Finite Group Databasebeta

← #17980222

Subgroups — #17980222

Order 4352.

1702 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545697C214yesmaximal, complemented
2#14545695C214yesmaximal, complemented
3#14545688C218yesmaximal, complemented
4#14545586C214yesmaximal, complemented
5Q8×#63243C214yesmaximal, complemented
6#14545465C214yesmaximal
7#14545461C212yesmaximal
8D17×#533C211yesmaximal, characteristic
9#11763861C22116yescomplemented
10#11763860C2218yescomplemented
11#11763859C2218yescomplemented
12#11763859C22116yescomplemented
13#11763858C2218yescomplemented
14Q8×#3624C2218yescomplemented
15#11763834C22116yes—
16#11763833C22116yes—
17#11763831C2214yes—
18#11763830C2214yes—
19#11763828C2214yes—
20#11763825C2214yes—
21#11763814C2214yes—
22#11763813C2212yes—
23#11763813C2212yes—
24#11763813C2214yes—
25#11763780C2214yes—
26C4×#63243C2214yes—
27C4×#63243C2218yes—
28#11763366C411yescharacteristic
29#11763365C2212yes—
30#11763364C2212yes—
31#11763363C414yes—
32#11763362C412yes—
33#11763361C2214yes—
34C4⋊Q8×D17C412yes—
35#125×D17C412yes—
36C22⋊Q8×D17C414yes—
37C4×Q8×D17C2214yes—
38#117×D17C2212yes—
39C2×C42×D17C2211yescharacteristic
40C17×#533C411yescharacteristic
41C2×Q8×D17C2×C414yes—
42C22×C4×D17C2×C412yes—
43C22×C4×D17C2311yescharacteristic
44C2×#63243C2314yes—
45C2×#63243C2314yes—
46C17×C4⋊Q8C2×C412yes—
47C17×#125C2×C412yes—
48C17×C22⋊Q8C2×C414yes—
49C4×Q8×C17C2×C414yes—
50C17×#117C2×C412yes—
51C2×C42×C17C2×C411yescharacteristic
52#10587632C2×C414yes—
53Q8×Dic17C2314yes—
54#10587630C2×C414yes—
55#10587612C2×C412yes—
56#10587610C2×C414yes—
57C2×C4×Dic17C2×C411yescharacteristic
58C2×C4×Dic17C2312yes—
59#10587583C2×C414yes—
60#10587582C2×C418yes—
61#10587578C2314yes—
62C4⋊C4×D17C2×C414yes—
63C4⋊C4×D17C2×C418yes—
64C4⋊C4×D17C2×C414yes—
65#10587576C2×C414yes—
66#10587575C2×C418yes—
67#10587574C2×C414yes—
68#10587574C2×C414yes—
69#10587573C2314yes—
70#10587573C2318yes—
71C22⋊C4×D17C2×C414yes—
72#10587564C2×C418yes—
73#10587563C2314yes—
74#10587558C2×C414yes—
75C42×D17C2×C411yescharacteristic
76C42×D17C2×C411yescharacteristic
77C42×D17C2×C412yes—
78C42×D17C2312yes—
79C42×D17C2311yescharacteristic
80C42×D17C2311yescharacteristic
81C42×D17C2314yes—
82#10587556C2×C412yes—
83#10587555C2×C412yes—
84C4×Dic34C2×C418yes—
85C4×Dic34C2×C414yes—
86C22⋊#3624C2318yes—
87#10587544C23116yes—
88#10587543C2318yes—
89#10587542C2318yes—
90#10587542C23116yes—
91C4⋊#3624C2314yes—
92C4⋊#3624C2314yes—
93C4⋊#3624C2314yes—
94C4⋊#3624C2314yes—
95C4⋊#3624C23116yes—
96C4×#3624C2318yes—
97C4×#3624C23116yes—
98C23×D17C22×C411yescharacteristic
99C2×Q8×C17C22×C414yes—
100C22×C4×C17C22×C412yes—
101C22×C4×C17C22×C411yescharacteristic
102C22×Dic17C22×C412yes—
103C22×Dic17C2411yescharacteristic
104Q8×D17—48no—
105C2×C4×D17—22no—
106C2×C4×D17—22no—
107C2×C4×D17—44no—
108C2×C4×D17C22×C412yes—
109C2×C4×D17C22×C412yes—
110C2×C4×D17C22×C411yescharacteristic
111C2×C4×D17C22×C411yescharacteristic
112C2×C4×D17C22×C412yes—
113C2×C4×D17C22×C412yes—
114C2×C4×D17C22×C418yes—
115C2×C4×D17C2411yescharacteristic
116C2×C4×D17C2411yescharacteristic
117C2×Dic34C22×C414yes—
118C2×Dic34C22×C418yes—
119C2×#3624C2418yes—
120C2×#3624C2418yes—
121#63243—28no—
