Group #16949093 — subgroups

Subgroups — #16949093

Order 4352.

274 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 91 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	#14531037	C₂	1	2	yes	maximal
2	#14531012	C₂	1	2	yes	maximal
3	#14528402	C₂	1	1	yes	maximal, characteristic
4	#14528381	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₃₆	—	2	4	no	—
7	#11763214	—	2	1	no	—
8	#11763214	—	2	1	no	—
9	#11763214	C₂²	1	1	yes	characteristic
10	#11763207	—	2	1	no	—
11	#11763189	C₂²	1	1	yes	characteristic
12	#11763189	C₂²	1	2	yes	—
13	#11763160	—	2	1	no	—
14	#11762614	—	2	4	no	—
15	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₈	—	2	2	no	—
18	C₂×Dic₆₈	—	2	2	no	—
19	C₂×Dic₆₈	—	2	2	no	—
20	#10587555	—	2	1	no	—
21	#10587555	—	2	1	no	—
22	#10587555	—	2	2	no	—
23	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
24	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	#10587489	—	2	2	no	—
28	#10587489	—	4	2	no	—
29	#10587489	—	4	2	no	—
30	#10587488	—	4	2	no	—
31	#10587487	—	4	2	no	—
32	Dic₁₃₆	—	4	8	no	—
33	C₂×Dic₃₄	—	4	1	no	—
34	C₂×Dic₃₄	—	4	2	no	—
35	C₂×Dic₃₄	—	4	1	no	—
36	C₂×C₈×C₁₇	D₈	1	2	yes	—
37	C₂×C₈×C₁₇	D₈	1	2	yes	—
38	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
39	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₁₇	—	4	1	no	—
41	C₄⋊Dic₁₇	—	4	1	no	—
42	C₄⋊Dic₁₇	—	4	1	no	—
43	C₄⋊Dic₁₇	—	4	1	no	—
44	C₄⋊Dic₁₇	—	4	2	no	—
45	C₄⋊Dic₁₇	—	4	2	no	—
46	Dic₆₈	—	4	4	no	—
47	Dic₆₈	—	4	4	no	—
48	Dic₆₈	—	4	4	no	—
49	C₁₆×C₁₇	—	2	4	no	—
50	#12528	—	17	1	no	maximal, Sylow
51	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₁₇	—	8	1	no	—
55	C₂×Dic₁₇	—	8	2	no	—
56	C₂×Dic₁₇	—	8	1	no	—
57	Dic₃₄	—	8	2	no	—
58	Dic₃₄	—	8	2	no	—
59	Dic₃₄	—	8	2	no	—
60	Dic₃₄	—	8	2	no	—
61	C₈×C₁₇	—	2	2	no	—
62	C₈×C₁₇	Q₃₂	1	4	yes	complemented
63	C₈×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
64	#2247	—	17	2	no	—
65	#2241	—	17	2	no	—
66	#1713	—	17	1	no	—
67	#1707	—	17	1	no	—
68	#1312	D₁₇	1	1	yes	characteristic
69	C₂²×C₁₇	#493	1	1	yes	characteristic
70	C₄×C₁₇	#496	1	2	yes	—
71	C₄×C₁₇	C₂×Q₃₂	1	2	yes	—
72	C₄×C₁₇	#510	1	2	yes	—
73	Dic₁₇	—	16	1	no	—
74	Dic₁₇	—	16	2	no	—
75	Dic₁₇	—	16	1	no	—
76	C₂×Q₃₂	—	34	4	no	—
77	#500	—	17	1	no	—
78	#500	—	34	1	no	—
79	#500	—	34	1	no	—
80	#498	—	34	1	no	—
81	#494	—	17	1	no	—
82	#494	—	17	2	no	—
83	#487	—	34	1	no	—
84	#358	—	34	4	no	—
85	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
86	C₈²	C₂×D₁₇	1	1	yes	characteristic
87	C₂×C₁₇	#1721	1	1	yes	characteristic
88	C₂×C₁₇	#2247	1	1	yes	characteristic
89	C₂×C₁₇	#2251	1	1	yes	characteristic
90	C₂×Q₁₆	—	34	2	no	—
91	C₂×Q₁₆	—	34	2	no	—
92	C₂×Q₁₆	—	34	2	no	—
93	C₄⋊Q₈	—	34	1	no	—
94	C₄⋊Q₈	—	34	1	no	—
95	C₄⋊Q₈	—	34	2	no	—
96	Q₃₂	—	68	8	no	—
97	C₂×C₁₆	D₆₈	1	4	yes	—
98	#107	—	34	2	no	—
99	#107	—	68	2	no	—
100	#107	—	68	2	no	—
101	#106	—	68	2	no	—
102	#103	—	68	2	no	—
103	C₄×C₈	D₆₈	1	1	yes	characteristic
104	C₄×C₈	D₆₈	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
106	C₁₇	#12528	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₈	—	68	1	no	—
108	C₂×Q₈	—	68	2	no	—
109	C₂×Q₈	—	68	1	no	—
110	Q₁₆	—	68	4	no	—
111	Q₁₆	—	68	4	no	—
112	Q₁₆	—	68	4	no	—
113	C₂×C₈	D₁₃₆	1	2	yes	—
114	C₂×C₈	D₁₃₆	1	2	yes	—
115	C₂×C₈	C₂×D₆₈	1	2	yes	—
116	C₄⋊C₄	—	68	1	no	—
117	C₄⋊C₄	—	68	1	no	—
118	C₄⋊C₄	—	68	1	no	—
119	C₄⋊C₄	—	68	1	no	—
120	C₄⋊C₄	—	68	2	no	—
121	C₄⋊C₄	—	68	2	no	—
122	C₄²	C₂×D₆₈	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	Q₈	—	136	2	no	—
125	Q₈	—	136	2	no	—
126	Q₈	—	136	2	no	—
127	Q₈	—	136	2	no	—
128	C₂×C₄	—	136	1	no	—
129	C₂×C₄	—	136	2	no	—
130	C₂×C₄	—	136	1	no	—
131	C₂×C₄	#10587560	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Dic₁₃₆	1	4	yes	complemented
136	C₈	#10587594	1	4	yes	—
137	C₂²	#11763186	1	1	yes	characteristic
138	C₄	—	272	1	no	—
139	C₄	—	272	2	no	—
140	C₄	—	272	1	no	—
141	C₄	#11763198	1	2	yes	—
142	C₄	C₂×Dic₁₃₆	1	2	yes	—
143	C₄	#11763252	1	2	yes	—
144	C₂	#14528450	1	1	yes	characteristic
145	C₂	#14531037	1	1	yes	characteristic
146	C₂	#14531055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531037

