Group #14528539 — subgroups

Subgroups — #14528539

Order 2176.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#11762764	C₂	1	2	yes	maximal
2	#11762567	C₂	1	1	yes	maximal, characteristic
3	#11762566	C₂	1	1	yes	maximal, characteristic
4	#11762565	C₂	1	2	yes	maximal
5	C₁₇×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#117	C₂²	1	1	yes	characteristic
9	#10587612	C₂²	1	2	yes	—
10	C₂×#63223	C₂²	1	2	yes	—
11	C₁₇×#97	C₄	1	2	yes	complemented
12	C₄×C₈×C₁₇	C₄	1	2	yes	complemented
13	#10587486	C₄	1	4	yes	complemented
14	C₈×Dic₁₇	C₄	1	4	yes	complemented
15	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₇	—	2	2	no	—
17	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₄×C₁₇	—	2	1	no	—
23	C₂²⋊C₄×C₁₇	—	2	1	no	—
24	C₄²×C₁₇	C₂×C₄	1	2	yes	—
25	C₂²⋊Dic₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₄×Dic₁₇	C₂×C₄	1	4	yes	—
28	#63223	—	2	2	no	—
29	#63223	—	2	2	no	—
30	C₂×#3622	C₂×C₄	1	4	yes	—
31	C₂³×C₁₇	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₇	—	2	2	no	—
33	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₇	#41	1	2	yes	—
37	C₂×Dic₁₇	—	2	4	no	—
38	C₈×C₁₇	—	2	2	no	—
39	C₈×C₁₇	C₄²	1	2	yes	—
40	C₈×C₁₇	C₄²	1	1	yes	characteristic
41	C₈×C₁₇	C₄²	1	1	yes	characteristic
42	#3622	—	2	4	no	—
43	#1747	—	17	1	no	maximal, Sylow
44	C₂²×C₁₇	—	4	1	no	—
45	C₂²×C₁₇	#117	1	2	yes	—
46	C₂²×C₁₇	#117	1	1	yes	characteristic
47	C₄×C₁₇	—	4	2	no	—
48	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
49	C₄×C₁₇	#117	1	2	yes	—
50	C₄×C₁₇	#117	1	1	yes	characteristic
51	Dic₁₇	—	4	4	no	—
52	#405	—	17	2	no	—
53	#405	D₁₇	1	1	yes	characteristic
54	#338	—	17	1	no	—
55	H(3,ℤ/4)	—	17	1	no	—
56	H(3,ℤ/4)	—	17	2	no	—
57	C₂×C₁₇	—	2	2	no	—
58	C₂×C₁₇	—	2	1	no	—
59	C₂×C₁₇	#376	1	1	yes	characteristic
60	C₂×M₁₆	—	17	2	no	—
61	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
63	#117	—	17	2	no	—
64	#117	C₂×D₁₇	1	1	yes	characteristic
65	#97	—	17	4	no	—
66	#97	Dic₁₇	1	2	yes	complemented
67	C₄×C₈	—	17	4	no	—
68	C₄×C₈	Dic₁₇	1	2	yes	complemented
69	C₁₇	#1747	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	34	2	no	—
73	M₁₆	—	34	2	no	—
74	C₂×C₈	—	17	4	no	—
75	C₂×C₈	C₄×D₁₇	1	2	yes	—
76	C₂×C₈	C₄×D₁₇	1	2	yes	—
77	C₂×C₈	C₂×Dic₁₇	1	2	yes	—
78	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	34	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	34	2	no	—
84	C₄²	—	17	4	no	—
85	C₄²	C₂×Dic₁₇	1	2	yes	—
86	C₂³	#63252	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	34	4	no	—
89	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
90	C₂×C₄	#63252	1	1	yes	characteristic
91	C₂×C₄	#63252	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	34	4	no	—
95	C₈	C₄×Dic₁₇	1	1	yes	characteristic
96	C₈	C₄×Dic₁₇	1	2	yes	—
97	C₈	C₄×Dic₁₇	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#10587558	1	2	yes	—
100	C₂²	#10587612	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	68	4	no	—
103	C₄	#10587558	1	2	yes	—
104	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
105	C₄	#10587612	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#11762655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762764

C₂

yes

maximal

#11762567

C₂

yes

maximal, characteristic

#11762566

C₂

yes

maximal, characteristic

#11762565

C₂

yes

maximal

C₁₇×#405

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

#10587612

C₂²

yes

—

C₂×#63223

C₂²

yes

—

C₁₇×#97

C₄

yes

complemented

C₄×C₈×C₁₇

C₄

yes

complemented

#10587486

C₄

yes

complemented

C₈×Dic₁₇

C₄

yes

complemented

C₂²×C₄×C₁₇

C₂³

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

—

#63223

—

#63223

—

C₂×#3622

C₂×C₄

yes

—

C₂³×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄²

yes

—

C₈×C₁₇

C₄²

yes

characteristic

C₈×C₁₇

C₄²

yes

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#117

yes

—

C₂²×C₁₇

#117

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄²

yes

characteristic

C₄×C₁₇

#117

yes

—

C₄×C₁₇

#117

yes

characteristic

Dic₁₇

—

#405

—

#405

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#376

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

#117

—

#117

C₂×D₁₇

yes

characteristic

#97

—

#97

Dic₁₇

yes

complemented

C₄×C₈

—

C₄×C₈

Dic₁₇

yes

complemented

C₁₇

#1747

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×D₁₇

yes

—

C₂×C₈

C₄×D₁₇

yes

—

C₂×C₈

C₂×Dic₁₇

yes

—

C₂×C₈

C₂×Dic₁₇

yes

characteristic

C₂×C₈

C₂×Dic₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₁₇

yes

—

C₂³

#63252

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₁₇

yes

—

C₂×C₄

#63252

yes

characteristic

C₂×C₄

#63252

yes

—

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

C₈

—

C₈

—

C₈

C₄×Dic₁₇

yes

characteristic

C₈

C₄×Dic₁₇

yes

—

C₈

C₄×Dic₁₇

yes

characteristic

C₂²

—

C₂²

#10587558

yes

—

100

C₂²

#10587612

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

#10587558

yes

—

104

C₄

C₂×C₄×Dic₁₇

yes

characteristic

105

C₄

#10587612

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#11762655

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#405	C₂
socle	C₂×C₁₇	#376
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#405	D₁₇
hypocenter	C₁₇	#1747
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₁₇

commutator

C₄×C₁₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#405

C₂

C₂×C₁₇

C₂³×C₁₇

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
11. C₁₇×#97	C₄	semidirect
12. C₄×C₈×C₁₇	C₄	semidirect
13. #10587486	C₄	semidirect
14. C₈×Dic₁₇	C₄	semidirect
66. #97	Dic₁₇	semidirect
68. C₄×C₈	Dic₁₇	semidirect
69. C₁₇	#1747	semidirect

H ≅ G/N

type

11. C₁₇×#97

C₄

semidirect

12. C₄×C₈×C₁₇

C₄

semidirect

13. #10587486

C₄

semidirect

14. C₈×Dic₁₇

C₄

semidirect

66. #97

Dic₁₇

semidirect

68. C₄×C₈

Dic₁₇

semidirect

69. C₁₇

#1747

semidirect

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

subgroup	index
#11762764	2
#11762567	2
#11762566	2
#11762565	2
C₁₇×#405	2
#1747	17

subgroup

index

C₁₇×#405

Subgroups — #14528539

Conjugacy classes of subgroups

Complements

Maximal subgroups