Finite Group Databasebeta

← #33241321

Subgroups — #33241321

Order 18688.

602 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
1#22620417C211yesmaximal, characteristic
2#22612174C212yesmaximal, complemented
3#22612170C212yesmaximal, complemented
4#22612165C212yesmaximal, complemented
5#22612156C212yesmaximal, complemented
6#22611646C211yesmaximal, characteristic, complemented
7#22611641C211yesmaximal, characteristic, complemented
8#22611636C211yesmaximal, characteristic, complemented
9C2×#17988387C211yesmaximal, characteristic, complemented
10#22611528C211yesmaximal, characteristic, complemented
11C73×#3164C211yesmaximal, characteristic, complemented
12C2×#14556619—21no—
13C2×#14556613C2211yescharacteristic
14C22×D584C2211yescharacteristic
15#17989156—21no—
16C2×#14556531C2211yescharacteristic
17C22×#11924003C2211yescharacteristic
18#17988976—22no—
19#17988971C2212yes—
20#17988964C2212yes—
21#17988390C2211yescharacteristic, complemented
22#17988390C2211yescharacteristic, complemented
23#17988390C2212yescomplemented
24#17988387C2212yes—
25#17988387C2212yes—
26#17988387C2212yes—
27#17988387C2212yes—
28#17988387C2211yescharacteristic, complemented
29#17988387C2211yescharacteristic, complemented
30#17988387C2212yescomplemented
31#17988363C2212yescomplemented
32C4×#11924003C2212yescomplemented
33#544×C73—21no—
34#501×C73—22no—
35#500×C73C2212yescomplemented
36#499×C73C2212yescomplemented
37#428×C73—21no—
38#427×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40#405×C73C2211yescharacteristic
41C2×#11924028—21no—
42C22×D292—21no—
43#14556619—44no—
44#14556613—22no—
45#14556613—22no—
46C2×Dic292C2311yescharacteristic
47C2×#11924017—22no—
48C2×#11924017—21no—
49C2×D584—22no—
50C2×D584—21no—
51C2×D584—22no—
52C2×D584—22no—
53C2×D584C2312yes—
54C2×D584C2311yescharacteristic
55#14556588—22no—
56#14556579—22no—
57#14556566—41no—
58#14556566—41no—
59#14556561—42no—
60#14556531—22no—
61C2×#11924003—22no—
62C2×#11924003C2311yescharacteristic
63C2×#11924003C2312yes—
64C2×#11924003C2311yescharacteristic
65C2×M16×C73D411yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C4⋊Q8×C73—22no—
68C4⋊Q8×C73—41no—
69#125×C73—22no—
70#125×C73—41no—
71C22⋊Q8×C73—42no—
72#117×C73—21no—
73#117×C73D411yescharacteristic
74C2×C4⋊C4×C73—21no—
75#107×C73—21no—
76#107×C73—22no—
77#107×C73C2312yes—
78#107×C73C2311yescharacteristic
79#107×C73C2311yescharacteristic
80#106×C73—21no—
81#106×C73—22no—
82#97×C73D412yes—
83C4×C8×C73C2312yes—
84C23×D73—41no—
85#11924028—42no—
86#11924028—42no—
87C2×Dic146—21no—
88C2×D292—22no—
89C2×D292—21no—
90C2×D292—42no—
91C2×D292—42no—
92C2×D292—41no—
93C2×D292—42no—
94C2×D292—42no—
95C2×C22⋊D73—41no—
96C2×C4×D73—41no—
97Dic292—21no—
98Dic292—21no—
99#11924017—42no—
100#11924017—44no—
101D584—22no—
102D584—22no—
103D584—21no—
104D584—21no—
105D584—42no—
106D584—44no—
107#11924005—42no—
108#11924005—42no—
109#11924003—21no—
110#11924003—21no—
111#11924003—22no—
112C2×Q8×C73—42no—
113C22×C4×C73—41no—
114C22×C4×C73C2×D411yescharacteristic
115M16×C73SD1614yes—
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73C2×D412yes—
118C2×C8×C73#4112yes—
