Finite Group Databasebeta

← #36572953

Subgroups — #36572953

Order 22784.

808 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029169C211yesmaximal, characteristic
2C2×#19151720C211yesmaximal, characteristic
3#26028861C212yesmaximal, complemented
4#26021042C211yesmaximal, characteristic, complemented
5#26021040C211yesmaximal, characteristic, complemented
6#26020492C212yesmaximal, complemented
7#26020490C212yesmaximal, complemented
8#26020425C211yesmaximal, characteristic, complemented
9#26020304C212yesmaximal, complemented
10#26020137C211yesmaximal, characteristic, complemented
11C89×#2957C211yesmaximal, characteristic
12#19152365C2212yes—
13C2×#15668019C2212yes—
14#19151720—22no—
15#19151720C414yescomplemented
16#19151720C414yescomplemented
17#19151715—22no—
18#19151646—24no—
19#19151646C2212yes—
20#19151646C2212yes—
21C2×#15667948C2211yescharacteristic
22C2×#15667948C2211yescharacteristic
23C2×#15667947—24no—
24C2×#15667947C2212yes—
25C2×#15667947C2211yescharacteristic
26#19151634C2211yescharacteristic
27#19151621C2212yes—
28#19151615C2212yes—
29C2×#15667939C2212yes—
30#19151553C2211yescharacteristic
31C2×#15667922C2211yescharacteristic
32#19151369C2212yes—
33#19151368C2212yes—
34#19151359C2211yescharacteristic
35#19151357C2211yescharacteristic
36C2×#135×C89C2211yescharacteristic
37#568×C89C2212yes—
38#444×C89—22no—
39#443×C89—22no—
40C2×#108×C89C2211yescharacteristic
41#421×C89C2212yes—
42#405×C89C2211yescharacteristic
43#15668019—22no—
44#15668019—22no—
45#15668019—24no—
46#15668019—42no—
47#15668019C2×C414yes—
48#15668019C2×C414yes—
49C2×#13083558—22no—
50C2×#13083558—24no—
51C2×#13083558C2312yes—
52C2×#13083558C2312yes—
53C2×#13083558C2311yescharacteristic
54C22×#10623494—22no—
55C22×#10623494C2311yescharacteristic
56#15667971C2311yescharacteristic
57C2×C4×Dic89C2311yescharacteristic
58#15667948—21no—
59#15667948—21no—
60#15667948C2×C412yes—
61#15667948C2×C412yes—
62#15667947—24no—
63#15667947—22no—
64#15667947—22no—
65#15667947—24no—
66#15667947D414yes—
67#15667945—21no—
68#15667943—21no—
69#15667941—42no—
70#15667939—24no—
71#15667939C2×C418yes—
72#15667922—21no—
73#15667922C2×C412yes—
74#15667922C2×C412yes—
75C8×Dic89—21no—
76C2×#41×C89C2312yes—
77#135×C89C2×C412yes—
78#135×C89C2×C412yes—
79#135×C89D412yes—
80#135×C89D412yes—
81C2×Q16×C89Q811yescharacteristic
82C2×SD16×C89Q812yes—
83C2×D8×C89Q811yescharacteristic
84#131×C89—24no—
85#131×C89—42no—
86C2×M16×C89—24no—
87C2×M16×C89C2312yes—
88C2×M16×C89C2311yescharacteristic
89C22×C8×C89C2311yescharacteristic
90#117×C89C2311yescharacteristic
91#108×C89—21no—
92#108×C89—21no—
93#104×C89—24no—
94#97×C89—21no—
95C4×C8×C89—21no—
96C22×Dic89—21no—
97#13083558—22no—
98#13083558—21no—
99#13083558—24no—
100#13083558—24no—
101#13083558—22no—
102#13083558—22no—
103#13083558—44no—
104#13083558C22×C414yes—
105#13083558C2×D412yes—
106C2×#10623494—22no—
107C2×#10623494—21no—
108C2×#10623494—24no—
109C2×#10623494—24no—
110C2×#10623494—22no—
111C2×#10623494—44no—
112C2×#10623494C22×C412yes—
113C2×#10623494C2×D412yes—
114C2×#10623494#4111yescharacteristic
115C2×#10623494#4111yescharacteristic
116C4⋊Dic89—21no—
117C22⋊Dic89—21no—
118C4×Dic89—21no—
119C4×Dic89—21no—
120C4×Dic89C22×C412yes—
121#41×C89—22no—
122#41×C89—22no—
123#41×C89C22×C414yes—
124#41×C89C2×D414yes—
125C2×Q8×C89C2×Q812yes—
126C2×D4×C89—22no—
127C2×D4×C89C2×Q812yes—
128C22×C4×C89—22no—
