Group #16948582 — subgroups

Subgroups — #16948582

Order 4352.

248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14531034	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#14528145	C₂	1	2	yes	maximal, complemented
4	#14526829	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1371	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₇₂	—	2	1	no	—
7	C₂×D₂₇₂	—	2	1	no	—
8	#11763186	C₂²	1	2	yes	—
9	#11763010	—	2	2	no	—
10	C₈×Dic₃₄	C₂²	1	1	yes	characteristic
11	C₄⋊#63214	C₂²	1	1	yes	characteristic
12	#11762627	—	2	1	no	—
13	C₁₆×Dic₁₇	—	2	1	no	—
14	#11762552	—	2	2	no	—
15	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#335	C₂²	1	2	yes	—
17	C₂×D₁₃₆	—	2	2	no	—
18	C₂×D₁₃₆	—	2	2	no	—
19	C₂×#63219	—	4	2	no	—
20	#10587560	—	2	2	no	—
21	C₄×Dic₃₄	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
24	C₁₇×C₄⋊C₈	—	2	2	no	—
25	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	#10587492	—	4	2	no	—
27	#10587485	—	2	1	no	—
28	C₈×Dic₁₇	—	2	1	no	—
29	C₂×#63214	—	2	1	no	—
30	C₄⋊#3622	D₄	1	1	yes	characteristic
31	D₂₇₂	—	4	2	no	—
32	D₂₇₂	—	4	2	no	—
33	C₂×D₆₈	—	4	2	no	—
34	C₂×D₆₈	—	4	2	no	—
35	C₂×D₆₈	—	4	2	no	—
36	C₂×Dic₃₄	D₈	1	1	yes	characteristic
37	C₂×C₈×C₁₇	—	4	2	no	—
38	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
40	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
42	#63225	—	2	1	no	—
43	C₄×Dic₁₇	—	2	1	no	—
44	C₂×#3622	—	2	1	no	—
45	D₁₃₆	—	4	2	no	—
46	D₁₃₆	—	4	2	no	—
47	D₁₃₆	—	8	2	no	—
48	#63219	—	8	4	no	—
49	C₁₆×C₁₇	—	2	1	no	—
50	C₁₆×C₁₇	SD₁₆	1	2	yes	complemented
51	#63214	—	4	1	no	—
52	#12439	—	17	1	no	maximal, Sylow
53	C₂²×D₁₇	—	8	2	no	—
54	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
57	C₂×Dic₁₇	—	2	1	no	—
58	C₂×Dic₁₇	—	2	1	no	—
59	D₆₈	—	8	2	no	—
60	D₆₈	—	8	2	no	—
61	D₆₈	—	8	2	no	—
62	D₆₈	—	8	2	no	—
63	D₆₈	—	16	2	no	—
64	Dic₃₄	—	2	1	no	—
65	Dic₃₄	D₁₆	1	2	yes	complemented
66	C₈×C₁₇	—	2	1	no	—
67	C₈×C₁₇	—	8	2	no	—
68	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₈×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	#3622	—	4	1	no	—
71	#2246	—	17	1	no	—
72	Q₈×C₁₆	—	17	1	no	—
73	#1668	—	17	2	no	—
74	#1371	D₁₇	1	1	yes	characteristic, complemented
75	#1361	—	17	2	no	—
76	C₂²×C₁₇	#466	1	1	yes	characteristic
77	C₂×D₁₇	—	16	2	no	—
78	C₂×D₁₇	—	16	2	no	—
79	C₂×D₁₇	—	16	2	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	#460	1	1	yes	characteristic
82	C₄×C₁₇	#472	1	1	yes	characteristic
83	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
