Group #14543597 — subgroups

Subgroups — #14543597

Order 2176.

5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 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	#11763645	C₂	1	15	yes	maximal, complemented
2	#11763644	C₂	1	20	yes	maximal, complemented
3	C₂×#10587697	C₂	1	15	yes	maximal, complemented
4	2₋¹⁺⁴×D₁₇	C₂	1	12	yes	maximal, complemented
5	C₁₇×#585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×2₋¹⁺⁴	C₂²	1	6	yes	complemented
7	C₁₇×H(5,𝔽₂)	C₂²	1	10	yes	complemented
8	C₂×#41×C₁₇	C₂²	1	15	yes	complemented
9	#10587697	C₂²	1	120	yes	complemented
10	#10587697	C₂²	1	60	yes	complemented
11	#10587696	C₂²	1	20	yes	complemented
12	#41×D₁₇	C₂²	1	120	yes	complemented
13	#10587694	C₂²	1	30	yes	complemented
14	C₂×Q₈×D₁₇	C₂²	1	30	yes	complemented
15	#10587691	C₂²	1	90	yes	complemented
16	C₂×#63254	C₂²	1	90	yes	complemented
17	C₂×#63252	C₂²	1	45	yes	complemented
18	C₂²×Dic₃₄	C₂²	1	15	yes	complemented
19	#41×C₁₇	C₂³	1	60	yes	—
20	#41×C₁₇	C₂³	1	20	yes	—
21	C₂×Q₈×C₁₇	C₂³	1	15	yes	—
22	C₂×D₄×C₁₇	C₂³	1	45	yes	—
23	C₂²×C₄×C₁₇	C₂³	1	15	yes	—
24	C₂×C₂²⋊D₁₇	C₂³	1	90	yes	—
25	C₂²×Dic₁₇	C₂³	1	30	yes	—
26	#63256	C₂³	1	40	yes	—
27	Q₈×D₁₇	C₂³	1	120	yes	—
28	#63254	C₂³	1	360	yes	—
29	D₄×D₁₇	C₂³	1	120	yes	—
30	#63252	C₂³	1	60	yes	—
31	#63252	C₂³	1	180	yes	—
32	C₂×D₆₈	C₂³	1	15	yes	—
33	C₂×C₄×D₁₇	C₂³	1	90	yes	—
34	C₂×Dic₃₄	C₂³	1	45	yes	—
35	C₂×Dic₃₄	C₂³	1	90	yes	—
36	C₂³×C₁₇	C₂⁴	1	15	yes	—
37	C₂²×D₁₇	C₂⁴	1	30	yes	—
38	Q₈×C₁₇	C₂⁴	1	20	yes	—
39	D₄×C₁₇	C₂⁴	1	60	yes	—
40	C₂×C₄×C₁₇	C₂⁴	1	15	yes	—
41	C₂×C₄×C₁₇	C₂⁴	1	45	yes	—
42	C₂²⋊D₁₇	C₂⁴	1	120	yes	—
43	C₂×Dic₁₇	C₂⁴	1	90	yes	—
44	D₆₈	C₂⁴	1	6	yes	—
45	D₆₈	C₂⁴	1	30	yes	—
46	C₄×D₁₇	C₂⁴	1	120	yes	—
47	Dic₃₄	C₂⁴	1	10	yes	—
48	Dic₃₄	C₂⁴	1	90	yes	—
49	2₋¹⁺⁶	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	—	4	15	no	—
51	C₂²×C₁₇	C₂⁵	1	15	yes	—
52	C₂×D₁₇	—	4	30	no	—
53	C₂×D₁₇	C₂⁵	1	12	yes	—
54	C₄×C₁₇	C₂⁵	1	1	yes	characteristic
55	C₄×C₁₇	C₂⁵	1	15	yes	—
56	Dic₁₇	C₂⁵	1	20	yes	—
57	#585	—	17	20	no	—
58	#585	—	17	15	no	—
59	#585	D₁₇	1	1	yes	characteristic, complemented
60	C₂×2₋¹⁺⁴	—	17	12	no	—
61	C₂×2₋¹⁺⁴	—	17	15	no	—
62	C₂×C₁₇	—	2	15	no	—
63	C₂×C₁₇	C₂⁶	1	1	yes	characteristic
64	D₁₇	—	2	12	no	—
65	2₋¹⁺⁴	—	17	120	no	—
66	2₋¹⁺⁴	—	17	60	no	—
67	2₋¹⁺⁴	—	17	30	no	—
68	2₋¹⁺⁴	C₂×D₁₇	1	6	yes	complemented
69	H(5,𝔽₂)	—	17	20	no	—
70	H(5,𝔽₂)	—	17	90	no	—
71	H(5,𝔽₂)	C₂×D₁₇	1	10	yes	complemented
72	C₂×#41	—	17	120	no	—
73	C₂×#41	—	17	90	no	—
74	C₂×#41	—	17	45	no	—
75	C₂×#41	C₂×D₁₇	1	15	yes	complemented
76	C₂²×Q₈	—	17	30	no	—
77	C₂²×Q₈	—	17	15	no	—
78	C₁₇	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
79	#41	—	17	40	no	—
80	#41	—	17	360	no	—
81	#41	—	17	180	no	—
82	#41	—	17	60	no	—
83	#41	C₂²×D₁₇	1	60	yes	—
84	#41	C₂²×D₁₇	1	20	yes	—
85	C₂×Q₈	—	17	120	no	—
86	C₂×Q₈	—	17	90	no	—
87	C₂×Q₈	—	17	45	no	—
88	C₂×Q₈	C₂²×D₁₇	1	15	yes	—
89	C₂×D₄	—	17	90	no	—
90	C₂×D₄	—	17	120	no	—
91	C₂×D₄	—	17	15	no	—
92	C₂×D₄	C₂²×D₁₇	1	45	yes	—
93	C₂²×C₄	—	17	30	no	—
94	C₂²×C₄	—	17	90	no	—
95	C₂²×C₄	C₂²×D₁₇	1	15	yes	—
96	C₂³	—	17	30	no	—
97	C₂³	C₂³×D₁₇	1	15	yes	—
98	Q₈	—	17	90	no	—
99	Q₈	—	17	10	no	—
100	Q₈	C₂³×D₁₇	1	20	yes	—
101	D₄	—	17	120	no	—
102	D₄	—	17	30	no	—
103	D₄	—	17	6	no	—
104	D₄	C₂³×D₁₇	1	60	yes	—
105	C₂×C₄	—	17	90	no	—
106	C₂×C₄	—	17	120	no	—
107	C₂×C₄	C₂³×D₁₇	1	45	yes	—
108	C₂×C₄	C₂³×D₁₇	1	15	yes	—
109	C₂²	—	4	15	no	—
110	C₂²	—	17	12	no	—
111	C₂²	—	68	30	no	—
112	C₂²	C₂⁴×D₁₇	1	15	yes	—
113	C₄	—	17	20	no	—
114	C₄	C₂⁴×D₁₇	1	15	yes	—
115	C₄	C₂⁴×D₁₇	1	1	yes	characteristic
116	C₂	—	2	15	no	—
117	C₂	—	34	12	no	—
118	C₂	C₂⁵×D₁₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763645

