Group #14542932 — subgroups

Subgroups — #14542932

Order 2176.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#11763563	C₂	1	10	yes	maximal, complemented
2	#11763557	C₂	1	5	yes	maximal, complemented
3	#11763365	C₂	1	5	yes	maximal, complemented
4	#11763344	C₂	1	10	yes	maximal, complemented
5	C₁₇×#564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#125	C₂²	1	5	yes	—
7	C₁₇×#125	C₂²	1	10	yes	—
8	#10587584	C₂²	1	20	yes	—
9	#10587583	C₂²	1	10	yes	—
10	#10587582	C₂²	1	20	yes	—
11	#10587581	C₂²	1	10	yes	—
12	#10587580	C₂²	1	20	yes	—
13	#10587579	C₂²	1	10	yes	—
14	#10587578	C₂²	1	10	yes	—
15	#10587574	C₂²	1	10	yes	—
16	#10587573	C₂²	1	10	yes	—
17	#10587561	C₂²	1	10	yes	—
18	C₄²×D₁₇	C₂²	1	5	yes	—
19	#10587556	C₂²	1	5	yes	—
20	C₂×D₆₈	C₂³	1	10	yes	—
21	C₂×C₄×D₁₇	C₂³	1	5	yes	—
22	C₂×C₄×D₁₇	C₂³	1	10	yes	—
23	C₂×Dic₃₄	C₂³	1	10	yes	—
24	C₄⋊C₄×C₁₇	C₂³	1	10	yes	—
25	C₄⋊C₄×C₁₇	C₂³	1	10	yes	—
26	C₄⋊C₄×C₁₇	C₂³	1	10	yes	—
27	C₄²×C₁₇	C₂³	1	5	yes	—
28	#63227	C₂³	1	20	yes	—
29	#63227	C₂³	1	20	yes	—
30	C₄⋊Dic₁₇	C₂³	1	10	yes	—
31	#63225	C₂³	1	20	yes	—
32	C₄×Dic₁₇	C₂³	1	10	yes	—
33	C₄×Dic₁₇	C₂³	1	5	yes	—
34	C₂²×D₁₇	C₂⁴	1	5	yes	—
35	C₂×C₄×C₁₇	C₂⁴	1	10	yes	—
36	C₂×C₄×C₁₇	C₂⁴	1	5	yes	—
37	C₂×Dic₁₇	C₂⁴	1	10	yes	—
38	C₂×Dic₁₇	C₂⁴	1	1	yes	characteristic
39	D₆₈	—	4	10	no	—
40	C₄×D₁₇	—	2	10	no	—
41	C₄×D₁₇	—	4	10	no	—
42	Dic₃₄	—	4	10	no	—
43	#3531	—	17	1	no	maximal, Sylow
44	C₂²×C₁₇	C₂⁵	1	1	yes	characteristic
45	C₂×D₁₇	—	2	5	no	—
46	C₂×D₁₇	—	2	10	no	—
47	C₄×C₁₇	—	2	5	no	—
48	C₄×C₁₇	—	2	10	no	—
49	Dic₁₇	—	2	10	no	—
50	Dic₁₇	2₋¹⁺⁴	1	2	yes	—
51	#564	D₁₇	1	1	yes	characteristic, complemented
52	#563	—	17	10	no	—
53	#562	—	17	5	no	—
54	#533	—	17	5	no	—
55	#529	—	17	10	no	—
56	C₂×C₁₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
57	C₂×C₁₇	#585	1	2	yes	—
58	D₁₇	—	4	5	no	—
59	C₄⋊Q₈	—	17	10	no	—
60	#126	—	17	20	no	—
61	#125	—	17	5	no	—
62	#125	C₂×D₁₇	1	10	yes	—
63	#125	C₂×D₁₇	1	5	yes	—
64	#124	—	17	10	no	—
65	#123	—	17	20	no	—
66	C₂²⋊Q₈	—	17	10	no	—
67	C₂²⋊Q₈	—	17	20	no	—
68	#121	—	17	10	no	—
69	C₄×Q₈	—	17	10	no	—
70	C₄×D₄	—	17	10	no	—
71	#117	—	17	10	no	—
72	C₂×C₄²	—	17	5	no	—
73	C₁₇	#3531	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	17	10	no	—
75	C₂×D₄	—	17	10	no	—
76	C₂²×C₄	—	17	5	no	—
77	C₂²×C₄	—	17	10	no	—
78	C₄⋊C₄	—	17	10	no	—
79	C₄⋊C₄	—	17	20	no	—
80	C₄⋊C₄	C₂²×D₁₇	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₇	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₇	1	10	yes	—
83	C₂²⋊C₄	—	17	20	no	—
84	C₂²⋊C₄	—	17	20	no	—
85	C₄²	—	17	10	no	—
86	C₄²	—	17	5	no	—
87	C₄²	C₂²×D₁₇	1	5	yes	—
88	C₂³	—	17	5	no	—
89	Q₈	—	68	10	no	—
90	D₄	—	68	10	no	—
91	C₂×C₄	—	17	1	no	—
92	C₂×C₄	—	17	10	no	—
93	C₂×C₄	—	34	10	no	—
94	C₂×C₄	—	68	10	no	—
95	C₂×C₄	C₂³×D₁₇	1	5	yes	—
96	C₂×C₄	C₂³×D₁₇	1	10	yes	—
97	C₂²	—	34	5	no	—
98	C₂²	—	34	10	no	—
99	C₂²	C₂⁴×D₁₇	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	17	2	no	—
103	C₄	—	34	10	no	—
104	C₂	—	68	5	no	—
105	C₂	2₋¹⁺⁴×D₁₇	1	1	yes	characteristic
106	C₂	#11763646	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763563

