Group #14531007 — subgroups

Subgroups — #14531007

Order 2176.

160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 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	#11763214	C₂	1	1	yes	maximal, characteristic, complemented
2	#11762610	C₂	1	4	yes	maximal, complemented
3	C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×D₈×C₁₇	—	2	1	no	—
6	C₂×D₈×C₁₇	C₂²	1	2	yes	—
7	C₁₇×#127	—	2	1	no	—
8	#10587555	—	2	1	no	—
9	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	#10587489	—	2	1	no	—
11	#10587489	C₂²	1	2	yes	—
12	C₂×#63214	C₂²	1	2	yes	—
13	C₂×D₄×C₁₇	—	2	2	no	—
14	C₂×D₄×C₁₇	—	4	1	no	—
15	C₂×D₄×C₁₇	—	4	1	no	—
16	C₂×Dic₃₄	—	4	1	no	—
17	D₈×C₁₇	—	2	4	no	—
18	D₈×C₁₇	—	4	2	no	—
19	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₇	D₄	1	1	yes	characteristic
22	C₄⋊Dic₁₇	—	2	2	no	—
23	C₄⋊Dic₁₇	—	4	1	no	—
24	#63214	—	2	2	no	—
25	C₂³×C₁₇	—	4	2	no	—
26	D₄×C₁₇	—	4	2	no	—
27	D₄×C₁₇	—	4	2	no	—
28	D₄×C₁₇	—	8	2	no	—
29	D₄×C₁₇	—	8	2	no	—
30	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₁₇	—	4	2	no	—
34	Dic₃₄	—	8	2	no	—
35	C₈×C₁₇	SD₁₆	1	2	yes	—
36	C₈×C₁₇	#41	1	2	yes	—
37	#2240	—	17	1	no	maximal, Sylow
38	C₂²×C₁₇	—	8	2	no	—
39	C₂²×C₁₇	—	8	2	no	—
40	C₂²×C₁₇	—	8	2	no	—
41	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
42	C₄×C₁₇	D₁₆	1	2	yes	—
43	C₄×C₁₇	SD₃₂	1	2	yes	—
44	C₄×C₁₇	#124	1	1	yes	characteristic
45	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
46	Dic₁₇	—	8	2	no	—
47	#500	—	17	1	no	—
48	#493	D₁₇	1	1	yes	characteristic
49	#357	—	17	4	no	—
50	C₄×C₁₆	—	17	1	no	—
51	C₂×C₁₇	—	16	2	no	—
52	C₂×C₁₇	#486	1	1	yes	characteristic
53	C₂×C₁₇	C₂×D₁₆	1	1	yes	characteristic
54	C₂×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
55	C₂×D₈	—	2	1	no	—
56	C₂×D₈	C₂×D₁₇	1	2	yes	—
57	C₄⋊Q₈	—	34	1	no	—
58	#127	—	2	1	no	—
59	C₂×C₁₆	—	17	2	no	—
60	#107	—	17	2	no	—
61	#107	—	34	1	no	—
62	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
63	C₁₇	#2240	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	68	1	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	4	1	no	—
67	C₂×D₄	—	4	1	no	—
68	D₈	—	2	4	no	—
69	D₈	—	4	2	no	—
70	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
72	C₄⋊C₄	—	34	2	no	—
73	C₄⋊C₄	—	68	1	no	—
74	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
75	C₁₆	—	34	2	no	—
76	C₂³	—	4	2	no	—
77	Q₈	—	136	2	no	—
78	D₄	—	4	2	no	—
79	D₄	—	4	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	2	no	—
82	C₂×C₄	—	68	2	no	—
83	C₂×C₄	#63228	1	1	yes	characteristic
84	C₂×C₄	#63228	1	1	yes	characteristic
85	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
86	C₈	#63229	1	2	yes	—
87	C₈	#63254	1	2	yes	—
88	C₂²	—	8	2	no	—
89	C₂²	—	8	2	no	—
90	C₂²	—	8	2	no	—
91	C₂²	C₂×#63228	1	1	yes	characteristic
92	C₄	—	136	2	no	—
93	C₄	#10587497	1	2	yes	—
94	C₄	#10587498	1	2	yes	—
95	C₄	C₂×#63229	1	1	yes	characteristic
96	C₄	#10587622	1	1	yes	characteristic
97	C₂	—	16	2	no	—
98	C₂	#11763155	1	1	yes	characteristic
99	C₂	C₂×#10587497	1	1	yes	characteristic
100	C₂	C₂×#10587498	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763214

C₂

yes

maximal, characteristic, complemented

#11762610

C₂

yes

maximal, complemented

C₄×#63214

C₂

yes

maximal, characteristic, complemented

C₁₇×#493

C₂

yes

maximal, characteristic

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

C₂²

yes

—

C₁₇×#127

—

#10587555

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587489

—

#10587489

C₂²

yes

—

C₂×#63214

C₂²

yes

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×Dic₃₄

—

D₈×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63214

—

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

D₈

yes

characteristic

C₂×C₄×C₁₇

D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

SD₁₆

yes

—

C₈×C₁₇

#41

yes

—

#2240

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×D₈

yes

characteristic

C₄×C₁₇

D₁₆

yes

—

C₄×C₁₇

SD₃₂

yes

—

C₄×C₁₇

#124

yes

characteristic

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

Dic₁₇

—

#500

—

#493

D₁₇

yes

characteristic

#357

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#486

yes

characteristic

C₂×C₁₇

C₂×D₁₆

yes

characteristic

C₂×C₁₇

C₂×SD₃₂

yes

characteristic

C₂×D₈

—

C₂×D₈

C₂×D₁₇

yes

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#2240

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

C₂×C₈

C₂²×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₇

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

136

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

C₈

#63229

yes

—

C₈

#63254

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#63228

yes

characteristic

C₄

—

136

—

C₄

#10587497

yes

—

C₄

#10587498

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

100

C₂

C₂×#10587498

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting, prime radical	C₁₇×#493	C₂
socle	C₂²×C₁₇	C₂×D₈
hypercenter	#493	D₁₇
hypocenter	C₁₇	#2240
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

C₂×#63228

commutator, cosocle

C₂×C₈×C₁₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₇

Fitting, prime radical

C₁₇×#493

C₂

socle

C₂²×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #11763214	C₂	semidirect
2. #11762610	C₂	semidirect
3. C₄×#63214	C₂	semidirect
63. C₁₇	#2240	semidirect

H ≅ G/N

type

1. #11763214

C₂

semidirect

2. #11762610

C₂

semidirect

3. C₄×#63214

C₂

semidirect

63. C₁₇

#2240

semidirect

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

subgroup	index
#11763214	2
#11762610	2
C₄×#63214	2
C₁₇×#493	2
#2240	17

subgroup

index

C₁₇×#493

Subgroups — #14531007

Conjugacy classes of subgroups

Complements

Maximal subgroups