Group #16948982 — subgroups

Subgroups — #16948982

Order 4352.

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

normal subgroups only

This group has 262 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	#14531059	C₂	1	1	yes	maximal, characteristic
2	#14531037	C₂	1	1	yes	maximal, characteristic
3	#14531012	C₂	1	2	yes	maximal
4	#14528381	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₃₆	—	2	2	no	—
7	#11763214	C₂²	1	2	yes	—
8	#11763207	—	2	2	no	—
9	#11763189	C₂²	1	2	yes	—
10	#11763160	—	2	2	no	—
11	#11762635	—	2	2	no	—
12	#11762614	—	2	4	no	—
13	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₈²×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₈	—	2	2	no	—
17	C₂×Dic₆₈	—	2	2	no	—
18	#10587555	—	2	2	no	—
19	#10587555	—	2	2	no	—
20	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
21	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
22	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587489	—	2	2	no	—
26	#10587489	—	2	2	no	—
27	#10587488	—	4	4	no	—
28	#10587487	—	4	4	no	—
29	Dic₁₃₆	—	4	4	no	—
30	C₂×Dic₃₄	—	4	2	no	—
31	C₂×Dic₃₄	—	4	2	no	—
32	C₂×C₈×C₁₇	D₈	1	2	yes	—
33	C₂×C₈×C₁₇	D₈	1	2	yes	—
34	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	4	2	no	—
38	C₄⋊Dic₁₇	—	4	2	no	—
39	C₄⋊Dic₁₇	—	4	2	no	—
40	C₄⋊Dic₁₇	—	4	2	no	—
41	Dic₆₈	—	4	4	no	—
42	Dic₆₈	—	4	4	no	—
43	C₁₆×C₁₇	—	2	2	no	—
44	C₁₆×C₁₇	SD₁₆	1	4	yes	—
45	#12504	—	17	1	no	maximal, Sylow
46	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
49	C₂×Dic₁₇	—	8	2	no	—
50	C₂×Dic₁₇	—	8	2	no	—
51	Dic₃₄	—	8	2	no	—
52	Dic₃₄	—	8	2	no	—
53	Dic₃₄	—	8	4	no	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	Q₃₂	1	4	yes	complemented
56	C₈×C₁₇	C₂×SD₁₆	1	2	yes	—
57	C₈×C₁₇	#135	1	2	yes	—
58	#2252	—	17	1	no	—
59	#2247	—	17	1	no	—
60	#2241	—	17	2	no	—
61	#1707	—	17	2	no	—
62	C₈×C₁₆	D₁₇	1	1	yes	characteristic
63	C₂²×C₁₇	#493	1	1	yes	characteristic
64	C₄×C₁₇	#492	1	1	yes	characteristic
65	C₄×C₁₇	#495	1	1	yes	characteristic
66	C₄×C₁₇	C₂×Q₃₂	1	2	yes	—
67	C₄×C₁₇	#508	1	2	yes	—
68	Dic₁₇	—	16	2	no	—
69	Dic₁₇	—	16	2	no	—
70	C₂×Q₃₂	—	34	2	no	—
71	#500	—	17	2	no	—
72	#498	—	34	2	no	—
73	#494	—	17	2	no	—
74	#487	—	34	2	no	—
75	#366	—	34	2	no	—
76	#358	—	34	4	no	—
77	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
78	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
79	C₈²	C₂×D₁₇	1	1	yes	characteristic
80	C₂×C₁₇	#1706	1	1	yes	characteristic
81	C₂×C₁₇	#2247	1	1	yes	characteristic
82	C₂×C₁₇	#2249	1	1	yes	characteristic
83	C₂×Q₁₆	—	34	2	no	—
84	C₂×Q₁₆	—	34	2	no	—
85	C₄⋊Q₈	—	34	2	no	—
86	C₄⋊Q₈	—	34	2	no	—
87	Q₃₂	—	68	4	no	—
88	C₂×C₁₆	D₆₈	1	2	yes	—
89	C₂×C₁₆	D₆₈	1	2	yes	—
90	#107	—	34	2	no	—
91	#107	—	34	2	no	—
92	#106	—	68	4	no	—
93	#103	—	68	4	no	—
94	C₄×C₈	D₆₈	1	1	yes	characteristic
95	C₄×C₈	D₆₈	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
97	C₁₇	#12504	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	—	68	2	no	—
99	C₂×Q₈	—	68	2	no	—
100	Q₁₆	—	68	4	no	—
101	Q₁₆	—	68	4	no	—
102	C₂×C₈	D₁₃₆	1	2	yes	—
103	C₂×C₈	D₁₃₆	1	2	yes	—
104	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
105	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
106	C₄⋊C₄	—	68	2	no	—
107	C₄⋊C₄	—	68	2	no	—
108	C₄⋊C₄	—	68	2	no	—
109	C₄⋊C₄	—	68	2	no	—
110	C₄²	C₂×D₆₈	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	#63219	1	4	yes	—
113	Q₈	—	136	4	no	—
114	Q₈	—	136	2	no	—
115	Q₈	—	136	2	no	—
116	C₂×C₄	—	136	2	no	—
117	C₂×C₄	—	136	2	no	—
118	C₂×C₄	#10587560	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
120	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Dic₁₃₆	1	4	yes	complemented
123	C₈	C₂×#63219	1	2	yes	—
124	C₈	#10587590	1	2	yes	—
125	C₂²	#11763186	1	1	yes	characteristic
126	C₄	—	272	2	no	—
127	C₄	—	272	2	no	—
128	C₄	#11763182	1	1	yes	characteristic
129	C₄	#11763193	1	1	yes	characteristic
130	C₄	C₂×Dic₁₃₆	1	2	yes	—
131	C₄	#11763241	1	2	yes	—
132	C₂	#14528376	1	1	yes	characteristic
133	C₂	#14531037	1	1	yes	characteristic
134	C₂	#14531045	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531059

