Finite Group Databasebeta

← #36707766

Subgroups — #36707766

Order 22784.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152033C211yesmaximal, characteristic
2#26029105C211yesmaximal, characteristic, complemented
3#26022126C212yesmaximal, complemented
4#26022114C212yesmaximal, complemented
5C4×#15667955C212yesmaximal, complemented
6C4×#15667953C212yesmaximal, complemented
7#26022038C211yesmaximal, characteristic, complemented
8#26021959C211yesmaximal, characteristic, complemented
9C2×#19151398C211yesmaximal, characteristic, complemented
10#26021849C211yesmaximal, characteristic, complemented
11C89×#2951C211yesmaximal, characteristic, complemented
12C22×D712C2211yescharacteristic
13#19152072—21no—
14#19152033C418yescomplemented
15C2×#15667955C2212yes—
16C2×#15667953C2212yes—
17C2×#15667952C2211yescharacteristic
18C22×#13083549C2211yescharacteristic
19#19151711—22no—
20C4×D712C2212yes—
21C4×D712C2212yes—
22#19151636C2211yescharacteristic
23#19151604—21no—
24#19151425C2212yescomplemented
25#19151423C2212yescomplemented
26#19151402C2211yescharacteristic, complemented
27#19151402C2211yescharacteristic, complemented
28#19151399C2211yescharacteristic, complemented
29#19151399C2211yescharacteristic, complemented
30#19151398C2212yes—
31#19151398C2212yes—
32#19151374C2212yescomplemented
33C4×#13083549C2212yescomplemented
34C2×#135×C89C2211yescharacteristic
35#520×C89—21no—
36#440×C89—22no—
37C4×Q16×C89C2212yescomplemented
38C4×D8×C89C2212yescomplemented
39C2×#107×C89C2211yescharacteristic
40#417×C89—21no—
41#405×C89C2211yescharacteristic
42C22×D356—21no—
43C2×D712—21no—
44C2×D712—22no—
45C2×D712C2×C414yescomplemented
46C2×D712C2312yes—
47C4×D356—22no—
48C4×D356—22no—
49#15667974—41no—
50#15667974—41no—
51#15667971—21no—
52C22⋊C4×D89—42no—
53#15667955C2×C418yescomplemented
54#15667953C2×C418yescomplemented
55#15667952C2×C414yescomplemented
56C2×#13083549—22no—
57C2×#13083549C2311yescharacteristic
58C2×#13083549C2311yescharacteristic
59C2×#13083549C2312yes—
60#15667946C2311yescharacteristic
61#15667946C2311yescharacteristic
62#15667944—21no—
63#15667934—22no—
64#15667934—22no—
65C2×#41×C89—21no—
66#135×C89—22no—
67#135×C89C2×C414yescomplemented
68C2×Q16×C89C2311yescharacteristic
69C2×SD16×C89—21no—
70C2×D8×C89C2311yescharacteristic
71C2×M16×C89D411yescharacteristic
72C22×C8×C89C2311yescharacteristic
73C4×Q8×C89—22no—
74C4×D4×C89—22no—
75C4×D4×C89—42no—
76#117×C89—41no—
77#117×C89D411yescharacteristic
78C2×C4⋊C4×C89—21no—
79C2×C4⋊C4×C89—41no—
80#107×C89—21no—
81#107×C89C2312yes—
82#103×C89—22no—
83#102×C89—22no—
84#97×C89D412yes—
85C4×C8×C89C2312yes—
86C23×D89—41no—
87C2×D356—24no—
88C2×D356—22no—
89C2×D356—41no—
90C2×D356—42no—
91C2×C4×D89—42no—
92C2×C4×D89—42no—
93D712—24no—
94D712—22no—
95D712C22×C414yes—
96C4⋊Dic89—21no—
97C4⋊Dic89—21no—
98C22⋊Dic89—41no—
99#13083551—42no—
100#13083551—42no—
101C4×Dic89—41no—
102#13083549—21no—
103#13083549—21no—
104#13083549C22×C414yes—
105#41×C89—24no—
106#41×C89—42no—
107C2×Q8×C89—21no—
108C2×D4×C89—21no—
109C2×D4×C89—41no—
110C22×C4×C89—41no—
111C22×C4×C89—41no—
112C22×C4×C89—42no—
113C22×C4×C89C2×D411yescharacteristic
114Q16×C89—21no—
115Q16×C89C22×C412yes—
116SD16×C89—24no—
117D8×C89—21no—
118D8×C89C22×C412yes—
119M16×C89—22no—
120C2×C8×C89C22×C412yes—
121C2×C8×C89C2×D412yes—
