Group #14536492 — subgroups

Subgroups — #14536492

Order 2176.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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	#11762726	C₂	1	3	yes	maximal
2	#11762720	C₂	1	2	yes	maximal, complemented
3	#11762701	C₂	1	6	yes	maximal, complemented
4	#11762674	C₂	1	3	yes	maximal, complemented
5	C₁₇×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×D₆₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
9	C₂×#63227	C₂²	1	6	yes	—
10	C₂×#63227	C₂²	1	6	yes	—
11	C₂×C₄⋊Dic₁₇	C₂²	1	3	yes	—
12	C₂×C₄×Dic₁₇	C₂²	1	3	yes	—
13	C₁₇×#95	C₂²	1	3	yes	—
14	#10587502	C₂²	1	6	yes	—
15	#10587502	C₂²	1	3	yes	—
16	C₂³×D₁₇	C₂³	1	2	yes	—
17	C₂×Q₈×C₁₇	—	4	3	no	—
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	#63227	—	2	12	no	—
25	#63227	—	4	6	no	—
26	C₄⋊Dic₁₇	—	4	3	no	—
27	C₄×Dic₁₇	—	2	6	no	—
28	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
29	C₂²×D₁₇	—	2	6	no	—
30	C₂²×D₁₇	—	2	6	no	—
31	C₂²×D₁₇	—	2	2	no	—
32	Q₈×C₁₇	—	4	4	no	—
33	C₂×C₄×C₁₇	—	2	3	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	6	no	—
38	C₂×C₄×C₁₇	#41	1	6	yes	—
39	C₂×Dic₁₇	—	2	6	no	—
40	C₂×Dic₁₇	—	2	6	no	—
41	C₂×Dic₁₇	—	2	6	no	—
42	D₆₈	—	4	4	no	—
43	#2785	—	17	1	no	maximal, Sylow
44	C₂²×C₁₇	C₂×#41	1	3	yes	—
45	C₂²×C₁₇	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
47	C₂×D₁₇	—	4	6	no	—
48	C₂×D₁₇	—	4	6	no	—
49	C₂×D₁₇	—	4	2	no	—
50	C₄×C₁₇	—	2	6	no	—
51	C₄×C₁₇	—	4	1	no	—
52	C₄×C₁₇	—	4	3	no	—
53	Dic₁₇	—	4	6	no	—
54	#395	D₁₇	1	1	yes	characteristic, complemented
55	#394	—	17	3	no	—
56	#393	—	17	2	no	—
57	#388	—	17	6	no	—
58	#382	—	17	3	no	—
59	C₂×C₁₇	#539	1	3	yes	—
60	C₂×C₁₇	#555	1	3	yes	—
61	C₂×C₁₇	H(3,𝔽₄)	1	1	yes	characteristic
62	D₁₇	—	8	2	no	—
63	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
64	C₂²×D₄	—	17	1	no	—
65	C₂×C₄⋊C₄	—	17	3	no	—
66	C₂×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
67	C₂×C₂²⋊C₄	—	17	6	no	—
68	C₂×C₂²⋊C₄	—	17	6	no	—
69	C₂×C₄²	—	17	3	no	—
70	#95	—	17	3	no	—
71	#95	—	17	6	no	—
72	#95	C₂×D₁₇	1	3	yes	—
73	C₁₇	#2785	1	1	yes	characteristic, complemented, Sylow
74	C₂⁴	—	17	2	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	68	3	no	—
77	C₂²×C₄	—	17	6	no	—
78	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
80	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	68	3	no	—
83	C₂²⋊C₄	—	34	12	no	—
84	C₂²⋊C₄	—	68	6	no	—
85	C₄²	—	34	6	no	—
86	C₂³	—	34	6	no	—
87	C₂³	—	34	6	no	—
88	C₂³	—	34	2	no	—
89	C₂³	C₂³×D₁₇	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	68	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	34	6	no	—
98	C₂×C₄	—	34	6	no	—
99	C₂×C₄	—	34	6	no	—
100	C₂×C₄	#63256	1	6	yes	—
101	C₂²	—	68	6	no	—
102	C₂²	—	68	2	no	—
103	C₂²	—	68	6	no	—
104	C₂²	#10587691	1	3	yes	—
105	C₂²	C₂×#63256	1	3	yes	—
106	C₂²	#10587697	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	68	6	no	—
111	C₂	—	136	2	no	—
112	C₂	#11763408	1	3	yes	—
113	C₂	#11763525	1	3	yes	—
114	C₂	#11763552	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762726

C₂

yes

maximal

#11762720

C₂

yes

maximal, complemented

#11762701

C₂

yes

maximal, complemented

#11762674

C₂

yes

maximal, complemented

C₁₇×#395

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₇

C₂²

yes

characteristic

C₂²×D₆₈

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×#63227

C₂²

yes

—

C₂×#63227

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

—

C₁₇×#95

C₂²

yes

—

#10587502

C₂²

yes

—

#10587502

C₂²

yes

—

C₂³×D₁₇

C₂³

yes

—

C₂×Q₈×C₁₇

—

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

—

#63227

—

#63227

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

#2785

—

maximal, Sylow

C₂²×C₁₇

C₂×#41

yes

—

C₂²×C₁₇

H(5,𝔽₂)

yes

—

C₂²×C₁₇

2₋¹⁺⁴

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

#395

D₁₇

yes

characteristic, complemented

—

—

—

—

C₂×C₁₇

#539

yes

—

C₂×C₁₇

#555

yes

—

C₂×C₁₇

H(3,𝔽₄)

yes

characteristic

D₁₇

—

C₂²×Q₈

C₂×D₁₇

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₁₇

yes

—

C₁₇

#2785

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₇

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#63256

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#10587691

yes

—

105

C₂²

C₂×#63256

yes

—

106

C₂²

#10587697

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

136

—

112

C₂

#11763408

yes

—

113

C₂

#11763525

yes

—

114

C₂

#11763552

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₁₇×#395	C₂
prime radical	C₂²×D₆₈	C₂²
hypercenter	#395	D₁₇
hypocenter	C₁₇	#2785
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₁₇×#395

C₂

prime radical

C₂²×D₆₈

C₂²

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
2. #11762720	C₂	semidirect
3. #11762701	C₂	semidirect
4. #11762674	C₂	semidirect
5. C₁₇×#395	C₂	semidirect
54. #395	D₁₇	semidirect
73. C₁₇	#2785	semidirect

H ≅ G/N

type

2. #11762720

C₂

semidirect

3. #11762701

C₂

semidirect

4. #11762674

C₂

semidirect

5. C₁₇×#395

C₂

semidirect

54. #395

D₁₇

semidirect

73. C₁₇

#2785

semidirect

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

subgroup	index
#11762726	2
#11762720	2
#11762701	2
#11762674	2
C₁₇×#395	2
#2785	17

subgroup

index

C₁₇×#395

Subgroups — #14536492

Conjugacy classes of subgroups

Complements

Maximal subgroups