Group #16949018 — subgroups

Subgroups — #16949018

Order 4352.

260 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 83 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	#14531012	C₂	1	2	yes	maximal, complemented
2	#14531008	C₂	1	2	yes	maximal
3	#14528394	C₂	1	1	yes	maximal, characteristic, complemented
4	#14528390	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763214	C₂²	1	2	yes	—
7	#11763189	C₂²	1	1	yes	characteristic
8	#11763186	C₂²	1	1	yes	characteristic
9	#11763160	—	2	2	no	—
10	#11763156	—	2	2	no	—
11	#11762614	—	2	4	no	—
12	#11762611	—	2	4	no	—
13	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
14	C₈²×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₈	—	2	2	no	—
16	C₂×D₁₃₆	—	2	2	no	—
17	#10587560	—	2	1	no	—
18	#10587555	—	2	2	no	—
19	#10587555	—	2	1	no	—
20	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
21	C₄×C₈×C₁₇	D₄	1	2	yes	—
22	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	#10587492	—	4	4	no	—
24	#10587489	—	2	2	no	—
25	#10587489	—	2	2	no	—
26	#10587487	—	4	4	no	—
27	C₂×D₆₈	—	4	1	no	—
28	C₂×D₆₈	—	4	2	no	—
29	C₂×Dic₃₄	—	4	1	no	—
30	C₂×Dic₃₄	—	4	2	no	—
31	C₂×C₈×C₁₇	D₈	1	4	yes	—
32	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
33	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₁₇	—	4	2	no	—
35	C₄⋊Dic₁₇	—	4	2	no	—
36	C₄⋊Dic₁₇	—	4	2	no	—
37	Dic₆₈	—	4	4	no	—
38	D₁₃₆	—	4	4	no	—
39	C₁₆×C₁₇	—	2	4	no	—
40	#12511	—	17	1	no	maximal, Sylow
41	C₂²×D₁₇	—	8	1	no	—
42	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
44	C₂×Dic₁₇	—	8	1	no	—
45	C₂×Dic₁₇	—	8	2	no	—
46	D₆₈	—	8	2	no	—
47	D₆₈	—	8	4	no	—
48	Dic₃₄	—	8	2	no	—
49	Dic₃₄	—	8	4	no	—
50	C₈×C₁₇	—	2	4	no	—
51	C₈×C₁₇	#135	1	4	yes	—
52	#2241	—	17	2	no	—
53	#2240	—	17	2	no	—
54	#1710	—	17	1	no	—
55	#1709	—	17	1	no	—
56	C₈×C₁₆	D₁₇	1	1	yes	characteristic, complemented
57	C₂²×C₁₇	#493	1	1	yes	characteristic
58	C₂×D₁₇	—	16	1	no	—
59	C₂×D₁₇	—	16	2	no	—
60	C₄×C₁₇	#495	1	2	yes	—
61	C₄×C₁₇	#508	1	4	yes	—
62	Dic₁₇	—	16	1	no	—
63	Dic₁₇	—	16	2	no	—
64	#500	—	17	2	no	—
65	#494	—	17	1	no	—
66	#493	—	17	1	no	—
67	#487	—	34	2	no	—
68	#486	—	34	2	no	—
69	#358	—	34	4	no	—
70	#357	—	34	4	no	—
71	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
72	C₈²	C₂×D₁₇	1	1	yes	characteristic
73	C₂×C₁₇	#1709	1	1	yes	characteristic
74	C₂×C₁₇	#2249	1	2	yes	—
75	D₁₇	—	32	1	no	—
76	C₂×Q₁₆	—	34	2	no	—
77	C₂×D₈	—	34	2	no	—
78	C₄⋊Q₈	—	34	1	no	—
79	C₄⋊Q₈	—	34	2	no	—
80	#127	—	34	1	no	—
81	C₂×C₁₆	D₆₈	1	4	yes	—
82	#107	—	34	2	no	—
83	#107	—	34	2	no	—
84	#103	—	68	4	no	—
85	#102	—	68	4	no	—
86	C₄×C₈	D₆₈	1	2	yes	—
87	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
88	C₁₇	#12511	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	68	1	no	—
90	C₂×Q₈	—	68	2	no	—
91	C₂×D₄	—	68	1	no	—
92	C₂×D₄	—	68	2	no	—
93	Q₁₆	—	68	4	no	—
94	D₈	—	68	4	no	—
95	C₂×C₈	D₁₃₆	1	4	yes	—
96	C₂×C₈	C₂×D₆₈	1	2	yes	—
97	C₄⋊C₄	—	68	2	no	—
98	C₄⋊C₄	—	68	2	no	—
99	C₄⋊C₄	—	68	2	no	—
100	C₄²	C₂×D₆₈	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₂³	—	136	1	no	—
103	Q₈	—	136	4	no	—
104	Q₈	—	136	2	no	—
105	D₄	—	136	4	no	—
106	D₄	—	136	2	no	—
107	C₂×C₄	—	136	1	no	—
108	C₂×C₄	—	136	2	no	—
109	C₂×C₄	#10587560	1	1	yes	characteristic
110	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#10587590	1	4	yes	—
113	C₂²	—	272	1	no	—
114	C₂²	—	272	2	no	—
115	C₂²	#11763186	1	1	yes	characteristic
116	C₄	—	272	1	no	—
117	C₄	—	272	2	no	—
118	C₄	#11763193	1	2	yes	—
119	C₄	#11763241	1	4	yes	—
120	C₂	—	544	1	no	—
121	C₂	#14528390	1	1	yes	characteristic
122	C₂	#14531045	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531012

