Finite Group Databasebeta

← #16927857

Subgroups — #16927857

Order 4352.

546 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14527111C212yesmaximal
2#14526690C214yesmaximal, complemented
3C17×#2315C211yesmaximal, characteristic
4#11762773C2214yes—
5#11762533—22no—
6#11762533—22no—
7C22×C22⋊C4×C17C411yescharacteristic
8C22×C22⋊C4×C17C411yescharacteristic
9C22×C22⋊C4×C17C2211yescharacteristic
10C17×#385—22no—
11C17×#385—22no—
12C17×#380C412yes—
13C17×#380C2212yes—
14C25×C17C2×C411yescharacteristic
15C23×C4×C17—21no—
16C23×C4×C17—21no—
17C23×C4×C17C2×C411yescharacteristic
18C23×C4×C17C2311yescharacteristic
19C2×C4⋊C4×C17—22no—
20C2×C4⋊C4×C17—22no—
21C2×C22⋊C4×C17—24no—
22C2×C22⋊C4×C17—21no—
23C2×C22⋊C4×C17—21no—
24C2×C22⋊C4×C17—42no—
25C2×C22⋊C4×C17—42no—
26C2×C22⋊C4×C17C2×C412yes—
27C2×C22⋊C4×C17C2×C412yes—
28C2×C22⋊C4×C17D412yes—
29C2×C22⋊C4×C17D412yes—
30C2×#63223—24no—
31C17×#95—22no—
32C17×#95—22no—
33C22⋊#3622—24no—
34C22⋊#3622—24no—
35C24×C17—22no—
36C24×C17—22no—
37C24×C17—22no—
38C24×C17—81no—
39C24×C17—81no—
40C24×C17C22⋊C411yescharacteristic
41C24×C17C22⋊C411yescharacteristic
42C24×C17C22×C411yescharacteristic
43C22×C4×C17—22no—
44C22×C4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17—42no—
47C22×C4×C17—42no—
48C22×C4×C17—81no—
49C22×C4×C17—81no—
50C22×C4×C17—81no—
51C22×C4×C17—81no—
52C22×C4×C17—81no—
53C22×C4×C17—81no—
54C22×C4×C17C22⋊C412yes—
55C22×C4×C17C2×D412yes—
56C22×C4×C17#4114yes—
57C4⋊C4×C17—82no—
58C4⋊C4×C17—82no—
59C22⋊C4×C17—24no—
60C22⋊C4×C17—24no—
61C22⋊C4×C17—42no—
62C22⋊C4×C17—42no—
63C22⋊C4×C17—42no—
64C22⋊C4×C17—42no—
65#63223—84no—
66C2×#3622—44no—
67#8756—171nomaximal, Sylow
68C23×C17—28no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17—21no—
73C23×C17—21no—
74C23×C17—21no—
75C23×C17—41no—
76C23×C17—41no—
77C23×C17—41no—
78C23×C17—41no—
79C23×C17—41no—
80C23×C17—41no—
81C23×C17—82no—
82C23×C17—82no—
83C23×C17—84no—
84C23×C17—84no—
85C23×C17#10012yes—
86C23×C17#10112yes—
87C23×C17C2×C22⋊C411yescharacteristic
88C23×C17#11712yes—
89C2×C4×C17—22no—
90C2×C4×C17—42no—
91C2×C4×C17—44no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—82no—
100C2×C4×C17—84no—
101C2×C4×C17—84no—
102C2×C4×C17#12314yes—
103#3622—84no—
104#2315D1711yescharacteristic
105#1462—172no—
106#1305—174no—
107C22×C17—21no—
108C22×C17—21no—
109C22×C17—22no—
110C22×C17—22no—
111C22×C17—22no—
112C22×C17—22no—
113C22×C17—22no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17—41no—
121C22×C17—41no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—88no—
125C22×C17#38111yescharacteristic
126C22×C17C2×#10011yescharacteristic
127C22×C17C2×#10111yescharacteristic
128C4×C17—24no—
129C4×C17—42no—
130C4×C17—82no—
131C4×C17—82no—
132C22×C22⋊C4Dic1711yescharacteristic
133C22×C22⋊C4Dic1711yescharacteristic
134C22×C22⋊C4C2×D1711yescharacteristic
135#407—174no—
136#385—22no—
137#385—22no—
138#380Dic1712yes—
139#380C2×D1712yes—
140#323—342no—
141#323—342no—
142C2×C17—21no—
143C2×C17—21no—
144C2×C17—42no—
145C2×C17—42no—
146C2×C17—42no—
147C2×C17#182111yescharacteristic
148C2×C17#182811yescharacteristic
149C2×C17#182911yescharacteristic
150C25C2×Dic1711yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×Dic1711yescharacteristic
154C23×C4C22×D1711yescharacteristic
155C2×M16—344no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4C2×Dic1712yes—
164C2×C22⋊C4C2×Dic1712yes—
165C2×C22⋊C4C22⋊D1712yes—
166C2×C22⋊C4C22⋊D1712yes—
167C22⋊C8—344no—
168C22⋊C8—344no—
169#95—22no—
170#95—22no—
171C17#875611yescharacteristic, complemented, Sylow
172C24—22no—
173C24—22no—
174C24—22no—
175C24—81no—
176C24—81no—
177C24C22⋊Dic1711yescharacteristic
178C24C22⋊Dic1711yescharacteristic
179C24C22×Dic1711yescharacteristic
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4C22⋊Dic1712yes—
192C22×C4#6325214yes—
193C22×C4C2×C22⋊D1712yes—
194M16—1364no—
195C2×C8—684no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C23—28no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—82no—
218C23—82no—
219C23—84no—
220C23—84no—
221C23#1058750412yes—
222C23#1058750712yes—
223C23#1058761212yes—
224C23C2×C22⋊Dic1711yescharacteristic
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4#1058761514yes—
239C8—1364no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—88no—
258C22#1176267211yescharacteristic
259C22C2×#1058750411yescharacteristic
260C22C2×#1058750711yescharacteristic
261C4—24no—
262C4—42no—
263C4—82no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—42no—
269C2—42no—
270C2#1452886411yescharacteristic
271C2#1452889211yescharacteristic
272C2#1452889511yescharacteristic
subgroupG/H
centerC22#11762672
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2315C2
socleC22×C17#381
prime radicalC25×C17C2×C4
cosocleC23×C4×C17C23
hypercenter#2315D17
hypocenterC17#8756
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14526690C2semidirect
171. C17#8756semidirect

Maximal subgroups

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

subgroupindex
#145271112
#145266902
C17×#23152
#875617