C₂

yes

maximal, complemented

#11763644

C₂

yes

maximal, complemented

C₂×#10587697

C₂

yes

maximal, complemented

2₋¹⁺⁴×D₁₇

C₂

yes

maximal, complemented

C₁₇×#585

C₂

yes

maximal, characteristic, complemented

C₁₇×2₋¹⁺⁴

C₂²

yes

complemented

C₁₇×H(5,𝔽₂)

C₂²

yes

complemented

C₂×#41×C₁₇

C₂²

yes

complemented

#10587697

C₂²

120

yes

complemented

#10587697

C₂²

yes

complemented

#10587696

C₂²

yes

complemented

#41×D₁₇

C₂²

120

yes

complemented

#10587694

C₂²

yes

complemented

C₂×Q₈×D₁₇

C₂²

yes

complemented

#10587691

C₂²

yes

complemented

C₂×#63254

C₂²

yes

complemented

C₂×#63252

C₂²

yes

complemented

C₂²×Dic₃₄

C₂²

yes

complemented

#41×C₁₇

C₂³

yes

—

#41×C₁₇

C₂³

yes

—

C₂×Q₈×C₁₇

C₂³

yes

—

C₂×D₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂×C₂²⋊D₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

#63256

C₂³

yes

—

Q₈×D₁₇

C₂³

120

yes

—

#63254

C₂³

360

yes

—

D₄×D₁₇

C₂³

120

yes

—

#63252

C₂³

yes

—

#63252

C₂³

180

yes

—

C₂×D₆₈

C₂³

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

C₂×Dic₃₄

C₂³

yes

—

C₂×Dic₃₄

C₂³

yes

—

C₂³×C₁₇

C₂⁴

yes

—

C₂²×D₁₇

C₂⁴

yes

—

Q₈×C₁₇

C₂⁴

yes

—

D₄×C₁₇

C₂⁴

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

—

C₂²⋊D₁₇

C₂⁴

120

yes

—

C₂×Dic₁₇

C₂⁴

yes

—

D₆₈

C₂⁴

yes

—

D₆₈

C₂⁴

yes

—

C₄×D₁₇

C₂⁴

120

yes

—

Dic₃₄

C₂⁴

yes

—

Dic₃₄

C₂⁴

yes

—

2₋¹⁺⁶

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₂⁵

yes

—

C₂×D₁₇

—

C₂×D₁₇

C₂⁵

yes

—

C₄×C₁₇

C₂⁵

yes

characteristic

C₄×C₁₇

C₂⁵

yes

—

Dic₁₇

C₂⁵

yes

—

#585

—

#585

—

#585

D₁₇

yes

characteristic, complemented

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

C₂×C₁₇

—

C₂×C₁₇

C₂⁶

yes

characteristic

D₁₇

—

2₋¹⁺⁴

—

120

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₁₇

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

C₁₇

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

#41

—

#41

—

360

—

#41

—

180

—

#41

—

#41

C₂²×D₁₇

yes

—

#41

C₂²×D₁₇

yes

—

C₂×Q₈

—

120

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₇

yes

—

C₂×D₄

—

C₂×D₄

—

120

—

C₂×D₄

—

C₂×D₄

C₂²×D₁₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂³

—

C₂³

C₂³×D₁₇

yes

—

Q₈

—

Q₈

—

100

Q₈

C₂³×D₁₇

yes

—

101

D₄

—

120

—

102

D₄

—

103

D₄

—

104

D₄

C₂³×D₁₇

