Finite Group Databasebeta

← #36717340

Subgroups — #36717340

Order 22784.

1898 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152154C211yesmaximal, characteristic
2#26023430C214yesmaximal, complemented
3#26023358C211yesmaximal, characteristic, complemented
4#26023357C212yesmaximal, complemented
5#26023206C214yesmaximal, complemented
6#26023205C214yesmaximal, complemented
7#26023006C214yesmaximal
8#26023003C214yesmaximal
9#26022858C214yesmaximal
10#26022644C212yesmaximal
11C89×#2371C211yesmaximal, characteristic, complemented
12#19152154C4116yescomplemented
13C2×#15668011C2212yes—
14C2×#15668004C2211yescharacteristic
15C2×#15667990C2214yes—
16C2×#15667977C2214yes—
17C2×#15667970C2212yes—
18C2×C42×D89C2211yescharacteristic
19#19151504C2214yes—
20#19151503C2214yes—
21#19151502C2214yes—
22#19151502C2214yes—
23#19151490C2214yes—
24#19151489C2214yes—
25#19151488C2214yes—
26#19151487C2214yes—
27#19151485C2214yes—
28#19151484C2214yes—
29#19151484C2214yes—
30#19151483C2214yes—
31#19151482C2214yes—
32#19151482C2214yes—
33#19151481C2214yes—
34#19151480C2214yes—
35#19151480C2214yes—
36#19151479C2214yes—
37#19151467C2214yes—
38#19151465C2214yes—
39#19151462C2212yes—
40#19151462C2214yes—
41#19151462C2212yes—
42#19151460C2212yes—
43#19151460C2212yes—
44#19151457C2214yes—
45C4⋊C4×Dic89C2214yes—
46C4×#13083553C2214yes—
47C4×#13083553C2214yes—
48C4×#13083553C2214yes—
49#19151449C2214yes—
50#19151449C2214yes—
51C4×#13083551C2214yes—
52#19151444C2212yes—
53C2×#125×C89C2211yescharacteristic
54#387×C89C2214yes—
55#387×C89C2214yes—
56C4×C4⋊C4×C89C2214yes—
57#376×C89C2212yes—
58C22×Dic178C2314yes—
59C22×C4×D89C2311yescharacteristic
60C22×C4×D89C2312yes—
61#15668011C2×C4116yes—
62#15668004C2×C418yes—
63#15667990C2×C4132yes—
64#15667977C2×C4132yes—
65#15667970C2×C4116yes—
66C2×C4⋊Dic89C2314yes—
67C2×C4⋊Dic89C2312yes—
68C2×#13083553C2314yes—
69C2×#13083553C2314yes—
70C2×#13083553C2314yes—
71C2×#13083553C2314yes—
72C2×#13083553C2314yes—
73C2×#13083553C2314yes—
74C2×#13083551C2314yes—
75C2×#13083551C2314yes—
76C2×#13083551C2314yes—
77C2×C4×Dic89C2314yes—
78C2×C4×Dic89C2314yes—
79C2×C4×Dic89C2314yes—
80C2×C4×Dic89C2314yes—
81C2×C4×Dic89C2311yescharacteristic
82C2×C4×Dic89C2314yes—
83C2×C4×Dic89C2312yes—
84C42×D89C2×C418yes—
85#15667919C2312yes—
86#15667919C2312yes—
87#15667919C2314yes—
88#15667919C2314yes—
89#15667919C2314yes—
90#15667919C2314yes—
91#15667919C2314yes—
92#15667919C2314yes—
93#15667919C2314yes—
94#15667919C2314yes—
95#15667919C2314yes—
96#15667919C2314yes—
97#15667919C2314yes—
98#125×C89C2×C418yes—
99C2×C4⋊C4×C89C2314yes—
100C2×C4⋊C4×C89C2314yes—
101C2×C4⋊C4×C89C2312yes—
102C2×C42×C89C2314yes—
103C2×C42×C89C2314yes—
104C2×C42×C89C2311yescharacteristic
105#95×C89C2312yes—
106#95×C89C2312yes—
107#95×C89C2314yes—
108#95×C89C2314yes—
109#95×C89C2314yes—
110C23×D89C2411yescharacteristic
111C2×Dic178—44no—
112C2×Dic178—44no—
113C2×Dic178C22×C4116yes—
114C22×Dic89C2414yes—
115C22×Dic89C2414yes—
116C22×Dic89C2412yes—
117C22×Dic89C2414yes—
118C22×Dic89C2411yescharacteristic
119C2×C4×D89—24no—
