Group #16955297 — subgroups

Subgroups — #16955297

Order 4352.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#14531059	C₂	1	1	yes	maximal, characteristic
2	#14531037	C₂	1	1	yes	maximal, characteristic
3	#14526966	C₂	1	4	yes	maximal
4	C₄×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₃₆	—	2	1	no	—
6	C₂×Dic₁₃₆	C₂²	1	2	yes	—
7	#11763214	—	2	1	no	—
8	#11763189	—	2	1	no	—
9	#11762635	—	2	1	no	—
10	#11762635	C₂²	1	2	yes	—
11	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
12	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₈	—	2	2	no	—
14	C₂×Dic₆₈	—	4	1	no	—
15	#10587555	—	4	1	no	—
16	#10587555	—	4	1	no	—
17	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
19	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
20	#10587489	—	2	2	no	—
21	#10587489	—	4	1	no	—
22	Dic₁₃₆	—	2	4	no	—
23	Dic₁₃₆	—	4	2	no	—
24	C₁₇×C₃₂	—	2	2	no	—
25	C₂×Dic₃₄	—	4	2	no	—
26	C₂×Dic₃₄	—	8	1	no	—
27	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
28	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₁₇	D₈	1	1	yes	characteristic
30	C₄⋊Dic₁₇	—	4	2	no	—
31	C₄⋊Dic₁₇	—	8	1	no	—
32	C₄⋊Dic₁₇	—	8	1	no	—
33	C₄⋊Dic₁₇	—	8	1	no	—
34	Dic₆₈	—	4	2	no	—
35	Dic₆₈	—	4	2	no	—
36	Dic₆₈	—	8	2	no	—
37	C₁₆×C₁₇	SD₁₆	1	2	yes	—
38	C₁₆×C₁₇	#41	1	2	yes	—
39	#13720	—	17	1	no	maximal, Sylow
40	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
43	C₂×Dic₁₇	—	8	2	no	—
44	C₂×Dic₁₇	—	8	2	no	—
45	Dic₃₄	—	8	2	no	—
46	Dic₃₄	—	8	2	no	—
47	Dic₃₄	—	16	2	no	—
48	C₈×C₁₇	SD₃₂	1	2	yes	—
49	C₈×C₁₇	#124	1	1	yes	characteristic
50	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
51	#2252	—	17	1	no	—
52	#2247	—	17	1	no	—
53	#1416	—	17	4	no	—
54	C₄×C₃₂	D₁₇	1	1	yes	characteristic
55	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₁₇	SD₆₄	1	2	yes	—
57	C₄×C₁₇	Q₆₄	1	2	yes	—
58	C₄×C₁₇	#486	1	1	yes	characteristic
59	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
60	Dic₁₇	—	16	2	no	—
61	Dic₁₇	—	16	2	no	—
62	C₂×Q₃₂	—	17	2	no	—
63	C₂×Q₃₂	—	34	1	no	—
64	#500	—	34	1	no	—
65	#494	—	34	1	no	—
66	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
67	#366	—	17	2	no	—
68	#366	—	34	1	no	—
69	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
70	C₂×C₁₇	#2240	1	1	yes	characteristic
71	C₂×C₁₇	C₂×SD₆₄	1	1	yes	characteristic
72	C₂×C₁₇	C₂×Q₆₄	1	1	yes	characteristic
73	C₂×Q₁₆	—	34	2	no	—
74	C₂×Q₁₆	—	68	1	no	—
75	C₄⋊Q₈	—	68	1	no	—
76	C₄⋊Q₈	—	68	1	no	—
77	Q₃₂	—	34	4	no	—
78	Q₃₂	—	68	2	no	—
79	C₂×C₁₆	D₆₈	1	1	yes	characteristic
80	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
81	#107	—	34	2	no	—
82	#107	—	68	1	no	—
83	C₄×C₈	D₆₈	1	1	yes	characteristic
84	C₃₂	—	2	2	no	—
85	C₁₇	#13720	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	68	2	no	—
87	C₂×Q₈	—	136	1	no	—
88	Q₁₆	—	68	2	no	—
89	Q₁₆	—	68	2	no	—
90	Q₁₆	—	136	2	no	—
91	C₂×C₈	D₁₃₆	1	1	yes	characteristic
92	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
93	C₄⋊C₄	—	68	2	no	—
94	C₄⋊C₄	—	136	1	no	—
95	C₄⋊C₄	—	136	1	no	—
96	C₄⋊C₄	—	136	1	no	—
97	C₄²	D₁₃₆	1	1	yes	characteristic
98	C₁₆	#63219	1	2	yes	—
99	C₁₆	#63252	1	2	yes	—
100	Q₈	—	272	2	no	—
101	Q₈	—	136	2	no	—
102	Q₈	—	136	2	no	—
103	C₂×C₄	—	136	2	no	—
104	C₂×C₄	—	136	2	no	—
105	C₂×C₄	D₂₇₂	1	1	yes	characteristic
106	C₂×C₄	D₂₇₂	1	1	yes	characteristic
107	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
108	C₈	#10587471	1	2	yes	—
109	C₈	#10587561	1	1	yes	characteristic
110	C₈	C₂×#63219	1	1	yes	characteristic
111	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
112	C₄	—	272	2	no	—
113	C₄	—	272	2	no	—
114	C₄	#11762648	1	2	yes	—
115	C₄	Dic₂₇₂	1	2	yes	—
116	C₄	#11763156	1	1	yes	characteristic
117	C₄	C₂×#10587471	1	1	yes	characteristic
118	C₂	#14531008	1	1	yes	characteristic
119	C₂	C₂×#11762648	1	1	yes	characteristic
120	C₂	C₂×Dic₂₇₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531059