122C2×C8×C89C2×D411yescharacteristic
123C2×C8×C89#4112yes—
124C2×C8×C89C2411yescharacteristic
125C4⋊C4×C89—22no—
126C4⋊C4×C89—41no—
127C4⋊C4×C89—42no—
128C4⋊C4×C89D811yescharacteristic
129C4⋊C4×C89D811yescharacteristic
130C22⋊C4×C89—42no—
131C22⋊C4×C89D812yes—
132C42×C89—42no—
133C42×C89C2×D412yes—
134C23×C89—41no—
135C23×C89C2×D811yescharacteristic
136C22×D89—42no—
137C22×D89—44no—
138C22×D89—161no—
139C22×D89—161no—
140Q8×C89—22no—
141Q8×C89—41no—
142D4×C89—22no—
143D4×C89—44no—
144D4×C89—41no—
145C2×C4×C89—41no—
146C2×C4×C89—44no—
147C2×C4×C89—42no—
148C2×C4×C89—162no—
149C2×C4×C89—161no—
150C2×C4×C89C4×D412yes—
151C2×C4×C89C2×D812yes—
152C2×C4×C89C2×D812yes—
153C2×C4×C89C2×D811yescharacteristic
154C2×C4×C89#13512yes—
155C2×C4×C89C22×D411yescharacteristic
156C2×Dic89—41no—
157C2×Dic89—41no—
158D356—24no—
159D356—42no—
160D356—44no—
161C4×D89—162no—
162C4×D89—162no—
163C8×C89—22no—
164C8×C89C4×D412yes—
165C8×C89C23×C411yescharacteristic
166C8×C89C2×#4111yescharacteristic
167C22×C89—21no—
168C22×C89—21no—
169C22×C89—42no—
170C22×C89—81no—
171C22×C89—81no—
172C22×C89C4×D812yes—
173C22×C89C22×D811yescharacteristic
174C2×D89—44no—
175C2×D89—82no—
176C2×D89—82no—
177C2×D89—164no—
178C4×C89—22no—
179C4×C89—22no—
180C4×C89—42no—
181C4×C89—82no—
182C4×C89C4×D812yes—
183C4×C89C2×C4×D411yescharacteristic
184C4×C89C2×#13511yescharacteristic
185Dic89—81no—
186Dic89—81no—
187#33454—891nomaximal, Sylow
188C2×C89—22no—
189C2×C89—82no—
190C2×C89C2×C4×D811yescharacteristic
191C2×C89#341511yescharacteristic
192C2×C89#341611yescharacteristic
193D89—84no—
194C2×#509—891no—
195#2951D8911yescharacteristic, complemented
196#2948—891no—
197#2177—892no—
198#2175—892no—
199C4×SD32—892no—
200C4×D16—892no—
201#2159—891no—
202#2138—891no—
203C2×#357—891no—
204#2108—891no—
205C89#3345411yescharacteristic, complemented, Sylow
206C2×#135C2×D8911yescharacteristic
207C22×D8—891no—
208#520—21no—
209#518—1781no—
210#509—898no—
211C2×SD32—892no—
212C2×D16—892no—
213C2×M32—891no—
214C22×C16—891no—
215#442—1782no—
216#440—22no—
217C4×Q16C2×D8912yescomplemented
218C4×D8—892no—
219C4×D8—892no—
220C4×D8C2×D8912yescomplemented
221#427—891no—
222C2×#107C2×D8911yescharacteristic
223#418—1781no—
224#417—21no—
225#405C2×D8911yescharacteristic
226#367—892no—
227#366—892no—
228#358—891no—
229#358—891no—
230#357—891no—
231#357—891no—
232#357—892no—
233#357—892no—
234#346—892no—
235C4×C16—892no—
236C2×#41—21no—
237C22×D4—1781no—
238#135—22no—
239#135C4×D8914yescomplemented
240C2×Q16C22×D8911yescharacteristic
241C2×SD16—21no—
242C2×D8—894no—
243C2×D8—892no—
244C2×D8—1781no—
245C2×D8—1782no—
246C2×D8C22×D8911yescharacteristic
247C2×M16C22⋊D8911yescharacteristic
248C22×C8C22×D8911yescharacteristic
249C4×Q8—22no—
250C4×D4—22no—
251C4×D4—42no—
252C4×D4—3561no—
253C4×D4—3561no—
254C4×D4—1782no—
255C4×D4—1782no—
256#117—41no—
257#117—1781no—
258#117C22⋊D8911yescharacteristic
259C2×C4⋊C4—21no—
260C2×C4⋊C4—41no—
261C2×C22⋊C4—3562no—
262SD32—898no—
263D16—898no—
264M32—894no—
265C2×C16—892no—
266C2×C16—891no—
267C2×C16—891no—
268C2×C16—1782no—
269#107—21no—
270#107—891no—
271#107—891no—
272#107C22×D8912yes—
273#106—1781no—
274#103—22no—
275#102—22no—
276#102—1782no—
277#102—1782no—
278#97C22⋊D8912yes—
279C4×C8C22×D8912yes—
280C24—3561no—
281#41—24no—