C₂

yes

maximal, characteristic

#14531037

C₂

yes

maximal, characteristic

#14531012

C₂

yes

maximal

#14528381

C₂

yes

maximal

C₈×C₁₆×C₁₇

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

—

#11763214

C₂²

yes

—

#11763207

—

#11763189

C₂²

yes

—

#11763160

—

#11762635

—

#11762614

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

Dic₁₃₆

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

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

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

C₁₆×C₁₇

SD₁₆

yes

—

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

Q₃₂

yes

complemented

C₈×C₁₇

C₂×SD₁₆

yes

—

C₈×C₁₇

#135

yes

—

—

—

—

—

C₈×C₁₆

D₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×Q₃₂

yes

—

C₄×C₁₇

#508

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×Q₃₂

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₆₈

yes

—

C₂×C₁₆

D₆₈

yes

—

—

—

—

—

C₄×C₈

D₆₈

yes

characteristic

C₄×C₈

D₆₈

yes

characteristic

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12504

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

D₁₃₆

yes

—

103

C₂×C₈

D₁₃₆

yes

—

104

C₂×C₈

C₂×D₆₈

yes

characteristic

105

C₂×C₈

C₂×D₆₈

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

C₂×D₆₈

yes

characteristic

111

C₁₆

—

112

C₁₆

#63219

yes

—

113

Q₈

—

136

—

114

Q₈

—

136

—

115

Q₈

—

136

—

116

C₂×C₄

—

136

—

117

C₂×C₄

—

136

—

118

C₂×C₄

#10587560

yes

characteristic

119

C₂×C₄

C₂×D₁₃₆

yes

characteristic

120

C₂×C₄

C₂×D₁₃₆

yes

characteristic

121

C₈

—

122

C₈

Dic₁₃₆

yes

complemented

123

C₈

C₂×#63219

yes

—

124

C₈

#10587590

yes

—

125

C₂²

#11763186

yes

characteristic

126

C₄

—

272

—

127

C₄

—

272

—

128

C₄

#11763182

yes

characteristic

129

C₄

#11763193

yes

characteristic

130

C₄

C₂×Dic₁₃₆

yes

—

131

C₄

#11763241

yes

—

132

C₂

#14528376

yes

characteristic

133

C₂

#14531037

yes

characteristic

134

C₂

#14531045

yes

characteristic

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₈×C₁₆×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	C₈×C₁₆	D₁₇
hypocenter	C₁₇	#12504
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

socle, prime radical

C₂²×C₁₇

#493

hypercenter

C₈×C₁₆

D₁₇

hypocenter

C₁₇

#12504

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
55. C₈×C₁₇	Q₃₂	semidirect
97. C₁₇	#12504	semidirect
122. C₈	Dic₁₃₆	semidirect

H ≅ G/N

type

55. C₈×C₁₇

Q₃₂

semidirect

97. C₁₇

#12504

semidirect

122. C₈

Dic₁₃₆

semidirect

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

subgroup	index
#14531059	2
#14531037	2
#14531012	2
#14528381	2
C₈×C₁₆×C₁₇	2
#12504	17

subgroup

index

C₈×C₁₆×C₁₇

Subgroups — #16948982

Conjugacy classes of subgroups

Complements

Maximal subgroups