Finite Group Databasebeta

← #36556529

Subgroups — #36556529

Order 22784.

906 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151626C211yesmaximal, characteristic
2C2×#19151556C211yesmaximal, characteristic
3#26022659C211yesmaximal, characteristic
4#26020815C214yesmaximal, complemented
5#26020808C214yesmaximal, complemented
6#26019950C211yesmaximal, characteristic
7#26019950C211yesmaximal, characteristic
8C2×#19151353C211yesmaximal, characteristic
9C89×#1728C211yesmaximal, characteristic
10C23×#10623494C2211yescharacteristic
11C23×#10623494C2211yescharacteristic
12C2×#15667971C2211yescharacteristic
13C2×#15667971C2211yescharacteristic
14#19151626—24no—
15C2×#15667941C2212yes—
16C2×#15667941C2212yes—
17#19151556—24no—
18C2×#15667922C2212yes—
19C2×C8×Dic89C2212yes—
20#19151468C411yescharacteristic
21#19151468C411yescharacteristic
22#19151464C411yescharacteristic
23#19151464C411yescharacteristic
24#19151458C414yes—
25C4×C4⋊Dic89C2214yes—
26#19151353C2212yes—
27#19151353C2212yes—
28#19151352C2214yes—
29#19151352C2214yes—
30C22×M16×C89C2211yescharacteristic
31C22×M16×C89C2211yescharacteristic
32C22×C42×C89C2211yescharacteristic
33#336×C89C2214yes—
34C22×#10623494—22no—
35C22×#10623494—22no—
36C22×#10623494—22no—
37C22×#10623494—22no—
38C22×#10623494—22no—
39C22×#10623494D414yes—
40C22×#10623494C2312yes—
41C22×#10623494C2311yescharacteristic
42C22×#10623494C2311yescharacteristic
43#15667971—22no—
44#15667971—22no—
45#15667971—22no—
46#15667971—22no—
47C2×C4⋊Dic89C2×C414yes—
48C2×C4⋊Dic89C2×C411yescharacteristic
49C2×C4⋊Dic89C2×C411yescharacteristic
50C2×C4⋊Dic89C2×C411yescharacteristic
51C2×C4⋊Dic89C2×C411yescharacteristic
52C2×C4⋊Dic89C2×C414yes—
53C2×C22⋊Dic89C2×C412yes—
54C2×C22⋊Dic89C2×C412yes—
55C2×C4×Dic89C2312yes—
56C2×C4×Dic89C2312yes—
57#15667941—24no—
58#15667941—24no—
59#15667922—24no—
60C8×Dic89—24no—
61#15667919C2×C414yes—
62#15667919C2×C414yes—
63C23×C4×C89C2×C412yes—
64C23×C4×C89C2311yescharacteristic
65C2×M16×C89—22no—
66C2×M16×C89—22no—
67C2×M16×C89—22no—
68C2×M16×C89—22no—
69C2×M16×C89—22no—
70C2×M16×C89Q814yes—
71C22×C8×C89C2312yes—
72C22×C8×C89C2312yes—
73C2×C42×C89—22no—
74C2×C42×C89—22no—
75C2×C42×C89C2×C411yescharacteristic
76C2×C42×C89C2×C411yescharacteristic
77C2×C42×C89C2312yes—
78C22×Dic89C22×C412yes—
79C22×Dic89C22×C412yes—
80C22×Dic89C22×C412yes—
81C22×Dic89C22×C412yes—
82C2×#10623494—28no—
83C2×#10623494—21no—
84C2×#10623494—21no—
85C2×#10623494—21no—
86C2×#10623494—21no—
87C2×#10623494—22no—
88C2×#10623494—22no—
89C2×#10623494—22no—
90C2×#10623494—44no—
91C2×#10623494C2×D412yes—
92C2×#10623494C2×D412yes—
93C4⋊Dic89—22no—
94C4⋊Dic89—22no—
95C4⋊Dic89—22no—
96C4⋊Dic89—22no—
97C4⋊Dic89—28no—
98C4⋊Dic89—44no—
99C22⋊Dic89—24no—
100C22⋊Dic89—24no—
101C4×Dic89—22no—
102C4×Dic89—22no—
103C4×Dic89—22no—
104C4×Dic89—22no—
105C24×C89C22×C411yescharacteristic
106C22×C4×C89—24no—
107C22×C4×C89—24no—
108C22×C4×C89—42no—
109C22×C4×C89C22×C412yes—
110C22×C4×C89C22×C412yes—
111C22×C4×C89C22×C411yescharacteristic
112C22×C4×C89C2×D412yes—
113C22×C4×C89C2×Q812yes—
114C22×C4×C89#4112yes—
115C22×C4×C89#4112yes—
116C22×C4×C89#4112yes—
117C22×C4×C89#4112yes—
118C22×C4×C89C2411yescharacteristic
