Finite Group Databasebeta

← #36554938

Subgroups — #36554938

Order 22784.

854 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151660C211yesmaximal, characteristic
2C2×#19151657C211yesmaximal, characteristic
3#26022591C211yesmaximal, characteristic
4#26020405C212yesmaximal, complemented
5#26020367C212yesmaximal, complemented
6#26020235C212yesmaximal
7#26020229C212yesmaximal
8#26020109C211yesmaximal, characteristic, complemented
9#26020106C211yesmaximal, characteristic, complemented
10C4×#15667941C211yesmaximal, characteristic, complemented
11C89×#1768C211yesmaximal, characteristic, complemented
12C2×#15667969C2211yescharacteristic
13C2×#15667969C2211yescharacteristic
14C2×C42×D89C2211yescharacteristic
15#19151660—24no—
16#19151657—24no—
17C2×C4⋊#10623494C2212yes—
18C2×#15667941C2212yes—
19C2×#15667941C2212yes—
20C2×#15667941C2212yes—
21C2×#15667941C2212yes—
22C2×#15667924C2212yes—
23C2×#15667924C2212yes—
24C2×#15667923C2211yescharacteristic
25C2×C4×#10623494C2211yescharacteristic
26#19151449C412yes—
27#19151449C412yes—
28#19151445C412yes—
29#19151445C412yes—
30#19151444C2211yescharacteristic
31C42×Dic89C2211yescharacteristic
32C42×Dic89C2211yescharacteristic
33#19151353C2212yes—
34#19151352C2212yes—
35#19151352C2212yes—
36C2×C4⋊C8×C89C2212yes—
37C2×#97×C89C2211yescharacteristic
38C2×C4×C8×C89C2211yescharacteristic
39#376×C89C2211yescharacteristic
40#336×C89C2212yes—
41C22×C4×D89C2×C412yes—
42C22×C4×D89C2311yescharacteristic
43C22×#10623494C2312yes—
44C22×#10623494C2312yes—
45#15667969—24no—
46#15667969D414yes—
47#15667969Q814yes—
48C2×#13083553C2×C411yescharacteristic
49C2×#13083553C2×C411yescharacteristic
50C2×#13083553C2×C412yes—
51C2×#13083551C2×C411yescharacteristic
52C2×#13083551C2×C411yescharacteristic
53C2×#13083551C2×C412yes—
54C2×C4×Dic89C2×C412yes—
55C2×C4×Dic89C2×C412yes—
56C2×C4×Dic89C2×C412yes—
57C2×C4×Dic89C2×C412yes—
58C2×C4×Dic89C2×C412yes—
59C2×C4×Dic89C2311yescharacteristic
60C2×C4×Dic89C2311yescharacteristic
61C2×C4×Dic89C2311yescharacteristic
62C42×D89—24no—
63C4⋊#10623494—24no—
64#15667941—24no—
65#15667941—24no—
66#15667941—24no—
67#15667941—24no—
68#15667924—24no—
69#15667924—24no—
70#15667923—22no—
71C4×#10623494—22no—
72#15667919C2×C412yes—
73#15667919C2×C412yes—
74C22×C8×C89C2312yes—
75C22×C8×C89C2312yes—
76C2×C42×C89C2311yescharacteristic
77C2×C42×C89C2311yescharacteristic
78C2×C42×C89C2311yescharacteristic
79C4⋊C8×C89—24no—
80#97×C89—22no—
81C4×C8×C89—22no—
82#95×C89C2×C412yes—
83#95×C89C2×C412yes—
84C23×D89C22×C411yescharacteristic
85C22×Dic89C22×C411yescharacteristic
86C22×Dic89C22×C412yes—
87C22×Dic89C22×C411yescharacteristic
88C22×Dic89C22×C411yescharacteristic
89C22×Dic89C22×C412yes—
90C2×C4×D89—22no—
91C2×C4×D89—22no—
92C2×C4×D89—24no—
93C2×C4×D89—24no—
94C2×C4×D89—42no—
95C2×C4×D89C2×D412yes—
96C2×C4×D89C2×Q812yes—
97C2×#10623494—22no—
98C2×#10623494—22no—
99C2×#10623494—22no—
100C2×#10623494—22no—
101C2×#10623494—22no—
102C2×#10623494—22no—
103#13083553—22no—
104#13083553—22no—
105#13083553C4⋊C418yes—
106#13083551—22no—
107#13083551—22no—
108#13083551C4⋊C418yes—
109C4×Dic89—24no—
110C4×Dic89—22no—
111C4×Dic89—24no—
112C4×Dic89—24no—