129C22×C4×C89C2411yescharacteristic
130Q16×C89C4⋊C412yes—
131Q16×C89C4⋊C412yes—
132SD16×C89C4⋊C414yes—
133SD16×C89C4⋊C414yes—
134D8×C89C4⋊C412yes—
135D8×C89C4⋊C412yes—
136M16×C89—21no—
137M16×C89—22no—
138M16×C89—22no—
139M16×C89—44no—
140M16×C89C2×D412yes—
141C2×C8×C89—21no—
142C2×C8×C89—22no—
143C2×C8×C89—44no—
144C2×C8×C89C22×C411yescharacteristic
145C2×C8×C89C22×C411yescharacteristic
146C2×C8×C89C2×D411yescharacteristic
147C2×C8×C89C2×D411yescharacteristic
148C2×C8×C89C2×Q811yescharacteristic
149C2×C8×C89C2×Q811yescharacteristic
150C4⋊C4×C89—21no—
151C22⋊C4×C89—21no—
152C42×C89—21no—
153C23×C89—22no—
154C23×C892−1+411yescharacteristic
155Q8×C89—22no—
156Q8×C89C2×C4⋊C414yes—
157D4×C89—24no—
158D4×C89—24no—
159D4×C89—22no—
160D4×C89C2×C4⋊C414yes—
161C2×C4×C89—22no—
162C2×C4×C89—24no—
163C2×C4×C89—24no—
164C2×C4×C89—41no—
165C2×C4×C89C23×C411yescharacteristic
166C2×C4×C89C22×D411yescharacteristic
167C2×C4×C89C22×D411yescharacteristic
168C2×C4×C89C22×Q811yescharacteristic
169C2×C4×C89C2×#4111yescharacteristic
170C2×C4×C89H(5,𝔽2)11yescharacteristic
171C2×Dic89—22no—
172C2×Dic89—41no—
173C2×Dic89—81no—
174C8×C89—22no—
175C8×C89—42no—
176C8×C89C2×C4⋊C411yescharacteristic
177C8×C89C2×C4⋊C411yescharacteristic
178C8×C89C2×C4⋊C411yescharacteristic
179C8×C89C2×C4⋊C411yescharacteristic
180#10623494—24no—
181#10623494—22no—
182#10623494—42no—
183#10623494C4×D412yes—
184#10623494C4×D412yes—
185C22×C89—24no—
186C22×C89—41no—
187C22×C89—82no—
188C22×C89#52011yescharacteristic
189C22×C89#54711yescharacteristic
190C22×C89D4×Q811yescharacteristic
191C4×C89—24no—
192C4×C89—81no—
193C4×C89C22×C4⋊C411yescharacteristic
194C4×C89C2×C4×D411yescharacteristic
195C4×C89D4211yescharacteristic
196C4×C89#55411yescharacteristic
197Dic89—42no—
198Dic89—81no—
199#22695—891nomaximal, Sylow
200C2×C89—21no—
201C2×C89—21no—
202C2×C89—21no—
203C2×C89—44no—
204C2×C89D4×C4⋊C411yescharacteristic
205#2957—891no—
206#2957D8911yescharacteristic
207C2×#444—891no—
208#2916—892no—
209#1953—891no—
210#1953—891no—
211#1861—892no—
212#1861—892no—
213#1849—891no—
214#1814—892no—
215#1773—891no—
216C89#2269511yescharacteristic, complemented, Sylow
217C2×#135C2×D8911yescharacteristic
218#568—892no—
219#568C2×D8912yes—
220C2×#131—892no—
221#444—22no—
222#444—894no—
223#444—894no—
224#444—1782no—
225#443—22no—
226#443—1782no—
227#430—892no—
228#430—892no—
229#430—1784no—
230C2×#108—891no—
231C2×#108—891no—
232C2×#108—892no—
233C2×#108—891no—
234C2×#108—1784no—
235C2×#108C2×D8911yescharacteristic
236#427—891no—
237#423—892no—
238#421—892no—
239#421C2×D8912yes—
240C2×#104—892no—
241#405—891no—
242#405C2×D8911yescharacteristic
243C2×#97—891no—
244#344—892no—
245#344—892no—
246#339—891no—
247#339—891no—
248C2×#41C22×D8912yes—
249#135C4×D8912yes—
250#135C4×D8912yes—
251#135C22⋊D8912yes—
252#135C22⋊D8912yes—
253C2×Q16Dic17811yescharacteristic
254C2×SD16Dic17812yes—
255C2×D8Dic17811yescharacteristic
256#131—24no—
257#131—42no—
258#131—894no—
259#131—894no—
260#131—3562no—
261#131—1782no—
262#131—1782no—
263#131—1784no—
264C2×M16—24no—
265C2×M16—892no—
266C2×M16—891no—
267C2×M16—892no—
268C2×M16—1782no—
269C2×M16—1784no—
270C2×M16C22×D8912yes—
271C2×M16C22×D8911yescharacteristic
272C22×C8—891no—
273C22×C8—1782no—
274C22×C8C22×D8911yescharacteristic
275#117—891no—
276#117C22×D8911yescharacteristic
277C2×C42—891no—
278#108—21no—
279#108—21no—
280#108—894no—
281#108—892no—
