Group #17953786 — subgroups

Subgroups — #17953786

Order 4352.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#14544347	C₂	1	2	yes	maximal
2	#14543641	C₂	1	4	yes	maximal
3	#14527066	C₂	1	1	yes	maximal, characteristic
4	#11763831	C₂²	1	4	yes	—
5	#11763673	—	2	2	no	—
6	#11763671	—	2	2	no	—
7	#11762860	C₂²	1	2	yes	—
8	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	#11762531	C₄	1	2	yes	—
10	C₈×#3622	C₄	1	1	yes	characteristic
11	C₈²×C₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×D₁₇	Q₈	1	2	yes	—
13	C₄²×D₁₇	C₂³	1	1	yes	characteristic
14	#10587543	—	2	4	no	—
15	#10587542	—	2	4	no	—
16	C₄⋊#3624	—	2	4	no	—
17	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	#10587491	—	2	2	no	—
20	#10587485	—	2	2	no	—
21	C₈×Dic₁₇	D₄	1	2	yes	—
22	C₄×#3622	C₂×C₄	1	2	yes	—
23	C₄×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×D₁₇	#41	1	2	yes	—
26	C₂×#3624	—	4	4	no	—
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
29	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
30	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
31	C₄×Dic₁₇	#41	1	2	yes	—
32	C₂×#3622	—	2	2	no	—
33	C₂×#3622	C₄⋊C₄	1	2	yes	—
34	C₈×D₁₇	—	2	4	no	—
35	#8797	—	17	1	no	maximal, Sylow
36	C₂²×D₁₇	#125	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	#117	1	2	yes	—
39	C₂×Dic₁₇	#124	1	2	yes	—
40	C₂×Dic₁₇	#125	1	1	yes	characteristic
41	C₄×D₁₇	—	2	2	no	—
42	C₄×D₁₇	—	4	2	no	—
43	#3624	—	8	4	no	—
44	C₈×C₁₇	—	2	2	no	—
45	C₈×C₁₇	—	4	2	no	—
46	#3622	—	4	2	no	—
47	#3622	#108	1	4	yes	—
48	#1568	—	17	2	no	—
49	#1450	—	17	1	no	—
50	#1287	—	17	4	no	—
51	C₂²×C₁₇	#382	1	1	yes	characteristic
52	C₂×D₁₇	—	2	2	no	—
53	C₂×D₁₇	—	2	1	no	—
54	C₄×C₁₇	—	2	2	no	—
55	C₄×C₁₇	C₂×#108	1	2	yes	—
56	Dic₁₇	—	2	2	no	—
57	Dic₁₇	—	2	1	no	—
58	#432	—	17	4	no	—
59	#431	—	17	2	no	—
60	C₂×C₄×C₈	—	17	1	no	—
61	#335	—	34	2	no	—
62	#334	—	34	2	no	—
63	#322	—	17	2	no	—
64	C₈²	—	17	1	no	—
65	C₈²	#595	1	1	yes	characteristic
66	C₂×C₁₇	#1833	1	2	yes	—
67	C₂×C₁₇	#1837	1	1	yes	characteristic
68	D₁₇	—	4	1	no	—
69	C₂²×C₈	—	17	2	no	—
70	C₂×C₄²	—	17	1	no	—
71	C₄⋊C₈	—	34	2	no	—
72	C₄⋊C₈	—	34	4	no	—
73	C₂²⋊C₈	—	34	2	no	—
74	C₂²⋊C₈	—	34	4	no	—
75	#97	—	34	4	no	—
76	C₄×C₈	—	17	2	no	—
77	C₄×C₈	—	17	1	no	—
78	C₄×C₈	—	17	2	no	—
79	C₄×C₈	C₂×#595	1	2	yes	—
80	C₄×C₈	C₂×#595	1	1	yes	characteristic
81	C₁₇	#8797	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	17	1	no	—
83	C₂²×C₄	—	17	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	17	2	no	—
86	C₂×C₈	—	34	4	no	—
87	C₂×C₈	—	34	2	no	—
88	C₂×C₈	—	68	4	no	—
89	C₂×C₈	C₄⋊#595	1	2	yes	—
90	C₄²	—	17	1	no	—
91	C₄²	—	17	2	no	—
92	C₄²	C₂²×#595	1	1	yes	characteristic
93	C₂³	—	17	1	no	—
94	C₂×C₄	—	17	2	no	—
95	C₂×C₄	—	17	1	no	—
96	C₂×C₄	—	34	2	no	—
97	C₂×C₄	—	68	2	no	—
98	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
99	C₂×C₄	#10587677	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	4	2	no	—
102	C₈	—	17	4	no	—
103	C₈	—	68	2	no	—
104	C₈	—	136	4	no	—
105	C₂²	—	34	2	no	—
106	C₂²	—	34	1	no	—
107	C₂²	#11763753	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	34	2	no	—
110	C₄	—	34	1	no	—
111	C₄	D₁₇⋊#108	1	2	yes	—
112	C₂	—	68	1	no	—
113	C₂	#14544703	1	2	yes	—
114	C₂	#14544709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544347

C₂

yes

maximal

#14543641

C₂

yes

maximal

#14527066

C₂

yes

maximal, characteristic

#11763831

C₂²

yes

—

#11763673

—

#11763671

—

#11762860

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762531

C₄

yes

—

C₈×#3622

C₄

yes

characteristic

C₈²×C₁₇

C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

—

C₄²×D₁₇

C₂³

yes

—

—

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

#10587485

—

C₈×Dic₁₇

D₄

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×D₁₇

#41

yes

—

C₂×#3624

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

#41

yes

—

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×Dic₁₇

#124

yes

—

C₂×Dic₁₇

#125

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

#108

yes

—

#1568

—

#1450

—

#1287

—

C₂²×C₁₇

#382

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#108

yes

—

Dic₁₇

—

Dic₁₇

—

#432

—

#431

—

C₂×C₄×C₈

—

#335

—

#334

—

#322

—

C₈²

—

C₈²

#595

yes

characteristic

C₂×C₁₇

#1833

yes

—

C₂×C₁₇

#1837

yes

characteristic

D₁₇

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#8797

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊#595

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#595

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

C₂×C₄

#10587677

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

136

—

105

C₂²

—

106

C₂²

—

107

C₂²

#11763753

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

D₁₇⋊#108

yes

—

112

C₂

—

113

C₂

#14544703

yes

—

114

C₂

#14544709

yes

characteristic

	subgroup	G/H
center	C₂²	#11763753
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	#382
prime radical	C₂²×D₁₇	#125
cosocle	C₄²×D₁₇	C₂³
hypercenter	C₈²	#595
hypocenter	C₁₇	#8797
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11763753

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₈²×C₁₇

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄²×D₁₇

C₂³

C₈²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
81. C₁₇	#8797	semidirect

H ≅ G/N

type

81. C₁₇

#8797

semidirect

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

subgroup	index
#14544347	2
#14543641	2
#14527066	2
#8797	17

subgroup

index

Subgroups — #17953786

Conjugacy classes of subgroups

Complements

Maximal subgroups