Group #16949047 — subgroups

Subgroups — #16949047

Order 4352.

330 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14531040	C₂	1	2	yes	maximal, complemented
2	#14531037	C₂	1	1	yes	maximal, characteristic, complemented
3	#14531034	C₂	1	1	yes	maximal, characteristic
4	#14528390	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₃₆	—	2	2	no	—
7	C₂×#10587471	—	2	4	no	—
8	C₂×D₂₇₂	—	2	2	no	—
9	#11763189	C₂²	1	2	yes	—
10	#11763186	C₂²	1	2	yes	—
11	#11763156	—	2	4	no	—
12	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₈²×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₈	—	2	2	no	—
16	C₂×Dic₆₈	—	2	2	no	—
17	C₂×D₁₃₆	—	2	2	no	—
18	C₂×D₁₃₆	—	2	2	no	—
19	#10587560	—	2	2	no	—
20	#10587555	—	2	2	no	—
21	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
22	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
23	C₄×C₈×C₁₇	D₄	1	2	yes	—
24	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587492	—	4	8	no	—
26	Dic₁₃₆	—	4	4	no	—
27	#10587471	—	4	8	no	—
28	D₂₇₂	—	4	4	no	—
29	C₂×D₆₈	—	4	2	no	—
30	C₂×D₆₈	—	4	4	no	—
31	C₂×Dic₃₄	—	4	2	no	—
32	C₂×C₈×C₁₇	D₈	1	4	yes	—
33	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₇	—	4	4	no	—
37	Dic₆₈	—	4	4	no	—
38	Dic₆₈	—	4	4	no	—
39	D₁₃₆	—	4	4	no	—
40	D₁₃₆	—	4	4	no	—
41	C₁₆×C₁₇	D₈	1	4	yes	—
42	C₁₆×C₁₇	D₈	1	4	yes	—
43	#12518	—	17	1	no	maximal, Sylow
44	C₂²×D₁₇	—	8	2	no	—
45	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
47	C₂×Dic₁₇	—	8	2	no	—
48	D₆₈	—	8	2	no	—
49	D₆₈	—	8	2	no	—
50	D₆₈	—	8	8	no	—
51	Dic₃₄	—	8	2	no	—
52	Dic₃₄	—	8	2	no	—
53	C₈×C₁₇	—	2	4	no	—
54	C₈×C₁₇	C₂×D₈	1	2	yes	—
55	C₈×C₁₇	C₂×D₈	1	2	yes	—
56	#2248	—	17	2	no	—
57	#2247	—	17	1	no	—
58	#2246	—	17	1	no	—
59	#1709	—	17	2	no	—
60	C₈×C₁₆	D₁₇	1	1	yes	characteristic, complemented
61	C₂²×C₁₇	#493	1	1	yes	characteristic
62	C₂×D₁₇	—	16	2	no	—
63	C₂×D₁₇	—	16	4	no	—
64	C₄×C₁₇	#493	1	1	yes	characteristic
65	C₄×C₁₇	#493	1	1	yes	characteristic
66	C₄×C₁₇	#508	1	4	yes	—
67	Dic₁₇	—	16	2	no	—
68	C₂×Q₃₂	—	34	2	no	—
69	C₂×SD₃₂	—	34	4	no	—
70	C₂×D₁₆	—	34	2	no	—
71	#494	—	17	2	no	—
72	#493	—	17	2	no	—
73	#486	—	34	4	no	—
74	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
75	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
76	C₈²	C₂×D₁₇	1	1	yes	characteristic
77	C₂×C₁₇	#1712	1	1	yes	characteristic
78	C₂×C₁₇	#2249	1	2	yes	—
79	D₁₇	—	32	2	no	—
80	C₂×Q₁₆	—	34	2	no	—
81	C₂×Q₁₆	—	34	2	no	—
82	C₂×D₈	—	34	2	no	—
83	C₂×D₈	—	34	2	no	—
84	C₄⋊Q₈	—	34	2	no	—
85	#127	—	34	2	no	—
86	Q₃₂	—	68	4	no	—
87	SD₃₂	—	68	8	no	—
88	D₁₆	—	68	4	no	—
89	C₂×C₁₆	D₆₈	1	2	yes	—
90	C₂×C₁₆	D₆₈	1	2	yes	—
91	#102	—	68	8	no	—
92	C₄×C₈	D₆₈	1	2	yes	—
93	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
94	C₁₇	#12518	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	68	2	no	—
96	C₂×D₄	—	68	2	no	—
97	C₂×D₄	—	68	4	no	—
98	Q₁₆	—	68	4	no	—
99	Q₁₆	—	68	4	no	—
100	D₈	—	68	4	no	—
101	D₈	—	68	4	no	—
102	C₂×C₈	D₁₃₆	1	4	yes	—
103	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
104	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
105	C₄⋊C₄	—	68	4	no	—
106	C₄²	C₂×D₆₈	1	1	yes	characteristic
107	C₁₆	D₁₃₆	1	4	yes	—
108	C₁₆	D₁₃₆	1	4	yes	—
109	C₂³	—	136	2	no	—
110	Q₈	—	136	2	no	—
111	Q₈	—	136	2	no	—
112	D₄	—	136	8	no	—
113	D₄	—	136	2	no	—
114	D₄	—	136	2	no	—
115	C₂×C₄	—	136	2	no	—
116	C₂×C₄	#10587560	1	1	yes	characteristic
117	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₁₃₆	1	2	yes	—
120	C₈	C₂×D₁₃₆	1	2	yes	—
121	C₂²	—	272	2	no	—
122	C₂²	—	272	4	no	—
123	C₂²	#11763186	1	1	yes	characteristic
124	C₄	—	272	2	no	—
125	C₄	#11763186	1	1	yes	characteristic
126	C₄	#11763186	1	1	yes	characteristic
127	C₄	#11763241	1	4	yes	—
128	C₂	—	544	2	no	—
129	C₂	#14528399	1	1	yes	characteristic
130	C₂	#14531045	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531040