C₂

yes

maximal, complemented

#11763557

C₂

yes

maximal, complemented

#11763365

C₂

yes

maximal, complemented

#11763344

C₂

yes

maximal, complemented

C₁₇×#564

C₂

yes

maximal, characteristic, complemented

C₁₇×#125

C₂²

yes

—

C₁₇×#125

C₂²

yes

—

#10587584

C₂²

yes

—

#10587583

C₂²

yes

—

#10587582

C₂²

yes

—

#10587581

C₂²

yes

—

#10587580

C₂²

yes

—

#10587579

C₂²

yes

—

#10587578

C₂²

yes

—

#10587574

C₂²

yes

—

#10587573

C₂²

yes

—

#10587561

C₂²

yes

—

C₄²×D₁₇

C₂²

yes

—

#10587556

C₂²

yes

—

C₂×D₆₈

C₂³

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

C₂×Dic₃₄

C₂³

yes

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₄²×C₁₇

C₂³

yes

—

#63227

C₂³

yes

—

#63227

C₂³

yes

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

C₂³

yes

—

C₄×Dic₁₇

C₂³

yes

—

C₄×Dic₁₇

C₂³

yes

—

C₂²×D₁₇

C₂⁴

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

—

C₂×Dic₁₇

C₂⁴

yes

—

C₂×Dic₁₇

C₂⁴

yes

characteristic

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

#3531

—

maximal, Sylow

C₂²×C₁₇

C₂⁵

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

2₋¹⁺⁴

yes

—

#564

D₁₇

yes

characteristic, complemented

—

—

—

—

C₂×C₁₇

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₇

#585

yes

—

D₁₇

—

C₄⋊Q₈

—

#126

—

#125

—

#125

C₂×D₁₇

yes

—

#125

C₂×D₁₇

yes

—

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₁₇

#3531

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

C₂²×D₁₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₇

yes

—

C₂³

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×D₁₇

yes

—

C₂×C₄

C₂³×D₁₇

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₁₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₂

—

105

C₂

2₋¹⁺⁴×D₁₇

yes

characteristic

106

C₂

#11763646

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁵
Fitting	C₁₇×#564	C₂
hypercenter	#564	D₁₇
hypocenter	C₁₇	#3531
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₇

commutator, socle, cosocle

C₂²×C₁₇

C₂⁵

Fitting

C₁₇×#564

C₂

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #11763563	C₂	semidirect
2. #11763557	C₂	semidirect
3. #11763365	C₂	semidirect
4. #11763344	C₂	semidirect
5. C₁₇×#564	C₂	semidirect
51. #564	D₁₇	semidirect
73. C₁₇	#3531	semidirect

H ≅ G/N

type

1. #11763563

C₂

semidirect

2. #11763557

C₂

semidirect

3. #11763365

C₂

semidirect

4. #11763344

C₂

semidirect

5. C₁₇×#564

C₂

semidirect

51. #564

D₁₇

semidirect

73. C₁₇

#3531

semidirect

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

subgroup	index
#11763563	2
#11763557	2
#11763365	2
#11763344	2
C₁₇×#564	2
#3531	17

subgroup

index

C₁₇×#564

Subgroups — #14542932

Conjugacy classes of subgroups

Complements

Maximal subgroups