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

120

—

107

C₂×C₄

C₂³×D₁₇

yes

—

108

C₂×C₄

C₂³×D₁₇

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂⁴×D₁₇

yes

—

113

C₄

—

114

C₄

C₂⁴×D₁₇

yes

—

115

C₄

C₂⁴×D₁₇

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

C₂⁵×D₁₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₁₇
commutator, socle, cosocle	C₂×C₁₇	C₂⁶
Fitting	C₁₇×#585	C₂
hypercenter	#585	D₁₇
hypocenter	C₁₇	2₋¹⁺⁶
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₁₇

commutator, socle, cosocle

C₂×C₁₇

C₂⁶

Fitting

C₁₇×#585

C₂

hypercenter

#585

D₁₇

hypocenter

C₁₇

2₋¹⁺⁶

LCS

G ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. #11763645	C₂	semidirect
2. #11763644	C₂	semidirect
3. C₂×#10587697	C₂	semidirect
4. 2₋¹⁺⁴×D₁₇	C₂	semidirect
5. C₁₇×#585	C₂	semidirect
6. C₁₇×2₋¹⁺⁴	C₂²	semidirect
7. C₁₇×H(5,𝔽₂)	C₂²	semidirect
8. C₂×#41×C₁₇	C₂²	semidirect
9. #10587697	C₂²	semidirect
10. #10587697	C₂²	semidirect
11. #10587696	C₂²	semidirect
12. #41×D₁₇	C₂²	semidirect
13. #10587694	C₂²	semidirect
14. C₂×Q₈×D₁₇	C₂²	semidirect
15. #10587691	C₂²	semidirect
16. C₂×#63254	C₂²	semidirect
17. C₂×#63252	C₂²	semidirect
18. C₂²×Dic₃₄	C₂²	semidirect
59. #585	D₁₇	semidirect
68. 2₋¹⁺⁴	C₂×D₁₇	semidirect
71. H(5,𝔽₂)	C₂×D₁₇	semidirect
75. C₂×#41	C₂×D₁₇	semidirect
78. C₁₇	2₋¹⁺⁶	semidirect

H ≅ G/N

type

1. #11763645

C₂

semidirect

2. #11763644

C₂

semidirect

3. C₂×#10587697

C₂

semidirect

4. 2₋¹⁺⁴×D₁₇

C₂

semidirect

5. C₁₇×#585

C₂

semidirect

6. C₁₇×2₋¹⁺⁴

C₂²

semidirect

7. C₁₇×H(5,𝔽₂)

C₂²

semidirect

8. C₂×#41×C₁₇

C₂²

semidirect

9. #10587697

C₂²

semidirect

10. #10587697

C₂²

semidirect

11. #10587696

C₂²

semidirect

12. #41×D₁₇

C₂²

semidirect

13. #10587694

C₂²

semidirect

14. C₂×Q₈×D₁₇

C₂²

semidirect

15. #10587691

C₂²

semidirect

16. C₂×#63254

C₂²

semidirect

17. C₂×#63252

C₂²

semidirect

18. C₂²×Dic₃₄

C₂²

semidirect

59. #585

D₁₇

semidirect

68. 2₋¹⁺⁴

C₂×D₁₇

semidirect

71. H(5,𝔽₂)

C₂×D₁₇

semidirect

75. C₂×#41

C₂×D₁₇

semidirect

78. C₁₇

2₋¹⁺⁶

semidirect

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

subgroup	index
#11763645	2
#11763644	2
C₂×#10587697	2
2₋¹⁺⁴×D₁₇	2
C₁₇×#585	2
2₋¹⁺⁶	17

subgroup

index

#11763645

#11763644

C₂×#10587697

2₋¹⁺⁴×D₁₇

C₁₇×#585

2₋¹⁺⁶

Subgroups — #14543597

Conjugacy classes of subgroups

Complements

Maximal subgroups