282#41—42no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—3561no—
287C2×D4—3562no—
288C2×D4—1784no—
289C2×D4—1782no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—3562no—
294C22×C4—3562no—
295C22×C4C2×C22⋊D8911yescharacteristic
296Q16—21no—
297Q16C2×C4×D8912yes—
298SD16—24no—
299D8—21no—
300D8—894no—
301D8—1784no—
302D8—1782no—
303D8C2×C4×D8912yes—
304M16—22no—
305C2×C8C2×C4×D8912yes—
306C2×C8C2×C22⋊D8912yes—
307C2×C8C2×C22⋊D8911yescharacteristic
308C2×C8#1308357412yes—
309C2×C8C23×D8911yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—1781no—
314C4⋊C4—1781no—
315C4⋊C4#1308355911yescharacteristic
316C4⋊C4#1308355911yescharacteristic
317C22⋊C4—42no—
318C22⋊C4—3561no—
319C22⋊C4—3562no—
320C22⋊C4—3562no—
321C22⋊C4#1308355912yes—
322C42—42no—
323C42—3561no—
324C42C2×C22⋊D8912yes—
325C16—894no—
326C16—1781no—
327C16—1781no—
328C23—41no—
329C23—3564no—
330C23—3562no—
331C23—14241no—
332C23—14241no—
333C23C2×#1308355911yescharacteristic
334Q8—22no—
335Q8—41no—
336D4—22no—
337D4—44no—
338D4—41no—
339D4—3564no—
340D4—3562no—
341D4—1784no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—162no—
346C2×C4—161no—
347C2×C4—3561no—
348C2×C4—3561no—
349C2×C4—14242no—
350C2×C4—14242no—
351C2×C4C4×C22⋊D8912yes—
352C2×C4C2×#1308355912yes—
353C2×C4C2×#1308355912yes—
354C2×C4C2×#1308355911yescharacteristic
355C2×C4#1566803212yes—
356C2×C4C22×C22⋊D8911yescharacteristic
357C8—22no—
358C8C4×C22⋊D8912yes—
359C8C22×C4×D8911yescharacteristic
360C8C2×#1308357411yescharacteristic
361C22—21no—
362C22—21no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—3564no—
367C22—14244no—
368C22—7122no—
369C22—7122no—
370C22C4×#1308355912yes—
371C22C22×#1308355911yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—82no—
376C4—7121no—
377C4—7121no—
378C4C4×#1308355912yes—
379C4C2×C4×C22⋊D8911yescharacteristic
380C4C2×#1566803211yescharacteristic
381C2—22no—
382C2—82no—
383C2—7124no—
384C2C2×C4×#1308355911yescharacteristic
385C2#2603339111yescharacteristic
386C2#2603339611yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutatorC8×C89C23×C4
FrattiniC2×C8C23×D89
FittingC89×#2951C2
socleC22×C89C22×D8
prime radicalC2×#19152033C2
cosocleC2×C8×C89C24
hypercenter#2951D89
hypocenterC89#33454
LCSG ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. #26029105C2semidirect
3. #26022126C2semidirect
4. #26022114C2semidirect
5. C4×#15667955C2semidirect
6. C4×#15667953C2semidirect
7. #26022038C2semidirect
8. #26021959C2semidirect
9. C2×#19151398C2semidirect
10. #26021849C2semidirect
11. C89×#2951C2semidirect
14. #19152033C4semidirect
24. #19151425C22semidirect
25. #19151423C22semidirect
26. #19151402C22semidirect
27. #19151402C22semidirect
28. #19151399C22semidirect
29. #19151399C22semidirect
32. #19151374C22semidirect
33. C4×#13083549C22semidirect
37. C4×Q16×C89C22semidirect
38. C4×D8×C89C22semidirect
45. C2×D712C2×C4semidirect
53. #15667955C2×C4semidirect
54. #15667953C2×C4semidirect
55. #15667952C2×C4semidirect
67. #135×C89C2×C4semidirect
195. #2951D89semidirect
205. C89#33454semidirect
217. C4×Q16C2×D89semidirect
220. C4×D8C2×D89semidirect
239. #135C4×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191520332
#260291052
#260221262
#260221142
C4×#156679552
C4×#156679532
#260220382
#260219592
C2×#191513982
#260218492
C89×#29512
#3345489