Finite Group Databasebeta

← #36709941

Subgroups — #36709941

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030789C211yesmaximal, characteristic
2#26030785C211yesmaximal, characteristic
3#26022538C211yesmaximal, characteristic, complemented
4#26022537C211yesmaximal, characteristic, complemented
5#26022536C211yesmaximal, characteristic, complemented
6#26022533C211yesmaximal, characteristic, complemented
7#26022532C211yesmaximal, characteristic, complemented
8#26022529C211yesmaximal, characteristic, complemented
9#26022528C211yesmaximal, characteristic, complemented
10#26022039C211yesmaximal, characteristic
11#26022037C211yesmaximal, characteristic
12#26022033C211yesmaximal, characteristic
13#367×D89C211yesmaximal, characteristic
14#26021850C211yesmaximal, characteristic
15C89×#2255C211yesmaximal, characteristic, complemented
16#19152228—21no—
17#19152225—21no—
18C2×#15667951C2211yescharacteristic
19C2×C16×D89C2211yescharacteristic
20#19152008—21no—
21#19152006—21no—
22#19152005—21no—
23#19152004—21no—
24#19152003C2211yescharacteristic
25#19152002C2211yescharacteristic
26#19152002C2211yescharacteristic
27#19152000C2211yescharacteristic
28#19151999C2211yescharacteristic
29#19151999C2211yescharacteristic
30#19151639—21no—
31#19151637—21no—
32#19151635C2211yescharacteristic
33#19151635C2211yescharacteristic
34#107×D89C2211yescharacteristic
35#107×D89C2211yescharacteristic
36#19151552C2211yescharacteristic
37#19151426C2211yescharacteristic
38#19151426C2211yescharacteristic
39#19151425C2211yescharacteristic
40#19151425C2211yescharacteristic
41#19151425C2211yescharacteristic
42#19151425C2211yescharacteristic
43#19151425C2211yescharacteristic
44#19151425C2211yescharacteristic
45#19151424C2211yescharacteristic
46#19151424C2211yescharacteristic
47#19151423C2211yescharacteristic
48#19151423C2211yescharacteristic
49#19151373C2211yescharacteristic
50C4×#13083549C2211yescharacteristic
51C16×Dic89C2211yescharacteristic
52#500×C89C2211yescharacteristic
53#499×C89C2211yescharacteristic
54#367×C89C2211yescharacteristic
55#367×C89C2211yescharacteristic
56#366×C89C2211yescharacteristic
57#366×C89C2211yescharacteristic
58#346×C89C2211yescharacteristic
59C2×#13083557D411yescharacteristic
60C2×C8×D89C2311yescharacteristic
61#15668012—21no—
62#15668011—21no—
63#15668011—21no—
64#15668011—41no—
65#15668010—41no—
66#15668004—21no—
67#15668003—21no—
68#15668003—21no—
69#15668002—41no—
70#15668002—41no—
71#15667992—41no—
72#15667990—41no—
73#15667990—41no—
74#15667988—41no—
75#15667970—21no—
76#15667970—21no—
77#15667969D411yescharacteristic
78C4⋊C4×D89—21no—
79C4⋊C4×D89—21no—
80#15667951Q814yescomplemented
81C2×#13083549C2311yescharacteristic
82C2×#13083549C2311yescharacteristic
83C16×D89Q814yescomplemented
84#15667946—21no—
85#15667946C2311yescharacteristic
86#15667946C2311yescharacteristic
87#15667946C2311yescharacteristic
88#15667946C2311yescharacteristic
89#15667945—21no—
90#15667945—21no—
91#15667945—21no—
92#15667945—21no—
93#15667944—21no—
94#15667943—21no—
95#15667943—21no—
96#15667943—21no—
97#15667943—21no—
98#15667923D411yescharacteristic
99#15667922D411yescharacteristic
100C8×Dic89C2311yescharacteristic
101C4⋊Q8×C89—21no—
102#125×C89—21no—
103C2×C16×C89C2311yescharacteristic
104C2×C16×C89C2311yescharacteristic
105#107×C89—21no—
106#107×C89C2311yescharacteristic
107#107×C89C2311yescharacteristic
108#107×C89C2311yescharacteristic
109#107×C89C2311yescharacteristic
110#106×C89—21no—
111C4×C8×C89C2311yescharacteristic
