Group #16933808 — subgroups

Subgroups — #16933808

Order 4352.

390 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×#11763207	C₂	1	1	yes	maximal, characteristic
2	#14527470	C₂	1	2	yes	maximal
3	#14527466	C₂	1	2	yes	maximal
4	#14527076	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1837	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	C₂²	1	2	yes	—
7	#11763207	—	2	2	no	—
8	#11763207	C₄	1	4	yes	—
9	#11762862	C₂²	1	2	yes	—
10	C₂×#10587488	—	2	2	no	—
11	#11762557	—	2	4	no	—
12	#11762554	—	2	4	no	—
13	#11762531	—	2	2	no	—
14	C₁₇×#423	C₂²	1	1	yes	characteristic
15	C₁₇×#423	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#341	—	2	2	no	—
18	C₂²×Dic₃₄	—	2	2	no	—
19	C₂×M₁₆×C₁₇	—	2	1	no	—
20	C₂×M₁₆×C₁₇	—	2	1	no	—
21	C₂²×C₈×C₁₇	D₄	1	2	yes	—
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	#10587555	—	2	2	no	—
26	#10587555	—	2	2	no	—
27	#10587555	—	2	2	no	—
28	#10587555	C₂×C₄	1	4	yes	—
29	C₁₇×C₄⋊C₈	—	2	2	no	—
30	C₁₇×C₄⋊C₈	—	2	2	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₇	D₄	1	2	yes	—
33	C₂²⋊#3622	—	2	2	no	—
34	#10587488	—	2	8	no	—
35	C₄⋊#3622	D₄	1	4	yes	—
36	C₄×#3622	—	2	2	no	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	#41	1	2	yes	—
39	C₂²×Dic₁₇	—	4	2	no	—
40	C₂×Dic₃₄	—	2	4	no	—
41	C₂×Dic₃₄	—	4	2	no	—
42	C₂×Dic₃₄	—	8	4	no	—
43	M₁₆×C₁₇	—	4	2	no	—
44	M₁₆×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	4	1	no	—
48	C₂×C₈×C₁₇	—	4	1	no	—
49	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
50	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂×D₄	1	2	yes	—
53	C₄⋊Dic₁₇	—	2	4	no	—
54	C₄⋊Dic₁₇	—	2	4	no	—
55	C₄⋊Dic₁₇	—	4	2	no	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₂×#3622	—	2	4	no	—
58	#9776	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	4	no	—
61	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₁₇	#121	1	2	yes	—
64	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#137	1	2	yes	—
66	C₂×C₄×C₁₇	#137	1	2	yes	—
67	C₂×Dic₁₇	—	4	4	no	—
68	C₂×Dic₁₇	—	8	4	no	—
69	Dic₃₄	—	8	4	no	—
70	Dic₃₄	—	8	4	no	—
71	C₈×C₁₇	—	2	4	no	—
72	C₈×C₁₇	—	8	1	no	—
73	C₈×C₁₇	—	8	1	no	—
74	#3622	—	4	4	no	—
75	C₂×#498	—	17	1	no	—
76	#1837	D₁₇	1	1	yes	characteristic
77	#1549	—	17	2	no	—
78	#1548	—	17	2	no	—
79	#1452	—	17	1	no	—
80	C₂²×C₁₇	—	2	2	no	—
81	C₂²×C₁₇	#390	1	1	yes	characteristic
82	C₂²×C₁₇	#491	1	2	yes	—
83	C₄×C₁₇	—	2	2	no	—
84	C₄×C₁₇	#419	1	2	yes	—
85	C₄×C₁₇	#471	1	4	yes	—
86	C₄×C₁₇	#497	1	1	yes	characteristic
87	C₄×C₁₇	#497	1	1	yes	characteristic
88	Dic₁₇	—	8	4	no	—
89	C₂×C₄⋊Q₈	—	17	2	no	—
90	#498	—	17	4	no	—
91	#498	—	34	2	no	—
92	#431	—	17	2	no	—
93	C₂×#106	—	34	2	no	—
94	#423	C₂×D₁₇	1	1	yes	characteristic
95	#423	C₂×D₁₇	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
97	#341	—	2	2	no	—
98	#333	—	34	4	no	—
99	#332	—	34	4	no	—
100	#322	—	34	2	no	—
101	C₂×C₁₇	—	2	2	no	—
102	C₂×C₁₇	#1960	1	1	yes	characteristic
103	C₂×C₁₇	#1977	1	2	yes	—
104	C₂²×Q₈	—	34	2	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
108	C₄⋊Q₈	—	17	4	no	—
109	C₄⋊Q₈	—	34	2	no	—
110	C₄⋊Q₈	—	34	2	no	—
111	C₄⋊Q₈	—	34	2	no	—
112	C₂×C₄⋊C₄	—	34	2	no	—
113	C₂×C₄⋊C₄	—	34	2	no	—
114	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
115	#106	—	34	8	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	17	4	no	—
119	C₂²⋊C₈	—	34	2	no	—
120	C₄×C₈	—	34	2	no	—
121	C₄×C₈	C₄×D₁₇	1	2	yes	—
122	C₄×C₈	D₆₈	1	2	yes	—
123	C₁₇	#9776	1	1	yes	characteristic, complemented, Sylow
124	C₂×Q₈	—	34	4	no	—
125	C₂×Q₈	—	68	2	no	—
126	C₂×Q₈	—	136	4	no	—
127	C₂²×C₄	—	68	2	no	—
128	C₂²×C₄	#63256	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
130	M₁₆	—	4	2	no	—
131	M₁₆	—	4	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	34	4	no	—
137	C₂×C₈	#63227	1	4	yes	—
138	C₄⋊C₄	—	34	4	no	—
139	C₄⋊C₄	—	34	4	no	—
140	C₄⋊C₄	—	68	2	no	—
141	C₄⋊C₄	—	68	2	no	—
142	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
143	C₄²	C₂×D₆₈	1	1	yes	characteristic
144	C₄²	D₄×D₁₇	1	2	yes	—
145	C₂³	#10587633	1	1	yes	characteristic
146	Q₈	—	136	4	no	—
147	Q₈	—	136	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	68	4	no	—
150	C₂×C₄	—	136	4	no	—
151	C₂×C₄	#10587579	1	2	yes	—
152	C₂×C₄	#10587581	1	2	yes	—
153	C₂×C₄	C₂×#63227	1	1	yes	characteristic
154	C₂×C₄	#10587626	1	1	yes	characteristic
155	C₂×C₄	#10587637	1	2	yes	—
156	C₂×C₄	#10587637	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	8	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	68	4	no	—
161	C₂²	—	2	2	no	—
162	C₂²	#11762710	1	1	yes	characteristic
163	C₂²	#11763179	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	136	4	no	—
166	C₄	#11762817	1	2	yes	—
167	C₄	#11763073	1	4	yes	—
168	C₄	#11763204	1	1	yes	characteristic
169	C₄	#11763204	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#14529603	1	1	yes	characteristic
172	C₂	#14529685	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763207