C₂

yes

maximal

#14531012

C₂

yes

maximal

#14528402

C₂

yes

maximal, characteristic

#14528381

C₂

yes

maximal, characteristic

C₁₇×#1312

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

—

#11763214

—

#11763214

—

#11763214

C₂²

yes

characteristic

#11763207

—

#11763189

C₂²

yes

characteristic

#11763189

C₂²

yes

—

#11763160

—

#11762614

—

C₄×C₁₆×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

#10587555

—

#10587555

—

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×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

Q₃₂

yes

complemented

C₈×C₁₇

AGL(1,ℤ/8)

yes

—

—

—

—

—

D₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

yes

—

C₄×C₁₇

C₂×Q₃₂

yes

—

C₄×C₁₇

#510

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂×Q₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₇

yes

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₆₈

yes

—

#107

—

#107

—

100

#107

—

101

#106

—

102

#103

—

103

C₄×C₈

D₆₈

yes

characteristic

104

C₄×C₈

D₆₈

yes

characteristic

105

C₄×C₈

C₂²×D₁₇

yes

characteristic

106

C₁₇

#12528

yes

characteristic, complemented, Sylow

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

Q₁₆

—

113

C₂×C₈

D₁₃₆

yes

—

114

C₂×C₈

D₁₃₆

yes

—

115

C₂×C₈

C₂×D₆₈

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

C₂×D₆₈

yes

characteristic

123

C₁₆

—

124

Q₈

—

136

—

125

Q₈

—

136

—

126

Q₈

—

136

—

127

Q₈

—

136

—

128

C₂×C₄

—

136

—

129

C₂×C₄

—

136

—

130

C₂×C₄

—

136

—

131

C₂×C₄

#10587560

yes

characteristic

132

C₂×C₄

C₂×D₁₃₆

yes

characteristic

133

C₂×C₄

C₂×D₁₃₆

yes

characteristic

134

C₈

—

135

C₈

Dic₁₃₆

yes

complemented

136

C₈

#10587594

yes

—

137

C₂²

#11763186

yes

characteristic

138

C₄

—

272

—

139

C₄

—

272

—

140

C₄

—

272

—

141

C₄

#11763198

yes

—

142

C₄

C₂×Dic₁₃₆

yes

—

143

C₄

#11763252

yes

—

144

C₂

#14528450

yes

characteristic

145

C₂

#14531037

yes

characteristic

146

C₂

#14531055

yes

characteristic

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1312	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	#1312	D₁₇
hypocenter	C₁₇	#12528
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#11763186

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1312

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
62. C₈×C₁₇	Q₃₂	semidirect
106. C₁₇	#12528	semidirect
135. C₈	Dic₁₃₆	semidirect

H ≅ G/N

type

62. C₈×C₁₇

Q₃₂

semidirect

106. C₁₇

#12528

semidirect

135. C₈

Dic₁₃₆

semidirect

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

subgroup	index
#14531037	2
#14531012	2
#14528402	2
#14528381	2
C₁₇×#1312	2
#12528	17

subgroup

index

C₁₇×#1312

Subgroups — #16949093

Conjugacy classes of subgroups

Complements

Maximal subgroups