Group #16949029 — subgroups

Subgroups — #16949029

Order 4352.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14531009	C₂	1	4	yes	maximal, complemented
2	#14528399	C₂	1	1	yes	maximal, characteristic
3	#14526701	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763186	—	2	2	no	—
6	#11763186	C₂²	1	2	yes	—
7	#11762609	—	2	8	no	—
8	C₄×#63214	C₂²	1	2	yes	—
9	C₁₇×#500	C₂²	1	2	yes	—
10	C₁₇×#487	—	2	2	no	—
11	C₈²×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊Q₈	—	2	2	no	—
13	C₂×D₁₃₆	—	2	4	no	—
14	C₂×D₁₃₆	—	4	4	no	—
15	#10587560	—	2	2	no	—
16	C₁₇×#107	—	2	4	no	—
17	C₁₇×#103	—	4	4	no	—
18	C₄×C₈×C₁₇	D₄	1	2	yes	—
19	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×#63214	D₄	1	4	yes	—
21	C₂×Q₈×C₁₇	—	4	2	no	—
22	C₂×D₆₈	—	4	2	no	—
23	C₂×D₆₈	—	4	4	no	—
24	C₂×C₈×C₁₇	D₈	1	4	yes	—
25	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₁₇	—	4	2	no	—
27	C₄⋊C₄×C₁₇	—	4	2	no	—
28	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
29	D₁₃₆	—	4	8	no	—
30	D₁₃₆	—	8	8	no	—
31	#63214	—	2	4	no	—
32	#12513	—	17	1	no	maximal, Sylow
33	C₂²×D₁₇	—	8	2	no	—
34	Q₈×C₁₇	—	8	4	no	—
35	C₂×C₄×C₁₇	—	8	2	no	—
36	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
38	D₆₈	—	8	4	no	—
39	D₆₈	—	8	8	no	—
40	C₈×C₁₇	—	2	4	no	—
41	C₈×C₁₇	#135	1	4	yes	—
42	#2240	—	17	4	no	—
43	#1712	—	17	1	no	—
44	#1710	D₁₇	1	1	yes	characteristic, complemented
45	#1311	—	17	1	no	—
46	C₂²×C₁₇	#493	1	1	yes	characteristic
47	C₂×D₁₇	—	16	2	no	—
48	C₂×D₁₇	—	16	4	no	—
49	C₄×C₁₇	—	16	2	no	—
50	C₄×C₁₇	#495	1	2	yes	—
51	C₄×C₁₇	#509	1	4	yes	—
52	#500	C₂×D₁₇	1	2	yes	—
53	#493	—	17	2	no	—
54	#493	—	34	2	no	—
55	#487	—	2	2	no	—
56	#357	—	34	8	no	—
57	C₄×C₁₆	—	17	2	no	—
58	C₈²	C₂×D₁₇	1	1	yes	characteristic
59	C₂×C₁₇	#1709	1	1	yes	characteristic
60	C₂×C₁₇	#2250	1	2	yes	—
61	D₁₇	—	32	2	no	—
62	C₂×D₈	—	34	4	no	—
63	C₂×D₈	—	68	4	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	#127	—	34	2	no	—
66	C₂×C₁₆	—	17	4	no	—
67	#107	—	2	4	no	—
68	#103	—	4	4	no	—
69	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
70	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
71	C₁₇	#12513	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	68	4	no	—
74	C₂×D₄	—	68	2	no	—
75	D₈	—	68	8	no	—
76	D₈	—	136	8	no	—
77	C₂×C₈	#63228	1	4	yes	—
78	C₂×C₈	D₄×D₁₇	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
82	C₁₆	—	34	4	no	—
83	C₂³	—	136	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	136	8	no	—
86	D₄	—	136	4	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	C₂×#63228	1	2	yes	—
89	C₂×C₄	#10587626	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	#10587605	1	4	yes	—
92	C₂²	—	272	2	no	—
93	C₂²	—	272	4	no	—
94	C₂²	#11763185	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#11763194	1	2	yes	—
97	C₄	#11763244	1	4	yes	—
98	C₂	—	544	2	no	—
99	C₂	#14528391	1	1	yes	characteristic
100	C₂	#14531051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531009

C₂

yes

maximal, complemented

#14528399

C₂

yes

maximal, characteristic

#14526701

C₂

yes

maximal, characteristic, complemented

C₁₇×#1710

C₂

yes

maximal, characteristic, complemented

#11763186

—

#11763186

C₂²

yes

—

#11762609

—

C₄×#63214

C₂²

yes

—

C₁₇×#500

C₂²

yes

—

C₁₇×#487

—

C₈²×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

#10587560

—

C₁₇×#107

—

C₁₇×#103

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

characteristic

C₂×#63214

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

D₁₃₆

—

D₁₃₆

—

#63214

—

#12513

—

maximal, Sylow

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

D₆₈

—

D₆₈

—

C₈×C₁₇

—

C₈×C₁₇

#135

yes

—

#2240

—

#1712

—

#1710

D₁₇

yes

characteristic, complemented

#1311

—

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#495

yes

—

C₄×C₁₇

#509

yes

—

#500

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

D₁₇

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₁₇

yes

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12513

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

136

—

C₂×C₈

#63228

yes

—

C₂×C₈

D₄×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

C₁₆

—

C₂³

—

136

—

Q₈

—

D₄

—

136

—

D₄

—

136

—

C₂×C₄

—

C₂×C₄

C₂×#63228

yes

—

C₂×C₄

#10587626

yes

characteristic

C₈

—

C₈

#10587605

yes

—

C₂²

—

272

—

C₂²

—

272

—

C₂²

#11763185

yes

characteristic

C₄

—

C₄

#11763194

yes

—

C₄

#11763244

yes

—

C₂

—

544

—

C₂

#14528391

yes

characteristic

100

C₂

#14531051

yes

—

	subgroup	G/H
center	C₂²	#11763185
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1710	C₂
socle	C₂²×C₁₇	#493
prime radical	#14528399	C₂
hypercenter	#1710	D₁₇
hypocenter	C₁₇	#12513
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#11763185

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1710

C₂

C₂²×C₁₇

C₂

D₁₇

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #14531009	C₂	semidirect
3. #14526701	C₂	semidirect
4. C₁₇×#1710	C₂	semidirect
44. #1710	D₁₇	semidirect
71. C₁₇	#12513	semidirect

H ≅ G/N

type

1. #14531009

C₂

semidirect

3. #14526701

C₂

semidirect

4. C₁₇×#1710

C₂

semidirect

44. #1710

D₁₇

semidirect

71. C₁₇

#12513

semidirect

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

subgroup	index
#14531009	2
#14528399	2
#14526701	2
C₁₇×#1710	2
#12513	17

subgroup

index

C₁₇×#1710

Subgroups — #16949029

Conjugacy classes of subgroups

Complements

Maximal subgroups