Group #14530294 — subgroups

Subgroups — #14530294

Order 2176.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#11762852	C₂	1	2	yes	maximal
2	C₂×#10587488	C₂	1	1	yes	maximal, characteristic
3	#11762577	C₂	1	2	yes	maximal
4	#11762574	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#424	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#117	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	C₂×#63223	C₂²	1	2	yes	—
11	C₁₇×C₄⋊C₈	—	2	2	no	—
12	#10587488	—	2	2	no	—
13	#10587479	—	2	2	no	—
14	#10587478	—	2	2	no	—
15	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₁₇	—	2	2	no	—
17	M₁₆×C₁₇	—	2	2	no	—
18	C₂×C₈×C₁₇	—	2	2	no	—
19	C₂×C₈×C₁₇	—	2	1	no	—
20	C₂×C₈×C₁₇	D₄	1	2	yes	—
21	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
22	C₂²⋊C₄×C₁₇	—	2	1	no	—
23	C₄²×C₁₇	—	2	1	no	—
24	C₄⋊Dic₁₇	—	2	2	no	—
25	C₄⋊Dic₁₇	—	2	2	no	—
26	#63223	—	2	2	no	—
27	#63223	Q₈	1	4	yes	—
28	C₂×#3622	—	2	2	no	—
29	C₂³×C₁₇	#41	1	1	yes	characteristic
30	C₂×C₄×C₁₇	—	2	2	no	—
31	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₇	#41	1	2	yes	—
35	C₂×Dic₁₇	—	4	2	no	—
36	C₂×Dic₁₇	—	8	2	no	—
37	C₈×C₁₇	—	2	2	no	—
38	C₈×C₁₇	—	4	1	no	—
39	#3622	—	2	4	no	—
40	#2090	—	17	1	no	maximal, Sylow
41	C₂²×C₁₇	—	2	2	no	—
42	C₂²×C₁₇	#121	1	1	yes	characteristic
43	C₂²×C₁₇	#125	1	2	yes	—
44	C₄×C₁₇	—	4	2	no	—
45	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
46	C₄×C₁₇	#123	1	1	yes	characteristic
47	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₁₇	—	8	2	no	—
49	#428	—	17	2	no	—
50	C₂×#106	—	17	1	no	—
51	#424	D₁₇	1	1	yes	characteristic
52	#343	—	17	2	no	—
53	#341	—	17	1	no	—
54	C₂×C₁₇	—	2	2	no	—
55	C₂×C₁₇	#398	1	1	yes	characteristic
56	C₂×C₁₇	#471	1	2	yes	—
57	C₂×M₁₆	—	17	2	no	—
58	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
60	#117	C₂×D₁₇	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	17	2	no	—
62	#107	—	34	2	no	—
63	#106	—	34	2	no	—
64	#106	—	34	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₁₇	#2090	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₄	—	34	2	no	—
68	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	17	4	no	—
71	M₁₆	—	34	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	34	2	no	—
75	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
76	C₄⋊C₄	—	34	2	no	—
77	C₄⋊C₄	—	34	2	no	—
78	C₄⋊C₄	D₆₈	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#63252	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	68	2	no	—
84	C₂×C₄	—	136	2	no	—
85	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
86	C₂×C₄	#63254	1	2	yes	—
87	C₂×C₄	Q₈×D₁₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	34	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#10587575	1	2	yes	—
94	C₂²	#10587616	1	1	yes	characteristic
95	C₄	—	4	2	no	—
96	C₄	—	136	2	no	—
97	C₄	#10587583	1	2	yes	—
98	C₄	#10587621	1	1	yes	characteristic
99	C₄	#10587630	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#11762739	1	1	yes	characteristic
102	C₂	#11763072	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11762852

C₂

yes

maximal

C₂×#10587488

C₂

yes

maximal, characteristic

#11762577

C₂

yes

maximal

#11762574

C₂

yes

maximal, characteristic

C₁₇×#424

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63223

C₂²

yes

—

C₁₇×C₄⋊C₈

—

#10587488

—

#10587479

—

#10587478

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63223

—

#63223

Q₈

yes

—

C₂×#3622

—

C₂³×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#2090

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

#125

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

#123

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

#428

—

C₂×#106

—

#424

D₁₇

yes

characteristic

#343

—

#341

—

C₂×C₁₇

—

C₂×C₁₇

#398

yes

characteristic

C₂×C₁₇

#471

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

#117

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₁₇

#2090

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₆₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#63252

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

136

—

C₂×C₄

C₂×D₆₈

yes

characteristic

C₂×C₄

#63254

yes

—

C₂×C₄

Q₈×D₁₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#10587575

yes

—

C₂²

#10587616

yes

characteristic

C₄

—

C₄

—

136

—

C₄

#10587583

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#11762739

yes

characteristic

102

C₂

#11763072

yes

—

	subgroup	G/H
center	C₂²	#10587616
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#424	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂³×C₁₇	#41
hypercenter	#424	D₁₇
hypocenter	C₁₇	#2090
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#10587616

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#424

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
66. C₁₇	#2090	semidirect

H ≅ G/N

type

66. C₁₇

#2090

semidirect

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

subgroup	index
#11762852	2
C₂×#10587488	2
#11762577	2
#11762574	2
C₁₇×#424	2
#2090	17

subgroup

index

C₁₇×#424

Subgroups — #14530294

Conjugacy classes of subgroups

Complements

Maximal subgroups