Group #14528413 — subgroups

Subgroups — #14528413

Order 2176.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11763186	C₂	1	1	yes	maximal, characteristic
2	#11763183	C₂	1	2	yes	maximal, complemented
3	#11763157	C₂	1	2	yes	maximal, complemented
4	#11762530	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#63230	—	2	4	no	—
8	C₂×D₁₃₆	—	2	2	no	—
9	#10587560	C₂²	1	1	yes	characteristic
10	#10587560	C₂²	1	1	yes	characteristic
11	C₁₇×#106	—	2	2	no	—
12	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	#10587480	—	2	4	no	—
14	C₄×#3622	C₂²	1	2	yes	—
15	C₂×Q₈×C₁₇	—	2	2	no	—
16	C₂×D₆₈	—	2	2	no	—
17	C₂×D₆₈	—	2	1	no	—
18	C₂×D₆₈	—	2	1	no	—
19	C₂×D₆₈	—	2	2	no	—
20	C₂×C₈×C₁₇	D₄	1	2	yes	—
21	C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	#63230	—	4	8	no	—
25	C₂×#3622	D₄	1	4	yes	—
26	D₁₃₆	—	4	4	no	—
27	C₂²×D₁₇	—	4	1	no	—
28	C₂²×D₁₇	—	4	1	no	—
29	Q₈×C₁₇	—	4	4	no	—
30	C₂×C₄×C₁₇	—	4	2	no	—
31	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	D₆₈	—	4	2	no	—
34	D₆₈	—	4	2	no	—
35	D₆₈	—	4	4	no	—
36	D₆₈	—	4	4	no	—
37	C₈×C₁₇	SD₁₆	1	4	yes	complemented
38	#3622	—	2	4	no	—
39	#1715	—	17	1	no	maximal, Sylow
40	C₂²×C₁₇	#127	1	1	yes	characteristic
41	C₂×D₁₇	—	8	1	no	—
42	C₂×D₁₇	—	8	1	no	—
43	C₂×D₁₇	—	8	2	no	—
44	C₂×D₁₇	—	8	2	no	—
45	C₄×C₁₇	—	8	2	no	—
46	C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
47	C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
48	#498	D₁₇	1	1	yes	characteristic, complemented
49	#493	—	17	1	no	—
50	#492	—	17	2	no	—
51	#486	—	17	2	no	—
52	#322	—	17	1	no	—
53	C₂×C₁₇	#492	1	1	yes	characteristic
54	C₂×C₁₇	#496	1	2	yes	—
55	D₁₇	—	16	1	no	—
56	D₁₇	—	16	1	no	—
57	C₂×SD₁₆	—	34	4	no	—
58	C₂×D₈	—	34	2	no	—
59	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
60	#127	—	17	1	no	—
61	#127	—	17	1	no	—
62	#106	—	2	2	no	—
63	#102	—	34	4	no	—
64	C₄×C₈	—	17	2	no	—
65	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
66	C₁₇	#1715	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	34	2	no	—
69	C₂×D₄	—	34	2	no	—
70	C₂×D₄	—	34	1	no	—
71	C₂×D₄	—	34	1	no	—
72	SD₁₆	—	68	8	no	—
73	D₈	—	68	4	no	—
74	C₂×C₈	—	17	4	no	—
75	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₁₇	1	1	yes	characteristic
79	C₂³	—	68	1	no	—
80	C₂³	—	68	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	68	4	no	—
83	D₄	—	68	4	no	—
84	D₄	—	68	2	no	—
85	D₄	—	68	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₁₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
89	C₈	—	34	4	no	—
90	C₈	#63230	1	4	yes	complemented
91	C₂²	—	136	1	no	—
92	C₂²	—	136	1	no	—
93	C₂²	—	136	2	no	—
94	C₂²	—	136	2	no	—
95	C₂²	#10587626	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	#10587600	1	4	yes	—
98	C₄	C₂×#63230	1	2	yes	—
99	C₂	—	272	1	no	—
100	C₂	—	272	1	no	—
101	C₂	#11763183	1	1	yes	characteristic
102	C₂	#11763195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763186

C₂

yes

maximal, characteristic

#11763183

C₂

yes

maximal, complemented

#11763157

C₂

yes

maximal, complemented

#11762530

C₂

yes

maximal, characteristic, complemented

C₁₇×#498

C₂

yes

maximal, characteristic, complemented

C₁₇×C₄⋊Q₈

C₂²

yes

—

C₂×#63230

—

C₂×D₁₃₆

—

#10587560

C₂²

yes

characteristic

#10587560

C₂²

yes

characteristic

C₁₇×#106

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587480

—

C₄×#3622

C₂²

yes

—

C₂×Q₈×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63230

—

C₂×#3622

D₄

yes

—

D₁₃₆

—

C₂²×D₁₇

—

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

C₈×C₁₇

SD₁₆

yes

—

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

—

C₄×C₁₇

AGL(1,ℤ/8)

yes

—

#498

D₁₇

yes

characteristic, complemented

—

—

—

—

C₂×C₁₇

#492

yes

characteristic

C₂×C₁₇

#496

yes

—

D₁₇

—

D₁₇

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1715

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₄×D₁₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

C₈

—

C₈

#63230

yes

complemented

C₂²

—

136

—

C₂²

—

136

—

C₂²

—

136

—

C₂²

—

136

—

C₂²

#10587626

yes

characteristic

C₄

—

C₄

#10587600

yes

—

C₄

C₂×#63230

yes

—

C₂

—

272

—

100

C₂

—

272

—

101

C₂

#11763183

yes

characteristic

102

C₂

#11763195

yes

—

	subgroup	G/H
center	C₂²	#10587626
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#498	C₂
socle	C₂²×C₁₇	#127
prime radical	#11763186	C₂
hypercenter	#498	D₁₇
hypocenter	C₁₇	#1715
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#10587626

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#498

C₂

C₂²×C₁₇

C₂

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #11763183	C₂	semidirect
3. #11763157	C₂	semidirect
4. #11762530	C₂	semidirect
5. C₁₇×#498	C₂	semidirect
37. C₈×C₁₇	SD₁₆	semidirect
48. #498	D₁₇	semidirect
66. C₁₇	#1715	semidirect
90. C₈	#63230	semidirect

H ≅ G/N

type

2. #11763183

C₂

semidirect

3. #11763157

C₂

semidirect

4. #11762530

C₂

semidirect

5. C₁₇×#498

C₂

semidirect

37. C₈×C₁₇

SD₁₆

semidirect

48. #498

D₁₇

semidirect

66. C₁₇

#1715

semidirect

90. C₈

#63230

semidirect

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

subgroup	index
#11763186	2
#11763183	2
#11763157	2
#11762530	2
C₁₇×#498	2
#1715	17

subgroup

index

C₁₇×#498

Subgroups — #14528413

Conjugacy classes of subgroups

Complements

Maximal subgroups