Finite Group Databasebeta

← #33241122

Subgroups — #33241122

Order 18688.

534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620590C211yesmaximal, characteristic, complemented
2#22612133C211yesmaximal, characteristic, complemented
3#22612128C211yesmaximal, characteristic, complemented
4#22612095C211yesmaximal, characteristic, complemented
5#22612088C211yesmaximal, characteristic, complemented
6#22611981C211yesmaximal, characteristic, complemented
7#22611976C211yesmaximal, characteristic, complemented
8#22611871C211yesmaximal, characteristic
9#22611871C211yesmaximal, characteristic
10#22611800C211yesmaximal, characteristic, complemented
11#22611795C211yesmaximal, characteristic, complemented
12C4×#14556537C211yesmaximal, characteristic, complemented
13C4×#14556535C211yesmaximal, characteristic, complemented
14#22611738C211yesmaximal, characteristic, complemented
15C73×#2951C211yesmaximal, characteristic
16#17989199—21no—
17C2×#14556537C2211yescharacteristic
18C2×#14556537C2211yescharacteristic
19C2×#14556535C2211yescharacteristic
20C2×#14556535C2211yescharacteristic
21#17988989C2211yescharacteristic
22#17988967C2211yescharacteristic
23#17988956—21no—
24#17988845—21no—
25#17988828C2211yescharacteristic
26#17988828C2211yescharacteristic
27#17988695—21no—
28C4×Dic292C2211yescharacteristic
29C4×Dic292C2211yescharacteristic
30#17988414C2211yescharacteristic
31#17988414C2211yescharacteristic
32#17988412C2211yescharacteristic
33#17988412C2211yescharacteristic
34C4⋊#11924003C2211yescharacteristic
35C4⋊#11924003C2211yescharacteristic
36#17988391C2211yescharacteristic
37#17988391C2211yescharacteristic
38#17988390C2211yescharacteristic
39#17988390C2211yescharacteristic
40#17988390C2211yescharacteristic
41#17988390C2211yescharacteristic
42#17988388C2211yescharacteristic
43#17988388C2211yescharacteristic
44C22⋊#11924003C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46#17988363C2211yescharacteristic
47C4×#11924003C2211yescharacteristic
48C2×#135×C73C2211yescharacteristic
49#520×C73—21no—
50#440×C73—21no—
51#440×C73—21no—
52C4×Q16×C73C2211yescharacteristic
53C4×Q16×C73C2211yescharacteristic
54C4×D8×C73C2211yescharacteristic
55C4×D8×C73C2211yescharacteristic
56C2×#107×C73C2211yescharacteristic
57#417×C73—21no—
58#405×C73C2211yescharacteristic
59C2×Dic292—21no—
60C2×Dic292C2311yescharacteristic
61C2×Dic292C2311yescharacteristic
62#14556591—21no—
63#14556584—41no—
64#14556584—41no—
65#14556583—21no—
66#14556581—41no—
67#14556581—41no—
68#14556570—21no—
69#14556570—21no—
70C4×Dic146—21no—
71C4×Dic146—21no—
72#14556537—22no—
73#14556537—22no—
74#14556535—22no—
75#14556535—22no—
76C2×#11924003C2311yescharacteristic
77C2×#11924003C2311yescharacteristic
78C2×#11924003C2311yescharacteristic
79C2×#11924003C2311yescharacteristic
80#14556525C2311yescharacteristic
81#14556525C2311yescharacteristic
82#14556523—21no—
83#14556516—21no—
84#14556516—21no—
