Group #17982814 — subgroups

Subgroups — #17982814

Order 4352.

166 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14545883	C₂	1	2	yes	maximal
2	C₄⋊#10587528	C₂	1	1	yes	maximal, characteristic
3	#14545867	C₂	1	2	yes	maximal
4	C₄×#10587528	C₂	1	1	yes	maximal, characteristic
5	#14545198	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587528	C₂²	1	1	yes	characteristic
7	C₂×#10587528	C₂²	1	2	yes	—
8	C₂×#10587528	C₂²	1	1	yes	characteristic
9	#11763730	C₄	1	2	yes	—
10	C₄⋊#63216	C₄	1	1	yes	characteristic
11	#11763695	C₂²	1	2	yes	—
12	C₄×#63216	C₂²	1	1	yes	characteristic
13	#11762933	C₄	1	1	yes	characteristic
14	C₄×Dic₃₄	C₈	1	1	yes	characteristic
15	C₂×#63216	C₂×C₄	1	2	yes	—
16	C₂×#63216	C₂×C₄	1	1	yes	characteristic
17	C₂×#63216	C₂³	1	1	yes	characteristic
18	#10587528	—	2	1	no	—
19	#10587528	—	2	2	no	—
20	#10587528	Q₈	1	2	yes	complemented
21	C₁₇×C₄⋊C₈	C₈	1	1	yes	characteristic
22	#10587485	C₈	1	2	yes	—
23	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
24	C₄×#3622	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₃₄	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
27	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₄⋊Dic₁₇	C₁₆	1	1	yes	characteristic
29	#63225	C₁₆	1	2	yes	—
30	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
31	C₂×#3622	C₂×C₈	1	1	yes	characteristic
32	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
33	#63216	—	2	2	no	—
34	#63216	—	2	1	no	—
35	#63216	C₂×Q₈	1	1	yes	characteristic
36	#63216	#41	1	1	yes	characteristic
37	Q₈×C₃₂	—	17	1	no	maximal, Sylow
38	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
41	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
42	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
43	Dic₃₄	C₃₂	1	4	yes	complemented
44	C₈×C₁₇	—	2	2	no	—
45	#3622	—	2	1	no	—
46	#3622	C₄×Q₈	1	1	yes	characteristic
47	#3622	#131	1	1	yes	characteristic
48	Q₈×C₁₆	—	17	1	no	—
49	C₄⋊C₃₂	—	17	1	no	—
50	C₄⋊C₃₂	—	17	2	no	—
51	C₄×C₃₂	—	17	1	no	—
52	C₄×C₃₂	—	17	2	no	—
53	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₁₇	—	2	1	no	—
55	C₄×C₁₇	C₂×C₃₂	1	2	yes	—
56	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
57	C₄×C₁₇	#504	1	1	yes	characteristic
58	Dic₁₇	—	2	2	no	—
59	Dic₁₇	C₂×C₃₂	1	4	yes	—
60	C₈×Q₈	—	17	1	no	—
61	C₂×C₃₂	—	17	1	no	—
62	C₂×C₃₂	—	17	1	no	—
63	C₂×C₃₂	—	17	2	no	—
64	C₄⋊C₁₆	—	17	1	no	—
65	C₄⋊C₁₆	—	17	2	no	—
66	C₄×C₁₆	—	17	1	no	—
67	C₄×C₁₆	—	17	2	no	—
68	C₂×C₁₇	Q₈×C₁₆	1	1	yes	characteristic
69	C₂×C₁₇	C₂²×C₃₂	1	1	yes	characteristic
70	C₂×C₁₇	#2258	1	1	yes	characteristic
71	C₄×Q₈	—	17	1	no	—
72	C₂×C₁₆	—	17	1	no	—
73	C₂×C₁₆	—	17	1	no	—
74	C₂×C₁₆	—	17	2	no	—
75	C₄⋊C₈	—	17	2	no	—
76	C₄⋊C₈	#3633	1	1	yes	characteristic
77	C₄×C₈	—	17	2	no	—
78	C₄×C₈	—	17	1	no	—
79	C₃₂	—	17	2	no	—
80	C₃₂	—	34	1	no	—
81	C₃₂	—	34	2	no	—
82	C₁₇	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	17	1	no	—
84	C₂×C₈	—	17	1	no	—
85	C₂×C₈	—	17	1	no	—
86	C₂×C₈	C₂×#3633	1	2	yes	—
87	C₄⋊C₄	—	17	2	no	—
88	C₄⋊C₄	—	17	1	no	—
89	C₄²	—	17	2	no	—
90	C₄²	C₂×#3633	1	1	yes	characteristic
91	C₁₆	—	17	1	no	—
92	C₁₆	—	17	1	no	—
93	C₁₆	—	34	1	no	—
94	C₁₆	—	34	2	no	—
95	Q₈	—	17	4	no	—
96	C₂×C₄	—	17	2	no	—
97	C₂×C₄	—	17	2	no	—
98	C₂×C₄	#10587665	1	1	yes	characteristic
99	C₂×C₄	#10587665	1	1	yes	characteristic
100	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	17	1	no	—
103	C₈	—	17	1	no	—
104	C₈	—	34	1	no	—
105	C₂²	C₂×#10587665	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	17	4	no	—
108	C₄	—	34	2	no	—
109	C₄	#11763916	1	2	yes	—
110	C₄	Q₈×#3633	1	1	yes	characteristic
111	C₄	#11763930	1	1	yes	characteristic
112	C₂	#14546008	1	1	yes	characteristic
113	C₂	C₂×#11763916	1	1	yes	characteristic
114	C₂	#14546018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545883

