Finite Group Databasebeta

← #36520151

Subgroups — #36520151

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029219C211yesmaximal, characteristic
2C4×#15668019C211yesmaximal, characteristic
3#26018555C212yesmaximal, complemented
4#26018547C212yesmaximal, complemented
5#26018542C211yesmaximal, characteristic, complemented
6#26018541C211yesmaximal, characteristic, complemented
7#26018536C211yesmaximal, characteristic, complemented
8#26018533C212yesmaximal, complemented
9M16×#10623494C212yesmaximal, complemented
10C2×C8×#10623494C211yesmaximal, characteristic, complemented
11C89×#2965C211yesmaximal, characteristic
12C2×#15668019C412yes—
13#19151729C412yes—
14#19151676C412yes—
15#19151670C414yes—
16#19151651C2212yes—
17#19151651C2211yescharacteristic
18#19151554C414yes—
19#19151554C412yes—
20C4×#13083558C2211yescharacteristic
21C4×#13083558C2212yes—
22C4×#13083558C2212yes—
23C2×C4×#10623494C2211yescharacteristic
24C2×C4×#10623494C2212yes—
25C2×C4×#10623494C2211yescharacteristic
26#19151320C2212yes—
27#19151320C2212yes—
28#19151320C2212yes—
29#19151319C2212yes—
30#19151319C2212yes—
31C8×#10623494C2211yescharacteristic
32C8×#10623494C2211yescharacteristic
33C8×#10623494C2211yescharacteristic
34C8×#10623494C2211yescharacteristic
35C8×#10623494C2212yes—
36C4×#41×C89C2211yescharacteristic
37#446×C89C412yes—
38#436×C89C412yes—
39#435×C89C414yes—
40#431×C89C2212yes—
41#431×C89C2211yescharacteristic
42C4×M16×C89C2212yes—
43C2×C4×C8×C89C2211yescharacteristic
44#15668019—28no—
45C2×#13083558C2×C414yes—
46C2×#13083558C2×C414yes—
47C2×#13083558C2×C412yes—
48C22×#10623494C2×C412yes—
49C22×#10623494C2×C414yes—
50C22×#10623494C2×C412yes—
51C4⋊#10623494C2×C414yes—
52C4⋊#10623494C2×C412yes—
53C22⋊#10623494C2×C412yes—
54C22⋊#10623494C2×C414yes—
55#15667923C2×C414yes—
56#15667923C2×C412yes—
57#15667923C2×C414yes—
58C4×#10623494C2×C412yes—
59C4×#10623494C2×C414yes—
60C4×#10623494C2×C412yes—
61C4×#10623494C2311yescharacteristic
62C4×#10623494C2312yes—
63C4×#10623494C2311yescharacteristic
64C4×#10623494C2311yescharacteristic
65C4×#10623494C2311yescharacteristic
66C4×#10623494C2312yes—
67C2×#41×C89C2×C411yescharacteristic
68C2×M16×C89C2×C414yes—
69C22×C8×C89C2×C412yes—
70C4×Q8×C89C2×C411yescharacteristic
71C4×Q8×C89C2×C411yescharacteristic
72C4×D4×C89C2×C412yes—
73C4×D4×C89C2×C411yescharacteristic
74C4×D4×C89C2×C412yes—
75C4×D4×C89C2×C411yescharacteristic
76#117×C89C2×C412yes—
77#117×C89C2×C411yescharacteristic
78C2×C42×C89C2311yescharacteristic
79C2×C42×C89C2312yes—
80C4⋊C8×C89C2×C414yes—
81C4⋊C8×C89C2×C412yes—
82C22⋊C8×C89C2×C412yes—
83C22⋊C8×C89C2×C414yes—
84#97×C89C2×C414yes—
85C4×C8×C89C2×C412yes—
86C4×C8×C89C2311yescharacteristic
87C4×C8×C89C2311yescharacteristic
88C4×C8×C89C2312yes—
89#13083558—24no—
90#13083558—28no—
91#13083558—28no—
92C2×#10623494—28no—
93C2×#10623494—24no—
94C2×#10623494M1618yescomplemented
95C2×#10623494C22×C414yes—
96C2×#10623494C22×C412yes—
97C2×#10623494C22×C414yes—
98C2×#10623494C22×C412yes—
99C2×#10623494C22×C412yes—
100C2×#10623494C22×C412yes—
101#41×C89—24no—
102C2×Q8×C89C4211yescharacteristic
103C2×D4×C89C4212yes—
104C2×D4×C89C4211yescharacteristic
105C22×C4×C89C22×C412yes—
106C22×C4×C89C22×C411yescharacteristic
107C22×C4×C89C22×C412yes—
