Group #14536970 — subgroups

Subgroups — #14536970

Order 2176.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₁₇⋊#401	C₂	1	2	yes	maximal, complemented
2	#11762742	C₂	1	6	yes	maximal, complemented
3	#11762734	C₂	1	3	yes	maximal
4	#11762681	C₂	1	3	yes	maximal
5	C₁₇×#399	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
8	C₂×C₂²⋊C₄×C₁₇	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₁₇	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₁₇	C₂²	1	3	yes	—
11	C₂×#63225	C₂²	1	6	yes	—
12	C₁₇×#95	C₂²	1	1	yes	characteristic
13	#10587502	C₂²	1	3	yes	—
14	#10587502	C₂²	1	6	yes	—
15	#10587502	C₂²	1	6	yes	—
16	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
19	C₂²×Dic₁₇	—	4	3	no	—
20	C₂²×Dic₁₇	C₂³	1	2	yes	—
21	C₂²×Dic₁₇	C₂³	1	6	yes	—
22	C₄⋊C₄×C₁₇	—	4	3	no	—
23	C₂²⋊C₄×C₁₇	—	4	3	no	—
24	C₂²⋊Dic₁₇	—	2	6	no	—
25	C₄⋊Dic₁₇	—	4	3	no	—
26	#63225	—	4	6	no	—
27	C₂³×C₁₇	—	2	3	no	—
28	C₂³×C₁₇	—	2	3	no	—
29	C₂³×C₁₇	#41	1	2	yes	—
30	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₁₇	—	2	3	no	—
32	C₂×C₄×C₁₇	—	2	3	no	—
33	C₂×C₄×C₁₇	—	2	6	no	—
34	C₂×C₄×C₁₇	—	2	6	no	—
35	C₂×Dic₁₇	—	2	6	no	—
36	C₂×Dic₁₇	—	2	6	no	—
37	C₂×Dic₁₇	—	2	6	no	—
38	C₂×Dic₁₇	—	2	6	no	—
39	C₂×Dic₁₇	—	4	4	no	—
40	#2834	—	17	1	no	maximal, Sylow
41	C₂²×C₁₇	—	2	6	no	—
42	C₂²×C₁₇	—	4	1	no	—
43	C₂²×C₁₇	—	4	3	no	—
44	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
45	C₂²×C₁₇	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₁₇	2₋¹⁺⁴	1	3	yes	—
47	C₄×C₁₇	—	4	3	no	—
48	C₄×C₁₇	—	4	3	no	—
49	Dic₁₇	—	4	6	no	—
50	Dic₁₇	—	4	2	no	—
51	#401	—	17	2	no	—
52	#399	D₁₇	1	1	yes	characteristic
53	#398	—	17	6	no	—
54	#397	—	17	3	no	—
55	#385	—	17	3	no	—
56	C₂×C₁₇	—	4	2	no	—
57	C₂×C₁₇	#539	1	3	yes	—
58	C₂×C₁₇	#560	1	1	yes	characteristic
59	C₂×C₁₇	#563	1	3	yes	—
60	C₂³×C₄	—	17	1	no	—
61	C₂×C₄⋊C₄	—	17	6	no	—
62	C₂×C₄⋊C₄	—	17	3	no	—
63	C₂×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
64	C₂×C₂²⋊C₄	—	17	3	no	—
65	C₂×C₂²⋊C₄	C₂×D₁₇	1	3	yes	—
66	#95	—	17	6	no	—
67	#95	—	17	3	no	—
68	#95	—	17	6	no	—
69	#95	C₂×D₁₇	1	1	yes	characteristic
70	C₁₇	#2834	1	1	yes	characteristic, complemented, Sylow
71	C₂⁴	C₂²×D₁₇	1	1	yes	characteristic
72	C₂²×C₄	—	17	6	no	—
73	C₂²×C₄	—	17	2	no	—
74	C₂²×C₄	—	68	3	no	—
75	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
76	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
77	C₄⋊C₄	—	4	3	no	—
78	C₄⋊C₄	—	68	6	no	—
79	C₄⋊C₄	—	68	3	no	—
80	C₂²⋊C₄	—	4	3	no	—
81	C₂²⋊C₄	—	34	6	no	—
82	C₂³	—	2	3	no	—
83	C₂³	—	2	3	no	—
84	C₂³	#63256	1	2	yes	—
85	C₂³	C₂³×D₁₇	1	1	yes	characteristic
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	6	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	34	6	no	—
91	C₂×C₄	—	34	6	no	—
92	C₂×C₄	—	34	6	no	—
93	C₂×C₄	—	34	6	no	—
94	C₂×C₄	—	68	4	no	—
95	C₂²	—	2	6	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	3	no	—
98	C₂²	#10587691	1	3	yes	—
99	C₂²	C₂×#63256	1	1	yes	characteristic
100	C₂²	#10587697	1	3	yes	—
101	C₄	—	4	3	no	—
102	C₄	—	4	3	no	—
103	C₄	—	68	2	no	—
104	C₄	—	68	6	no	—
105	C₂	—	4	2	no	—
106	C₂	#11763408	1	3	yes	—
107	C₂	#11763546	1	1	yes	characteristic
108	C₂	#11763562	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇⋊#401

C₂

yes

maximal, complemented

#11762742

C₂

yes

maximal, complemented

#11762734

C₂

yes

maximal

#11762681

C₂

yes

maximal

C₁₇×#399

C₂

yes

maximal, characteristic

C₂³×Dic₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₇

C₂²

yes

—

C₂×C₂²⋊Dic₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

C₂²

yes

—

C₁₇×#95

C₂²

yes

characteristic

#10587502

C₂²

yes

—

#10587502

C₂²

yes

—

#10587502

C₂²

yes

—

C₂⁴×C₁₇

C₂³

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

#41

yes

—

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

#2834

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#41

yes

characteristic

C₂²×C₁₇

H(5,𝔽₂)

yes

—

C₂²×C₁₇

2₋¹⁺⁴

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

—

#401

—

#399

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#539

yes

—

C₂×C₁₇

#560

yes

characteristic

C₂×C₁₇

#563

yes

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₇

yes

—

—

—

—

C₂×D₁₇

yes

characteristic

C₁₇

#2834

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₁₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

#63256

yes

—

C₂³

C₂³×D₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#10587691

yes

—

C₂²

C₂×#63256

yes

characteristic

100

C₂²

#10587697

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₂

—

106

C₂

#11763408

yes

—

107

C₂

#11763546

yes

characteristic

108

C₂

#11763562

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₁₇×#399	C₂
prime radical	C₂⁴×C₁₇	C₂³
hypercenter	#399	D₁₇
hypocenter	C₁₇	#2834
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₁₇×#399

C₂

prime radical

C₂⁴×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. C₁₇⋊#401	C₂	semidirect
2. #11762742	C₂	semidirect
70. C₁₇	#2834	semidirect

H ≅ G/N

type

1. C₁₇⋊#401

C₂

semidirect

2. #11762742

C₂

semidirect

70. C₁₇

#2834

semidirect

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

subgroup	index
C₁₇⋊#401	2
#11762742	2
#11762734	2
#11762681	2
C₁₇×#399	2
#2834	17

subgroup

index

C₁₇⋊#401

C₁₇×#399

Subgroups — #14536970

Conjugacy classes of subgroups

Complements

Maximal subgroups