120C2×C4×D89—44no—
121C2×C4×D89C22×C414yes—
122C2×C4×D89C22×C418yes—
123C4⋊Dic89—44no—
124C4⋊Dic89C22×C418yes—
125#13083553—28no—
126#13083553—28no—
127#13083553—44no—
128#13083553—44no—
129#13083553C22×C4116yes—
130#13083553C22×C4116yes—
131#13083551—44no—
132#13083551—44no—
133#13083551C22×C4116yes—
134C4×Dic89—44no—
135C4×Dic89—44no—
136C4×Dic89—44no—
137C4×Dic89—44no—
138C4×Dic89C22×C4116yes—
139C4×Dic89C22×C418yes—
140C4×Dic89C22×C414yes—
141C22×C4×C89C2414yes—
142C22×C4×C89C2414yes—
143C22×C4×C89C2411yescharacteristic
144C22×C4×C89C2412yes—
145C22×C4×C89C2414yes—
146C4⋊C4×C89—44no—
147C4⋊C4×C89C22×C418yes—
148C4⋊C4×C89C22×C4116yes—
149C42×C89—44no—
150C42×C89—44no—
151C42×C89C22×C414yes—
152C23×C89C2511yescharacteristic
153C22×D89—22no—
154C22×D89—24no—
155C22×D89C23×C412yes—
156C2×C4×C89—22no—
157C2×C4×C89—24no—
158C2×C4×C89—24no—
159C2×C4×C89—24no—
160C2×C4×C89—24no—
161C2×C4×C89—24no—
162C2×C4×C89—24no—
163C2×C4×C89—24no—
164C2×C4×C89—24no—
165C2×C4×C89—24no—
166C2×C4×C89C23×C414yes—
167C2×C4×C89C23×C412yes—
168C2×C4×C89C23×C418yes—
169C2×Dic89—24no—
170C2×Dic89—24no—
171C2×Dic89—24no—
172C2×Dic89—24no—
173C2×Dic89—24no—
174C2×Dic89—24no—
175C2×Dic89—24no—
176C2×Dic89—24no—
177C2×Dic89—24no—
178C2×Dic89C23×C414yes—
179C2×Dic89C23×C418yes—
180C2×Dic89C23×C412yes—
181C2×Dic892−1+414yes—
182C4×D89—28no—
183C4×D89—48no—
184Dic178—416no—
185C22×C89C24×C411yescharacteristic
186C22×C89C2×2−1+412yes—
187C22×C89#58514yes—
188C2×D89—22no—
189C2×D89—24no—
190C2×D89—44no—
191C4×C89—22no—
192C4×C89—28no—
193C4×C89—24no—
194C4×C89—44no—
195C4×C89—44no—
196Dic89—24no—
197Dic89—28no—
198Dic89—44no—
199Dic89—44no—
200Dic89#51914yes—
201#34375—891nomaximal, Sylow
202C2×C89C2×#51911yescharacteristic
203C2×C89#342812yes—
204C2×C89#355714yes—
205D89—42no—
206C2×#533—891no—
207#2381—894no—
208#2371—892no—
209#2371—891no—
210#2371D8911yescharacteristic, complemented
211#2355—894no—
212#2355—894no—
213#2332—894no—
214#2332—894no—
215#2314—894no—
216#2283—892no—
217C89#3437511yescharacteristic, complemented, Sylow
218#533—8916no—
219C2×C4⋊Q8—892no—
220C2×#125—891no—
221C2×#125C2×D8911yescharacteristic
222C2×C22⋊Q8—894no—
223C2×C4×Q8—894no—
224C2×#117—892no—
225C22×C42—891no—
226#391—894no—
227#391—894no—
228#391—894no—
229#391—894no—
230#388—894no—
231#388—894no—
232#388—894no—
233#388—894no—
234#387—894no—
235#387—894no—
236#387—894no—
237#387—894no—
238#387—894no—
239#387—894no—
240#387—894no—
241#387—894no—
242#387—894no—
243#387—894no—
244#387C2×D8914yes—
245#387C2×D8914yes—
246#384—894no—
247#383—894no—
248#382—894no—
249#382—892no—
250#382—892no—
251#381—892no—
252#381—892no—
253#380—894no—
254C4×C4⋊C4—894no—
255C4×C4⋊C4—894no—
256C4×C4⋊C4—894no—
257C4×C4⋊C4—894no—
258C4×C4⋊C4—894no—
259C4×C4⋊C4—894no—
260C4×C4⋊C4C2×D8914yes—
261C4×C22⋊C4—894no—
262#376—892no—
263#376C2×D8912yes—
264C22×Q8—894no—
265C23×C4—892no—
266C23×C4—891no—
267C4⋊Q8—8916no—
268#125—898no—
269#125C4×D8918yes—
270C22⋊Q8—8932no—
271C4×Q8—8932no—
272#117—8916no—
273C2×C4⋊C4—892no—
274C2×C4⋊C4—894no—
275C2×C4⋊C4—894no—