282#108—892no—
283#108—1782no—
284#108—1782no—
285#108—1784no—
286#108—1784no—
287#108—1781no—
288#108—1781no—
289#107—1781no—
290#106—1781no—
291C4⋊C8—3562no—
292#104—24no—
293#104—898no—
294#104—1784no—
295#97—21no—
296#97—892no—
297#97—892no—
298#97—1781no—
299C4×C8—21no—
300C4×C8—1781no—
301#41—22no—
302#41—22no—
303#41C2×C4×D8914yes—
304#41C2×C22⋊D8914yes—
305C2×Q8C2×Dic17812yes—
306C2×D4—22no—
307C2×D4C2×Dic17812yes—
308C22×C4—22no—
309C22×C4—1781no—
310C22×C4C23×D8911yescharacteristic
311Q16#1308355312yes—
312Q16#1308355312yes—
313SD16#1308355314yes—
314SD16#1308355314yes—
315D8#1308355312yes—
316D8#1308355312yes—
317M16—21no—
318M16—22no—
319M16—22no—
320M16—44no—
321M16—894no—
322M16—892no—
323M16—3564no—
324M16—1784no—
325M16—1784no—
326M16—1782no—
327M16—1782no—
328M16—1782no—
329M16—1781no—
330M16D4×D8912yes—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—44no—
334C2×C8—892no—
335C2×C8—892no—
336C2×C8—891no—
337C2×C8—891no—
338C2×C8—3564no—
339C2×C8—1784no—
340C2×C8—1784no—
341C2×C8—1782no—
342C2×C8—1782no—
343C2×C8—1781no—
344C2×C8C2×C4×D8911yescharacteristic
345C2×C8C2×C4×D8911yescharacteristic
346C2×C8C2×C22⋊D8911yescharacteristic
347C2×C8C2×C22⋊D8911yescharacteristic
348C2×C8C2×Dic17811yescharacteristic
349C2×C8C2×Dic17811yescharacteristic
350C4⋊C4—21no—
351C4⋊C4—1781no—
352C22⋊C4—21no—
353C22⋊C4—1781no—
354C42—21no—
355C42—892no—
356C42—1781no—
357C42—1781no—
358C23—22no—
359C23#1566805911yescharacteristic
360Q8—22no—
361Q8C2×#1308355314yes—
362D4—24no—
363D4—24no—
364D4—22no—
365D4C2×#1308355314yes—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—3561no—
371C2×C4—1782no—
372C2×C4—7121no—
373C2×C4C22×C4×D8911yescharacteristic
374C2×C4C2×D4×D8911yescharacteristic
375C2×C4C22×C22⋊D8911yescharacteristic
376C2×C4C22×Dic17811yescharacteristic
377C2×C4#41×D8911yescharacteristic
378C2×C4#1566805711yescharacteristic
379C8—22no—
380C8—42no—
381C8—892no—
382C8—892no—
383C8—3562no—
384C8—1782no—
385C8—1784no—
386C8C2×#1308355311yescharacteristic
387C8C2×#1308355311yescharacteristic
388C8C2×#1308355311yescharacteristic
389C8C2×#1308355311yescharacteristic
390C22—24no—
391C22—41no—
392C22—82no—
393C22#1915207711yescharacteristic
394C22#1915224711yescharacteristic
395C22D4×Dic17811yescharacteristic
396C4—24no—
397C4—81no—
398C4—3562no—
399C4—7121no—
400C4C22×#1308355311yescharacteristic
401C4C4×D4×D8911yescharacteristic
402C4D4×C22⋊D8911yescharacteristic
403C4#1915230711yescharacteristic
404C2—21no—
405C2—21no—
406C2—21no—
407C2—44no—
408C2D4×#1308355311yescharacteristic
subgroupG/H
centerC4D4×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC89×#2957C2
socleC2×C89D4×C4⋊C4
prime radicalC2×#135×C89C22
cosocleC22×C4×C89C24
hypercenter#2957D89
hypocenterC89#22695
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2957

Complements

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

NH ≅ G/Ntype
3. #26028861C2semidirect
4. #26021042C2semidirect
5. #26021040C2semidirect
6. #26020492C2semidirect
7. #26020490C2semidirect
8. #26020425C2semidirect
9. #26020304C2semidirect
10. #26020137C2semidirect
15. #19151720C4semidirect
16. #19151720C4semidirect
216. C89#22695semidirect

Maximal subgroups

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

subgroupindex
#260291692
C2×#191517202
#260288612
#260210422
#260210402
#260204922
#260204902
#260204252
#260203042
#260201372
C89×#29572
#2269589