Finite Group Databasebeta

← #36555030

Subgroups — #36555030

Order 22784.

842 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151550C211yesmaximal, characteristic
2#26022648C212yesmaximal
3#26020415C211yesmaximal, characteristic, complemented
4#26020413C211yesmaximal, characteristic, complemented
5#26020381C212yesmaximal, complemented
6#26020286C211yesmaximal, characteristic
7#26020284C212yesmaximal
8#26020282C211yesmaximal, characteristic
9#26020150C212yesmaximal, complemented
10#26020149C211yesmaximal, characteristic, complemented
11C89×#1776C211yesmaximal, characteristic, complemented
12C22×#13083557C2211yescharacteristic
13C22×#13083557C2211yescharacteristic
14C2×C42×D89C2211yescharacteristic
15C2×#15667946C2211yescharacteristic
16C2×#15667945C2212yes—
17C2×#15667944C2211yescharacteristic
18C2×#15667943C2212yes—
19#19151550C418yescomplemented
20C2×#15667923C2211yescharacteristic
21C2×C8×Dic89C2211yescharacteristic
22C2×C8×Dic89C2211yescharacteristic
23#19151465—22no—
24#19151463C2212yes—
25#19151463C2212yes—
26#19151460—22no—
27#19151455C2212yes—
28#19151455C2212yes—
29C4×C4⋊Dic89C2212yes—
30C4×#13083551—22no—
31#19151444—22no—
32#19151361C2212yes—
33#19151360C2212yes—
34#19151360C2212yes—
35C2×#107×C89C2211yescharacteristic
36C2×#106×C89C2211yescharacteristic
37C2×#97×C89C2211yescharacteristic
38C4×C4⋊C4×C89C2212yes—
39#340×C89C2212yes—
40C22×C4×D89D412yes—
41C22×C4×D89C2311yescharacteristic
42C2×#13083557—22no—
43C2×#13083557—22no—
44C2×#13083557—22no—
45C2×#13083557—22no—
46C2×#13083557C2×C414yes—
47C2×#13083557C2×C414yes—
48C22×#10623494C2311yescharacteristic
49C22×#10623494C2311yescharacteristic
50C2×C4⋊Dic89C2312yes—
51C2×C4⋊Dic89C2312yes—
52C2×#13083551—22no—
53C2×#13083551—22no—
54C2×C4×Dic89—22no—
55C2×C4×Dic89D412yes—
56C2×C4×Dic89C2311yescharacteristic
57C42×D89C2×C414yes—
58C42×D89Q814yes—
59#15667946—22no—
60#15667945—24no—
61#15667944—22no—
62#15667943—24no—
63#15667923C2×C414yes—
64C8×Dic89C2×C414yes—
65C8×Dic89C2×C414yes—
66#15667919—22no—
67#15667919—22no—
68#15667919—22no—
69#15667919—22no—
70#15667919—22no—
71#15667919—22no—
72#15667919—22no—
73C22×C8×C89C2311yescharacteristic
74C22×C8×C89C2311yescharacteristic
75C2×C4⋊C4×C89C2312yes—
76C2×C4⋊C4×C89C2312yes—
77C2×C42×C89—22no—
78C2×C42×C89C2311yescharacteristic
79#107×C89—22no—
80#106×C89—22no—
81#97×C89C2×C414yes—
82#95×C89—22no—
83C23×D89C2×D411yescharacteristic
84C22×Dic89—22no—
85C22×Dic89—22no—
86C22×Dic89C2×D412yes—
87C22×Dic89C2×D411yescharacteristic
88C2×C4×D89—21no—
89C2×C4×D89—21no—
90C2×C4×D89—21no—
91C2×C4×D89—21no—
92C2×C4×D89—42no—
93C2×C4×D89—42no—
94C2×C4×D89C4⋊C418yes—
95C2×C4×D89C22×C412yes—
96C2×C4×D89C2×Q812yes—
97#13083557—28no—
98#13083557—28no—
99C2×#10623494—21no—
100C2×#10623494—21no—
101C2×#10623494C22×C412yes—
102C2×#10623494C22×C412yes—
103C2×#10623494#4112yes—
104C2×#10623494#4112yes—
105C4⋊Dic89—22no—
106C4⋊Dic89—22no—
107C4⋊Dic89—22no—
108C4⋊Dic89—22no—
109#13083551—44no—
110#13083551—44no—
111C4×Dic89—44no—
112C4×Dic89C4⋊C418yes—
113C4×Dic89C22×C412yes—
114C4×Dic89C2×Q812yes—
115C22×C4×C89—22no—
116C22×C4×C89—22no—
117C22×C4×C89C2×D412yes—
118C22×C4×C89C2411yescharacteristic
119C2×C8×C89—21no—