122#63243—28no—
123C4⋊C4×C17C22×C418yes—
124C4⋊C4×C17C22×C414yes—
125C4⋊C4×C17C22×C414yes—
126C22⋊C4×C17C22×C414yes—
127C42×C17C22×C411yescharacteristic
128C42×C17C22×C411yescharacteristic
129C42×C17C22×C412yes—
130C42×C17C22×C414yes—
131C22⋊Dic17C22×C414yes—
132#63227C22×C418yes—
133C4⋊Dic17C22×C414yes—
134C4⋊Dic17C22×C418yes—
135C4⋊Dic17C22×C414yes—
136#63225C22×C418yes—
137#63225C22×C418yes—
138#63225C22×C418yes—
139#63225C22×C418yes—
140C4×Dic17C22×C412yes—
141C4×Dic17C22×C414yes—
142C4×Dic17C22×C412yes—
143C4×Dic17C22×C412yes—
144C4×Dic17C22×C411yescharacteristic
145C4×Dic17C22×C411yescharacteristic
146C4×Dic17C2412yes—
147C4×Dic17C2418yes—
148C4×Dic17C2412yes—
149#60249—171nomaximal, Sylow
150C23×C17C23×C411yescharacteristic
151C22×D17—22no—
152C22×D17—42no—
153C22×D17C23×C411yescharacteristic
154C22×D17C23×C411yescharacteristic
155Q8×C17—28no—
156C2×C4×C17—22no—
157C2×C4×C17—24no—
158C2×C4×C17C23×C412yes—
159C2×C4×C17C23×C411yescharacteristic
160C2×C4×C17C23×C418yes—
161C2×C4×C17C23×C411yescharacteristic
162C2×C4×C17C23×C412yes—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—42no—
166C2×Dic17C23×C418yes—
167C2×Dic17C23×C412yes—
168C2×Dic17C23×C412yes—
169C2×Dic17C23×C411yescharacteristic
170C2×Dic17C2511yescharacteristic
171C4×D17—22no—
172C4×D17—22no—
173C4×D17—24no—
174C4×D17—24no—
175C4×D17—21no—
176C4×D17—21no—
177C4×D17—42no—
178C4×D17—42no—
179C4×D17—48no—
180C4×D172−1+412yes—
181C4×D172−1+412yes—
182Dic34—28no—
183Dic34—48no—
184#3624—28no—
185#3624—28no—
186C2×#533—171no—
187#2987—174no—
188#2986—174no—
189#2985—178no—
190Q8×M16—174no—
191Q8×M16—174no—
192#2921—174no—
193#2920—172no—
194C22×C17—21no—
195C22×C17—22no—
196C22×C17C24×C411yescharacteristic
197C2×D17—22no—
198C2×D17—21no—
199C2×D17—22no—
200C2×D17—44no—
201C2×D17#51912yes—
202C4×C17—21no—
203C4×C17—28no—
204C4×C17—22no—
205C4×C17#51912yes—
206C4×C17#56814yes—
207Dic17—21no—
208Dic17—28no—
209Dic17—22no—
210Dic17—22no—
211Dic17C2×2−1+411yescharacteristic
212Dic17C2×2−1+411yescharacteristic
213#533—172no—
214#533—172no—
215#533—174no—
216#533—174no—
217#533—172no—
218#533—171no—
219#533#59511yescharacteristic
220C2×C4⋊Q8—172no—
221C2×#125—172no—
222C2×C22⋊Q8—174no—
223C2×C4×Q8—174no—
224C2×#117—172no—
225C22×C42—171no—
226#446—178no—
227#446—1716no—
228#446—178no—
229#446—1716no—
230C8×Q8—178no—
231C8×Q8—178no—
232#433—1716no—
233#433—1716no—
234#432—174no—
235#432—174no—
236#431—174no—
237#431—174no—
238#423—172no—
239#423—172no—
240#423—174no—
241#423—174no—
242#407—174no—
243C4×M16—174no—
244C4×M16—178no—
245C2×C17—41no—
246C2×C17C2×#51911yescharacteristic
247C2×C17C2×#56812yes—
248D17—22no—
249D17—41no—
250C22×Q8—174no—
251C23×C4—171no—
252C23×C4—172no—
253C2×M16—174no—
254C2×M16—174no—
255C4⋊Q8—174no—
256C4⋊Q8—174no—
257C4⋊Q8—174no—
258C4⋊Q8—172no—
259C4⋊Q8C2×#59512yes—
260#125—178no—
261#125—174no—
262#125—172no—
263#125C2×#59512yes—
264C22⋊Q8—174no—
265C22⋊Q8—174no—
266C22⋊Q8—178no—
267C22⋊Q8—174no—
268C22⋊Q8—178no—
269C22⋊Q8C2×#59514yes—
270C4×Q8—174no—
271C4×Q8—174no—
272C4×Q8—178no—
273C4×Q8—178no—
274C4×Q8—174no—
275C4×Q8C2×#59514yes—
276#117—172no—
277#117—174no—
278#117—174no—
279#117—174no—
280#117C2×#59512yes—