C₂

yes

maximal, complemented

#14531037

C₂

yes

maximal, characteristic, complemented

#14531034

C₂

yes

maximal, characteristic

#14528390

C₂

yes

maximal, complemented

C₈×C₁₆×C₁₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₃₆

—

C₂×#10587471

—

C₂×D₂₇₂

—

#11763189

C₂²

yes

—

#11763186

C₂²

yes

—

#11763156

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

#10587560

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587492

—

Dic₁₃₆

—

#10587471

—

D₂₇₂

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

D₁₃₆

—

D₁₃₆

—

C₁₆×C₁₇

D₈

yes

—

C₁₆×C₁₇

D₈

yes

—

#12518

—

maximal, Sylow

C₂²×D₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂×D₈

yes

—

C₈×C₁₇

C₂×D₈

yes

—

—

—

—

—

C₈×C₁₆

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

—

#493

—

#486

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

D₁₇

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₆₈

yes

—

C₂×C₁₆

D₆₈

yes

—

#102

—

C₄×C₈

D₆₈

yes

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12518

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

100

D₈

—

101

D₈

—

102

C₂×C₈

D₁₃₆

yes

—

103

C₂×C₈

C₂×D₆₈

yes

characteristic

104

C₂×C₈

C₂×D₆₈

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

C₂×D₆₈

yes

characteristic

107

C₁₆

D₁₃₆

yes

—

108

C₁₆

D₁₃₆

yes

—

109

C₂³

—

136

—

110

Q₈

—

136

—

111

Q₈

—

136

—

112

D₄

—

136

—

113

D₄

—

136

—

114

D₄

—

136

—

115

C₂×C₄

—

136

—

116

C₂×C₄

#10587560

yes

characteristic

117

C₂×C₄

C₂×D₁₃₆

yes

—

118

C₈

—

119

C₈

C₂×D₁₃₆

yes

—

120

C₈

C₂×D₁₃₆

yes

—

121

C₂²

—

272

—

122

C₂²

—

272

—

123

C₂²

#11763186

yes

characteristic

124

C₄

—

272

—

125

C₄

#11763186

yes

characteristic

126

C₄

#11763186

yes

characteristic

127

C₄

#11763241

yes

—

128

C₂

—

544

—

129

C₂

#14528399

yes

characteristic

130

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	#14531034	C₂
hypercenter	C₈×C₁₆	D₁₇
hypocenter	C₁₇	#12518
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

C₂

C₈×C₁₆

D₁₇

hypocenter

C₁₇

#12518

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #14531040	C₂	semidirect
2. #14531037	C₂	semidirect
4. #14528390	C₂	semidirect
5. C₈×C₁₆×C₁₇	C₂	semidirect
60. C₈×C₁₆	D₁₇	semidirect
94. C₁₇	#12518	semidirect

H ≅ G/N

type

1. #14531040

C₂

semidirect

2. #14531037

C₂

semidirect

4. #14528390

C₂

semidirect

5. C₈×C₁₆×C₁₇

C₂

semidirect

60. C₈×C₁₆

D₁₇

semidirect

94. C₁₇

#12518

semidirect

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

subgroup	index
#14531040	2
#14531037	2
#14531034	2
#14528390	2
C₈×C₁₆×C₁₇	2
#12518	17

subgroup

index

C₈×C₁₆×C₁₇

Subgroups — #16949047

Conjugacy classes of subgroups

Complements

Maximal subgroups