112C2×Dic178—41no—
113C2×Dic178—41no—
114C2×Dic178—41no—
115C2×C4×D89—41no—
116C2×C4×D89—41no—
117C2×C4×D89C2×D411yescharacteristic
118#13083557Q1614yes—
119C2×#10623494C2×D411yescharacteristic
120C2×#10623494C2×D411yescharacteristic
121C8×D89C2×Q812yes—
122C8×D89C2×Q812yes—
123C4⋊Dic89—21no—
124C4⋊Dic89—21no—
125C4⋊Dic89—21no—
126C4⋊Dic89—21no—
127C4⋊Dic89—41no—
128C4⋊Dic89—41no—
129C4⋊Dic89—41no—
130#13083553—41no—
131#13083553—41no—
132#13083553—41no—
133#13083553—41no—
134#13083553—41no—
135#13083553—41no—
136#13083553—41no—
137#13083553—41no—
138#13083553D811yescharacteristic
139#13083553D811yescharacteristic
140#13083551—41no—
141#13083551—41no—
142#13083551D811yescharacteristic
143#13083551D811yescharacteristic
144C4×Dic89—41no—
145C4×Dic89—41no—
146C4×Dic89C2×D411yescharacteristic
147#13083549C2×Q812yes—
148#13083549C2×Q812yes—
149C2×Q8×C89—41no—
150C2×C8×C89C2×D411yescharacteristic
151C2×C8×C89C2411yescharacteristic
152C4⋊C4×C89—21no—
153C4⋊C4×C89—21no—
154C4⋊C4×C89—21no—
155C4⋊C4×C89—21no—
156C4⋊C4×C89—41no—
157C4⋊C4×C89—41no—
158C4⋊C4×C89—41no—
159C42×C89C2×D411yescharacteristic
160C16×C89C2×Q812yes—
161C16×C89C2×Q812yes—
162C22×D89C2×D811yescharacteristic
163Q8×C89—161no—
164C2×C4×C89—41no—
165C2×C4×C89—41no—
166C2×C4×C89—41no—
167C2×C4×C89—41no—
168C2×C4×C89C2×D811yescharacteristic
169C2×C4×C89C2×D811yescharacteristic
170C2×C4×C89C22×D411yescharacteristic
171C2×Dic89—41no—
172C2×Dic89—41no—
173C2×Dic89—41no—
174C2×Dic89—41no—
175C2×Dic89C2×D811yescharacteristic
176C2×Dic89C2×D811yescharacteristic
177C2×Dic89C2×D811yescharacteristic
178C4×D89—161no—
179C4×D89—161no—
180C4×D89C4⋊Q812yes—
181C4×D89C2×Q1612yes—
182Dic178—161no—
183Dic178—161no—
184Dic178—161no—
185C8×C89C2×Q1612yes—
186C8×C89C22×Q811yescharacteristic
187C8×C89C22×Q811yescharacteristic
188#10623494C4⋊Q812yes—
189#10623494C2×Q1612yes—
190C22×C89C22×D811yescharacteristic
191C2×D89—21no—
192C2×D89—21no—
193C2×D89#50012yes—
194C4×C89—21no—
195C4×C89—21no—
196C4×C89—81no—
197C4×C89—81no—
198C4×C89—81no—
199C4×C89—81no—
200C4×C89C2×C4⋊Q811yescharacteristic
201C4×C89C22×Q1611yescharacteristic
202Dic89—21no—
203Dic89—21no—
204Dic89—81no—
205Dic89—81no—
206Dic89—81no—
207Dic89—81no—
208Dic89#50012yes—
209#33730—891nomaximal, Sylow
210C2×C89C2×#50011yescharacteristic
211C2×C89#341611yescharacteristic
212C2×C89#341611yescharacteristic
213D89—22no—
214#3164—891no—
215#3164—891no—
216#2255—891no—
217#2255—891no—
218#2255—891no—
219#2255D8911yescharacteristic, complemented
220#2254—891no—
221#2254—891no—
222#2253—891no—
223#2253—891no—
224#2159—891no—
225#2159—891no—
226#2158—891no—
227C2×#367—891no—
228#2108—891no—
229C89#3373011yescharacteristic, complemented, Sylow
230#544—1781no—
231#544—1781no—
232C2×M32—891no—
233C22×C16—891no—
234#501—1781no—
235#501—1781no—
236#501—1781no—
237#501—1781no—
238#500—891no—
239#500—891no—
240#500—891no—
241#500C2×D8911yescharacteristic
242#499—891no—
243#499—891no—
244#499—891no—
245#499C2×D8911yescharacteristic
246#428—1781no—
247#428—1781no—
248#427—891no—
249#427—891no—
250C2×#107—891no—
251C2×#107—891no—
252#405—891no—
253#367—891no—
254#367—891no—
255#367—891no—
256#367—891no—
257#367—891no—
258#367—891no—
259#367—891no—
260#367—891no—
261#367C2×D8911yescharacteristic
262#367C2×D8911yescharacteristic
263#366—891no—
264#366—891no—
265#366—891no—
266#366—891no—
267#366C2×D8911yescharacteristic
268#366C2×D8911yescharacteristic
269#346—891no—
270#346C2×D8911yescharacteristic
271C4×C16—891no—
272C4×C16—891no—
273C2×M16—891no—