85#14556516—21no—
86#14556516—21no—
87C2×#41×C73—21no—
88#135×C73—22no—
89#135×C73—22no—
90C2×Q16×C73C2311yescharacteristic
91C2×SD16×C73—21no—
92C2×D8×C73C2311yescharacteristic
93C2×M16×C73D411yescharacteristic
94C22×C8×C73C2311yescharacteristic
95C4×Q8×C73—21no—
96C4×Q8×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73—21no—
99C4×D4×C73—41no—
100C4×D4×C73—41no—
101#117×C73—41no—
102#117×C73D411yescharacteristic
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—41no—
105#107×C73—21no—
106#107×C73C2311yescharacteristic
107#107×C73C2311yescharacteristic
108#103×C73—21no—
109#103×C73—21no—
110#102×C73—21no—
111#102×C73—21no—
112#97×C73D411yescharacteristic
113#97×C73D411yescharacteristic
114C4×C8×C73C2311yescharacteristic
115C4×C8×C73C2311yescharacteristic
116C2×Dic146—21no—
117C2×Dic146—21no—
118Dic292—21no—
119Dic292—21no—
120Dic292—21no—
121Dic292—21no—
122Dic292—42no—
123C4⋊Dic73—21no—
124C4⋊Dic73—21no—
125C4⋊Dic73—41no—
126C4⋊Dic73—41no—
127#11924007—41no—
128#11924007—41no—
129#11924007—41no—
130#11924007—41no—
131#11924007—41no—
132#11924007—41no—
133C22⋊Dic73—41no—
134C22⋊Dic73—41no—
135C4×Dic73—41no—
136C4×Dic73—41no—
137#11924003—21no—
138#11924003—21no—
139#11924003—21no—
140#11924003—21no—
141#41×C73—24no—
142#41×C73—42no—
143C2×Q8×C73—21no—
144C2×D4×C73—21no—
145C2×D4×C73—41no—
146C22×C4×C73—41no—
147C22×C4×C73—41no—
148C22×C4×C73—41no—
149C22×C4×C73—41no—
150C22×C4×C73C2×D411yescharacteristic
151Q16×C73—21no—
152Q16×C73#4112yes—
153SD16×C73—24no—
154D8×C73—21no—
155D8×C73#4112yes—
156M16×C73—41no—
157C2×C8×C73—21no—
158C2×C8×C73—21no—
159C2×C8×C73C2×D411yescharacteristic
160C2×C8×C73C2×D411yescharacteristic
161C2×C8×C73C2×D411yescharacteristic
162C2×C8×C73C2411yescharacteristic
163C4⋊C4×C73—21no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73—41no—
167C4⋊C4×C73—41no—
168C4⋊C4×C73D811yescharacteristic
169C4⋊C4×C73D811yescharacteristic
170C22⋊C4×C73—41no—
171C22⋊C4×C73—41no—
172C22⋊C4×C73D811yescharacteristic
173C22⋊C4×C73D811yescharacteristic
174C42×C73—41no—
175C42×C73—41no—
176C42×C73C2×D411yescharacteristic
177C42×C73C2×D411yescharacteristic
178C23×C73—41no—
179C23×C73C2×D811yescharacteristic
180Q8×C73—22no—
181Q8×C73—41no—
182D4×C73—22no—
183D4×C73—44no—
184D4×C73—41no—
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—44no—
188C2×C4×C73—41no—
189C2×C4×C73—41no—
190C2×C4×C73—41no—
191C2×C4×C73—161no—
192C2×C4×C73—161no—
193C2×C4×C73—161no—
194C2×C4×C73C2×D811yescharacteristic
195C2×C4×C73C2×D811yescharacteristic
196C2×C4×C73C2×D811yescharacteristic
197C2×C4×C73C2×D811yescharacteristic
198C2×C4×C73C2×D811yescharacteristic
199C2×C4×C73C22×D411yescharacteristic
200C2×Dic73—41no—
201C2×Dic73—41no—
202C2×Dic73—41no—
203C2×Dic73—41no—
204Dic146—41no—
205Dic146—41no—
206Dic146—41no—
207Dic146—41no—
208C8×C73—21no—
209C8×C73—21no—
210C8×C73—21no—
211C8×C73C2×#4111yescharacteristic
212C8×C73C2×#4111yescharacteristic
