Finite Group Databasebeta

← #16943949

Subgroups — #16943949

Order 4352.

444 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 95 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#14527122C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526718C214yesmaximal, complemented
4C17×#2547C211yesmaximal, characteristic
5#11762773—21no—
6C2×C22⋊#3622—21no—
7#11762535C2212yes—
8#11762535C2212yes—
9#11762533—21no—
10#11762533—21no—
11C2×C17×C4⋊Q8C411yescharacteristic
12C2×C17×#124C412yes—
13C2×C4×D4×C17C2211yescharacteristic
14C17×#397—21no—
15C17×#397—21no—
16C17×#391C2212yes—
17C17×#380—21no—
18C17×#380—21no—
19C4×C4⋊C4×C17C411yescharacteristic
20C22×Q8×C17C2×C412yes—
21C22×D4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C17×C4⋊Q8—41no—
25C17×C4⋊Q8—41no—
26C17×#124—44no—
27C4×D4×C17—41no—
28C4×D4×C17—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2311yescharacteristic
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C42×C17C2×C411yescharacteristic
38C2×C42×C17C2×C412yes—
39C2×#63223—41no—
40C22×#3622—41no—
41C17×#95—21no—
42C17×#95—22no—
43C17×#95—22no—
44C22⋊#3622—22no—
45C22⋊#3622—22no—
46C22⋊#3622—41no—
47C22⋊#3622—41no—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C22⋊C411yescharacteristic
50C2×Q8×C17—24no—
51C2×Q8×C17—42no—
52C2×Q8×C17—42no—
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—41no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C4⋊C4×C17—41no—
70C4⋊C4×C17—42no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—44no—
77C22⋊C4×C17—44no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—42no—
81#63223—82no—
82C2×#3622—42no—
83C2×#3622—42no—
84C2×#3622—82no—
85#11564—171nomaximal, Sylow
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17#10012yes—
96C23×C17C2×C22⋊C411yescharacteristic
97Q8×C17—84no—
98D4×C17—44no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—81no—
117C2×C4×C17—81no—
118C2×C4×C17#9911yescharacteristic
119C2×C4×C17#9911yescharacteristic
120C2×C4×C17#13112yes—
121#3622—82no—
122#3622—82no—
123#2547D1711yescharacteristic
124#1465—171no—
125#1459—171no—
126#1319—174no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—22no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17—81no—
138C22×C17—81no—
139C22×C17#40811yescharacteristic
140C22×C17C2×#9911yescharacteristic
141C22×C17C2×#10011yescharacteristic
142C4×C17—21no—
143C4×C17—22no—
144C4×C17—41no—
145C4×C17—41no—
146C4×C17—41no—
147C4×C17—82no—
148C4×C17—82no—
149C4×C17#35412yes—
150C2×C4⋊Q8Dic1711yescharacteristic
151C2×#124Dic1712yes—
152C2×C4×D4C2×D1711yescharacteristic
153#407—341no—
154C2×C22⋊C8—341no—
155#397—21no—
156#397—21no—
157#391C2×D1712yes—
158#380—21no—
159#380—21no—
160C4×C4⋊C4Dic1711yescharacteristic
161#324—172no—
162#324—172no—
163#323—341no—
164#323—341no—
165C2×C17—21no—
166C2×C17—21no—
167C2×C17—42no—
168C2×C17—81no—
169C2×C17#147111yescharacteristic
170C2×C17C2×#35411yescharacteristic
171C2×C17#213511yescharacteristic
172C22×Q8C2×Dic1712yes—
173C22×D4C2×Dic1711yescharacteristic
174C23×C4C22⋊D1711yescharacteristic
175C23×C4C22⋊D1711yescharacteristic
176C2×M16—681no—
177C22×C8—681no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180#124—44no—
181C4×D4—41no—
182C4×D4—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22×D1711yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C22⋊D1711yescharacteristic
190C2×C22⋊C4C22⋊D1711yescharacteristic
191C2×C42C2×Dic1711yescharacteristic
192C2×C42C2×Dic1712yes—
193C22⋊C8—342no—
194C22⋊C8—342no—
195C22⋊C8—681no—
196C22⋊C8—681no—
197#95—21no—
198#95—22no—
199#95—22no—
200C17#1156411yescharacteristic, complemented, Sylow
201C24C22⋊Dic1711yescharacteristic
202C24C22⋊Dic1711yescharacteristic
203C2×Q8—24no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C22⋊Dic1711yescharacteristic
218C22×C4C22⋊Dic1711yescharacteristic
219C22×C4C22×Dic1711yescharacteristic
220C22×C4C2×C22⋊D1711yescharacteristic
221C22×C4C2×C22⋊D1711yescharacteristic
222M16—1362no—
223C2×C8—682no—
224C2×C8—682no—
225C2×C8—1362no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42—21no—
236C42—21no—
237C42—42no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1058750412yes—
248C23C2×C22⋊Dic1711yescharacteristic
249Q8—84no—
250D4—44no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4#1058750511yescharacteristic
271C2×C4#1058750511yescharacteristic
272C2×C4#1058763412yes—
273C8—1362no—
274C8—1362no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22#1176277611yescharacteristic
288C22C2×#1058750511yescharacteristic
289C22C2×#1058750411yescharacteristic
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4#1176260412yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#1452714411yescharacteristic
303C2C2×#1176260411yescharacteristic
304C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2547C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2547D17
hypocenterC17#11564
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #14526718C2semidirect
200. C17#11564semidirect

Maximal subgroups

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

subgroupindex
#145271222
#145271022
#145267182
C17×#25472
#1156417