119M16×C89—28no—
120M16×C89—44no—
121C2×C8×C89—22no—
122C2×C8×C89—22no—
123C2×C8×C89—22no—
124C2×C8×C89—22no—
125C2×C8×C89—22no—
126C2×C8×C89C2×Q814yes—
127C42×C89—22no—
128C42×C89—22no—
129C42×C89—22no—
130C42×C89—22no—
131C23×C89—22no—
132C23×C89—22no—
133C23×C89#11712yes—
134C23×C89C4×D412yes—
135C23×C89C4×Q812yes—
136C23×C89C23×C411yescharacteristic
137C2×C4×C89—22no—
138C2×C4×C89—22no—
139C2×C4×C89—22no—
140C2×C4×C89—22no—
141C2×C4×C89—22no—
142C2×C4×C89—48no—
143C2×C4×C89#11712yes—
144C2×C4×C89C4×D412yes—
145C2×C4×C89C4×Q812yes—
146C2×C4×C89C22⋊Q814yes—
147C2×C4×C89#12314yes—
148C2×C4×C89#12514yes—
149C2×C4×C89#12614yes—
150C2×C4×C89#13114yes—
151C2×C4×C89C22×D411yescharacteristic
152C2×C4×C89C22×Q811yescharacteristic
153C2×C4×C89C2×#4111yescharacteristic
154C2×C4×C89C2×#4111yescharacteristic
155C2×C4×C89C2×#4111yescharacteristic
156C2×C4×C89C2×#4111yescharacteristic
157C2×Dic89—22no—
158C2×Dic89—22no—
159C2×Dic89—22no—
160C2×Dic89—22no—
161C2×Dic89—24no—
162C2×Dic89—24no—
163C2×Dic89—24no—
164C2×Dic89—24no—
165C8×C89—24no—
166C8×C89—42no—
167#10623494—22no—
168#10623494—22no—
169#10623494—42no—
170C22×C89—24no—
171C22×C89—22no—
172C22×C89—22no—
173C22×C89—24no—
174C22×C89#38714yes—
175C22×C89C2×#11711yescharacteristic
176C22×C89C2×C4×D411yescharacteristic
177C22×C89C2×C4×Q811yescharacteristic
178C22×C89C2×#13112yes—
179C22×C89#56812yes—
180C4×C89—24no—
181C4×C89—42no—
182C4×C89#38714yes—
183C4×C89C2×C22⋊Q811yescharacteristic
184C4×C89C2×#12311yescharacteristic
185C4×C89C2×#12511yescharacteristic
186C4×C89C2×#12611yescharacteristic
187Dic89—42no—
188Dic89—42no—
189Dic89—42no—
190Dic89—42no—
191#21025—891nomaximal, Sylow
192C2×C89—24no—
193C2×C89C2×#38711yescharacteristic
194C2×C89#292412yes—
195C2×C89#293111yescharacteristic
196C2×C89#293211yescharacteristic
197C2×C89#296212yes—
198C2×#424—891no—
199C2×#405—891no—
200#2285—891no—
201#1917—894no—
202#1916—894no—
203#1729—891no—
204#1729—891no—
205#1728D8911yescharacteristic
206C2×#336—891no—
207C89#2102511yescharacteristic, complemented, Sylow
208C22×M16C2×D8911yescharacteristic
209C22×M16C2×D8911yescharacteristic
210C23×C8—891no—
211C23×C8—891no—
212C2×#117—891no—
213C2×#117—891no—
214C22×C42C2×D8911yescharacteristic
215#424—1784no—
216C2×C4⋊C8—892no—
217C2×C4⋊C8—892no—
218#405—1784no—
219C2×#97—892no—
220C2×C4×C8—892no—
221#384—891no—
222#384—891no—
223#382—891no—
224#382—891no—
225#380—894no—
226C4×C4⋊C4—894no—
227#336—894no—
228#336—894no—
229#336—892no—
230#336—892no—
231#336C2×D8914yes—
232C23×C4C4×D8912yes—
233C23×C4C22×D8911yescharacteristic
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—22no—
239C2×M16Dic17814yes—
240C22×C8—894no—
241C22×C8—891no—
242C22×C8—891no—
243C22×C8—892no—
244C22×C8—1782no—
245C22×C8—1782no—
246C22×C8—1782no—
247C22×C8—1782no—
248C22×C8—1782no—
249C22×C8C22×D8912yes—
250C22×C8C22×D8912yes—
251#117—1782no—
252#117—1782no—
253#117—1782no—
254#117—1782no—
255C2×C4⋊C4—894no—
256C2×C4⋊C4—891no—
257C2×C4⋊C4—891no—
258C2×C4⋊C4—891no—
259C2×C4⋊C4—891no—
260C2×C4⋊C4—894no—
261C2×C22⋊C4—892no—
262C2×C22⋊C4—892no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—892no—
266C2×C42—892no—
