Group #14540583 — subgroups

Subgroups — #14540583

Order 2176.

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

normal subgroups only

This group has 476 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	#11763610	C₂	1	4	yes	maximal, complemented
2	#11763076	C₂	1	4	yes	maximal, complemented
3	#11762976	C₂	1	4	yes	maximal, complemented
4	#11762927	C₂	1	2	yes	maximal, complemented
5	C₁₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	#10587691	C₂²	1	4	yes	complemented
7	C₂×D₈×C₁₇	C₂²	1	4	yes	complemented
8	C₁₇×#127	C₂²	1	2	yes	complemented
9	#10587618	C₂²	1	8	yes	complemented
10	#10587598	—	2	4	no	—
11	#10587597	—	2	8	no	—
12	D₈×D₁₇	—	2	4	no	—
13	#10587590	C₂²	1	2	yes	—
14	#10587587	C₂²	1	4	yes	complemented
15	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
16	#10587504	C₂²	1	4	yes	complemented
17	#10587490	C₂²	1	2	yes	complemented
18	#10587476	C₂²	1	4	yes	complemented
19	C₂×D₄×C₁₇	—	4	2	no	—
20	C₂×D₄×C₁₇	C₂³	1	4	yes	—
21	C₂×C₂²⋊D₁₇	—	4	4	no	—
22	#63254	—	2	8	no	—
23	D₄×D₁₇	—	2	8	no	—
24	#63252	C₂³	1	4	yes	—
25	D₈×C₁₇	—	2	4	no	—
26	D₈×C₁₇	—	2	4	no	—
27	C₂×C₈×C₁₇	C₂³	1	2	yes	—
28	C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	#63229	—	2	8	no	—
30	#63228	—	2	8	no	—
31	#63223	C₂³	1	4	yes	—
32	Dic₆₈	D₄	1	2	yes	complemented
33	D₁₃₆	D₄	1	2	yes	complemented
34	#63219	D₄	1	4	yes	complemented
35	#63218	—	2	4	no	—
36	C₈×D₁₇	—	2	4	no	—
37	C₂³×C₁₇	—	2	4	no	—
38	C₂²×D₁₇	—	4	4	no	—
39	D₄×C₁₇	—	2	8	no	—
40	D₄×C₁₇	—	8	4	no	—
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
43	C₂²⋊D₁₇	—	2	4	no	—
44	C₂²⋊D₁₇	—	4	8	no	—
45	C₂×Dic₁₇	—	4	4	no	—
46	D₆₈	C₂×D₄	1	4	yes	—
47	C₄×D₁₇	—	2	4	no	—
48	Dic₃₄	C₂×D₄	1	4	yes	—
49	C₈×C₁₇	C₂×D₄	1	2	yes	—
50	C₈×C₁₇	C₂×D₄	1	2	yes	—
51	#3622	—	2	4	no	—
52	#3292	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	—	4	4	no	—
54	C₂²×C₁₇	—	8	4	no	—
55	C₂²×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
56	C₂×D₁₇	—	2	4	no	—
57	C₂×D₁₇	—	16	4	no	—
58	C₄×C₁₇	—	2	2	no	—
59	C₄×C₁₇	C₂²×D₄	1	2	yes	—
60	Dic₁₇	—	2	4	no	—
61	#576	—	17	4	no	—
62	#493	D₁₇	1	1	yes	characteristic, complemented
63	#472	—	17	4	no	—
64	#453	—	17	4	no	—
65	#443	—	17	2	no	—
66	C₂×C₁₇	—	2	1	no	—
67	C₂×C₁₇	—	8	4	no	—
68	C₂×C₁₇	D₄²	1	1	yes	characteristic
69	D₁₇	—	4	4	no	—
70	H(5,𝔽₂)	—	17	4	no	—
71	AGL(1,ℤ/8)	—	17	8	no	—
72	AGL(1,ℤ/8)	—	34	8	no	—
73	#135	—	17	2	no	—
74	#135	—	34	4	no	—
75	C₂×D₈	—	34	4	no	—
76	C₂×D₈	C₂×D₁₇	1	4	yes	complemented
77	#131	—	17	4	no	—
78	#127	C₂×D₁₇	1	2	yes	complemented
79	#108	—	17	2	no	—
80	#104	—	17	4	no	—
81	#100	—	17	4	no	—
82	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
83	C₁₇	#3292	1	1	yes	characteristic, complemented, Sylow
84	#41	—	17	4	no	—
85	#41	—	34	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	34	8	no	—
88	C₂×D₄	—	68	4	no	—
89	C₂×D₄	C₂²×D₁₇	1	4	yes	—
90	Q₁₆	—	17	2	no	—
91	SD₁₆	—	17	4	no	—
92	SD₁₆	—	34	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	17	2	no	—
96	D₈	—	34	8	no	—
97	M₁₆	—	17	4	no	—
98	M₁₆	—	34	4	no	—
99	C₂×C₈	—	34	4	no	—
100	C₂×C₈	C₂²×D₁₇	1	2	yes	—
101	C₄²	C₂²×D₁₇	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	68	4	no	—
104	Q₈	—	17	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	17	4	no	—
108	D₄	—	34	4	no	—
109	D₄	—	68	8	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	34	4	no	—
112	C₂×C₄	—	68	4	no	—
113	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
114	C₈	—	34	4	no	—
115	C₈	D₄×D₁₇	1	2	yes	—
116	C₈	D₄×D₁₇	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	272	4	no	—
120	C₂²	—	34	4	no	—
121	C₂²	#10587691	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	34	4	no	—
124	C₄	C₂×D₄×D₁₇	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	68	4	no	—
128	C₂	#11763444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763610