C₂

yes

maximal, characteristic

#14531037

C₂

yes

maximal, characteristic

#14526966

C₂

yes

maximal

C₄×C₁₇×C₃₂

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

—

C₂×Dic₁₃₆

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₁₇×C₃₂

C₂²

yes

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

#10587489

—

#10587489

—

Dic₁₃₆

—

Dic₁₃₆

—

C₁₇×C₃₂

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

D₈

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

SD₁₆

yes

—

C₁₆×C₁₇

#41

yes

—

#13720

—

maximal, Sylow

C₂×C₄×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

SD₃₂

yes

—

C₈×C₁₇

#124

yes

characteristic

C₈×C₁₇

C₂×SD₁₆

yes

—

—

—

C₄×C₃₂

D₁₇

yes

characteristic

C₂²×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

SD₆₄

yes

—

C₄×C₁₇

Q₆₄

yes

—

C₄×C₁₇

#486

yes

characteristic

C₄×C₁₇

C₂×SD₃₂

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

C₂×C₃₂

C₂×D₁₇

yes

—

#366

—

#366

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₂×C₁₇

#2240

yes

characteristic

C₂×C₁₇

C₂×SD₆₄

yes

characteristic

C₂×C₁₇

C₂×Q₆₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

D₆₈

yes

characteristic

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

#107

—

#107

—

C₄×C₈

D₆₈

yes

characteristic

C₃₂

—

C₁₇

#13720

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

136

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

136

—

C₂×C₈

D₁₃₆

yes

characteristic

C₂×C₈

C₂×D₆₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

136

—

C₄⋊C₄

—

136

—

C₄⋊C₄

—

136

—

C₄²

D₁₃₆

yes

characteristic

C₁₆

#63219

yes

—

C₁₆

#63252

yes

—

100

Q₈

—

272

—

101

Q₈

—

136

—

102

Q₈

—

136

—

103

C₂×C₄

—

136

—

104

C₂×C₄

—

136

—

105

C₂×C₄

D₂₇₂

yes

characteristic

106

C₂×C₄

D₂₇₂

yes

characteristic

107

C₂×C₄

C₂×D₁₃₆

yes

characteristic

108

C₈

#10587471

yes

—

109

C₈

#10587561

yes

characteristic

110

C₈

C₂×#63219

yes

characteristic

111

C₂²

C₂×D₂₇₂

yes

characteristic

112

C₄

—

272

—

113

C₄

—

272

—

114

C₄

#11762648

yes

—

115

C₄

Dic₂₇₂

yes

—

116

C₄

#11763156

yes

characteristic

117

C₄

C₂×#10587471

yes

characteristic

118

C₂

#14531008

yes

characteristic

119

C₂

C₂×#11762648

yes

characteristic

120

C₂

C₂×Dic₂₇₂

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₇₂
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₄×C₁₇×C₃₂	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₁₆
hypercenter	C₄×C₃₂	D₁₇
hypocenter	C₁₇	#13720
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₂₇₂

commutator, cosocle

C₂×C₁₆×C₁₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₇

Fitting

C₄×C₁₇×C₃₂

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₁₆

hypercenter

C₄×C₃₂

D₁₇

hypocenter

C₁₇

#13720

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
85. C₁₇	#13720	semidirect

H ≅ G/N

type

85. C₁₇

#13720

semidirect

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

subgroup	index
#14531059	2
#14531037	2
#14526966	2
C₄×C₁₇×C₃₂	2
#13720	17

subgroup

index

C₄×C₁₇×C₃₂

Subgroups — #16955297

Conjugacy classes of subgroups

Complements

Maximal subgroups