Group #14544377 — subgroups

Subgroups — #14544377

Order 2176.

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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	Q₈×#3624	C₂	1	2	yes	maximal
2	#11763670	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#3624	C₂	1	2	yes	maximal
4	#11763650	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
7	Q₈×Dic₁₇	C₂²	1	2	yes	—
8	C₄⋊#3624	—	2	2	no	—
9	C₄⋊#3624	C₂²	1	2	yes	—
10	C₄×#3624	—	2	2	no	—
11	C₄×#3624	C₂²	1	1	yes	characteristic
12	C₄×#3624	C₂²	1	1	yes	characteristic
13	#10587506	C₄	1	2	yes	—
14	#10587489	C₄	1	1	yes	characteristic
15	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
17	C₂×#3624	—	2	2	no	—
18	C₂×#3624	—	2	1	no	—
19	C₂×#3624	D₄	1	2	yes	—
20	Q₁₆×C₁₇	C₈	1	4	yes	complemented
21	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₁₇	—	2	2	no	—
23	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₇	—	2	2	no	—
25	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂×#3622	C₂×C₄	1	1	yes	characteristic
27	Q₈×C₁₇	—	2	2	no	—
28	Q₈×C₁₇	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₁₇	—	2	2	no	—
30	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₇	—	2	2	no	—
32	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₁₇	#41	1	1	yes	characteristic
34	#3624	—	2	2	no	—
35	#3624	—	4	1	no	—
36	#3624	—	4	2	no	—
37	C₈×C₁₇	C₂×C₈	1	2	yes	—
38	#3622	—	2	1	no	—
39	#1579	—	17	1	no	maximal, Sylow
40	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
41	C₄×C₁₇	—	2	4	no	—
42	C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
43	C₄×C₁₇	#131	1	1	yes	characteristic
44	Dic₁₇	—	2	1	no	—
45	Dic₁₇	—	4	2	no	—
46	Dic₁₇	#137	1	2	yes	—
47	C₈×Q₈	—	17	2	no	—
48	C₄×Q₁₆	—	17	1	no	—
49	#334	—	17	1	no	—
50	Q₈⋊C₈	—	17	2	no	—
51	#322	—	17	1	no	—
52	C₂×C₁₇	C₈×D₄	1	1	yes	characteristic
53	C₂×C₁₇	#441	1	1	yes	characteristic
54	C₂×C₁₇	#444	1	1	yes	characteristic
55	C₂×Q₁₆	#595	1	1	yes	characteristic
56	C₄×Q₈	—	17	2	no	—
57	#107	—	17	1	no	—
58	C₄⋊C₈	—	17	2	no	—
59	C₄⋊C₈	—	34	2	no	—
60	#103	—	17	2	no	—
61	C₄×C₈	—	17	1	no	—
62	C₄×C₈	—	17	1	no	—
63	C₄×C₈	—	17	1	no	—
64	C₄×C₈	—	34	2	no	—
65	C₁₇	#1579	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₈	C₂×#595	1	2	yes	—
67	Q₁₆	#3624	1	4	yes	complemented
68	C₂×C₈	—	17	2	no	—
69	C₂×C₈	—	17	1	no	—
70	C₂×C₈	—	34	1	no	—
71	C₂×C₈	—	34	2	no	—
72	C₂×C₈	C₂×#595	1	1	yes	characteristic
73	C₄⋊C₄	—	17	2	no	—
74	C₄⋊C₄	—	34	2	no	—
75	C₄²	—	17	1	no	—
76	C₄²	—	34	2	no	—
77	Q₈	—	2	2	no	—
78	Q₈	C₂×#3624	1	4	yes	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	17	1	no	—
81	C₂×C₄	—	17	1	no	—
82	C₂×C₄	—	34	2	no	—
83	C₂×C₄	C₂²×#595	1	1	yes	characteristic
84	C₈	—	34	2	no	—
85	C₈	—	34	1	no	—
86	C₈	—	68	2	no	—
87	C₈	—	68	1	no	—
88	C₈	C₂×#3624	1	2	yes	—
89	C₂²	D₄×#595	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	17	2	no	—
92	C₄	—	34	1	no	—
93	C₄	—	68	2	no	—
94	C₄	#10587678	1	1	yes	characteristic
95	C₄	C₂²×#3624	1	1	yes	characteristic
96	C₂	D₄×#3624	1	1	yes	characteristic
97	C₂	#11763849	1	1	yes	characteristic
98	C₂	#11763855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#3624

C₂

yes

maximal

#11763670

C₂

yes

maximal, characteristic

Q₈⋊#3624

C₂

yes

maximal

#11763650

C₂

yes

maximal, characteristic

Q₁₆×Dic₁₇

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₇

C₄

yes

characteristic

Q₈×Dic₁₇

C₂²

yes

—

C₄⋊#3624

—

C₄⋊#3624

C₂²

yes

—

C₄×#3624

—

C₄×#3624

C₂²

yes

characteristic

C₄×#3624

C₂²

yes

characteristic

#10587506

C₄

yes

—

#10587489

C₄

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₂×Q₈×C₁₇

C₂×C₄

yes

—

C₂×#3624

—

C₂×#3624

—

C₂×#3624

D₄

yes

—

Q₁₆×C₁₇

C₈

yes

complemented

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂³

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

C₂×Dic₁₇

yes

—

—

—

C₈×C₁₇

C₂×C₈

yes

—

#3622

—

#1579

—

maximal, Sylow

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂²×C₈

yes

characteristic

C₄×C₁₇

#131

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#137

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₂×C₁₇

C₈×D₄

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

yes

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1579

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×#595

yes

—

Q₁₆

#3624

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#595

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

Q₈

—

Q₈

C₂×#3624

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×#595

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×#3624

yes

—

C₂²

D₄×#595

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#10587678

yes

characteristic

C₄

C₂²×#3624

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×#595
commutator	C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#595
Fitting	C₂×Q₁₆×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	C₄×D₄
cosocle	C₄×Dic₁₇	C₂³
hypercenter	C₂×Q₁₆	#595
hypocenter	C₁₇	#1579
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#595

commutator

C₄×C₁₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#595

Fitting

C₂×Q₁₆×C₁₇

C₄

socle, prime radical

C₂²×C₁₇

C₄×D₄

cosocle

C₄×Dic₁₇

C₂³

C₂×Q₁₆

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
20. Q₁₆×C₁₇	C₈	semidirect
65. C₁₇	#1579	semidirect
67. Q₁₆	#3624	semidirect

H ≅ G/N

type

20. Q₁₆×C₁₇

C₈

semidirect

65. C₁₇

#1579

semidirect

67. Q₁₆

#3624

semidirect

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

subgroup	index
Q₈×#3624	2
#11763670	2
Q₈⋊#3624	2
#11763650	2
Q₁₆×Dic₁₇	2
#1579	17

subgroup

index

Q₁₆×Dic₁₇

Subgroups — #14544377

Conjugacy classes of subgroups

Complements

Maximal subgroups