C₂

yes

maximal

C₄⋊#10587528

C₂

yes

maximal, characteristic

#14545867

C₂

yes

maximal

C₄×#10587528

C₂

yes

maximal, characteristic

#14545198

C₂

yes

maximal, characteristic

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587528

C₂²

yes

—

C₂×#10587528

C₂²

yes

characteristic

#11763730

C₄

yes

—

C₄⋊#63216

C₄

yes

characteristic

#11763695

C₂²

yes

—

C₄×#63216

C₂²

yes

characteristic

#11762933

C₄

yes

characteristic

C₄×Dic₃₄

C₈

yes

characteristic

C₂×#63216

C₂×C₄

yes

—

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂³

yes

—

—

Q₈

yes

complemented

C₁₇×C₄⋊C₈

C₈

yes

characteristic

#10587485

C₈

yes

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂×Dic₃₄

C₁₆

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄⋊Dic₁₇

C₁₆

yes

characteristic

#63225

C₁₆

yes

—

C₄×Dic₁₇

C₂×C₈

yes

—

C₂×#3622

C₂×C₈

yes

characteristic

C₂×#3622

C₂²×C₄

yes

—

—

C₂×Q₈

yes

yes

Q₈×C₃₂

—

maximal, Sylow

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

—

C₂×Dic₁₇

C₂×C₁₆

yes

—

Dic₃₄

C₃₂

yes

complemented

C₈×C₁₇

—

#3622

—

#3622

C₄×Q₈

yes

yes

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

—

C₂²×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂×C₃₂

yes

—

C₄×C₁₇

C₈×Q₈

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

Dic₁₇

C₂×C₃₂

yes

—

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂×C₁₇

Q₈×C₁₆

yes

characteristic

C₂×C₁₇

C₂²×C₃₂

yes

characteristic

C₂×C₁₇

#2258

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

#3633

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

—

C₁₇

Q₈×C₃₂

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3633

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×#3633

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²×#3633

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₂²

C₂×#10587665

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#11763916

yes

—

110

C₄

Q₈×#3633

yes

characteristic

111

C₄

#11763930

yes

characteristic

112

C₂

#14546008

yes

characteristic

113

C₂

C₂×#11763916

yes

characteristic

114

C₂

#14546018

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator	C₂×C₁₇	C₂²×C₃₂
Fitting	C₁₇×C₄⋊C₈	C₈
socle, prime radical	C₂²×C₁₇	C₂²×C₁₆
cosocle	C₂×#63216	C₂³
hypercenter	C₄⋊C₈	#3633
hypocenter	C₁₇	Q₈×C₃₂
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3633

commutator

C₂×C₁₇

C₂²×C₃₂

Fitting

C₁₇×C₄⋊C₈

C₈

socle, prime radical

C₂²×C₁₇

C₂²×C₁₆

C₂³

C₄⋊C₈

C₁₇

Q₈×C₃₂

LCS

G ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
20. #10587528	Q₈	semidirect
43. Dic₃₄	C₃₂	semidirect
82. C₁₇	Q₈×C₃₂	semidirect

H ≅ G/N

type

20. #10587528

Q₈

semidirect

43. Dic₃₄

C₃₂

semidirect

82. C₁₇

Q₈×C₃₂

semidirect

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

subgroup	index
#14545883	2
C₄⋊#10587528	2
#14545867	2
C₄×#10587528	2
#14545198	2
Q₈×C₃₂	17

subgroup

index

Q₈×C₃₂

Subgroups — #17982814

Conjugacy classes of subgroups

Complements

Maximal subgroups