84	C₄×C₁₇	#508	1	1	yes	characteristic
85	Dic₁₇	—	2	2	no	—
86	Dic₁₇	—	4	1	no	—
87	C₂×D₁₆	—	34	1	no	—
88	C₂×D₁₆	—	34	1	no	—
89	#493	—	17	2	no	—
90	#460	—	34	2	no	—
91	C₈×Q₈	—	17	1	no	—
92	C₄⋊C₁₆	—	17	1	no	—
93	C₄⋊C₁₆	—	34	1	no	—
94	C₄×C₁₆	—	34	1	no	—
95	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
96	#335	C₂×D₁₇	1	2	yes	—
97	#331	—	34	2	no	—
98	C₂×C₁₇	#1668	1	1	yes	characteristic
99	C₂×C₁₇	#2215	1	1	yes	characteristic
100	C₂×C₁₇	#2221	1	1	yes	characteristic
101	D₁₇	—	32	2	no	—
102	C₂×SD₁₆	—	68	2	no	—
103	C₂×D₈	—	34	2	no	—
104	C₂×D₈	—	34	2	no	—
105	#127	—	34	2	no	—
106	C₄×Q₈	—	17	1	no	—
107	D₁₆	—	68	2	no	—
108	D₁₆	—	68	2	no	—
109	C₂×C₁₆	—	34	1	no	—
110	C₂×C₁₆	D₆₈	1	1	yes	characteristic
111	C₂×C₁₆	D₆₈	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	17	1	no	—
114	C₄⋊C₈	—	34	1	no	—
115	#102	—	68	2	no	—
116	C₄×C₈	—	34	1	no	—
117	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
118	C₁₇	#12439	1	1	yes	characteristic, complemented, Sylow
119	C₂×Q₈	—	17	1	no	—
120	C₂×D₄	—	68	2	no	—
121	C₂×D₄	—	68	2	no	—
122	C₂×D₄	—	68	2	no	—
123	SD₁₆	—	136	4	no	—
124	D₈	—	68	2	no	—
125	D₈	—	68	2	no	—
126	D₈	—	136	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	34	1	no	—
129	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
130	C₂×C₈	#63256	1	1	yes	characteristic
131	C₄⋊C₄	—	17	1	no	—
132	C₄⋊C₄	—	34	1	no	—
133	C₄²	—	34	1	no	—
134	C₄²	D₄×D₁₇	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	68	1	no	—
137	C₁₆	#63219	1	2	yes	complemented
138	C₂³	—	136	2	no	—
139	Q₈	—	17	2	no	—
140	Q₈	—	34	1	no	—
141	D₄	—	272	2	no	—
142	D₄	—	136	2	no	—
143	D₄	—	136	2	no	—
144	D₄	—	136	2	no	—
145	D₄	—	136	2	no	—
146	C₂×C₄	—	34	1	no	—
147	C₂×C₄	—	34	1	no	—
148	C₂×C₄	#10587581	1	1	yes	characteristic
149	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
150	C₂×C₄	#10587605	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	8	2	no	—
153	C₈	—	68	1	no	—
154	C₈	C₂×#63219	1	1	yes	characteristic
155	C₈	#10587594	1	1	yes	characteristic
156	C₂²	—	272	2	no	—
157	C₂²	—	272	2	no	—
158	C₂²	—	272	2	no	—
159	C₂²	#11763046	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	34	2	no	—
162	C₄	—	68	1	no	—
163	C₄	#11763010	1	1	yes	characteristic
164	C₄	#11763080	1	1	yes	characteristic
165	C₄	D₁₆×D₁₇	1	1	yes	characteristic
166	C₄	#11763242	1	1	yes	characteristic
167	C₂	—	544	2	no	—
168	C₂	#14528145	1	1	yes	characteristic
169	C₂	#14530871	1	1	yes	characteristic
170	C₂	#14530905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531034

