented
2	#31770402	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575501	C₂²	1	1	yes	characteristic, complemented
5	#22575479	—	2	3	no	—
6	#22575381	—	2	3	no	—
7	#22575372	C₂²	1	3	yes	complemented
8	#124×C₂₆₉	—	2	3	no	—
9	#124×C₂₆₉	C₂²	1	3	yes	—
10	#120×C₂₆₉	—	2	3	no	—
11	#16864403	—	4	1	no	—
12	#16864402	—	2	3	no	—
13	Q₈×D₂₆₉	—	4	1	no	—
14	#16864390	—	4	3	no	—
15	#16864387	—	4	3	no	—
16	#16864386	—	2	3	no	—
17	C₂⁴×C₂₆₉	—	2	1	no	—
18	C₂×Q₈×C₂₆₉	—	2	1	no	—
19	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₂₆₉	—	2	3	no	—
21	C₂×D₄×C₂₆₉	D₄	1	3	yes	—
22	C₂²⋊C₄×C₂₆₉	—	2	3	no	—
23	C₂²⋊C₄×C₂₆₉	—	2	6	no	—
24	C₄²×C₂₆₉	D₄	1	3	yes	—
25	Dic₅₃₈	—	2	3	no	—
26	D₁₀₇₆	—	2	3	no	—
27	C₂²⋊D₂₆₉	—	4	1	no	—
28	C₄×D₂₆₉	—	4	3	no	—
29	#14524130	—	4	3	no	—
30	C₂³×C₂₆₉	—	2	3	no	—
31	C₂³×C₂₆₉	—	8	2	no	—
32	C₂³×C₂₆₉	—	8	1	no	—
33	Q₈×C₂₆₉	—	4	1	no	—
34	Q₈×C₂₆₉	—	8	1	no	—
35	D₄×C₂₆₉	—	4	3	no	—
36	D₄×C₂₆₉	—	8	3	no	—
37	C₂×C₄×C₂₆₉	—	2	3	no	—
38	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
39	C₂²×C₂₆₉	—	4	6	no	—
40	C₂²×C₂₆₉	—	4	3	no	—
41	C₂²×C₂₆₉	—	8	4	no	—
42	C₂²×C₂₆₉	#120	1	1	yes	characteristic
43	C₂×D₂₆₉	—	4	1	no	—
44	C₄×C₂₆₉	—	2	3	no	—
45	C₄×C₂₆₉	—	4	3	no	—
46	Dic₂₆₉	—	4	1	no	—
47	C₂×C₂₆₉	—	2	1	no	—
48	C₂×C₂₆₉	—	8	3	no	—
49	C₂×C₂₆₉	B(4,𝔽₂)	1	1	yes	characteristic
50	D₂₆₉	—	8	1	no	—
51	C₂₆₉	#2202	1	1	yes	characteristic, complemented, Sylow
52	#2202	—	269	1	no	maximal, Sylow
53	H(3,𝔽₄)	D₂₆₉	1	1	yes	characteristic, complemented
54	#454	—	269	3	no	—
55	#355	—	269	3	no	—
56	2₋¹⁺⁴	—	269	1	no	—
57	AGL(1,ℤ/8)	—	538	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₂₆₉	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	538	3	no	—
62	#101	—	269	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	538	3	no	—
65	#41	—	1076	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	1076	1	no	—
68	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₂₆₉	1	3	yes	—
71	SD₁₆	—	1076	3	no	—
72	D₈	—	1076	3	no	—
73	M₁₆	—	538	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₂₆₉	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	538	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	538	3	no	—
86	D₄	—	1076	1	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	1076	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
90	C₈	—	1076	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	1076	1	no	—
95	C₂²	#22575422	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	1076	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	2152	1	no	—
102	C₂	#31770792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31770777

