Group #14534744 — subgroups

Subgroups — #14534744

Order 2176.

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

normal subgroups only

This group has 420 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	#11762738	C₂	1	6	yes	maximal, complemented
2	#11762733	C₂	1	2	yes	maximal, complemented
3	#11762711	C₂	1	3	yes	maximal
4	#11762707	C₂	1	3	yes	maximal, complemented
5	C₁₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₈	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₁₇	C₂²	1	3	yes	—
8	C₂×#63227	C₂²	1	6	yes	—
9	C₂×#63227	C₂²	1	6	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	C₁₇×#95	C₂²	1	3	yes	—
14	#10587502	C₂²	1	6	yes	—
15	C₂³×D₁₇	C₂³	1	2	yes	—
16	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
17	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₇	C₂³	1	6	yes	—
20	C₂×D₆₈	—	2	6	no	—
21	C₄²×C₁₇	—	2	6	no	—
22	#63227	—	4	6	no	—
23	#63227	—	4	6	no	—
24	C₄⋊Dic₁₇	—	2	6	no	—
25	#63225	—	4	6	no	—
26	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
27	C₂²×D₁₇	—	2	6	no	—
28	C₂²×D₁₇	—	2	6	no	—
29	C₂²×D₁₇	—	2	2	no	—
30	C₂×C₄×C₁₇	—	2	3	no	—
31	C₂×C₄×C₁₇	—	2	3	no	—
32	C₂×C₄×C₁₇	—	2	3	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	#41	1	6	yes	—
37	C₂×Dic₁₇	—	2	6	no	—
38	C₂×Dic₁₇	—	2	6	no	—
39	C₂×Dic₁₇	—	2	6	no	—
40	D₆₈	—	4	4	no	—
41	#2644	—	17	1	no	maximal, Sylow
42	C₂²×C₁₇	C₂×#41	1	3	yes	—
43	C₂²×C₁₇	H(5,𝔽₂)	1	3	yes	—
44	C₂²×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
45	C₂×D₁₇	—	4	2	no	—
46	C₂×D₁₇	—	4	6	no	—
47	C₂×D₁₇	—	4	6	no	—
48	C₄×C₁₇	—	4	3	no	—
49	C₄×C₁₇	—	4	3	no	—
50	C₄×C₁₇	#126	1	4	yes	—
51	Dic₁₇	—	4	6	no	—
52	#397	—	17	6	no	—
53	#396	—	17	2	no	—
54	#390	—	17	3	no	—
55	#389	—	17	3	no	—
56	#376	D₁₇	1	1	yes	characteristic, complemented
57	C₂×C₁₇	C₂×#126	1	1	yes	characteristic
58	C₂×C₁₇	#540	1	3	yes	—
59	C₂×C₁₇	#542	1	3	yes	—
60	D₁₇	—	8	2	no	—
61	C₂²×D₄	—	17	1	no	—
62	C₂×C₄⋊C₄	—	17	3	no	—
63	C₂×C₄⋊C₄	—	17	6	no	—
64	C₂×C₂²⋊C₄	—	17	6	no	—
65	C₂×C₂²⋊C₄	—	17	6	no	—
66	C₂×C₄²	C₂×D₁₇	1	3	yes	—
67	#95	—	17	6	no	—
68	#95	C₂×D₁₇	1	1	yes	characteristic
69	#95	C₂×D₁₇	1	3	yes	—
70	C₁₇	#2644	1	1	yes	characteristic, complemented, Sylow
71	C₂⁴	—	17	2	no	—
72	C₂×D₄	—	34	6	no	—
73	C₂²×C₄	—	17	6	no	—
74	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
75	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
76	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
77	C₄⋊C₄	—	34	6	no	—
78	C₄⋊C₄	—	68	6	no	—
79	C₂²⋊C₄	—	68	6	no	—
80	C₂²⋊C₄	—	68	6	no	—
81	C₄²	—	2	6	no	—
82	C₂³	—	34	6	no	—
83	C₂³	—	34	6	no	—
84	C₂³	—	34	2	no	—
85	C₂³	C₂³×D₁₇	1	1	yes	characteristic
86	D₄	—	68	4	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	34	6	no	—
94	C₂×C₄	—	34	6	no	—
95	C₂×C₄	—	34	6	no	—
96	C₂×C₄	#63252	1	6	yes	—
97	C₂²	—	68	2	no	—
98	C₂²	—	68	6	no	—
99	C₂²	—	68	6	no	—
100	C₂²	C₂×#63252	1	3	yes	—
101	C₂²	#10587696	1	3	yes	—
102	C₂²	#10587697	1	1	yes	characteristic
103	C₄	—	4	3	no	—
104	C₄	—	4	3	no	—
105	C₄	—	68	6	no	—
106	C₄	#10587562	1	4	yes	—
107	C₂	—	136	2	no	—
108	C₂	C₂×#10587562	1	1	yes	characteristic
109	C₂	#11763414	1	3	yes	—
110	C₂	#11763430	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11762738

C₂

yes

maximal, complemented

#11762733

C₂

yes

maximal, complemented

#11762711

C₂

yes

maximal

#11762707

C₂

yes

maximal, complemented

C₁₇×#376

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₈

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

—

C₂×#63227

C₂²

yes

—

C₂×#63227

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

C₂²

yes

—

C₁₇×#95

C₂²

yes

characteristic

C₁₇×#95

C₂²

yes

—

#10587502

C₂²

yes

—

C₂³×D₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

C₂³

yes

—

C₂×D₆₈

—

C₄²×C₁₇

—

#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×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

#2644

—

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

#126

yes

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

C₂×#126

yes

characteristic

C₂×C₁₇

#540

yes

—

C₂×C₁₇

#542

yes

—

D₁₇

—

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

—

#95

—

#95

C₂×D₁₇

yes

characteristic

#95

C₂×D₁₇

yes

—

C₁₇

#2644

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#63252

yes

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×#63252

yes

—

101

C₂²

#10587696

yes

—

102

C₂²

#10587697

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#10587562

yes

—

107

C₂

—

136

—

108

C₂

C₂×#10587562

yes

characteristic

109

C₂

#11763414

yes

—

110

C₂

#11763430

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₁₇×#376	C₂
prime radical	C₂²×D₆₈	C₂²
hypercenter	#376	D₁₇
hypocenter	C₁₇	#2644
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₁₇×#376

C₂

prime radical

C₂²×D₆₈

C₂²

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
1. #11762738	C₂	semidirect
2. #11762733	C₂	semidirect
4. #11762707	C₂	semidirect
5. C₁₇×#376	C₂	semidirect
56. #376	D₁₇	semidirect
70. C₁₇	#2644	semidirect

H ≅ G/N

type

1. #11762738

C₂

semidirect

2. #11762733

C₂

semidirect

4. #11762707

C₂

semidirect

5. C₁₇×#376

C₂

semidirect

56. #376

D₁₇

semidirect

70. C₁₇

#2644

semidirect

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

subgroup	index
#11762738	2
#11762733	2
#11762711	2
#11762707	2
C₁₇×#376	2
#2644	17

subgroup

index

C₁₇×#376

Subgroups — #14534744

Conjugacy classes of subgroups

Complements

Maximal subgroups