C₂

yes

maximal, characteristic

#14527470

C₂

yes

maximal

#14527466

C₂

yes

maximal

#14527076

C₂

yes

maximal, characteristic

C₁₇×#1837

C₂

yes

maximal, characteristic

C₂×#10587555

C₂²

yes

—

#11763207

—

#11763207

C₄

yes

—

#11762862

C₂²

yes

—

—

—

—

—

C₁₇×#423

C₂²

yes

characteristic

C₁₇×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#341

—

C₂²×Dic₃₄

—

C₂×M₁₆×C₁₇

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

—

—

—

C₂×C₄

yes

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₂²⋊#3622

—

#10587488

—

C₄⋊#3622

D₄

yes

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

#137

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

C₂×#498

—

#1837

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#491

yes

—

C₄×C₁₇

—

C₄×C₁₇

#419

yes

—

C₄×C₁₇

#471

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂×C₄⋊Q₈

—

—

—

—

—

C₂×D₁₇

yes

characteristic

#423

C₂×D₁₇

yes

characteristic

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

100

#322

—

101

C₂×C₁₇

—

102

C₂×C₁₇

#1960

yes

characteristic

103

C₂×C₁₇

#1977

yes

—

104

C₂²×Q₈

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂²×C₈

C₂²⋊D₁₇

yes

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄²

C₂²×D₁₇

yes

characteristic

115

#106

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₂²⋊C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₄×D₁₇

yes

—

122

C₄×C₈

D₆₈

yes

—

123

C₁₇

#9776

yes

characteristic, complemented, Sylow

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

136

—

127

C₂²×C₄

—

128

C₂²×C₄

#63256

yes

—

129

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

#63227

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

C₂×C₄×D₁₇

yes

characteristic

143

C₄²

C₂×D₆₈

yes

characteristic

144

C₄²

D₄×D₁₇

yes

—

145

C₂³

#10587633

yes

characteristic

146

Q₈

—

136

—

147

Q₈

—

136

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

136

—

151

C₂×C₄

#10587579

yes

—

152

C₂×C₄

#10587581

yes

—

153

C₂×C₄

C₂×#63227

yes

characteristic

154

C₂×C₄

#10587626

yes

characteristic

155

C₂×C₄

#10587637

yes

—

156

C₂×C₄

#10587637

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₂²

—

162

C₂²

#11762710

yes

characteristic

163

C₂²

#11763179

yes

—

164

C₄

—

165

C₄

—

136

—

166

C₄

#11762817

yes

—

167

C₄

#11763073

yes

—

168

C₄

#11763204

yes

characteristic

169

C₄

#11763204

yes

characteristic

170

C₂

—

171

C₂

#14529603

yes

characteristic

172

C₂

#14529685

yes

—

	subgroup	G/H
center	C₂²	#11762710
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1837	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1837	D₁₇
hypocenter	C₁₇	#9776
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#11762710

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1837

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
123. C₁₇	#9776	semidirect

H ≅ G/N

type

123. C₁₇

#9776

semidirect

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

subgroup	index
C₂×#11763207	2
#14527470	2
#14527466	2
#14527076	2
C₁₇×#1837	2
#9776	17

subgroup

index

C₁₇×#1837

Subgroups — #16933808

Conjugacy classes of subgroups

Complements

Maximal subgroups