Group #17949423 — subgroups

Subgroups — #17949423

Order 4352.

162 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₁₇⋊#368	C₂	1	2	yes	maximal
2	C₂×#11762642	C₂	1	1	yes	maximal, characteristic
3	D₁₇⋊#108	—	2	2	no	—
4	#11763738	C₄	1	4	yes	—
5	#11763737	—	2	2	no	—
6	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
7	#11762642	—	2	2	no	—
8	C₂×#10587465	C₄	1	1	yes	characteristic
9	C₂×C₁₇×C₃₂	C₄	1	1	yes	characteristic
10	D₁₇⋊M₁₆	—	4	2	no	—
11	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
12	#10587536	—	2	4	no	—
13	#10587535	—	4	2	no	—
14	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×#63214	Q₈	1	1	yes	characteristic
16	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
17	C₁₆×D₁₇	D₄	1	2	yes	—
18	C₁₇×C₃₂	—	2	1	no	—
19	#10587465	—	2	1	no	—
20	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
21	#63247	—	8	2	no	—
22	#63243	—	4	4	no	—
23	#63242	—	8	2	no	—
24	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×#3622	Q₁₆	1	1	yes	characteristic
26	C₈×D₁₇	C₂²⋊C₄	1	2	yes	—
27	C₈×D₁₇	SD₁₆	1	2	yes	—
28	C₁₆×C₁₇	C₄²	1	1	yes	characteristic
29	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
30	#63214	C₄⋊C₄	1	2	yes	—
31	#7476	—	17	1	no	maximal, Sylow
32	C₂²×D₁₇	Q₃₂	1	1	yes	characteristic
33	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
34	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
35	C₄×D₁₇	#102	1	2	yes	—
36	C₄×D₁₇	SD₃₂	1	2	yes	—
37	#3624	—	8	4	no	—
38	C₈×C₁₇	#95	1	1	yes	characteristic
39	C₈×C₁₇	#106	1	1	yes	characteristic
40	#3622	#103	1	2	yes	—
41	C₂×M₆₄	—	17	1	no	—
42	C₂×#368	—	17	2	no	—
43	C₂²×C₁₇	#366	1	1	yes	characteristic
44	C₂×D₁₇	—	4	1	no	—
45	C₂×D₁₇	#358	1	2	yes	—
46	C₄×C₁₇	#340	1	1	yes	characteristic
47	C₄×C₁₇	#367	1	1	yes	characteristic
48	Dic₁₇	#357	1	2	yes	—
49	C₂²×C₁₆	—	17	1	no	—
50	C₂×#108	—	34	2	no	—
51	M₆₄	—	34	2	no	—
52	C₂×C₃₂	—	17	1	no	—
53	C₂×C₃₂	#595	1	1	yes	characteristic
54	#368	—	17	4	no	—
55	#368	—	34	2	no	—
56	C₂×C₁₇	—	2	1	no	—
57	C₂×C₁₇	#1380	1	1	yes	characteristic
58	D₁₇	—	2	2	no	—
59	C₂×M₁₆	—	68	2	no	—
60	C₂²×C₈	—	17	1	no	—
61	C₂×C₁₆	—	17	1	no	—
62	C₂×C₁₆	—	17	2	no	—
63	C₂×C₁₆	—	17	2	no	—
64	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
65	#108	—	34	4	no	—
66	#108	—	68	2	no	—
67	C₃₂	—	2	1	no	—
68	C₃₂	—	34	1	no	—
69	C₁₇	#7476	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₄	—	17	1	no	—
71	M₁₆	—	68	4	no	—
72	M₁₆	—	136	2	no	—
73	C₂×C₈	—	17	1	no	—
74	C₂×C₈	—	17	2	no	—
75	C₂×C₈	—	17	2	no	—
76	C₂×C₈	—	136	2	no	—
77	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
78	C₁₆	—	17	2	no	—
79	C₁₆	C₄×#595	1	1	yes	characteristic
80	C₁₆	C₂²⋊#595	1	1	yes	characteristic
81	C₂³	—	17	1	no	—
82	C₂×C₄	—	17	1	no	—
83	C₂×C₄	—	17	2	no	—
84	C₂×C₄	—	17	2	no	—
85	C₂×C₄	#10587534	1	1	yes	characteristic
86	C₈	—	17	2	no	—
87	C₈	—	136	4	no	—
88	C₈	#10587533	1	1	yes	characteristic
89	C₈	(C₂×#595)⋊C₄	1	1	yes	characteristic
90	C₂²	—	17	2	no	—
91	C₂²	—	68	1	no	—
92	C₂²	#11763735	1	1	yes	characteristic
93	C₄	—	17	2	no	—
94	C₄	#11763683	1	1	yes	characteristic
95	C₄	#11763736	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	34	2	no	—
98	C₂	#14543842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₇⋊#368

