Group #14528451 — subgroups

Subgroups — #14528451

Order 2176.

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

normal subgroups only

This group has 242 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	#11763185	C₂	1	2	yes	maximal, complemented
2	#11763182	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763155	C₂	1	2	yes	maximal, complemented
4	#11762530	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×C₁₇	—	2	2	no	—
7	C₁₇×#127	C₂²	1	2	yes	—
8	C₂×#63228	—	2	4	no	—
9	C₂×#63219	—	2	2	no	—
10	#10587560	C₂²	1	1	yes	characteristic
11	#10587555	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	#10587503	—	2	4	no	—
14	C₄×#3622	C₂²	1	2	yes	—
15	C₂×D₄×C₁₇	—	2	2	no	—
16	C₂×D₄×C₁₇	—	2	2	no	—
17	C₂×D₄×C₁₇	—	2	2	no	—
18	C₂×D₆₈	—	2	1	no	—
19	C₂×D₆₈	—	2	2	no	—
20	C₂×Dic₃₄	—	2	1	no	—
21	D₈×C₁₇	—	4	4	no	—
22	C₂×C₈×C₁₇	D₄	1	2	yes	—
23	C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	#63228	—	4	8	no	—
25	C₄⋊Dic₁₇	—	2	2	no	—
26	C₂×#3622	D₄	1	4	yes	—
27	#63219	—	4	4	no	—
28	C₂³×C₁₇	—	4	2	no	—
29	C₂²×D₁₇	—	4	1	no	—
30	D₄×C₁₇	—	4	4	no	—
31	D₄×C₁₇	—	4	4	no	—
32	D₄×C₁₇	—	4	4	no	—
33	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₇	—	4	1	no	—
36	D₆₈	—	4	2	no	—
37	D₆₈	—	4	4	no	—
38	Dic₃₄	—	4	2	no	—
39	C₈×C₁₇	D₈	1	4	yes	complemented
40	#3622	—	2	4	no	—
41	#1721	—	17	1	no	maximal, Sylow
42	C₂²×C₁₇	—	8	2	no	—
43	C₂²×C₁₇	—	8	2	no	—
44	C₂²×C₁₇	—	8	2	no	—
45	C₂²×C₁₇	#127	1	1	yes	characteristic
46	C₂×D₁₇	—	8	1	no	—
47	C₂×D₁₇	—	8	2	no	—
48	C₄×C₁₇	C₂×D₈	1	2	yes	—
49	C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
50	Dic₁₇	—	8	1	no	—
51	#493	—	17	2	no	—
52	#493	D₁₇	1	1	yes	characteristic, complemented
53	#492	—	17	1	no	—
54	#486	—	17	2	no	—
55	#322	—	17	1	no	—
56	C₂×C₁₇	—	16	2	no	—
57	C₂×C₁₇	#493	1	1	yes	characteristic
58	C₂×C₁₇	#496	1	2	yes	—
59	D₁₇	—	16	1	no	—
60	C₂×SD₁₆	—	34	2	no	—
61	C₂×D₈	—	2	2	no	—
62	C₂×D₈	—	34	4	no	—
63	C₄⋊Q₈	—	17	1	no	—
64	#127	—	17	1	no	—
65	#127	C₂×D₁₇	1	2	yes	—
66	#102	—	34	4	no	—
67	C₄×C₈	—	17	2	no	—
68	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
69	C₁₇	#1721	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₈	—	34	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	34	2	no	—
75	C₂×D₄	—	34	1	no	—
76	SD₁₆	—	68	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	68	8	no	—
79	C₂×C₈	—	17	4	no	—
80	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
81	C₄⋊C₄	—	34	2	no	—
82	C₄²	C₂²×D₁₇	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	68	1	no	—
85	Q₈	—	68	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	68	4	no	—
90	D₄	—	68	2	no	—
91	C₂×C₄	—	68	1	no	—
92	C₂×C₄	D₄×D₁₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
94	C₈	—	34	4	no	—
95	C₈	#63228	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	136	1	no	—
100	C₂²	—	136	2	no	—
101	C₂²	#10587626	1	1	yes	characteristic
102	C₄	—	136	1	no	—
103	C₄	#10587597	1	4	yes	—
104	C₄	C₂×#63228	1	2	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	272	1	no	—
107	C₂	#11763185	1	1	yes	characteristic
108	C₂	#11763197	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763185

C₂

yes

maximal, complemented

#11763182

C₂

yes

maximal, characteristic, complemented

#11763155

C₂

yes

maximal, complemented

#11762530

C₂

yes

maximal, characteristic, complemented

C₁₇×#493

C₂

yes

maximal, characteristic, complemented

C₂×D₈×C₁₇

—

C₁₇×#127

C₂²

yes

—

C₂×#63228

—

C₂×#63219

—

#10587560

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587503

—

C₄×#3622

C₂²

yes

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

D₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

#63228

—

C₄⋊Dic₁₇

—

C₂×#3622

D₄

yes

—

#63219

—

C₂³×C₁₇

—

C₂²×D₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

Dic₃₄

—

C₈×C₁₇

D₈

yes

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#127

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

C₂×D₈

yes

—

C₄×C₁₇

AGL(1,ℤ/8)

yes

—

Dic₁₇

—

#493

—

#493

D₁₇

yes

characteristic, complemented

#492

—

#486

—

#322

—

C₂×C₁₇

—

C₂×C₁₇

#493

yes

characteristic

C₂×C₁₇

#496

yes

—

D₁₇

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₁₇

yes

—

#102

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1721

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₄×D₁₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

C₈

—

C₈

#63228

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

136

—

100

C₂²

—

136

—

101

C₂²

#10587626

yes

characteristic

102

C₄

—

136

—

103

C₄

#10587597

yes

—

104

C₄

C₂×#63228

yes

—

105

C₂

—

106

C₂

—

272

—

107

C₂

#11763185

yes

characteristic

108

C₂

#11763197

yes

—

	subgroup	G/H
center	C₂²	#10587626
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#493	C₂
socle	C₂²×C₁₇	#127
hypercenter	#493	D₁₇
hypocenter	C₁₇	#1721
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#10587626

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#493

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #11763185	C₂	semidirect
2. #11763182	C₂	semidirect
3. #11763155	C₂	semidirect
4. #11762530	C₂	semidirect
5. C₁₇×#493	C₂	semidirect
39. C₈×C₁₇	D₈	semidirect
52. #493	D₁₇	semidirect
69. C₁₇	#1721	semidirect
95. C₈	#63228	semidirect

H ≅ G/N

type

1. #11763185

C₂

semidirect

2. #11763182

C₂

semidirect

3. #11763155

C₂

semidirect

4. #11762530

C₂

semidirect

5. C₁₇×#493

C₂

semidirect

39. C₈×C₁₇

D₈

semidirect

52. #493

D₁₇

semidirect

69. C₁₇

#1721

semidirect

95. C₈

#63228

semidirect

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

subgroup	index
#11763185	2
#11763182	2
#11763155	2
#11762530	2
C₁₇×#493	2
#1721	17

subgroup

index

C₁₇×#493

Subgroups — #14528451

Conjugacy classes of subgroups

Complements

Maximal subgroups