281C2×C4⋊C4—174no—
282C2×C4⋊C4—178no—
283C2×C4⋊C4—174no—
284C2×C22⋊C4—174no—
285C2×C42—172no—
286C2×C42—171no—
287C2×C42—174no—
288C2×C42—172no—
289C2×C42—171no—
290C2×C42—171no—
291C2×C42—172no—
292C2×C42—171no—
293C2×C42—171no—
294C2×C42C2×#59511yescharacteristic
295C4⋊C8—174no—
296C4⋊C8—174no—
297C4⋊C8—174no—
298C4⋊C8—174no—
299C4⋊C8—1716no—
300C4⋊C8—1716no—
301C22⋊C8—178no—
302C22⋊C8—178no—
303#97—178no—
304#97—1716no—
305C4×C8—178no—
306C4×C8—1716no—
307C17#6024911yescharacteristic, complemented, Sylow
308C24—171no—
309C2×Q8—174no—
310C2×Q8—178no—
311C2×Q8—688no—
312C2×Q8C22×#59514yes—
313C22×C4—171no—
314C22×C4—171no—
315C22×C4—172no—
316C22×C4—171no—
317C22×C4—172no—
318C22×C4—171no—
319C22×C4—172no—
320C22×C4—171no—
321C22×C4—172no—
322C22×C4—172no—
323C22×C4—178no—
324C22×C4—342no—
325C22×C4—342no—
326C22×C4—684no—
327C22×C4C22×#59511yescharacteristic
328C22×C4C22×#59512yes—
329M16—348no—
330M16—348no—
331C2×C8—178no—
332C2×C8—178no—
333C4⋊C4—178no—
334C4⋊C4—174no—
335C4⋊C4—178no—
336C4⋊C4—178no—
337C4⋊C4—178no—
338C4⋊C4—178no—
339C4⋊C4—174no—
340C4⋊C4C22×#59514yes—
341C4⋊C4C22×#59518yes—
342C4⋊C4C22×#59514yes—
343C22⋊C4—174no—
344C22⋊C4—178no—
345C22⋊C4C22×#59514yes—
346C42—172no—
347C42—178no—
348C42—172no—
349C42—172no—
350C42—172no—
351C42—171no—
352C42—172no—
353C42—171no—
354C42—174no—
355C42C22×#59512yes—
356C42C22×#59511yescharacteristic
357C42C22×#59511yescharacteristic
358C42C22×#59514yes—
359C23—171no—
360C23—171no—
361C23—342no—
362C23—682no—
363C23C23×#59511yescharacteristic
364Q8—28no—
365Q8—348no—
366Q8—688no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—171no—
370C2×C4—172no—
371C2×C4—172no—
372C2×C4—171no—
373C2×C4—172no—
374C2×C4—172no—
375C2×C4—178no—
376C2×C4—341no—
377C2×C4—341no—
378C2×C4—342no—
379C2×C4—341no—
380C2×C4—341no—
381C2×C4—342no—
382C2×C4—344no—
383C2×C4—344no—
384C2×C4—682no—
385C2×C4—682no—
386C2×C4—688no—
387C2×C4—682no—
388C2×C4C23×#59511yescharacteristic
389C2×C4C23×#59512yes—
390C2×C4C23×#59518yes—
391C2×C4C23×#59511yescharacteristic
392C2×C4C23×#59512yes—
393C8—348no—
394C8—348no—
395C22—21no—
396C22—22no—
397C22—172no—
398C22—342no—
399C22—341no—
400C22—342no—
401C22—684no—
402C22C24×#59511yescharacteristic
403C4—21no—
404C4—22no—
405C4—28no—
406C4—171no—
407C4—171no—
408C4—341no—
409C4—342no—
410C4—342no—
411C4—348no—
412C4#1176387712yes—
413C4#1176388814yes—
414C2—41no—
415C2—342no—
416C2—681no—
417C2C2×#1176387711yescharacteristic
418C2C2×#1176388812yes—
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#533C4
prime radicalC23×D17C22×C4
cosocleC2×Dic17C25
hypercenter#533#595
hypocenterC17#60249
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #14545697C2semidirect
2. #14545695C2semidirect
3. #14545688C2semidirect
4. #14545586C2semidirect
5. Q8×#63243C2semidirect
9. #11763861C22semidirect
10. #11763860C22semidirect
11. #11763859C22semidirect
12. #11763859C22semidirect
13. #11763858C22semidirect
14. Q8×#3624C22semidirect
307. C17#60249semidirect

Maximal subgroups

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

subgroupindex
#145456972
#145456952
#145456882
#145455862
Q8×#632432
#145454652
#145454612
D17×#5332
#6024917