Group #14536858 — subgroups

Subgroups — #14536858

Order 2176.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#10587562	C₂	1	2	yes	maximal, complemented
2	#11762739	C₂	1	3	yes	maximal, complemented
3	#11762731	C₂	1	6	yes	maximal, complemented
4	#11762726	C₂	1	3	yes	maximal
5	C₁₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
8	C₂×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×#63227	C₂²	1	6	yes	—
11	C₂×#63227	C₂²	1	6	yes	—
12	C₂×C₄⋊Dic₁₇	C₂²	1	3	yes	—
13	C₂×#63225	C₂²	1	6	yes	—
14	#10587562	—	2	8	no	—
15	#10587502	C₂²	1	3	yes	—
16	#10587502	C₂²	1	3	yes	—
17	C₂³×D₁₇	C₂³	1	2	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
20	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
21	C₂²×Dic₁₇	C₂³	1	6	yes	—
22	C₂×D₆₈	—	4	3	no	—
23	C₄⋊C₄×C₁₇	—	4	3	no	—
24	C₄⋊C₄×C₁₇	—	4	3	no	—
25	C₄²×C₁₇	D₄	1	4	yes	complemented
26	#63227	—	2	12	no	—
27	#63227	—	4	6	no	—
28	C₄⋊Dic₁₇	—	4	3	no	—
29	#63225	—	2	12	no	—
30	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
31	C₂²×D₁₇	—	2	2	no	—
32	C₂²×D₁₇	—	2	6	no	—
33	C₂²×D₁₇	—	2	6	no	—
34	C₂×C₄×C₁₇	—	2	3	no	—
35	C₂×C₄×C₁₇	—	2	3	no	—
36	C₂×C₄×C₁₇	—	2	3	no	—
37	C₂×C₄×C₁₇	—	2	3	no	—
38	C₂×C₄×C₁₇	—	2	6	no	—
39	C₂×C₄×C₁₇	C₂×D₄	1	6	yes	—
40	C₂×Dic₁₇	—	2	6	no	—
41	C₂×Dic₁₇	—	2	6	no	—
42	C₂×Dic₁₇	—	2	6	no	—
43	D₆₈	—	4	4	no	—
44	#2823	—	17	1	no	maximal, Sylow
45	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
46	C₂²×C₁₇	H(5,𝔽₂)	1	3	yes	—
47	C₂²×C₁₇	2₋¹⁺⁴	1	3	yes	—
48	C₂×D₁₇	—	4	6	no	—
49	C₂×D₁₇	—	4	2	no	—
50	C₂×D₁₇	—	4	6	no	—
51	C₄×C₁₇	—	2	6	no	—
52	C₄×C₁₇	—	4	1	no	—
53	C₄×C₁₇	—	4	3	no	—
54	Dic₁₇	—	4	6	no	—
55	C₂×#126	—	17	2	no	—
56	#398	—	17	3	no	—
57	#396	—	17	6	no	—
58	#394	—	17	3	no	—
59	#384	D₁₇	1	1	yes	characteristic, complemented
60	C₂×C₁₇	#537	1	3	yes	—
61	C₂×C₁₇	#560	1	1	yes	characteristic
62	C₂×C₁₇	#563	1	3	yes	—
63	D₁₇	—	8	2	no	—
64	C₂²×D₄	—	17	1	no	—
65	#126	—	34	8	no	—
66	C₂×C₄⋊C₄	—	17	6	no	—
67	C₂×C₄⋊C₄	—	17	3	no	—
68	C₂×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
69	C₂×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
70	C₂×C₂²⋊C₄	—	17	6	no	—
71	C₂×C₂²⋊C₄	—	17	6	no	—
72	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
73	#95	—	17	3	no	—
74	#95	—	17	3	no	—
75	C₁₇	#2823	1	1	yes	characteristic, complemented, Sylow
76	C₂⁴	—	17	2	no	—
77	C₂×D₄	—	68	3	no	—
78	C₂²×C₄	—	17	6	no	—
79	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
81	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
82	C₄⋊C₄	—	4	3	no	—
83	C₄⋊C₄	—	4	3	no	—
84	C₄⋊C₄	—	34	12	no	—
85	C₄⋊C₄	—	68	3	no	—
86	C₂²⋊C₄	—	34	12	no	—
87	C₂²⋊C₄	—	68	6	no	—
88	C₄²	D₆₈	1	4	yes	complemented
89	C₂³	—	34	2	no	—
90	C₂³	—	34	6	no	—
91	C₂³	—	34	6	no	—
92	C₂³	C₂³×D₁₇	1	1	yes	characteristic
93	D₄	—	68	4	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	34	6	no	—
100	C₂×C₄	—	34	6	no	—
101	C₂×C₄	—	34	6	no	—
102	C₂×C₄	C₂×D₆₈	1	6	yes	—
103	C₂²	—	68	2	no	—
104	C₂²	—	68	6	no	—
105	C₂²	—	68	6	no	—
106	C₂²	C₂²×D₆₈	1	1	yes	characteristic
107	C₂²	#10587691	1	3	yes	—
108	C₂²	#10587694	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	3	no	—
112	C₄	—	68	6	no	—
113	C₂	—	136	2	no	—
114	C₂	#11763390	1	3	yes	—
115	C₂	#11763546	1	1	yes	characteristic
116	C₂	#11763558	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587562