213C22×C73—21no—
214C22×C73—21no—
215C22×C73—21no—
216C22×C73—42no—
217C22×C73—81no—
218C22×C73—81no—
219C22×C73C22×D811yescharacteristic
220C4×C73—21no—
221C4×C73—21no—
222C4×C73—21no—
223C4×C73—21no—
224C4×C73—21no—
225C4×C73—42no—
226C4×C73—81no—
227C4×C73—81no—
228C4×C73#52511yescharacteristic
229C4×C73#57511yescharacteristic
230Dic73—81no—
231Dic73—81no—
232Dic73—81no—
233Dic73—81no—
234#33680—731nomaximal, Sylow
235C2×C73—41no—
236C2×C73—82no—
237C2×C73#310011yescharacteristic
238C2×C73#341611yescharacteristic
239C2×C73#341711yescharacteristic
240#3177—731no—
241#2951D7311yescharacteristic
242#2236—731no—
243#2235—731no—
244#2229—731no—
245#2228—731no—
246#2209—731no—
247#2208—731no—
248#2189—731no—
249#2189—731no—
250#2176—731no—
251#2175—731no—
252C4×Q32—731no—
253C4×SD32—731no—
254#2164—731no—
255C73#3368011yescharacteristic, complemented, Sylow
256C2×#135C2×D7311yescharacteristic
257#541—1461no—
258#520—21no—
259C2×Q32—731no—
260C2×Q32—731no—
261C2×SD32—731no—
262C2×SD32—731no—
263#499—731no—
264#494—731no—
265#491—1461no—
266#470—1461no—
267#467—731no—
268#467—731no—
269#441—1461no—
270#440—21no—
271#440—21no—
272C4×Q16—731no—
273C4×Q16—731no—
274C4×Q16C2×D7311yescharacteristic
275C4×Q16C2×D7311yescharacteristic
276C4×D8C2×D7311yescharacteristic
277C4×D8C2×D7311yescharacteristic
278C2×#107C2×D7311yescharacteristic
279#417—21no—
280#405C2×D7311yescharacteristic
281#367—731no—
282#367—731no—
283#366—731no—
284#366—731no—
285C4⋊C16—731no—
286C4⋊C16—731no—
287#358—731no—
288#358—731no—
289#358—731no—
290#358—731no—
291#358—731no—
292#358—731no—
293#357—731no—
294#357—731no—
295C22⋊C16—731no—
296C22⋊C16—731no—
297#346—731no—
298C4×C16—731no—
299C2×#41—21no—
300#135—22no—
301#135—22no—
302C2×Q16—731no—
303C2×Q16—731no—
304C2×Q16—1461no—
305C2×Q16C22×D7311yescharacteristic
306C2×SD16—21no—
307C2×D8C22×D7311yescharacteristic
308C2×M16C22⋊D7311yescharacteristic
309C22×C8C22×D7311yescharacteristic
310C4⋊Q8—1461no—
311#126—2921no—
312#126—2921no—
313#125—2921no—
314#125—2921no—
315#125—1461no—
316C22⋊Q8—1461no—
317C22⋊Q8—1461no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—1461no—
321C4×Q8—1461no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—41no—
326#117—41no—
327#117C22⋊D7311yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—41no—
330Q32—1462no—
331Q32—1462no—
332SD32—1462no—
333SD32—1462no—
334C2×C16—731no—
335C2×C16—731no—
336C2×C16—731no—
337C2×C16—731no—
338#107—21no—
339#107—731no—
340#107—731no—
341#107C22×D7311yescharacteristic
342#107C22×D7311yescharacteristic
343#106—1461no—
344#103—21no—
345#103—21no—
346#103—1461no—
347#103—1461no—
348#103—1461no—
349#103—1461no—
350#102—21no—
351#102—21no—
352#97C22⋊D7311yescharacteristic
353#97C22⋊D7311yescharacteristic
354C4×C8C22×D7311yescharacteristic
355C4×C8C22×D7311yescharacteristic
356#41—24no—
357#41—42no—
358C2×Q8—21no—
359C2×Q8—1461no—