119C2×C8×C73#4112yes—
120C2×C8×C73C2411yescharacteristic
121C4⋊C4×C73—21no—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73—42no—
126C4⋊C4×C73—42no—
127C4⋊C4×C73—42no—
128C4⋊C4×C73D811yescharacteristic
129C4⋊C4×C73D811yescharacteristic
130C22⋊C4×C73—41no—
131C22⋊C4×C73D812yes—
132C42×C73—41no—
133C42×C73C2×D412yes—
134C23×C73C2×D811yescharacteristic
135C22×D73—42no—
136C22×D73—41no—
137C22×D73—82no—
138C22×D73—161no—
139C22×D73—161no—
140Q8×C73—162no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—42no—
144C2×C4×C73—161no—
145C2×C4×C73#12412yes—
146C2×C4×C73C2×D812yes—
147C2×C4×C73C2×D812yes—
148C2×C4×C73C2×D811yescharacteristic
149C2×C4×C73C2×SD1612yes—
150C2×C4×C73C22×D411yescharacteristic
151C22⋊D73—82no—
152C2×Dic73—41no—
153D292—42no—
154D292—41no—
155D292—41no—
156D292—42no—
157D292—82no—
158D292—162no—
159D292—162no—
160C4×D73—82no—
161Dic146—41no—
162Dic146—41no—
163C8×C73#12412yes—
164C8×C73C2×SD1614yes—
165C8×C73C2×#4111yescharacteristic
166C8×C73C2×#4111yescharacteristic
167C22×C73—21no—
168C22×C73—21no—
169C22×C73#48612yes—
170C22×C73C22×D811yescharacteristic
171C2×D73—81no—
172C2×D73—82no—
173C2×D73—81no—
174C2×D73—82no—
175C2×D73—81no—
176C2×D73—82no—
177C2×D73—164no—
178C4×C73—22no—
179C4×C73—22no—
180C4×C73—82no—
181C4×C73—81no—
182C4×C73—81no—
183C4×C73#48612yes—
184C4×C73C2×#12411yescharacteristic
185C4×C73C22×SD1611yescharacteristic
186Dic73—81no—
187#33702—731nomaximal, Sylow
188C2×C73—22no—
189C2×C73C2×#48611yescharacteristic
190C2×C73#341511yescharacteristic
191C2×C73#341611yescharacteristic
192D73—161no—
193D73—162no—
194#3164D7311yescharacteristic, complemented
195#3155—731no—
196#2244—732no—
197#2243—732no—
198#2242—732no—
199#2240—732no—
200#2140—731no—
201#2139—731no—
202#2138—731no—
203C2×#357—731no—
204#2108—731no—
205C73#3370211yescharacteristic, complemented, Sylow
206C2×AGL(1,ℤ/8)—1461no—
207C2×#135—731no—
208C22×D8—731no—
209#544—21no—
210#535—1461no—
211C2×M32—731no—
212C22×C16—731no—
213#501—22no—
214#500C2×D7312yescomplemented
215#499C2×D7312yescomplemented
216#496—1462no—
217#495—732no—
218#493—732no—
219#428—21no—
220#427C2×D7311yescharacteristic
221C2×#107C2×D7311yescharacteristic
222#405C2×D7311yescharacteristic
223#358—731no—
224#358—731no—
225#358—732no—
226#357—732no—
227#357—732no—
228#357—732no—
229#357—732no—
230#357—731no—
231#357—731no—
232#357—732no—
233#346—732no—
234C4×C16—732no—
235C2×#41—1461no—
236C22×D4—1461no—
237AGL(1,ℤ/8)—2924no—
238#135—1462no—
239#135—1462no—
240C2×Q16—731no—
241C2×SD16—1461no—
242C2×SD16—1462no—
243C2×D8—732no—
244C2×D8—731no—
245C2×D8—1461no—
246C2×D8—1462no—
247C2×D8—1462no—
248C2×D8—1462no—
249C2×M16C22⋊D7311yescharacteristic
250C22×C8C22×D7311yescharacteristic
251C4⋊Q8—22no—
252C4⋊Q8—41no—
253#127—1462no—
254#125—22no—
255#125—41no—
256#124—1462no—
257C22⋊Q8—42no—
258#121—2921no—
259#121—2921no—
260#120—2922no—
261#117—21no—
262#117C22⋊D7311yescharacteristic
263C2×C4⋊C4—21no—
264M32—1462no—
265C2×C16—731no—
266C2×C16—732no—
267C2×C16—731no—
268C2×C16—1462no—
269#107—21no—
270#107—22no—
271#107C22×D7312yes—
272#107C22×D7311yescharacteristic