C₂

yes

maximal, complemented

#14531008

C₂

yes

maximal

#14528394

C₂

yes

maximal, characteristic, complemented

#14528390

C₂

yes

maximal, characteristic

C₈×C₁₆×C₁₇

C₂

yes

maximal, characteristic, complemented

#11763214

C₂²

yes

—

#11763189

C₂²

yes

characteristic

#11763186

C₂²

yes

—

—

—

—

C₄×C₁₆×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×D₁₃₆

—

#10587560

—

#10587555

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

D₁₃₆

—

C₁₆×C₁₇

—

#12511

—

maximal, Sylow

C₂²×D₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

#135

yes

—

—

—

—

—

C₈×C₁₆

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#495

yes

—

C₄×C₁₇

#508

yes

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₇

yes

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

D₁₇

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₆₈

yes

—

—

—

—

—

C₄×C₈

D₆₈

yes

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12511

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

D₁₃₆

yes

—

C₂×C₈

C₂×D₆₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

C₂×D₆₈

yes

characteristic

101

C₁₆

—

102

C₂³

—

136

—

103

Q₈

—

136

—

104

Q₈

—

136

—

105

D₄

—

136

—

106

D₄

—

136

—

107

C₂×C₄

—

136

—

108

C₂×C₄

—

136

—

109

C₂×C₄

#10587560

yes

characteristic

110

C₂×C₄

C₂×D₁₃₆

yes

—

111

C₈

—

112

C₈

#10587590

yes

—

113

C₂²

—

272

—

114

C₂²

—

272

—

115

C₂²

#11763186

yes

characteristic

116

C₄

—

272

—

117

C₄

—

272

—

118

C₄

#11763193

yes

—

119

C₄

#11763241

yes

—

120

C₂

—

544

—

121

C₂

#14528390

yes

characteristic

122

C₂

#14531045

yes

—

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₈×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	#493
prime radical	#11763186	C₂²
hypercenter	C₈×C₁₆	D₁₇
hypocenter	C₁₇	#12511
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#11763186

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₈×C₁₆×C₁₇

C₂

C₂²×C₁₇

C₂²

C₈×C₁₆

D₁₇

hypocenter

C₁₇

#12511

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #14531012	C₂	semidirect
3. #14528394	C₂	semidirect
5. C₈×C₁₆×C₁₇	C₂	semidirect
56. C₈×C₁₆	D₁₇	semidirect
88. C₁₇	#12511	semidirect

H ≅ G/N

type

1. #14531012

C₂

semidirect

3. #14528394

C₂

semidirect

5. C₈×C₁₆×C₁₇

C₂

semidirect

56. C₈×C₁₆

D₁₇

semidirect

88. C₁₇

#12511

semidirect

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

subgroup	index
#14531012	2
#14531008	2
#14528394	2
#14528390	2
C₈×C₁₆×C₁₇	2
#12511	17

subgroup

index

C₈×C₁₆×C₁₇

Subgroups — #16949018

Conjugacy classes of subgroups

Complements

Maximal subgroups