113C4×Dic89—42no—
114C4×Dic89—42no—
115C4×Dic89C2×D412yes—
116C4×Dic89C2×Q812yes—
117C4×Dic89#4114yes—
118C22×C4×C89C22×C411yescharacteristic
119C22×C4×C89C22×C411yescharacteristic
120C22×C4×C89C22×C412yes—
121C22×C4×C89C22×C412yes—
122C22×C4×C89C2411yescharacteristic
123C2×C8×C89—22no—
124C2×C8×C89—22no—
125C2×C8×C89—22no—
126C2×C8×C89—22no—
127C2×C8×C89—22no—
128C2×C8×C89—22no—
129C42×C89—22no—
130C42×C89C2×D412yes—
131C42×C89C2×Q812yes—
132C42×C89#4114yes—
133C23×C89C23×C411yescharacteristic
134C22×D89—21no—
135C22×D89—21no—
136C22×D89—22no—
137C22×D89—22no—
138C22×D89C2×C4⋊C412yes—
139C2×C4×C89—22no—
140C2×C4×C89—22no—
141C2×C4×C89—22no—
142C2×C4×C89—22no—
143C2×C4×C89—21no—
144C2×C4×C89—21no—
145C2×C4×C89—22no—
146C2×C4×C89—22no—
147C2×C4×C89C2×C4⋊C414yes—
148C2×C4×C89#11714yes—
149C2×C4×C89C22×D411yescharacteristic
150C2×C4×C89C22×Q811yescharacteristic
151C2×C4×C89C2×#4112yes—
152C2×C4×C89C2×#4112yes—
153C2×Dic89—22no—
154C2×Dic89—21no—
155C2×Dic89—21no—
156C2×Dic89—22no—
157C2×Dic89—22no—
158C2×Dic89—22no—
159C2×Dic89—22no—
160C2×Dic89—22no—
161C2×Dic89C2×C4⋊C414yes—
162C2×Dic89C2×C4⋊C412yes—
163C2×Dic89#11712yes—
164C2×Dic89#11712yes—
165C2×Dic89#13114yes—
166C4×D89—24no—
167C4×D89—24no—
168C4×D89—48no—
169C8×C89—42no—
170C8×C89—42no—
171#10623494—42no—
172#10623494—42no—
173C22×C89C22×C4⋊C411yescharacteristic
174C22×C89C2×#11711yescharacteristic
175C22×C89C2×#11711yescharacteristic
176C22×C89C2×#13112yes—
177C22×C89#56812yes—
178C2×D89—24no—
179C2×D89—22no—
180C2×D89—41no—
181C2×D89—41no—
182C2×D89—42no—
183C4×C89—24no—
184C4×C89—24no—
185C4×C89—41no—
186C4×C89—41no—
187C4×C89#53212yes—
188C4×C89#53312yes—
189Dic89—24no—
190Dic89—22no—
191Dic89—22no—
192Dic89—24no—
193Dic89—22no—
194#20813—891nomaximal, Sylow
195C2×C89#228511yescharacteristic
196C2×C89#290612yes—
197C2×C89#292211yescharacteristic
198C2×C89#292311yescharacteristic
199C2×C89#292412yes—
200D89—42no—
201C2×#433—891no—
202C2×#433—891no—
203C4×#117—891no—
204#1843—892no—
205#1831—892no—
206#1798—892no—
207#1797—892no—
208#1768—891no—
209#1768D8911yescharacteristic, complemented
210#1767—891no—
211C4×C4⋊C8—891no—
212C89#2081311yescharacteristic, complemented, Sylow
213C2×#117—891no—
214C2×#117—891no—
215C22×C42—891no—
216#433—1784no—
217#433—1784no—
218C2×C4⋊C8—892no—
219C2×C4⋊C8—892no—
220C2×C4⋊C8—892no—
221C2×C4⋊C8—892no—
222C2×C4⋊C8—892no—
223C2×C4⋊C8C2×D8912yes—
224C2×C22⋊C8—892no—
225C2×C22⋊C8—892no—
226C2×#97—891no—
227C2×#97C2×D8911yescharacteristic
228C2×C4×C8—891no—
229C2×C4×C8C2×D8911yescharacteristic
230C4×C4⋊C4—892no—
231C4×C4⋊C4—892no—
232C4×C22⋊C4—892no—
233C4×C22⋊C4—892no—
234#376—891no—
235#376C2×D8911yescharacteristic
236C43—891no—
237C43—891no—
238#336—892no—
239#336—892no—
240#336—892no—
241#336C2×D8912yes—
242C23×C4—891no—
243C23×C4—892no—
244C22×C8—892no—
245C22×C8—892no—
246C22×C8C22×D8912yes—
247C22×C8C22×D8912yes—
248#117—894no—
249#117—894no—
250#117—1784no—
251C2×C4⋊C4—892no—
252C2×C4⋊C4—891no—
253C2×C4⋊C4—891no—
254C2×C22⋊C4—892no—
255C2×C22⋊C4—891no—
256C2×C22⋊C4—891no—
257C2×C42—891no—
258C2×C42—891no—
259C2×C42—892no—
260C2×C42—892no—
261C2×C42—892no—
262C2×C42—892no—
263C2×C42—892no—