273#107C22×D7311yescharacteristic
274#106—21no—
275#106—22no—
276#97C22⋊D7312yes—
277C4×C8C22×D7312yes—
278C24—2921no—
279#41—2922no—
280#41—2922no—
281C2×Q8—42no—
282C2×Q8—1461no—
283C2×D4—2921no—
284C2×D4—2922no—
285C2×D4—2921no—
286C2×D4—2922no—
287C2×D4—2922no—
288C2×D4—2922no—
289C2×D4—1462no—
290C2×D4—1461no—
291C22×C4—41no—
292C22×C4—2921no—
293C22×C4C2×C22⋊D7311yescharacteristic
294Q16—1461no—
295Q16—1461no—
296SD16—2922no—
297SD16—2924no—
298D8—2922no—
299D8—2924no—
300D8—1462no—
301D8—1462no—
302D8—1461no—
303D8—1461no—
304M16#1192401514yes—
305C2×C8C2×C22⋊D7311yescharacteristic
306C2×C8C2×C22⋊D7312yes—
307C2×C8#1192402912yes—
308C2×C8#1192402912yes—
309C2×C8C23×D7311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4#1192401311yescharacteristic
318C4⋊C4#1192401311yescharacteristic
319C22⋊C4—41no—
320C22⋊C4—2922no—
321C22⋊C4—2922no—
322C22⋊C4#1192401312yes—
323C42—41no—
324C42C2×C22⋊D7312yes—
325C16—1461no—
326C16—1461no—
327C16—1462no—
328C23—2922no—
329C23—2921no—
330C23—5842no—
331C23—11681no—
332C23—11681no—
333C23C2×#1192401311yescharacteristic
334Q8—162no—
335Q8—2921no—
336Q8—2921no—
337D4—2921no—
338D4—2922no—
339D4—2922no—
340D4—2921no—
341D4—5842no—
342D4—5842no—
343D4—11682no—
344D4—11682no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—161no—
349C2×C4—2921no—
350C2×C4—5842no—
351C2×C4#1455658012yes—
352C2×C4C2×#1192401312yes—
353C2×C4C2×#1192401312yes—
354C2×C4C2×#1192401311yescharacteristic
355C2×C4C2×#1192401512yes—
356C2×C4C22×C22⋊D7311yescharacteristic
357C8#1455658012yes—
358C8C2×#1192401514yes—
359C8C2×#1192402911yescharacteristic
360C8C2×#1192402911yescharacteristic
361C22—21no—
362C22—21no—
363C22—5841no—
364C22—5842no—
365C22—5841no—
366C22—5842no—
367C22—5841no—
368C22—5842no—
369C22—11684no—
370C22#1798893512yes—
371C22C22×#1192401311yescharacteristic
372C4—22no—
373C4—22no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—5841no—
378C4#1798893512yes—
379C4C2×#1455658011yescharacteristic
380C4C22×#1192401511yescharacteristic
381C2—22no—
382C2—11682no—
383C2—11681no—
384C2C2×#1798893511yescharacteristic
385C2#2262307011yescharacteristic
386C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3164C2
socleC22×C73C22×D8
prime radical#22620417C2
hypercenter#3164D73
hypocenterC73#33702
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #22612174C2semidirect
3. #22612170C2semidirect
4. #22612165C2semidirect
5. #22612156C2semidirect
6. #22611646C2semidirect
7. #22611641C2semidirect
8. #22611636C2semidirect
9. C2×#17988387C2semidirect
10. #22611528C2semidirect
11. C73×#3164C2semidirect
21. #17988390C22semidirect
22. #17988390C22semidirect
23. #17988390C22semidirect
28. #17988387C22semidirect
29. #17988387C22semidirect
30. #17988387C22semidirect
31. #17988363C22semidirect
32. C4×#11924003C22semidirect
35. #500×C73C22semidirect
36. #499×C73C22semidirect
194. #3164D73semidirect
205. C73#33702semidirect
214. #500C2×D73semidirect
215. #499C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204172
#226121742
#226121702
#226121652
#226121562
#226116462
#226116412
#226116362
C2×#179883872
#226115282
C73×#31642
#3370273