120C2×C8×C89—21no—
121C2×C8×C89C22×C412yes—
122C2×C8×C89C22×C412yes—
123C2×C8×C89#4112yes—
124C2×C8×C89#4112yes—
125C4⋊C4×C89—22no—
126C4⋊C4×C89—22no—
127C4⋊C4×C89—22no—
128C4⋊C4×C89—22no—
129C42×C89—44no—
130C42×C89C22×C412yes—
131C42×C89C2×Q812yes—
132C23×C89C22×D411yescharacteristic
133C22×D89—21no—
134C22×D89—21no—
135C22×D89—22no—
136C22×D89—22no—
137C22×D89C2×C4⋊C412yes—
138C2×C4×C89—22no—
139C2×C4×C89—22no—
140C2×C4×C89—42no—
141C2×C4×C89—42no—
142C2×C4×C89—42no—
143C2×C4×C89—42no—
144C2×C4×C89—42no—
145C2×C4×C89—42no—
146C2×C4×C89C2×C4⋊C414yes—
147C2×C4×C89C23×C411yescharacteristic
148C2×C4×C89C22×Q811yescharacteristic
149C2×C4×C89C2×#4111yescharacteristic
150C2×C4×C89C2×#4111yescharacteristic
151C2×C4×C89C2×#4111yescharacteristic
152C2×C4×C89C2×#4111yescharacteristic
153C2×Dic89—21no—
154C2×Dic89—21no—
155C2×Dic89—22no—
156C2×Dic89—22no—
157C2×Dic89—42no—
158C2×Dic89—42no—
159C2×Dic89—42no—
160C2×Dic89—42no—
161C2×Dic89—42no—
162C2×Dic89—42no—
163C2×Dic89C2×C4⋊C412yes—
164C2×Dic89C2×C4⋊C414yes—
165C4×D89—22no—
166C4×D89—22no—
167C4×D89—22no—
168C4×D89—22no—
169C4×D89—48no—
170C8×C89—22no—
171C8×C89#11714yes—
172#10623494—22no—
173#10623494#11714yes—
174C22×C89C22×C4⋊C411yescharacteristic
175C22×C89#53211yescharacteristic
176C22×C89#53211yescharacteristic
177C22×C89#57512yes—
178C22×C89#57512yes—
179C2×D89—24no—
180C2×D89—22no—
181C2×D89—42no—
182C2×D89—41no—
183C2×D89—41no—
184C4×C89—24no—
185C4×C89—82no—
186C4×C89—82no—
187C4×C89C2×#11711yescharacteristic
188C4×C89C2×#11711yescharacteristic
189C4×C89#53311yescharacteristic
190C4×C89#53311yescharacteristic
191Dic89—24no—
192Dic89—22no—
193Dic89—82no—
194Dic89—82no—
195#20825—891nomaximal, Sylow
196C2×C89#228511yescharacteristic
197C2×C89#291112yes—
198C2×C89#318312yes—
199C2×C89#318311yescharacteristic
200C2×C89#318311yescharacteristic
201D89—42no—
202C2×C4×M16—891no—
203#2283—892no—
204#1846—891no—
205#1846—891no—
206#1835—892no—
207#1809—891no—
208#1809—891no—
209#1809—892no—
210#1776—891no—
211#1776—892no—
212#1776D8911yescharacteristic, complemented
213C89#2082511yescharacteristic, complemented, Sylow
214C22×M16—891no—
215C22×M16—891no—
216C22×C42—891no—
217C2×#107—891no—
218C2×#107—892no—
219C2×#107C2×D8911yescharacteristic
220C2×#106—891no—
221C2×#106—892no—
222C2×#106C2×D8911yescharacteristic
223C4×M16—898no—
224C2×#97—891no—
225C2×#97C2×D8911yescharacteristic
226C2×C4×C8—891no—
227C2×C4×C8—891no—
228#383—1782no—
229#382—892no—
230#382—892no—
231#381—1782no—
232#380—892no—
233#380—892no—
234C4×C4⋊C4—892no—
235C4×C4⋊C4C2×D8912yes—
236C4×C22⋊C4—1782no—
237#376—1782no—
238#340—892no—
239#340—892no—
240#340—892no—
241#340C2×D8912yes—
242C23×C4—892no—
243C23×C4—891no—
244C2×M16—894no—
245C2×M16—894no—
246C2×M16—1782no—
247C2×M16—1782no—
248C2×M16—1782no—
249C2×M16—1782no—
250C22×C8—891no—
251C22×C8—891no—
252C22×C8C22×D8911yescharacteristic
253C22×C8C22×D8911yescharacteristic
254C2×C4⋊C4—892no—
255C2×C4⋊C4—892no—
256C2×C4⋊C4C22×D8912yes—
257C2×C4⋊C4C22×D8912yes—
258C2×C22⋊C4—1782no—
259C2×C22⋊C4—1782no—
260C2×C42—22no—
261C2×C42—892no—
262C2×C42—891no—
263C2×C42—894no—
264C2×C42—894no—
265C2×C42—1782no—
266C2×C42C22×D8911yescharacteristic
267#107—22no—