C₂

yes

maximal, characteristic

C₁₆×Dic₃₄

C₂

yes

maximal, characteristic, complemented

#14528145

C₂

yes

maximal, complemented

#14526829

C₂

yes

maximal, complemented

C₁₇×#1371

C₂

yes

maximal, characteristic, complemented

C₂×D₂₇₂

—

C₂×D₂₇₂

—

#11763186

C₂²

yes

—

#11763010

—

C₈×Dic₃₄

C₂²

yes

characteristic

C₄⋊#63214

C₂²

yes

characteristic

#11762627

—

C₁₆×Dic₁₇

—

#11762552

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#335

C₂²

yes

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×#63219

—

#10587560

—

C₄×Dic₃₄

D₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587492

—

#10587485

—

C₈×Dic₁₇

—

C₂×#63214

—

C₄⋊#3622

D₄

yes

characteristic

D₂₇₂

—

D₂₇₂

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

D₈

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

D₈

yes

characteristic

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

D₁₃₆

—

D₁₃₆

—

D₁₃₆

—

#63219

—

C₁₆×C₁₇

—

C₁₆×C₁₇

SD₁₆

yes

—

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

#135

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

Dic₃₄

—

Dic₃₄

D₁₆

yes

complemented

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×SD₁₆

yes

C₈×C₁₇

yes

—

—

Q₈×C₁₆

—

#1668

—

#1371

D₁₇

yes

characteristic, complemented

#1361

—

C₂²×C₁₇

#466

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

#508

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×D₁₆

—

C₂×D₁₆

—

#493

—

#460

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

#335

C₂×D₁₇

yes

—

#331

—

C₂×C₁₇

yes

C₂×C₁₇

yes

100

C₂×C₁₇

#2221

yes

characteristic

101

D₁₇

—

102

C₂×SD₁₆

—

103

C₂×D₈

—

104

C₂×D₈

—

105

#127

—

106

C₄×Q₈

—

107

D₁₆

—

108

D₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

D₆₈

yes

characteristic

111

C₂×C₁₆

D₆₈

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

#102

—

116

C₄×C₈

—

117

C₄×C₈

C₂²×D₁₇

yes

characteristic

118

C₁₇

#12439

yes

characteristic, complemented, Sylow

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

SD₁₆

—

136

—

124

D₈

—

125

D₈

—

126

D₈

—

136

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×D₆₈

yes

characteristic

130

C₂×C₈

#63256

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄²

—

134

C₄²

D₄×D₁₇

yes

characteristic

135

C₁₆

—

136

C₁₆

—

137

C₁₆

#63219

yes

complemented

138

C₂³

—

136

—

139

Q₈

—

140

Q₈

—

141

D₄

—

272

—

142

D₄

—

136

—

143

D₄

—

136

—

144

D₄

—

136

—

145

D₄

—

136

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#10587581

yes

characteristic

149

C₂×C₄

D₈×D₁₇

yes

characteristic

150

C₂×C₄

#10587605

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₂×#63219

yes

characteristic

155

C₈

#10587594

yes

characteristic

156

C₂²

—

272

—

157

C₂²

—

272

—

158

C₂²

—

272

—

159

C₂²

#11763046

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#11763010

yes

characteristic

164

C₄

#11763080

yes

characteristic

165

C₄

D₁₆×D₁₇

yes

characteristic

166

C₄

#11763242

yes

characteristic

167

C₂

—

544

—

168

C₂

#14528145

yes

characteristic

169

C₂

#14530871

yes

characteristic

170

C₂

#14530905

yes

characteristic

	subgroup	G/H
center	C₂²	#11763046
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1371	C₂
socle	C₂²×C₁₇	#466
prime radical	#14531034	C₂
hypercenter	#1371	D₁₇
hypocenter	C₁₇	#12439
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#11763046

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1371

C₂

C₂²×C₁₇

C₂

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
2. C₁₆×Dic₃₄	C₂	semidirect
3. #14528145	C₂	semidirect
4. #14526829	C₂	semidirect
5. C₁₇×#1371	C₂	semidirect
50. C₁₆×C₁₇	SD₁₆	semidirect
65. Dic₃₄	D₁₆	semidirect
74. #1371	D₁₇	semidirect
118. C₁₇	#12439	semidirect
137. C₁₆	#63219	semidirect

H ≅ G/N

type

2. C₁₆×Dic₃₄

C₂

semidirect

3. #14528145

C₂

semidirect

4. #14526829

C₂

semidirect

5. C₁₇×#1371

C₂

semidirect

50. C₁₆×C₁₇

SD₁₆

semidirect

65. Dic₃₄

D₁₆

semidirect

74. #1371

D₁₇

semidirect

118. C₁₇

#12439

semidirect

137. C₁₆

#63219

semidirect

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

subgroup	index
#14531034	2
C₁₆×Dic₃₄	2
#14528145	2
#14526829	2
C₁₇×#1371	2
#12439	17

subgroup

index

C₁₆×Dic₃₄

C₁₇×#1371

Subgroups — #16948582

Conjugacy classes of subgroups

Complements

Maximal subgroups