274C22×C8—891no—
275C4⋊Q8—21no—
276C4⋊Q8—3561no—
277C4⋊Q8—3561no—
278C4⋊Q8—1781no—
279C4⋊Q8—1781no—
280C4⋊Q8—1781no—
281#125—21no—
282#125—3561no—
283#125—3561no—
284#125—1781no—
285#125—1781no—
286#125—1781no—
287C22⋊Q8—3561no—
288C22⋊Q8—3561no—
289C22⋊Q8—3561no—
290C22⋊Q8—3561no—
291#117—891no—
292#117—1781no—
293#117—1781no—
294C2×C4⋊C4—1781no—
295C2×C4⋊C4—1781no—
296M32—894no—
297C2×C16—891no—
298C2×C16—891no—
299C2×C16—894no—
300C2×C16C22×D8911yescharacteristic
301C2×C16C22×D8911yescharacteristic
302#107—21no—
303#107—891no—
304#107—891no—
305#107—891no—
306#107—891no—
307#107—1781no—
308#107—1781no—
309#107—1781no—
310#107—1781no—
311#107—1781no—
312#107C22×D8911yescharacteristic
313#107C22×D8911yescharacteristic
314#107C22×D8911yescharacteristic
315#107C22×D8911yescharacteristic
316#106—21no—
317#106—1781no—
318#106—1781no—
319#106—1781no—
320#106—1781no—
321#106—1781no—
322#97—891no—
323#97—891no—
324C4×C8—891no—
325C4×C8C22×D8911yescharacteristic
326C2×Q8—41no—
327C2×Q8—3561no—
328C2×Q8—3561no—
329C2×Q8—3561no—
330C22×C4—891no—
331C22×C4—3561no—
332C22×C4—3561no—
333M16—894no—
334C2×C8—891no—
335C2×C8—891no—
336C2×C8—892no—
337C2×C8—892no—
338C2×C8D4×D8911yescharacteristic
339C2×C8C23×D8911yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—891no—
348C4⋊C4—891no—
349C4⋊C4—3561no—
350C4⋊C4—3561no—
351C4⋊C4—3561no—
352C4⋊C4—3561no—
353C4⋊C4—3561no—
354C4⋊C4—3561no—
355C4⋊C4—3561no—
356C4⋊C4—3561no—
357C4⋊C4—3561no—
358C4⋊C4—3561no—
359C4⋊C4—3561no—
360C4⋊C4—1781no—
361C4⋊C4—1781no—
362C4⋊C4—1781no—
363C4⋊C4—1781no—
364C22⋊C4—891no—
365C22⋊C4—891no—
366C22⋊C4—3561no—
367C22⋊C4—3561no—
368C42—891no—
369C42—3561no—
370C42—3561no—
371C42D4×D8911yescharacteristic
372C16—892no—
373C16—892no—
374C16Q8×D8912yes—
375C16Q8×D8912yes—
376C23—891no—
377Q8—161no—
378Q8—14241no—
379Q8—14241no—
380Q8—14241no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—891no—
386C2×C4—892no—
387C2×C4—892no—
388C2×C4—891no—
389C2×C4—891no—
390C2×C4—3561no—
391C2×C4—3561no—
392C2×C4—3561no—
393C2×C4—3561no—
394C2×C4—14241no—
395C2×C4—14241no—
396C2×C4D8×D8911yescharacteristic
397C2×C4D8×D8911yescharacteristic
398C2×C4C2×D4×D8911yescharacteristic
399C8—892no—
400C8—892no—
401C8Q16×D8912yes—
402C8C2×Q8×D8911yescharacteristic
403C8C2×Q8×D8911yescharacteristic
404C22—892no—
405C22—1781no—
406C22—1781no—
407C22C2×D8×D8911yescharacteristic
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—892no—
415C4—1781no—
416C4—1781no—
417C4—7121no—
418C4—7121no—
419C4—7121no—
420C4—7121no—
421C4C4⋊Q8×D8911yescharacteristic
422C4C2×Q16×D8911yescharacteristic
423C2—1782no—
424C2#500×D8911yescharacteristic
425C2#2603339711yescharacteristic
426C2#2603339811yescharacteristic
subgroupG/H
centerC22C2×D8×D89
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#2255C2
socleC22×C89C22×D8
prime radicalC22×D89C2×D8
hypercenter#2255D89
hypocenterC89#33730
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #26022538C2semidirect
4. #26022537C2semidirect
5. #26022536C2semidirect
6. #26022533C2semidirect
7. #26022532C2semidirect
8. #26022529C2semidirect
9. #26022528C2semidirect
15. C89×#2255C2semidirect
80. #15667951Q8semidirect
83. C16×D89Q8semidirect
219. #2255D89semidirect
229. C89#33730semidirect

Maximal subgroups

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

subgroupindex
#260307892
#260307852
#260225382
#260225372
#260225362
#260225332
#260225322
#260225292
#260225282
#260220392
#260220372
#260220332
#367×D892
#260218502
C89×#22552
#3373089