Group #14530801 — subgroups

Subgroups — #14530801

Order 2176.

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	#11762981	C₂	1	3	yes	maximal, complemented
2	#11762606	C₂	1	3	yes	maximal, complemented
3	C₁₇×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587694	C₂²	1	1	yes	characteristic, complemented
5	C₁₇×#124	—	2	3	no	—
6	C₁₇×#124	C₂²	1	3	yes	—
7	C₁₇×#120	—	2	3	no	—
8	#10587618	—	2	3	no	—
9	#10587507	C₂²	1	3	yes	complemented
10	#10587476	—	2	3	no	—
11	C₂⁴×C₁₇	—	2	1	no	—
12	C₂×Q₈×C₁₇	—	2	1	no	—
13	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
14	C₂×D₄×C₁₇	—	2	3	no	—
15	C₂×D₄×C₁₇	D₄	1	3	yes	—
16	#63256	—	4	1	no	—
17	Q₈×D₁₇	—	4	1	no	—
18	#63252	—	2	3	no	—
19	C₂²⋊C₄×C₁₇	—	2	3	no	—
20	C₂²⋊C₄×C₁₇	—	2	6	no	—
21	C₄²×C₁₇	D₄	1	3	yes	—
22	#63229	—	4	3	no	—
23	#63228	—	4	3	no	—
24	#63223	—	2	3	no	—
25	C₂³×C₁₇	—	2	3	no	—
26	C₂³×C₁₇	—	8	2	no	—
27	C₂³×C₁₇	—	8	1	no	—
28	Q₈×C₁₇	—	4	1	no	—
29	Q₈×C₁₇	—	8	1	no	—
30	D₄×C₁₇	—	4	3	no	—
31	D₄×C₁₇	—	8	3	no	—
32	C₂×C₄×C₁₇	—	2	3	no	—
33	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
34	C₂²⋊D₁₇	—	4	1	no	—
35	D₆₈	—	2	3	no	—
36	C₄×D₁₇	—	4	3	no	—
37	Dic₃₄	—	2	3	no	—
38	#3622	—	4	3	no	—
39	#2202	—	17	1	no	maximal, Sylow
40	C₂²×C₁₇	—	4	6	no	—
41	C₂²×C₁₇	—	4	3	no	—
42	C₂²×C₁₇	—	8	4	no	—
43	C₂²×C₁₇	#120	1	1	yes	characteristic
44	C₂×D₁₇	—	4	1	no	—
45	C₄×C₁₇	—	2	3	no	—
46	C₄×C₁₇	—	4	3	no	—
47	Dic₁₇	—	4	1	no	—
48	H(3,𝔽₄)	D₁₇	1	1	yes	characteristic, complemented
49	#454	—	17	3	no	—
50	#355	—	17	3	no	—
51	C₂×C₁₇	—	2	1	no	—
52	C₂×C₁₇	—	8	3	no	—
53	C₂×C₁₇	B(4,𝔽₂)	1	1	yes	characteristic
54	D₁₇	—	8	1	no	—
55	2₋¹⁺⁴	—	17	1	no	—
56	AGL(1,ℤ/8)	—	34	3	no	—
57	#124	—	2	3	no	—
58	#124	C₂×D₁₇	1	3	yes	—
59	#120	—	2	3	no	—
60	#104	—	34	3	no	—
61	#101	—	17	3	no	—
62	C₁₇	#2202	1	1	yes	characteristic, complemented, Sylow
63	C₂⁴	—	2	1	no	—
64	#41	—	34	3	no	—
65	#41	—	68	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	68	1	no	—
68	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₁₇	1	3	yes	—
71	SD₁₆	—	68	3	no	—
72	D₈	—	68	3	no	—
73	M₁₆	—	34	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₁₇	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	—	34	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	34	3	no	—
86	D₄	—	68	1	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	68	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
90	C₈	—	68	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	68	1	no	—
95	C₂²	#10587639	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	68	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	136	1	no	—
102	C₂	#11762996	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762981

C₂

yes

maximal, complemented

#11762606

C₂

yes

maximal, complemented

C₁₇×H(3,𝔽₄)

C₂

yes

maximal, characteristic, complemented

#10587694

C₂²

yes

characteristic, complemented

C₁₇×#124

—

C₁₇×#124

C₂²

yes

—

C₁₇×#120

—

#10587618

—

#10587507

C₂²

yes

complemented

#10587476

—

C₂⁴×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

C₂³

yes

characteristic

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

D₄

yes

—

#63256

—

Q₈×D₁₇

—

#63252

—

C₂²⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

D₄

yes

—

#63229

—

#63228

—

#63223

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂²⋊D₁₇

—

D₆₈

—

C₄×D₁₇

—

Dic₃₄

—

#3622

—

#2202

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#120

yes

characteristic

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

H(3,𝔽₄)

D₁₇

yes

characteristic, complemented

#454

—

#355

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

B(4,𝔽₂)

yes

characteristic

D₁₇

—

2₋¹⁺⁴

—

AGL(1,ℤ/8)

—

#124

—

#124

C₂×D₁₇

yes

—

#120

—

#104

—

#101

—

C₁₇

#2202

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₇

yes

—

SD₁₆

—

D₈

—

M₁₆

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₁₇

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#10587639

yes

characteristic

C₄

—

C₄

—

C₄

—

C₂

—

100

C₂

—

101

C₂

—

136

—

102

C₂

#11762996

yes

characteristic

	subgroup	G/H
center	C₂	#11762996
commutator, cosocle	C₂×Q₈×C₁₇	C₂³
Frattini	C₂×Q₈	C₂²×D₁₇
Fitting	C₁₇×H(3,𝔽₄)	C₂
socle	C₂×C₁₇	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₁₇
hypocenter	C₁₇	#2202
LCS	G ▷ C₂×Q₈×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#11762996

commutator, cosocle

C₂×Q₈×C₁₇

C₂³

Frattini

C₂×Q₈

C₂²×D₁₇

Fitting

C₁₇×H(3,𝔽₄)

C₂

C₂×C₁₇

D₁₇

C₁₇

G ▷ C₂×Q₈×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #11762981	C₂	semidirect
2. #11762606	C₂	semidirect
3. C₁₇×H(3,𝔽₄)	C₂	semidirect
4. #10587694	C₂²	semidirect
9. #10587507	C₂²	semidirect
48. H(3,𝔽₄)	D₁₇	semidirect
62. C₁₇	#2202	semidirect

H ≅ G/N

type

1. #11762981

C₂

semidirect

2. #11762606

C₂

semidirect

3. C₁₇×H(3,𝔽₄)

C₂

semidirect

4. #10587694

C₂²

semidirect

9. #10587507

C₂²

semidirect

48. H(3,𝔽₄)

D₁₇

semidirect

62. C₁₇

#2202

semidirect

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

subgroup	index
#11762981	2
#11762606	2
C₁₇×H(3,𝔽₄)	2
#2202	17

subgroup

index

#11762981

#11762606

C₁₇×H(3,𝔽₄)

#2202

Subgroups — #14530801

Conjugacy classes of subgroups

Complements

Maximal subgroups