Finite Group Databasebeta

← #16944018

Subgroups — #16944018

Order 4352.

552 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 87 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#14532149C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526733C214yesmaximal, complemented
4C17×#1460C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic
6C2×#10587610—21no—
7#11762773—21no—
8#11762729—21no—
9#11762721—21no—
10#11762715—21no—
11#11762689C412yes—
12#11762681C2212yes—
13#11762669C411yescharacteristic
14#11762535—21no—
15#11762533C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#431—21no—
18C17×#323—21no—
19C17×#323C2212yes—
20C22×D4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17C2×C412yes—
24C22×Dic34—21no—
25C22×Dic34—21no—
26C22×Dic34—44no—
27C22×Dic34—41no—
28C22×Dic34—41no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17C2311yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17C2×C412yes—
37C2×C4⋊Dic17—21no—
38#10587610—44no—
39C2×#63225—22no—
40C2×#63223—41no—
41C17×C4⋊C8—41no—
42C17×C22⋊C8—22no—
43C4×C8×C17—41no—
44#10587502—22no—
45#10587502—21no—
46#10587502—22no—
47C22⋊#3622—22no—
48C22⋊#3622—41no—
49C24×C17C22⋊C411yescharacteristic
50C24×C17C22×C411yescharacteristic
51C2×D4×C17—22no—
52C2×D4×C17—22no—
53C2×D4×C17—41no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—44no—
68C22×Dic17—42no—
69C2×Dic34—24no—
70C2×Dic34—44no—
71C2×Dic34—42no—
72C2×Dic34—42no—
73C2×Dic34—48no—
74C2×Dic34—44no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34—44no—
78C2×C8×C17—42no—
79C4⋊C4×C17—41no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C42×C17—21no—
83C42×C17—21no—
84C22⋊Dic17—24no—
85C22⋊Dic17—44no—
86C4⋊Dic17—42no—
87#63225—44no—
88#63223—82no—
89C2×#3622—42no—
90#11580—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17C2×C22⋊C411yescharacteristic
101C23×C17#13112yes—
102D4×C17—44no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—22no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—81no—
116C2×C4×C17#9911yescharacteristic
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17#10112yes—
119C2×Dic17—24no—
120C2×Dic17—42no—
121C2×Dic17—44no—
122C2×Dic17—42no—
123C2×Dic17—44no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127Dic34—44no—
128Dic34—48no—
129Dic34—44no—
130C8×C17—82no—
131#3622—82no—
132#2409—171no—
133#1465—171no—
134#1460D1711yescharacteristic
135#1326—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—81no—
148C22×C17#40811yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17C2×#10111yescharacteristic
151C4×C17—21no—
152C4×C17—22no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17AΣL(1,𝔽16)12yes—
157Dic17—44no—
158Dic17—82no—
159C23×Q8—171no—
160C2×C22⋊Q8—341no—
161C2×C4×D4C2×D1711yescharacteristic
162#431—21no—
163#407—341no—
164#395—341no—
165#393—341no—
166#391—341no—
167#386—172no—
168#385—172no—
169#380—171no—
170#324—341no—
171#323—21no—
172#323—172no—
173#323C2×D1712yes—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—42no—
177C2×C17—81no—
178C2×C17#147211yescharacteristic
179C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
180C2×C17#212311yescharacteristic
181C22×Q8—341no—
182C22×Q8—341no—
183C22×Q8—681no—
184C22×Q8—681no—
185C22×Q8—684no—
186C22×D4C4×D1711yescharacteristic
187C23×C4—172no—
188C23×C4C4×D1711yescharacteristic
189C23×C4D6811yescharacteristic
190C2×M16—681no—
191C22⋊Q8—684no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4D6811yescharacteristic
197C2×C22⋊C4—172no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4C22⋊D1711yescharacteristic
200C2×C22⋊C4C22×D1711yescharacteristic
201C2×C42C22⋊D1711yescharacteristic
202C4⋊C8—41no—
203C22⋊C8—22no—
204C22⋊C8—342no—
205C22⋊C8—681no—
206C4×C8—41no—
207#95—342no—
208#95—341no—
209#95—342no—
210C17#1158011yescharacteristic, complemented, Sylow
211C24#6322711yescharacteristic
212C24C2×C4×D1711yescharacteristic
213C2×Q8—344no—
214C2×Q8—682no—
215C2×Q8—682no—
216C2×Q8—684no—
217C2×Q8—684no—
218C2×Q8—688no—
219C2×Q8—684no—
220C2×Q8—681no—
221C2×Q8—681no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4—684no—
234C22×C4—682no—
235C22×C4#6322711yescharacteristic
236C22×C4#6322711yescharacteristic
237C22×C4#6322711yescharacteristic
238C22×C4C2×D6811yescharacteristic
239C22×C4C2×C22⋊D1711yescharacteristic
240M16—1362no—
241C2×C8—42no—
242C2×C8—682no—
243C4⋊C4—41no—
244C4⋊C4—682no—
245C4⋊C4—684no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—344no—
249C22⋊C4—684no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1058758712yes—
262C23C2×#6322711yescharacteristic
263Q8—684no—
264Q8—688no—
265Q8—684no—
266D4—44no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—344no—
281C2×C4—682no—
282C2×C4—684no—
283C2×C4—682no—
284C2×C4—684no—
285C2×C4—682no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4#1058747711yescharacteristic
289C2×C4#1058747711yescharacteristic
290C2×C4#1058749512yes—
291C8—82no—
292C8—1362no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1176277511yescharacteristic
306C22C2×#1058747711yescharacteristic
307C22C2×#1058749511yescharacteristic
308C4—21no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—684no—
314C4—1362no—
315C4#1176259712yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—81no—
320C2#1452714711yescharacteristic
321C2C2×#1176259711yescharacteristic
322C2#1453043711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1460C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊C4×C17C23
hypercenter#1460D17
hypocenterC17#11580
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #14526733C2semidirect
210. C17#11580semidirect

Maximal subgroups

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

subgroupindex
#145321492
#145271222
#145267332
C17×#14602
#1158017