Finite Group Databasebeta

← #16934903

Subgroups — #16934903

Order 4352.

772 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#11763123C212yesmaximal, complemented
2#14534886C211yesmaximal, characteristic
3C2×#11762544C212yesmaximal, complemented
4#14527051C211yesmaximal, characteristic, complemented
5C17×#1280C211yesmaximal, characteristic, complemented
6C2×#10587616—22no—
7C2×#10587616C2212yes—
8C22×C4⋊Dic17C2211yescharacteristic
9#11763123—24no—
10#11763123—24no—
11C2×#10587489—22no—
12C2×#10587492—22no—
13#11762746—22no—
14#11762719—22no—
15C2×#10587502—22no—
16C2×#10587502C2211yescharacteristic
17#11762544—24no—
18C2×C17×C22⋊C8C2212yes—
19C2×C17×#95C2211yescharacteristic
20C23×C4×C17D412yes—
21C23×C4×C17C2311yescharacteristic
22C22×C22⋊D17—22no—
23C23×Dic17—21no—
24C23×Dic17—21no—
25C22×D68—22no—
26C22×D68D412yes—
27C22×C8×C17—22no—
28#10587616—22no—
29#10587616—22no—
30#10587616—22no—
31#10587616—22no—
32#10587616—44no—
33#10587616—44no—
34C2×#63227—24no—
35C2×#63227—22no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17—22no—
40C2×C4⋊Dic17D412yes—
41C2×C4⋊Dic17D412yescomplemented
42C17×C22⋊C8—22no—
43C17×C22⋊C8D414yescomplemented
44C17×#95—22no—
45#10587502—22no—
46#10587502—44no—
47#10587492—28no—
48#10587489—28no—
49C24×C17C2×D411yescharacteristic
50C23×D17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17—42no—
55C22×C4×C17—42no—
56C22×C4×C17D814yes—
57C22×C4×C17C2×D412yes—
58C22×C4×C17C2×D412yes—
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17#4111yescharacteristic
61C2×C22⋊D17—42no—
62C2×C22⋊D17—42no—
63C2×C22⋊D17—44no—
64C2×C22⋊D17—44no—
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—41no—
69C22×Dic17—42no—
70C22×Dic17—41no—
71C22×Dic17—41no—
72C22×Dic17—41no—
73C22×Dic17—42no—
74C22×Dic17—42no—
75C22×Dic17—42no—
76C2×D68—22no—
77C2×D68—22no—
78C2×D68—44no—
79C2×D68—44no—
80C2×D68—82no—
81C2×D68—82no—
82C2×C8×C17—24no—
83C2×C8×C17—44no—
84#63227—48no—
85#63227—42no—
86#63227—42no—
87C4⋊Dic17—22no—
88C4⋊Dic17—24no—
89C4⋊Dic17—22no—
90C4⋊Dic17—44no—
91C4⋊Dic17—42no—
92#9970—171nomaximal, Sylow
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—22no—
96C23×C17#12011yescharacteristic
97C23×C17#12111yescharacteristic
98C23×C17#12711yescharacteristic
99C23×C17C2×D812yes—
100C23×C17AGL(1,ℤ/8)12yes—
101C22×D17—42no—
102C22×D17—44no—
103C22×D17—42no—
104C22×D17—42no—
105C22×D17—44no—
106C2×C4×C17—24no—
107C2×C4×C17—24no—
108C2×C4×C17—42no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—44no—
114C2×C4×C17—82no—
115C2×C4×C17#12012yes—
116C2×C4×C17#12112yes—
117C22⋊D17—84no—
118C22⋊D17—82no—
119C22⋊D17—82no—
120C2×Dic17—44no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—44no—
125C2×Dic17—44no—
126C2×Dic17—82no—
127C2×Dic17—81no—
128C2×Dic17—82no—
129C2×Dic17—81no—
130C2×Dic17—81no—
131C2×Dic17—81no—
132D68—44no—
133D68—82no—
134D68—88no—
135C8×C17—44no—
136C2×#480—172no—
137#2656—171no—
138C2×#327—172no—
139#1445—171no—
140#1280D1711yescharacteristic, complemented
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17#39211yescharacteristic
150C22×C17#44712yes—
151C22×C17#45511yescharacteristic
152C22×C17#45611yescharacteristic
153C22×C17#45812yes—
154C22×C17#45912yes—
155C22×C17#49311yescharacteristic
156C22×C17#49611yescharacteristic
157C2×D17—84no—
158C2×D17—84no—
159C2×D17—82no—
160C2×D17—82no—
161C2×D17—82no—
162C4×C17—22no—
163C4×C17—22no—
164C4×C17—44no—
165Dic17—82no—
166Dic17—82no—
167Dic17—82no—
168C2×#121—172no—
169C2×#121—342no—
170C22×C4⋊C4—171no—
171#480—344no—
172#480—344no—
173C2×#107—342no—