276C2×C4⋊C4—894no—
277C2×C4⋊C4—894no—
278C2×C4⋊C4—894no—
279C2×C4⋊C4—894no—
280C2×C4⋊C4—894no—
281C2×C4⋊C4C22×D8914yes—
282C2×C4⋊C4C22×D8912yes—
283C2×C4⋊C4C22×D8914yes—
284C2×C22⋊C4—894no—
285C2×C22⋊C4—894no—
286C2×C22⋊C4—894no—
287C2×C42—894no—
288C2×C42—892no—
289C2×C42—891no—
290C2×C42—898no—
291C2×C42—894no—
292C2×C42—894no—
293C2×C42—894no—
294C2×C42—894no—
295C2×C42C22×D8914yes—
296C2×C42C22×D8914yes—
297C2×C42C22×D8911yescharacteristic
298#95—892no—
299#95—892no—
300#95—894no—
301#95—894no—
302#95—894no—
303#95—894no—
304#95—894no—
305#95—894no—
306#95—894no—
307#95—894no—
308#95—894no—
309#95—894no—
310#95—894no—
311#95C22×D8912yes—
312#95C22×D8912yes—
313#95C22×D8914yes—
314#95C22×D8914yes—
315#95C22×D8914yes—
316C24—891no—
317C2×Q8—8916no—
318C2×Q8—3564no—
319C2×Q8—3564no—
320C22×C4—891no—
321C22×C4—892no—
322C22×C4—894no—
323C22×C4—898no—
324C22×C4—894no—
325C22×C4—894no—
326C22×C4—894no—
327C22×C4—3564no—
328C22×C4—1784no—
329C22×C4C23×D8914yes—
330C22×C4C23×D8914yes—
331C22×C4C23×D8911yescharacteristic
332C22×C4C23×D8912yes—
333C22×C4C23×D8914yes—
334C4⋊C4—44no—
335C4⋊C4—898no—
336C4⋊C4—8916no—
337C4⋊C4—8916no—
338C4⋊C4—3564no—
339C4⋊C4—3564no—
340C4⋊C4—3564no—
341C4⋊C4—1788no—
342C4⋊C4—1788no—
343C4⋊C4C2×C4×D8918yes—
344C4⋊C4C2×C4×D89116yes—
345C22⋊C4—8916no—
346C22⋊C4—3564no—
347C22⋊C4—3564no—
348C42—44no—
349C42—44no—
350C42—898no—
351C42—894no—
352C42—8916no—
353C42—3564no—
354C42—3564no—
355C42—3564no—
356C42—3564no—
357C42C2×C4×D8914yes—
358C23—892no—
359C23—1782no—
360C23—1784no—
361C23C24×D8911yescharacteristic
362Q8—35616no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—892no—
374C2×C4—894no—
375C2×C4—894no—
376C2×C4—898no—
377C2×C4—3568no—
378C2×C4—1788no—
379C2×C4—1784no—
380C2×C4—1784no—
381C2×C4—1784no—
382C2×C4—1784no—
383C2×C4—1784no—
384C2×C4—1784no—
385C2×C4—1784no—
386C2×C4—1784no—
387C2×C4—1784no—
388C2×C4C22×C4×D8912yes—
389C2×C4C22×C4×D8914yes—
390C2×C4C22×C4×D8918yes—
391C22—3564no—
392C22—1782no—
393C22—1784no—
394C22C23×C4×D8911yescharacteristic
395C222−1+4×D8912yes—
396C22#1915243414yes—
397C4—22no—
398C4—24no—
399C4—28no—
400C4—44no—
401C4—44no—
402C4—894no—
403C4—3564no—
404C4—3564no—
405C4—1784no—
406C4—1788no—
407C2—3562no—
408C2#519×D8911yescharacteristic
409C2#2603345112yes—
410C2#2603474814yes—
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2371C2
socle, cosocleC23×C89C25
prime radicalC23×D89C24
hypercenter#2371D89
hypocenterC89#34375
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2371

Complements

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

NH ≅ G/Ntype
2. #26023430C2semidirect
3. #26023358C2semidirect
4. #26023357C2semidirect
5. #26023206C2semidirect
6. #26023205C2semidirect
11. C89×#2371C2semidirect
12. #19152154C4semidirect
210. #2371D89semidirect
217. C89#34375semidirect

Maximal subgroups

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

subgroupindex
C2×#191521542
#260234302
#260233582
#260233572
#260232062
#260232052
#260230062
#260230032
#260228582
#260226442
C89×#23712
#3437589