Group #14528313 — subgroups

Subgroups — #14528313

Order 2176.

140 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 55 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
2	#11763105	C₂	1	2	yes	maximal
3	#11762939	C₂	1	1	yes	maximal, characteristic
4	#11762557	C₂	1	2	yes	maximal
5	C₁₇×#334	C₂	1	1	yes	maximal, characteristic
6	#10587555	C₂²	1	2	yes	—
7	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
8	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	#10587489	—	2	2	no	—
11	#10587489	—	2	1	no	—
12	#10587489	—	2	1	no	—
13	#10587487	—	2	2	no	—
14	#10587486	—	2	1	no	—
15	#10587485	—	2	1	no	—
16	C₄⋊#3622	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₄	—	2	2	no	—
18	C₂×Dic₃₄	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇	—	2	2	no	—
20	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₁₇	—	2	2	no	—
24	C₄⋊Dic₁₇	—	2	2	no	—
25	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
26	#63225	—	2	1	no	—
27	C₄×Dic₁₇	—	2	1	no	—
28	C₂×#3622	—	2	1	no	—
29	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
32	C₂×Dic₁₇	—	2	1	no	—
33	C₂×Dic₁₇	—	2	1	no	—
34	C₂×Dic₁₇	—	4	2	no	—
35	Dic₃₄	—	2	2	no	—
36	Dic₃₄	—	8	2	no	—
37	C₈×C₁₇	—	2	1	no	—
38	C₈×C₁₇	—	4	2	no	—
39	C₈×C₁₇	Q₁₆	1	2	yes	—
40	#3622	—	4	1	no	—
41	#1696	—	17	1	no	maximal, Sylow
42	C₂²×C₁₇	#121	1	1	yes	characteristic
43	C₄×C₁₇	—	2	1	no	—
44	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
45	C₄×C₁₇	#137	1	1	yes	characteristic
46	C₄×C₁₇	#137	1	2	yes	—
47	Dic₁₇	—	4	1	no	—
48	Dic₁₇	—	4	1	no	—
49	Dic₁₇	—	8	2	no	—
50	#500	—	17	1	no	—
51	#477	—	17	2	no	—
52	#446	—	17	1	no	—
53	#334	D₁₇	1	1	yes	characteristic
54	#333	—	17	2	no	—
55	C₂×C₁₇	#462	1	1	yes	characteristic
56	C₂×C₁₇	#470	1	1	yes	characteristic
57	C₂×C₁₇	#472	1	1	yes	characteristic
58	C₄⋊Q₈	—	17	2	no	—
59	C₄×Q₈	—	17	1	no	—
60	#107	—	34	2	no	—
61	#107	—	34	1	no	—
62	#107	—	34	1	no	—
63	C₄⋊C₈	—	17	1	no	—
64	C₄⋊C₈	—	34	1	no	—
65	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
66	#103	—	34	2	no	—
67	#97	—	34	1	no	—
68	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
69	C₁₇	#1696	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₈	—	17	1	no	—
71	C₂×Q₈	—	34	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	34	1	no	—
74	C₂×C₈	D₆₈	1	1	yes	characteristic
75	C₂×C₈	D₆₈	1	1	yes	characteristic
76	C₄⋊C₄	—	17	1	no	—
77	C₄⋊C₄	—	34	1	no	—
78	C₄⋊C₄	—	34	2	no	—
79	C₄⋊C₄	—	34	2	no	—
80	C₄²	—	34	1	no	—
81	C₄²	C₂²×D₁₇	1	1	yes	characteristic
82	Q₈	—	34	2	no	—
83	Q₈	—	136	2	no	—
84	C₂×C₄	—	34	1	no	—
85	C₂×C₄	—	34	1	no	—
86	C₂×C₄	—	68	2	no	—
87	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
88	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
89	C₂×C₄	#63256	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	4	2	no	—
92	C₈	—	68	1	no	—
93	C₈	Dic₆₈	1	2	yes	—
94	C₂²	#10587581	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	68	1	no	—
97	C₄	—	68	1	no	—
98	C₄	—	136	2	no	—
99	C₄	C₂×Dic₆₈	1	1	yes	characteristic
100	C₄	#10587595	1	1	yes	characteristic
101	C₄	#10587601	1	2	yes	—
102	C₂	#11763020	1	1	yes	characteristic
103	C₂	#11763068	1	1	yes	characteristic
104	C₂	#11763080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763214

C₂

yes

maximal, characteristic

#11763105

C₂

yes

maximal

#11762939

C₂

yes

maximal, characteristic

#11762557

C₂

yes

maximal

C₁₇×#334

C₂

yes

maximal, characteristic

#10587555

C₂²

yes

—

C₄×Dic₃₄

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

—

—

—

—

—

—

C₂²

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

characteristic

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

Q₁₆

yes

—

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

D₁₇

yes

characteristic

#333

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

—

#103

—

#97

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1696

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₈

yes

characteristic

C₂×C₈

D₆₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₇

yes

characteristic

Q₈

—

Q₈

—

136

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₆₈

yes

characteristic

C₂×C₄

D₄×D₁₇

yes

characteristic

C₂×C₄

#63256

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

Dic₆₈

yes

—

C₂²

#10587581

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

136

—

C₄

C₂×Dic₆₈

yes

characteristic

100

C₄

#10587595

yes

characteristic

101

C₄

#10587601

yes

—

102

C₂

#11763020

yes

characteristic

103

C₂

#11763068

yes

characteristic

104

C₂

#11763080

yes

characteristic

	subgroup	G/H
center	C₂²	#10587581
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#334	C₂
socle, prime radical	C₂²×C₁₇	#121
hypercenter	#334	D₁₇
hypocenter	C₁₇	#1696
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#10587581

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#334

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
69. C₁₇	#1696	semidirect

H ≅ G/N

type

69. C₁₇

#1696

semidirect

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

subgroup	index
#11763214	2
#11763105	2
#11762939	2
#11762557	2
C₁₇×#334	2
#1696	17

subgroup

index

C₁₇×#334

Subgroups — #14528313

Conjugacy classes of subgroups

Complements

Maximal subgroups