360C2×Q8—1461no—
361C2×D4—21no—
362C2×D4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4C2×C22⋊D7311yescharacteristic
368Q16—21no—
369Q16—2922no—
370Q16—1461no—
371Q16—1461no—
372Q16—1461no—
373Q16—1461no—
374Q16#1192402812yes—
375SD16—24no—
376D8—21no—
377D8#1192402812yes—
378M16—41no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8C2×C22⋊D7311yescharacteristic
382C2×C8C2×C22⋊D7311yescharacteristic
383C2×C8C2×C22⋊D7311yescharacteristic
384C2×C8C23×D7311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—2921no—
391C4⋊C4—2921no—
392C4⋊C4—2921no—
393C4⋊C4—2921no—
394C4⋊C4—2921no—
395C4⋊C4—2921no—
396C4⋊C4—2921no—
397C4⋊C4—2921no—
398C4⋊C4—1461no—
399C4⋊C4—1461no—
400C4⋊C4#1192401311yescharacteristic
401C4⋊C4#1192401311yescharacteristic
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—2921no—
405C22⋊C4—2921no—
406C22⋊C4#1192401311yescharacteristic
407C22⋊C4#1192401311yescharacteristic
408C42—41no—
409C42—41no—
410C42—2921no—
411C42—2921no—
412C42C2×C22⋊D7311yescharacteristic
413C42C2×C22⋊D7311yescharacteristic
414C16—1461no—
415C16—1461no—
416C16—1461no—
417C16—1461no—
418C23—41no—
419C23C2×#1192401311yescharacteristic
420Q8—22no—
421Q8—41no—
422Q8—2921no—
423Q8—2921no—
424Q8—2921no—
425Q8—2921no—
426D4—22no—
427D4—44no—
428D4—41no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—161no—
436C2×C4—161no—
437C2×C4—161no—
438C2×C4—2921no—
439C2×C4—2921no—
440C2×C4—2921no—
441C2×C4—2921no—
442C2×C4C2×#1192401311yescharacteristic
443C2×C4C2×#1192401311yescharacteristic
444C2×C4C2×#1192401311yescharacteristic
445C2×C4C2×#1192401311yescharacteristic
446C2×C4C2×#1192401311yescharacteristic
447C2×C4C22×C22⋊D7311yescharacteristic
448C8—21no—
449C8—21no—
450C8—21no—
451C8C2×#1192402811yescharacteristic
452C8C2×#1192402811yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22C22×#1192401311yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—5841no—
469C4—5841no—
470C4—5841no—
471C4—5841no—
472C4#1798909811yescharacteristic
473C4#1798938511yescharacteristic
474C2—41no—
475C2—82no—
476C2#2261993111yescharacteristic
477C2#2262307811yescharacteristic
478C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2951C2
socleC22×C73C22×D8
prime radicalC2×#135×C73C22
hypercenter#2951D73
hypocenterC73#33680
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #22620590C2semidirect
2. #22612133C2semidirect
3. #22612128C2semidirect
4. #22612095C2semidirect
5. #22612088C2semidirect
6. #22611981C2semidirect
7. #22611976C2semidirect
10. #22611800C2semidirect
11. #22611795C2semidirect
12. C4×#14556537C2semidirect
13. C4×#14556535C2semidirect
14. #22611738C2semidirect
255. C73#33680semidirect

Maximal subgroups

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

subgroupindex
#226205902
#226121332
#226121282
#226120952
#226120882
#226119812
#226119762
#226118712
#226118002
#226117952
C4×#145565372
C4×#145565352
#226117382
C73×#29512
#3368073