C₂

yes

maximal, complemented

#11763076

C₂

yes

maximal, complemented

#11762976

C₂

yes

maximal, complemented

#11762927

C₂

yes

maximal, complemented

C₁₇×#493

C₂

yes

maximal, characteristic, complemented

#10587691

C₂²

yes

complemented

C₂×D₈×C₁₇

C₂²

yes

complemented

C₁₇×#127

C₂²

yes

complemented

#10587618

C₂²

yes

complemented

#10587598

—

#10587597

—

D₈×D₁₇

—

#10587590

C₂²

yes

—

#10587587

C₂²

yes

complemented

C₄×C₈×C₁₇

C₂²

yes

characteristic, complemented

#10587504

C₂²

yes

complemented

#10587490

C₂²

yes

complemented

#10587476

C₂²

yes

complemented

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

C₂³

yes

—

C₂×C₂²⋊D₁₇

—

#63254

—

D₄×D₁₇

—

#63252

C₂³

yes

—

D₈×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄²×C₁₇

C₂³

yes

—

—

C₂³

yes

—

Dic₆₈

D₄

yes

complemented

D₁₃₆

D₄

yes

complemented

#63219

D₄

yes

complemented

#63218

—

C₈×D₁₇

—

C₂³×C₁₇

—

C₂²×D₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂²⋊D₁₇

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

C₂×D₄

yes

—

C₄×D₁₇

—

Dic₃₄

C₂×D₄

yes

—

C₈×C₁₇

C₂×D₄

yes

—

C₈×C₁₇

C₂×D₄

yes

—

#3622

—

#3292

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²×D₄

yes

—

Dic₁₇

—

#576

—

#493

D₁₇

yes

characteristic, complemented

#472

—

#453

—

#443

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

D₄²

yes

characteristic

D₁₇

—

—

—

—

—

—

C₂×D₈

—

C₂×D₈

C₂×D₁₇

yes

complemented

#131

—

#127

C₂×D₁₇

yes

—

—

—

C₄×C₈

C₂×D₁₇

yes

characteristic, complemented

C₁₇

#3292

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₁₇

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

C₂²×D₁₇

yes

—

101

C₄²

C₂²×D₁₇

yes

characteristic

102

C₂³

—

103

C₂³

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂³×D₁₇

yes

characteristic

114

C₈

—

115

C₈

D₄×D₁₇

yes

—

116

C₈

D₄×D₁₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

272

—

120

C₂²

—

121

C₂²

#10587691

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₂×D₄×D₁₇

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

128

C₂

#11763444

yes

characteristic

	subgroup	G/H
center	C₂	#11763444
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#493	C₂
socle	C₂×C₁₇	D₄²
hypercenter	#493	D₁₇
hypocenter	C₁₇	#3292
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#11763444

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#493

C₂

socle

C₂×C₁₇

D₄²

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #11763610	C₂	semidirect
2. #11763076	C₂	semidirect
3. #11762976	C₂	semidirect
4. #11762927	C₂	semidirect
5. C₁₇×#493	C₂	semidirect
6. #10587691	C₂²	semidirect
7. C₂×D₈×C₁₇	C₂²	semidirect
8. C₁₇×#127	C₂²	semidirect
9. #10587618	C₂²	semidirect
14. #10587587	C₂²	semidirect
15. C₄×C₈×C₁₇	C₂²	semidirect
16. #10587504	C₂²	semidirect
17. #10587490	C₂²	semidirect
18. #10587476	C₂²	semidirect
32. Dic₆₈	D₄	semidirect
33. D₁₃₆	D₄	semidirect
34. #63219	D₄	semidirect
62. #493	D₁₇	semidirect
76. C₂×D₈	C₂×D₁₇	semidirect
78. #127	C₂×D₁₇	semidirect
82. C₄×C₈	C₂×D₁₇	semidirect
83. C₁₇	#3292	semidirect

H ≅ G/N

type

1. #11763610

C₂

semidirect

2. #11763076

C₂

semidirect

3. #11762976

C₂

semidirect

4. #11762927

C₂

semidirect

5. C₁₇×#493

C₂

semidirect

6. #10587691

C₂²

semidirect

7. C₂×D₈×C₁₇

C₂²

semidirect

8. C₁₇×#127

C₂²

semidirect

9. #10587618

C₂²

semidirect

14. #10587587

C₂²

semidirect

15. C₄×C₈×C₁₇

C₂²

semidirect

16. #10587504

C₂²

semidirect

17. #10587490

C₂²

semidirect

18. #10587476

C₂²

semidirect

32. Dic₆₈

D₄

semidirect

33. D₁₃₆

D₄

semidirect

34. #63219

D₄

semidirect

62. #493

D₁₇

semidirect

76. C₂×D₈

C₂×D₁₇

semidirect

78. #127

C₂×D₁₇

semidirect

82. C₄×C₈

C₂×D₁₇

semidirect

83. C₁₇

#3292

semidirect

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

subgroup	index
#11763610	2
#11763076	2
#11762976	2
#11762927	2
C₁₇×#493	2
#3292	17

subgroup

index

C₁₇×#493

Subgroups — #14540583

Conjugacy classes of subgroups

Complements

Maximal subgroups