264C2×C42—891no—
265C2×C42—1784no—
266C2×C42C22×D8911yescharacteristic
267C2×C42C22×D8911yescharacteristic
268C2×C42C22×D8911yescharacteristic
269C4⋊C8—24no—
270C4⋊C8—1784no—
271C4⋊C8—1784no—
272C4⋊C8—1784no—
273C4⋊C8—1784no—
274C4⋊C8—1784no—
275C22⋊C8—1784no—
276C22⋊C8—1784no—
277#97—22no—
278#97—1782no—
279C4×C8—22no—
280C4×C8—1782no—
281#95—892no—
282#95—892no—
283#95C4×D8912yes—
284#95C4×D8912yes—
285C24—891no—
286C22×C4—892no—
287C22×C4—891no—
288C22×C4—891no—
289C22×C4—892no—
290C22×C4—891no—
291C22×C4—892no—
292C22×C4—892no—
293C22×C4—3562no—
294C22×C4—1784no—
295C22×C4—1784no—
296C22×C4—1782no—
297C22×C4—1782no—
298C22×C4C2×C4×D8911yescharacteristic
299C22×C4C2×C4×D8911yescharacteristic
300C22×C4C2×C4×D8912yes—
301C22×C4C2×C4×D8912yes—
302C22×C4C23×D8911yescharacteristic
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—1782no—
310C2×C8—1782no—
311C2×C8—1782no—
312C2×C8—1782no—
313C2×C8—1782no—
314C2×C8—1782no—
315C4⋊C4—898no—
316C4⋊C4—1782no—
317C4⋊C4—1782no—
318C22⋊C4—898no—
319C22⋊C4—1782no—
320C22⋊C4—1782no—
321C42—22no—
322C42—892no—
323C42—892no—
324C42—894no—
325C42—3562no—
326C42—3562no—
327C42—1784no—
328C42—1784no—
329C42—1784no—
330C42—1782no—
331C42D4×D8912yes—
332C42Q8×D8912yes—
333C42#1308357414yes—
334C23—892no—
335C23—1781no—
336C23—1781no—
337C23—1782no—
338C23—1782no—
339C23C22×C4×D8911yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—894no—
349C2×C4—892no—
350C2×C4—892no—
351C2×C4—892no—
352C2×C4—894no—
353C2×C4—3568no—
354C2×C4—1781no—
355C2×C4—1781no—
356C2×C4—1782no—
357C2×C4—1782no—
358C2×C4—1782no—
359C2×C4—1782no—
360C2×C4—1782no—
361C2×C4—1782no—
362C2×C4—1784no—
363C2×C4—1784no—
364C2×C4C4⋊C4×D8914yes—
365C2×C4#1566796914yes—
366C2×C4C2×D4×D8911yescharacteristic
367C2×C4C2×Q8×D8911yescharacteristic
368C2×C4#41×D8912yes—
369C2×C4C2×#1308357412yes—
370C8—42no—
371C8—42no—
372C8—3562no—
373C8—3562no—
374C22—3561no—
375C22—3561no—
376C22—3562no—
377C22—1782no—
378C22—1784no—
379C22C2×C4⋊C4×D8911yescharacteristic
380C22#117×D8911yescharacteristic
381C22C2×#1566796911yescharacteristic
382C22#131×D8912yes—
383C22#1915236312yes—
384C4—24no—
385C4—24no—
386C4—41no—
387C4—41no—
388C4—1782no—
389C4—1784no—
390C4—1782no—
391C4—1782no—
392C4—1784no—
393C4#1915214512yes—
394C4#1915215212yes—
395C2—3562no—
396C2#2602265611yescharacteristic
397C2#2602881912yes—
398C2#2602889311yescharacteristic
399C2#2602890011yescharacteristic
400C2#2602890712yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC89×#1768C2
prime radicalC23×D89C22×C4
cosocleC22×C4×C89C24
hypercenter#1768D89
hypocenterC89#20813
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ #1768

Complements

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

NH ≅ G/Ntype
4. #26020405C2semidirect
5. #26020367C2semidirect
8. #26020109C2semidirect
9. #26020106C2semidirect
10. C4×#15667941C2semidirect
11. C89×#1768C2semidirect
209. #1768D89semidirect
212. C89#20813semidirect

Maximal subgroups

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

subgroupindex
C2×#191516602
C2×#191516572
#260225912
#260204052
#260203672
#260202352
#260202292
#260201092
#260201062
C4×#156679412
C89×#17682
#2081389