108C22×C4×C89C22×C411yescharacteristic
109C22×C4×C89C22×C412yes—
110C22×C4×C89C22×C411yescharacteristic
111M16×C89—28no—
112C2×C8×C89—24no—
113C2×C8×C89C22×C412yes—
114C2×C8×C89C22×C414yes—
115C2×C8×C89C22×C412yes—
116C4⋊C4×C89C4212yes—
117C4⋊C4×C89C4212yes—
118C4⋊C4×C89C4211yescharacteristic
119C4⋊C4×C89C4211yescharacteristic
120C22⋊C4×C89C4212yes—
121C22⋊C4×C89C4212yes—
122C22⋊C4×C89C4212yes—
123C42×C89C22×C411yescharacteristic
124C42×C89C22×C412yes—
125C42×C89C22×C411yescharacteristic
126C42×C89C22×C412yes—
127C42×C89C22×C411yescharacteristic
128C42×C89C22×C412yes—
129C42×C89C2411yescharacteristic
130C23×C89C2×C4212yes—
131C23×C89C2×C4211yescharacteristic
132Q8×C89—22no—
133D4×C89—24no—
134D4×C89—22no—
135C2×C4×C89—24no—
136C2×C4×C89—22no—
137C2×C4×C89—24no—
138C2×C4×C89—22no—
139C2×C4×C89—24no—
140C2×C4×C89C2×C4211yescharacteristic
141C2×C4×C89C2×C4212yes—
142C2×C4×C89C2×C4211yescharacteristic
143C2×C4×C89C2×C4212yes—
144C2×C4×C89C2×C4212yes—
145C2×C4×C89C2×C4211yescharacteristic
146C2×C4×C89C2×M1614yes—
147C2×C4×C89C23×C411yescharacteristic
148C2×C4×C89C23×C411yescharacteristic
149C2×C4×C89C23×C411yescharacteristic
150C8×C89—22no—
151C8×C89—24no—
152C8×C89#13114yescomplemented
153#10623494—24no—
154#10623494—22no—
155#10623494—24no—
156#10623494C2×M1614yes—
157#10623494C2×M1614yes—
158#10623494#13114yescomplemented
159C22×C89—21no—
160C22×C89—21no—
161C22×C89—22no—
162C22×C89—22no—
163C22×C89—22no—
164C22×C89C4×M1612yes—
165C22×C89C22×C4211yescharacteristic
166C4×C89—22no—
167C4×C89—21no—
168C4×C89—22no—
169C4×C89—21no—
170C4×C89—22no—
171C4×C89C4×M1612yes—
172C4×C89C22×M1612yes—
173C4×C89C2×#13112yes—
174C4×C89C2×#13112yes—
175#17335—891nomaximal, Sylow
176C2×C89—22no—
177C2×C89—42no—
178C2×C89C2×C4×M1611yescharacteristic
179C2×C89C4×#13111yescharacteristic
180C2×C89C4×#13111yescharacteristic
181#2965—891no—
182#2965D8911yescharacteristic
183C4×#131—891no—
184#1455—892no—
185#1453—892no—
186#1451—891no—
187#1451—891no—
188#1450—891no—
189C8×M16—892no—
190C8×M16—892no—
191C2×C82—891no—
192C89#1733511yescharacteristic, complemented, Sylow
193C2×#131—892no—
194C4×#41C2×D8911yescharacteristic
195#446—892no—
196#446Dic8912yes—
197#436—892no—
198#436Dic8912yes—
199#435—894no—
200#435Dic8914yes—
201#431—892no—
202#431—891no—
203#431C2×D8912yes—
204#431C2×D8911yescharacteristic
205#405—892no—
206#405—894no—
207C4×M16—891no—
208C4×M16—892no—
209C4×M16—892no—
210C4×M16C2×D8912yes—
211C2×C4×C8—891no—
212C2×C4×C8—892no—
213C2×C4×C8—891no—
214C2×C4×C8C2×D8911yescharacteristic
215#322—892no—
216#322—892no—
217#322—892no—
218#322—892no—
219#322—892no—
220C82—891no—
221C82—891no—
222C82—891no—
223C82—891no—
224C82—892no—
225C2×#41C4×D8911yescharacteristic
226#131—1788no—
227C2×M16—894no—
228C2×M16—892no—
229C2×M16—894no—
230C2×M16C2×Dic8914yes—
231C22×C8—892no—
232C22×C8—894no—
233C22×C8—892no—
234C22×C8C2×Dic8912yes—
235C4×Q8C4×D8911yescharacteristic
236C4×Q8C2×Dic8911yescharacteristic
237C4×D4C4×D8912yes—
238C4×D4C4×D8911yescharacteristic
239C4×D4C2×Dic8912yes—
240C4×D4C2×Dic8911yescharacteristic
241#117C4×D8912yes—
242#117C4×D8911yescharacteristic
243C2×C42C22×D8911yescharacteristic
244C2×C42C22×D8912yes—
245C4⋊C8—894no—
246C4⋊C8—892no—
247C4⋊C8C2×Dic8914yes—