C₂

yes

maximal, complemented

#31770402

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₂₆₉

C₂

yes

maximal, characteristic, complemented

#22575501

C₂²

yes

characteristic, complemented

#22575479

—

#22575381

—

#22575372

C₂²

yes

complemented

#124×C₂₆₉

—

#124×C₂₆₉

C₂²

yes

—

#120×C₂₆₉

—

#16864403

—

#16864402

—

Q₈×D₂₆₉

—

#16864390

—

#16864387

—

#16864386

—

C₂⁴×C₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

C₂³

yes

characteristic

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

D₄

yes

—

C₂²⋊C₄×C₂₆₉

—

C₂²⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

D₄

yes

—

Dic₅₃₈

—

D₁₀₇₆

—

C₂²⋊D₂₆₉

—

C₄×D₂₆₉

—

#14524130

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

Q₈×C₂₆₉

—

Q₈×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#120

yes

characteristic

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

B(4,𝔽₂)

yes

characteristic

D₂₆₉

—

C₂₆₉

#2202

yes

characteristic, complemented, Sylow

#2202

—

269

maximal, Sylow

H(3,𝔽₄)

D₂₆₉

yes

characteristic, complemented

#454

—

269

—

#355

—

269

—

2₋¹⁺⁴

—

269

—

AGL(1,ℤ/8)

—

538

—

#124

—

#124

C₂×D₂₆₉

yes

—

#120

—

#104

—

538

—

#101

—

269

—

C₂⁴

—

#41

—

538

—

#41

—

1076

—

C₂×Q₈

—

C₂×Q₈

—

1076

—

C₂×Q₈

C₂²×D₂₆₉

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₆₉

yes

—

SD₁₆

—

1076

—

D₈

—

1076

—

M₁₆

—

538

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₂₆₉

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

538

—

D₄

—

D₄

—

D₄

—

538

—

D₄

—

1076

—

C₂×C₄

—

C₂×C₄

—

1076

—

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

C₈

—

1076

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

1076

—

C₂²

#22575422

yes

characteristic

C₄

—

C₄

—

C₄

—

1076

—

C₂

—

100

C₂

—

101

C₂

—

2152

—

102

C₂

#31770792

yes

characteristic

	subgroup	G/H
center	C₂	#31770792
commutator, cosocle	C₂×Q₈×C₂₆₉	C₂³
Frattini	C₂×Q₈	C₂²×D₂₆₉
Fitting	H(3,𝔽₄)×C₂₆₉	C₂
socle	C₂×C₂₆₉	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₂₆₉
hypocenter	C₂₆₉	#2202
LCS	G ▷ C₂×Q₈×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#31770792

commutator, cosocle

C₂×Q₈×C₂₆₉

C₂³

Frattini

C₂×Q₈

C₂²×D₂₆₉

Fitting

H(3,𝔽₄)×C₂₆₉

C₂

C₂×C₂₆₉

D₂₆₉

C₂₆₉

G ▷ C₂×Q₈×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #31770777	C₂	semidirect
2. #31770402	C₂	semidirect
3. H(3,𝔽₄)×C₂₆₉	C₂	semidirect
4. #22575501	C₂²	semidirect
7. #22575372	C₂²	semidirect
51. C₂₆₉	#2202	semidirect
53. H(3,𝔽₄)	D₂₆₉	semidirect

H ≅ G/N

type

1. #31770777

C₂

semidirect

2. #31770402

C₂

semidirect

3. H(3,𝔽₄)×C₂₆₉

C₂

semidirect

4. #22575501

C₂²

semidirect

7. #22575372

C₂²

semidirect

51. C₂₆₉

#2202

semidirect

53. H(3,𝔽₄)

D₂₆₉

semidirect

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

subgroup	index
#31770777	2
#31770402	2
H(3,𝔽₄)×C₂₆₉	2
#2202	269

subgroup

index

#31770777

#31770402

H(3,𝔽₄)×C₂₆₉

#2202

269

Subgroups — #46812902

Conjugacy classes of subgroups

Complements

Maximal subgroups