268#107—1784no—
269#107—1782no—
270#106—22no—
271#106—1784no—
272#106—1782no—
273#97—894no—
274#97C4×D8914yes—
275C4×C8—894no—
276C4×C8—894no—
277#95—22no—
278#95—1782no—
279#95—1782no—
280#95—1782no—
281#95—1782no—
282#95—1782no—
283#95—1782no—
284#95—1782no—
285C24—891no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—892no—
289C22×C4—891no—
290C22×C4—898no—
291C22×C4—892no—
292C22×C4—892no—
293C22×C4—3562no—
294C22×C4—3562no—
295C22×C4—1782no—
296C22×C4—1782no—
297C22×C4—1781no—
298C22×C4—1781no—
299C22×C4—1781no—
300C22×C4—1781no—
301C22×C4D4×D8912yes—
302C22×C4C23×D8911yescharacteristic
303M16—1788no—
304M16—1788no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—892no—
308C2×C8—892no—
309C2×C8—892no—
310C2×C8—892no—
311C2×C8—1781no—
312C2×C8—1781no—
313C2×C8C2×C4×D8912yes—
314C2×C8C2×C4×D8912yes—
315C2×C8#1308357412yes—
316C2×C8#1308357412yes—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—1782no—
322C4⋊C4—1782no—
323C4⋊C4—1782no—
324C4⋊C4—1782no—
325C22⋊C4—3564no—
326C22⋊C4—3564no—
327C42—44no—
328C42—892no—
329C42—892no—
330C42—898no—
331C42—3564no—
332C42C2×C4×D8912yes—
333C42Q8×D8912yes—
334C23—892no—
335C23—1782no—
336C23—1782no—
337C23—1781no—
338C23—1781no—
339C23C2×D4×D8911yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—892no—
349C2×C4—894no—
350C2×C4—3568no—
351C2×C4—3562no—
352C2×C4—3562no—
353C2×C4—3562no—
354C2×C4—3562no—
355C2×C4—3562no—
356C2×C4—3562no—
357C2×C4—1782no—
358C2×C4—1782no—
359C2×C4—1781no—
360C2×C4—1781no—
361C2×C4—1782no—
362C2×C4—1782no—
363C2×C4—1782no—
364C2×C4—1782no—
365C2×C4C4⋊C4×D8914yes—
366C2×C4C22×C4×D8911yescharacteristic
367C2×C4C2×Q8×D8911yescharacteristic
368C2×C4#41×D8911yescharacteristic
369C2×C4#41×D8911yescharacteristic
370C2×C4C2×#1308357411yescharacteristic
371C2×C4C2×#1308357411yescharacteristic
372C8—22no—
373C8—894no—
374C8—1782no—
375C8#1566796914yes—
376C22—3562no—
377C22—3561no—
378C22—3561no—
379C22—1782no—
380C22—1784no—
381C22C2×C4⋊C4×D8911yescharacteristic
382C22#1915214511yescharacteristic
383C22#1915214511yescharacteristic
384C22#1915239412yes—
385C22#1915239712yes—
386C4—24no—
387C4—82no—
388C4—82no—
389C4—1782no—
390C4—1784no—
391C4—7122no—
392C4—7122no—
393C4#117×D8911yescharacteristic
394C4C2×#1566796911yescharacteristic
395C4#1915215211yescharacteristic
396C4#1915215211yescharacteristic
397C2—3562no—
398C2#2602265611yescharacteristic
399C2#2602883912yes—
400C2#2603097712yes—
401C2#2603097911yescharacteristic
402C2#2603098411yescharacteristic
subgroupG/H
centerC23C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC89×#1776C2
socleC23×C89C22×D4
prime radicalC23×D89C2×D4
cosocleC22×C4×C89C24
hypercenter#1776D89
hypocenterC89#20825
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
3. #26020415C2semidirect
4. #26020413C2semidirect
5. #26020381C2semidirect
9. #26020150C2semidirect
10. #26020149C2semidirect
11. C89×#1776C2semidirect
19. #19151550C4semidirect
212. #1776D89semidirect
213. C89#20825semidirect

Maximal subgroups

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

subgroupindex
C2×#191515502
#260226482
#260204152
#260204132
#260203812
#260202862
#260202842
#260202822
#260201502
#260201492
C89×#17762
#2082589