248C4⋊C8C2×Dic8912yes—
249C22⋊C8—892no—
250C22⋊C8—894no—
251C22⋊C8C2×Dic8912yes—
252C22⋊C8C2×Dic8914yes—
253#97—894no—
254#97—892no—
255#97—894no—
256#97C2×Dic8914yes—
257C4×C8—892no—
258C4×C8—894no—
259C4×C8—892no—
260C4×C8—891no—
261C4×C8—891no—
262C4×C8—891no—
263C4×C8—891no—
264C4×C8—892no—
265C4×C8—892no—
266C4×C8C2×Dic8912yes—
267C4×C8C22×D8911yescharacteristic
268C4×C8C22×D8911yescharacteristic
269C4×C8C22×D8912yes—
270#41—24no—
271C2×Q8C4×Dic8911yescharacteristic
272C2×D4C4×Dic8912yes—
273C2×D4C4×Dic8911yescharacteristic
274C22×C4C2×C4×D8911yescharacteristic
275C22×C4C2×C4×D8912yes—
276C22×C4C2×C4×D8912yes—
277C22×C4C2×C4×D8911yescharacteristic
278C22×C4C22×Dic8911yescharacteristic
279C22×C4C22×Dic8912yes—
280M16—28no—
281M16—1788no—
282M16—1788no—
283M16—1784no—
284C2×C8—24no—
285C2×C8—898no—
286C2×C8—892no—
287C2×C8—892no—
288C2×C8—892no—
289C2×C8—894no—
290C2×C8—892no—
291C2×C8—894no—
292C2×C8—1788no—
293C2×C8—1784no—
294C2×C8C22×Dic8912yes—
295C2×C8C22×Dic8914yes—
296C2×C8C22×Dic8912yes—
297C4⋊C4C4×Dic8912yes—
298C4⋊C4C4×Dic8911yescharacteristic
299C4⋊C4C4×Dic8912yes—
300C4⋊C4C4×Dic8911yescharacteristic
301C22⋊C4C4×Dic8912yes—
302C22⋊C4C4×Dic8912yes—
303C22⋊C4C4×Dic8912yes—
304C42C2×C4×D8911yescharacteristic
305C42C2×C4×D8912yes—
306C42C2×C4×D8911yescharacteristic
307C42C2×C4×D8912yes—
308C42C22×Dic8911yescharacteristic
309C42C22×Dic8912yes—
310C42C23×D8911yescharacteristic
311C23C2×C4×Dic8912yes—
312C23C2×C4×Dic8911yescharacteristic
313Q8—22no—
314D4—24no—
315D4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4C2×C4×Dic8911yescharacteristic
322C2×C4C2×C4×Dic8912yes—
323C2×C4C2×C4×Dic8911yescharacteristic
324C2×C4C2×C4×Dic8912yes—
325C2×C4C2×C4×Dic8912yes—
326C2×C4C2×C4×Dic8911yescharacteristic
327C2×C4M16×D8914yes—
328C2×C4C22×C4×D8911yescharacteristic
329C2×C4C22×C4×D8911yescharacteristic
330C2×C4C23×Dic8911yescharacteristic
331C8—22no—
332C8—24no—
333C8—894no—
334C8—894no—
335C8—894no—
336C8—1784no—
337C8—1782no—
338C8—1784no—
339C8#1566801914yescomplemented
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22M16×Dic8912yes—
346C22C22×C4×Dic8911yescharacteristic
347C4—22no—
348C4—21no—
349C4—22no—
350C4—21no—
351C4—22no—
352C4M16×Dic8912yes—
353C4C2×M16×D8912yes—
354C4#131×D8912yes—
355C4C2×#1566801912yes—
356C2—22no—
357C2—42no—
358C2C2×M16×Dic8911yescharacteristic
359C2#131×Dic8911yescharacteristic
360C2C4×#1566801911yescharacteristic
subgroupG/H
center, FrattiniC42C23×D89
commutator, socleC22×C89C22×C42
FittingC89×#2965C2
prime radicalC2×#41×C89C2×C4
cosocleC42×C89C24
hypercenter#2965D89
hypocenterC89#17335
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
3. #26018555C2semidirect
4. #26018547C2semidirect
5. #26018542C2semidirect
6. #26018541C2semidirect
7. #26018536C2semidirect
8. #26018533C2semidirect
9. M16×#10623494C2semidirect
10. C2×C8×#10623494C2semidirect
94. C2×#10623494M16semidirect
152. C8×C89#131semidirect
158. #10623494#131semidirect
192. C89#17335semidirect
339. C8#15668019semidirect

Maximal subgroups

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

subgroupindex
#260292192
C4×#156680192
#260185552
#260185472
#260185422
#260185412
#260185362
#260185332
M16×#106234942
C2×C8×#106234942
C89×#29652
#1733589