C₂

yes

maximal, complemented

#11762739

C₂

yes

maximal, complemented

#11762731

C₂

yes

maximal, complemented

#11762726

C₂

yes

maximal

C₁₇×#384

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₈

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×#63227

C₂²

yes

—

C₂×#63227

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

C₂²

yes

—

#10587562

—

#10587502

C₂²

yes

—

#10587502

C₂²

yes

—

C₂³×D₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₂×D₆₈

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

D₄

yes

complemented

#63227

—

#63227

—

C₄⋊Dic₁₇

—

#63225

—

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

#2823

—

maximal, Sylow

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂²×C₁₇

H(5,𝔽₂)

yes

—

C₂²×C₁₇

2₋¹⁺⁴

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

#537

yes

—

C₂×C₁₇

#560

yes

characteristic

C₂×C₁₇

#563

yes

—

D₁₇

—

C₂²×D₄

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#95

—

#95

—

C₁₇

#2823

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

D₆₈

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂×D₆₈

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×D₆₈

yes

characteristic

107

C₂²

#10587691

yes

—

108

C₂²

#10587694

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₂

—

136

—

114

C₂

#11763390

yes

—

115

C₂

#11763546

yes

characteristic

116

C₂

#11763558

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₁₇×#384	C₂
prime radical	C₂²×D₆₈	C₂²
hypercenter	#384	D₁₇
hypocenter	C₁₇	#2823
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₁₇×#384

C₂

prime radical

C₂²×D₆₈

C₂²

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#10587562	C₂	semidirect
2. #11762739	C₂	semidirect
3. #11762731	C₂	semidirect
5. C₁₇×#384	C₂	semidirect
25. C₄²×C₁₇	D₄	semidirect
59. #384	D₁₇	semidirect
75. C₁₇	#2823	semidirect
88. C₄²	D₆₈	semidirect

H ≅ G/N

type

1. C₂×#10587562

C₂

semidirect

2. #11762739

C₂

semidirect

3. #11762731

C₂

semidirect

5. C₁₇×#384

C₂

semidirect

25. C₄²×C₁₇

D₄

semidirect

59. #384

D₁₇

semidirect

75. C₁₇

#2823

semidirect

88. C₄²

D₆₈

semidirect

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

subgroup	index
C₂×#10587562	2
#11762739	2
#11762731	2
#11762726	2
C₁₇×#384	2
#2823	17

subgroup

index

C₁₇×#384

Subgroups — #14536858

Conjugacy classes of subgroups

Complements

Maximal subgroups