Group #17981406 — subgroups

Subgroups — #17981406

Order 4352.

136 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 29 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	#14545823	C₂	1	2	yes	maximal
2	#14544709	C₂	1	1	yes	maximal, characteristic
3	#11763908	—	2	2	no	—
4	#11763814	C₄	1	1	yes	characteristic
5	#11763814	C₂²	1	1	yes	characteristic
6	#11763685	—	2	2	no	—
7	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
8	C₂×#63243	—	2	1	no	—
9	C₂×#63243	D₄	1	2	yes	—
10	#10587670	—	4	2	no	—
11	C₂×C₈×D₁₇	—	2	1	no	—
12	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
13	#10587544	—	4	1	no	—
14	#10587544	—	4	1	no	—
15	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
16	C₈×Dic₁₇	—	2	1	no	—
17	C₄×#3622	C₈	1	1	yes	characteristic
18	C₂×C₄×D₁₇	—	2	1	no	—
19	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×#3624	—	2	2	no	—
21	C₂×#3624	—	4	1	no	—
22	#63243	—	4	2	no	—
23	#63243	—	4	2	no	—
24	#63241	—	8	2	no	—
25	C₂×C₈×C₁₇	—	2	1	no	—
26	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₄×Dic₁₇	—	2	1	no	—
28	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
29	C₂×#3622	—	2	1	no	—
30	C₈×D₁₇	—	4	1	no	—
31	C₈×D₁₇	—	4	1	no	—
32	#7368	—	17	1	no	maximal, Sylow
33	C₂²×D₁₇	#101	1	1	yes	characteristic
34	C₂×C₄×C₁₇	—	2	1	no	—
35	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×Dic₁₇	—	2	1	no	—
37	C₂×Dic₁₇	#99	1	1	yes	characteristic
38	C₄×D₁₇	—	2	1	no	—
39	C₄×D₁₇	—	2	1	no	—
40	C₄×D₁₇	—	4	2	no	—
41	#3624	—	4	2	no	—
42	#3624	—	8	1	no	—
43	C₈×C₁₇	—	4	1	no	—
44	#3622	—	4	1	no	—
45	#1837	—	17	1	no	—
46	#1327	—	17	2	no	—
47	C₂²×C₁₇	#324	1	1	yes	characteristic
48	C₂×D₁₇	—	2	1	no	—
49	C₂×D₁₇	—	4	1	no	—
50	C₄×C₁₇	—	2	1	no	—
51	C₄×C₁₇	—	4	1	no	—
52	Dic₁₇	—	2	2	no	—
53	Dic₁₇	#354	1	1	yes	characteristic
54	Dic₁₇	#355	1	1	yes	characteristic
55	#423	—	17	1	no	—
56	#423	—	17	1	no	—
57	C₂×C₄×C₈	—	17	1	no	—
58	#349	—	34	2	no	—
59	#341	—	34	2	no	—
60	C₂×C₁₇	—	2	1	no	—
61	C₂×C₁₇	#1324	1	1	yes	characteristic
62	D₁₇	—	4	1	no	—
63	C₂×M₁₆	—	17	2	no	—
64	C₂×M₁₆	—	34	1	no	—
65	C₂²×C₈	—	34	1	no	—
66	C₂×C₄²	—	17	1	no	—
67	M₃₂	—	68	2	no	—
68	C₄⋊C₈	—	68	1	no	—
69	C₄⋊C₈	—	68	1	no	—
70	C₄×C₈	—	17	1	no	—
71	C₄×C₈	—	34	1	no	—
72	C₄×C₈	#3633	1	1	yes	characteristic
73	C₁₇	#7368	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₄	—	17	1	no	—
75	C₂²×C₄	—	34	1	no	—
76	M₁₆	—	68	2	no	—
77	M₁₆	—	68	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	34	2	no	—
80	C₂×C₈	—	34	1	no	—
81	C₂×C₈	—	68	1	no	—
82	C₂×C₈	—	68	1	no	—
83	C₂×C₈	—	68	1	no	—
84	C₄²	—	17	1	no	—
85	C₄²	—	34	1	no	—
86	C₄²	C₂×#3633	1	1	yes	characteristic
87	C₁₆	—	136	2	no	—
88	C₂³	—	17	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	17	1	no	—
91	C₂×C₄	—	34	1	no	—
92	C₂×C₄	—	34	1	no	—
93	C₂×C₄	—	34	1	no	—
94	C₂×C₄	—	68	2	no	—
95	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
96	C₈	—	4	1	no	—
97	C₈	—	68	1	no	—
98	C₈	—	68	2	no	—
99	C₈	—	136	1	no	—
100	C₂²	—	34	1	no	—
101	C₂²	—	68	1	no	—
102	C₂²	#11763895	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	17	1	no	—
106	C₄	—	17	1	no	—
107	C₄	—	34	2	no	—
108	C₂	—	2	1	no	—
109	C₂	—	68	1	no	—
110	C₂	#14545820	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545823

C₂

yes

maximal

#14544709

C₂

yes

maximal, characteristic

#11763908

—

#11763814

C₄

yes

characteristic

#11763814

C₂²

yes

characteristic

#11763685

—

C₄×C₈×D₁₇

C₄

yes

characteristic

C₂×#63243

—

C₂×#63243

D₄

yes

—

#10587670

—

C₂×C₈×D₁₇

—

C₄²×D₁₇

C₂×C₄

yes

characteristic

#10587544

—

#10587544

—

C₄×C₈×C₁₇

C₈

yes

characteristic

C₈×Dic₁₇

—

C₄×#3622

C₈

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

—

—

—

—

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

—

C₈×D₁₇

—

C₈×D₁₇

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#99

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

C₈×C₁₇

—

#3622

—

#1837

—

#1327

—

C₂²×C₁₇

#324

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

—

—

C₂×C₄×C₈

—

#349

—

#341

—

C₂×C₁₇

—

C₂×C₁₇

#1324

yes

characteristic

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3633

yes

characteristic

C₁₇

#7368

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

C₂×#3633

yes

characteristic

C₁₆

—

136

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#3633

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

136

—

100

C₂²

—

101

C₂²

—

102

C₂²

#11763895

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₂

—

109

C₂

—

110

C₂

#14545820

yes

characteristic

	subgroup	G/H
center	C₂	#14545820
commutator	C₄²×C₁₇	C₂×C₈
Frattini	C₄²	C₂×#3633
Fitting	C₄×C₈×C₁₇	C₈
socle	C₂×C₁₇	#1324
prime radical	C₂²×D₁₇	#101
cosocle	#11763814	C₂²
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#7368
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂

#14545820

commutator

C₄²×C₁₇

C₂×C₈

Frattini

C₄²

C₂×#3633

Fitting

C₄×C₈×C₁₇

C₈

C₂×C₁₇

C₂²×D₁₇

C₂²

C₄×C₈

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
73. C₁₇	#7368	semidirect

H ≅ G/N

type

73. C₁₇

#7368

semidirect

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

subgroup	index
#14545823	2
#14544709	2
#7368	17

subgroup

index

#14545823

#14544709

#7368

Subgroups — #17981406

Conjugacy classes of subgroups

Complements

Maximal subgroups