267C2×C42C4×D8911yescharacteristic
268C2×C42C4×D8911yescharacteristic
269C2×C42C22×D8912yes—
270C4⋊C8—1784no—
271C4⋊C8—1784no—
272#97—1784no—
273C4×C8—1784no—
274#95—894no—
275#95—894no—
276C24C2×C4×D8911yescharacteristic
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—892no—
281C22×C4—892no—
282C22×C4—892no—
283C22×C4—892no—
284C22×C4C2×C4×D8912yes—
285C22×C4C2×C4×D8912yes—
286C22×C4C2×C4×D8911yescharacteristic
287C22×C4D4×D8912yes—
288C22×C4C2×Dic17812yes—
289C22×C4#1308357312yes—
290C22×C4#1308357312yes—
291C22×C4#1308357412yes—
292C22×C4#1308357512yes—
293C22×C4C23×D8911yescharacteristic
294M16—28no—
295M16—44no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—892no—
302C2×C8—892no—
303C2×C8—3564no—
304C2×C8—1782no—
305C2×C8—1782no—
306C2×C8—1782no—
307C2×C8—1788no—
308C2×C8—1781no—
309C2×C8—1781no—
310C2×C8—1781no—
311C2×C8—1781no—
312C2×C8C2×Dic17814yes—
313C4⋊C4—3564no—
314C4⋊C4—1782no—
315C4⋊C4—1782no—
316C4⋊C4—1782no—
317C4⋊C4—1782no—
318C4⋊C4—1788no—
319C22⋊C4—1784no—
320C22⋊C4—1784no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—1782no—
326C42—1782no—
327C42—1782no—
328C42—1782no—
329C23—22no—
330C23—22no—
331C23#1566796812yes—
332C23#1566797412yes—
333C23C4×Dic17812yes—
334C23C22×C4×D8911yescharacteristic
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—48no—
341C2×C4—1782no—
342C2×C4—1782no—
343C2×C4—1782no—
344C2×C4—1782no—
345C2×C4—1784no—
346C2×C4—1784no—
347C2×C4—1784no—
348C2×C4—1784no—
349C2×C4#1566796812yes—
350C2×C4#1566797412yes—
351C2×C4C4×Dic17812yes—
352C2×C4#1566798914yes—
353C2×C4#1566799414yes—
354C2×C4#1566800314yes—
355C2×C4#1566800714yes—
356C2×C4#1566802014yes—
357C2×C4C2×D4×D8911yescharacteristic
358C2×C4C22×Dic17811yescharacteristic
359C2×C4C2×#1308357311yescharacteristic
360C2×C4C2×#1308357311yescharacteristic
361C2×C4C2×#1308357411yescharacteristic
362C2×C4C2×#1308357511yescharacteristic
363C8—24no—
364C8—42no—
365C8—3562no—
366C8—1782no—
367C8—1782no—
368C22—24no—
369C22—24no—
370C22—22no—
371C22—22no—
372C22#1915148114yes—
373C22C2×#1566796811yescharacteristic
374C22C2×#1566797411yescharacteristic
375C22C2×C4×Dic17811yescharacteristic
376C22C2×#1566802012yes—
377C22#1915236412yes—
378C4—24no—
379C4—42no—
380C4—3562no—
381C4—3562no—
382C4—3562no—
383C4—3562no—
384C4#1915148114yes—
385C4C2×#1566798911yescharacteristic
386C4C2×#1566799411yescharacteristic
387C4C2×#1566800311yescharacteristic
388C4C2×#1566800711yescharacteristic
389C2—24no—
390C2C2×#1915148111yescharacteristic
391C2#2602891212yes—
392C2#2602896311yescharacteristic
393C2#2602896911yescharacteristic
394C2#2602919912yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC89×#1728C2
prime radicalC24×C89C22×C4
cosocleC22×C4×C89C24
hypercenter#1728D89
hypocenterC89#21025
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ #1728

Complements

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

NH ≅ G/Ntype
4. #26020815C2semidirect
5. #26020808C2semidirect
207. C89#21025semidirect

Maximal subgroups

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

subgroupindex
C2×#191516262
C2×#191515562
#260226592
#260208152
#260208082
#260199502
C2×#191513532
C89×#17282
#2102589