174C2×#102—342no—
175C2×C22⋊C8C2×D1712yes—
176#399—342no—
177#392—342no—
178C2×#95—171no—
179C2×#95—342no—
180C2×#95C2×D1711yescharacteristic
181#327—344no—
182C2×C17—21no—
183C2×C17—22no—
184C2×C17—21no—
185C2×C17#160712yes—
186C2×C17#161012yes—
187C2×C17#200711yescharacteristic
188C2×C17#201111yescharacteristic
189C2×C17#202211yescharacteristic
190D17—162no—
191C22×D4—172no—
192C22×D4—342no—
193C22×D4—342no—
194C23×C4—341no—
195C23×C4—341no—
196C23×C4D6812yes—
197C23×C4C22×D1711yescharacteristic
198C22×C8—22no—
199#121—342no—
200#121—342no—
201#121—342no—
202#121—342no—
203#121—684no—
204#121—684no—
205C2×C4⋊C4—172no—
206C2×C4⋊C4—172no—
207C2×C4⋊C4—342no—
208C2×C4⋊C4—342no—
209C2×C4⋊C4—342no—
210C2×C4⋊C4—341no—
211C2×C22⋊C4—344no—
212C2×C22⋊C4—342no—
213#107—348no—
214#102—348no—
215C22⋊C8—22no—
216C22⋊C8D6814yescomplemented
217#95—22no—
218#95—342no—
219#95—684no—
220C17#997011yescharacteristic, complemented, Sylow
221C24—342no—
222C24C2×D6811yescharacteristic
223C2×D4—342no—
224C2×D4—342no—
225C2×D4—684no—
226C2×D4—684no—
227C2×D4—682no—
228C2×D4—682no—
229C2×D4—684no—
230C2×D4—684no—
231C2×D4—1362no—
232C2×D4—1362no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—342no—
239C22×C4—342no—
240C22×C4—342no—
241C22×C4—681no—
242C22×C4—682no—
243C22×C4—681no—
244C22×C4—681no—
245C22×C4—681no—
246C22×C4—682no—
247C22×C4—682no—
248C22×C4—682no—
249C22×C4D13614yes—
250C22×C4C2×D6812yes—
251C22×C4D4×D1712yes—
252C22×C4D4×D1711yescharacteristic
253C22×C4#6325611yescharacteristic
254C2×C8—24no—
255C2×C8—44no—
256C4⋊C4—344no—
257C4⋊C4—342no—
258C4⋊C4—342no—
259C4⋊C4—684no—
260C4⋊C4—682no—
261C22⋊C4—688no—
262C22⋊C4—682no—
263C22⋊C4—682no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—684no—
268C23—682no—
269C23—682no—
270C23—682no—
271C23—684no—
272C23#1058756011yescharacteristic
273C23#1058756811yescharacteristic
274C23#1058758111yescharacteristic
275C23C2×D13612yes—
276C23#1058759412yes—
277D4—684no—
278D4—1362no—
279D4—1368no—
280D4—1364no—
281D4—1362no—
282D4—1362no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—682no—
293C2×C4—684no—
294C2×C4—684no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—684no—
298C2×C4—1362no—
299C2×C4—1361no—
300C2×C4—1362no—
301C2×C4—1361no—
302C2×C4—1361no—
303C2×C4—1361no—
304C2×C4#1058756812yes—
305C2×C4#1058758112yes—
306C8—44no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—1364no—
316C22—1362no—
317C22—1364no—
318C22—1362no—
319C22—1362no—
320C22#1176271911yescharacteristic
321C22#1176294512yes—
322C22#1176298911yescharacteristic
323C22#1176299411yescharacteristic
324C22#1176300012yes—
325C22#1176300512yes—
326C22#1176318611yescharacteristic
327C22#1176319811yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—1362no—
332C4—1362no—
333C4—1362no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—2722no—
338C2#1452777212yes—
339C2#1452778912yes—
340C2#1452983311yescharacteristic
341C2#1452985711yescharacteristic
342C2#1452991611yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1280C2
socleC23×C17#120
prime radical#14534886C2
hypercenter#1280D17
hypocenterC17#9970
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
1. C2×#11763123C2semidirect
3. C2×#11762544C2semidirect
4. #14527051C2semidirect
5. C17×#1280C2semidirect
41. C2×C4⋊Dic17D4semidirect
43. C17×C22⋊C8D4semidirect
140. #1280D17semidirect
216. C22⋊C8D68semidirect
220. C17#9970semidirect

Maximal subgroups

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

subgroupindex
C2×#117631232
#145348862
C2×#117625442
#145270512
C17×#12802
#997017