C₂

yes

maximal

C₂×#11762642

C₂

yes

maximal, characteristic

D₁₇⋊#108

—

#11763738

C₄

yes

—

#11763737

—

C₂×C₁₆×D₁₇

C₂²

yes

characteristic

#11762642

—

C₂×#10587465

C₄

yes

characteristic

C₂×C₁₇×C₃₂

C₄

yes

characteristic

D₁₇⋊M₁₆

—

C₂×C₈×D₁₇

D₄

yes

characteristic

#10587536

—

#10587535

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×#63214

Q₈

yes

characteristic

C₁₆×D₁₇

C₂×C₄

yes

—

C₁₆×D₁₇

D₄

yes

—

C₁₇×C₃₂

—

#10587465

—

C₂×C₄×D₁₇

D₈

yes

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

Q₁₆

yes

characteristic

C₈×D₁₇

C₂²⋊C₄

yes

—

C₈×D₁₇

SD₁₆

yes

—

C₁₆×C₁₇

C₄²

yes

characteristic

C₁₆×C₁₇

C₂²⋊C₄

yes

characteristic

#63214

C₄⋊C₄

yes

—

#7476

—

maximal, Sylow

C₂²×D₁₇

Q₃₂

yes

characteristic

C₂×C₄×C₁₇

#107

yes

characteristic

C₂×Dic₁₇

D₁₆

yes

characteristic

C₄×D₁₇

#102

yes

—

C₄×D₁₇

SD₃₂

yes

—

#3624

—

C₈×C₁₇

yes

C₈×C₁₇

yes

yes

—

C₂×M₆₄

—

C₂×#368

—

C₂²×C₁₇

#366

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

#358

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

yes

—

C₂²×C₁₆

—

C₂×#108

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

yes

—

—

C₂×C₁₇

—

C₂×C₁₇

#1380

yes

characteristic

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

—

—

C₃₂

—

C₃₂

—

C₁₇

#7476

yes

characteristic, complemented, Sylow

C₂²×C₄

—

M₁₆

—

M₁₆

—

136

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

136

—

C₂×C₈

C₄⋊#595

yes

characteristic

C₁₆

—

C₁₆

C₄×#595

yes

characteristic

C₁₆

C₂²⋊#595

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#10587534

yes

characteristic

C₈

—

C₈

—

136

—

C₈

#10587533

yes

characteristic

C₈

(C₂×#595)⋊C₄

yes

characteristic

C₂²

—

C₂²

—

C₂²

#11763735

yes

characteristic

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

#14543842

yes

characteristic

	subgroup	G/H
center	C₄	#11763736
commutator	C₁₆×C₁₇	C₄²
Frattini	C₂×C₁₆	C₂×#595
Fitting	C₂×C₁₇×C₃₂	C₄
socle	C₂×C₁₇	#1380
prime radical	C₂²×D₁₇	Q₃₂
cosocle	C₂×C₁₆×D₁₇	C₂²
hypercenter	C₂×C₃₂	#595
hypocenter	C₁₇	#7476
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

subgroup

G/H

center

C₄

#11763736

commutator

C₁₆×C₁₇

C₄²

Frattini

C₂×C₁₆

C₂×#595

Fitting

C₂×C₁₇×C₃₂

C₄

socle

C₂×C₁₇

#1380

prime radical

C₂²×D₁₇

Q₃₂

cosocle

C₂×C₁₆×D₁₇

C₂²

C₂×C₃₂

C₁₇

G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

N	H ≅ G/N	type
69. C₁₇	#7476	semidirect

H ≅ G/N

type

69. C₁₇

#7476

semidirect

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

subgroup	index
D₁₇⋊#368	2
C₂×#11762642	2
#7476	17

subgroup

index

D₁₇⋊#368

C₂×#11762642

#7476

Subgroups — #17949423

Conjugacy classes of subgroups

Complements

Maximal subgroups