Finite Group Databasebeta

← #33068290

Subgroups — #33068290

Order 18688.

716 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619935C211yesmaximal, characteristic, complemented
2#22619927C211yesmaximal, characteristic
3C2×C8×D292C211yesmaximal, characteristic, complemented
4#22609379C212yesmaximal, complemented
5#22609365C212yesmaximal, complemented
6#22609351C212yesmaximal, complemented
7#22609337C212yesmaximal, complemented
8#22608679C211yesmaximal, characteristic, complemented
9#22608588C211yesmaximal, characteristic, complemented
10#22608580C211yesmaximal, characteristic, complemented
11C73×#3131C211yesmaximal, characteristic, complemented
12C22×C8×D73—21no—
13C2×C4×D292C2211yescharacteristic
14#17988915—21no—
15#17988898—21no—
16#17988874C2212yes—
17#17988864C2212yes—
18#17988789C2212yes—
19#17988779C2212yes—
20#17988737—21no—
21#17988720—21no—
22C4×#11924015—22no—
23C4×#11924013—22no—
24C8×D292—21no—
25C8×D292—21no—
26C8×D292C2212yes—
27C8×D292C2212yes—
28#17988640C2211yescharacteristic
29#17988640C2211yescharacteristic
30C2×C4⋊#10596831C2211yescharacteristic
31C2×#14556503—21no—
32#17988340C2212yescomplemented
33#17988338C2212yescomplemented
34#17988336C2212yescomplemented
35#17988333C2212yescomplemented
36#17988333C2212yescomplemented
37#17988331C2212yescomplemented
38#533×C73C2211yescharacteristic
39#532×C73C2211yescharacteristic
40#488×C73—21no—
41#488×C73—21no—
42#487×C73C2211yescharacteristic, complemented
43#487×C73C2211yescharacteristic, complemented
44#486×C73C2211yescharacteristic, complemented
45#486×C73C2211yescharacteristic, complemented
46#416×C73—21no—
47#416×C73—21no—
48C2×C4×C8×C73C2211yescharacteristic
49C22×D292D411yescharacteristic
50C22×C4×D73—21no—
51C2×#11924015—22no—
52C2×#11924013—22no—
53C22×#10596831—21no—
54C2×C8×D73—22no—
55C2×C8×D73—22no—
56C2×C8×D73—44no—
57C4×D292—21no—
58C4×D292—21no—
59C4×D292D412yes—
60C4×D292C2312yes—
61C2×C4⋊Dic73D411yescharacteristic
62C2×#11924005—21no—
63#14556524—22no—
64#14556522—22no—
65C4⋊#10596831—21no—
66C4⋊#10596831C2312yes—
67C4⋊#10596831C2312yes—
68C4⋊#10596831C2312yes—
69#14556515—22no—
70#14556512—22no—
71#14556511—22no—
72#14556511—22no—
73C22⋊#10596831—21no—
74C22⋊#10596831—21no—
75#14556503—22no—
76#14556503—22no—
77C4×#10596831—21no—
78C4×#10596831—21no—
79C2×#41×C73—21no—
80C22×C8×C73D411yescharacteristic
81C22×C8×C73D411yescharacteristic
82C4⋊Q8×C73C2311yescharacteristic
83C4⋊Q8×C73C2311yescharacteristic
84C4⋊Q8×C73C2311yescharacteristic
85#127×C73C2311yescharacteristic
86#125×C73—21no—
87#124×C73—21no—
88C22⋊Q8×C73—21no—
89C22⋊Q8×C73—21no—
90#121×C73—21no—
91#121×C73—21no—
92C4×Q8×C73—21no—
93C4×Q8×C73—21no—
94C4×D4×C73—21no—
95C4×D4×C73—21no—
96#117×C73—21no—
97C2×C42×C73C2311yescharacteristic
98#103×C73—21no—
99#103×C73—21no—
100#103×C73—21no—
101#103×C73—21no—
102#102×C73—21no—
103#102×C73—21no—
104#102×C73—21no—
105#102×C73—21no—
106C4×C8×C73D411yescharacteristic
107C4×C8×C73D411yescharacteristic
108C4×C8×C73C2311yescharacteristic
109C4×C8×C73C2311yescharacteristic
110C23×D73—21no—
111C2×D292—21no—
112C2×D292—24no—
113C2×D292C2×D412yes—
114C22×Dic73—21no—
115C2×C4×D73—22no—
116C2×C4×D73—22no—
117C2×C4×D73—42no—
118C2×C4×D73—42no—
119#11924015—44no—
120#11924013—44no—
121C2×#10596831—22no—
122C2×#10596831—22no—
123C2×#10596831—22no—
124C2×#10596831—42no—
125C8×D73—44no—
126C8×D73—44no—
127C4⋊Dic73—21no—
128C4⋊Dic73C2×D412yes—
129#11924005—22no—
130#11924005—22no—
131#41×C73—42no—
132#41×C73—42no—
133C2×Q8×C73—21no—
134C2×Q8×C73—21no—
135C2×Q8×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73—41no—
140C22×C4×C73—41no—
141C22×C4×C73C2×D411yescharacteristic
142C22×C4×C73C2×D411yescharacteristic
143C22×C4×C73#4111yescharacteristic
144C2×C8×C73—22no—
145C2×C8×C73—22no—
146C2×C8×C73C2×D411yescharacteristic
147C2×C8×C73C2×D411yescharacteristic
148C2×C8×C73C2×D411yescharacteristic
149C2×C8×C73C2×D411yescharacteristic
150C4⋊C4×C73—21no—
151C4⋊C4×C73—21no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—41no—
157C4⋊C4×C73—41no—
158C22⋊C4×C73—41no—
159C22⋊C4×C73—41no—
160C22⋊C4×C73—41no—
161C42×C73—41no—
162C42×C73C2×D411yescharacteristic
163C42×C73C2×D411yescharacteristic
164C42×C73#4111yescharacteristic
165C42×C73C2411yescharacteristic
166C23×C73—41no—
167C23×C73#12111yescharacteristic
168C22×D73—24no—
169C22×D73—22no—
170C22×D73—42no—
171C22×D73—42no—
172Q8×C73—41no—
173Q8×C73—42no—
174Q8×C73—41no—
175Q8×C73—81no—
176D4×C73—41no—
177D4×C73—42no—
178D4×C73—41no—
179D4×C73—81no—
180D4×C73—82no—
181C2×C4×C73—22no—
182C2×C4×C73—41no—
183C2×C4×C73—41no—
184C2×C4×C73—41no—
185C2×C4×C73—41no—
186C2×C4×C73—82no—
187C2×C4×C73#12111yescharacteristic
188C2×C4×C73#12111yescharacteristic
189C2×C4×C73#12111yescharacteristic
190C2×C4×C73#13512yes—
191C2×C4×C73AGL(1,ℤ/8)12yes—
192C2×C4×C73C22×D411yescharacteristic
193C2×C4×C73C22×D411yescharacteristic
194C2×C4×C73C2×#4111yescharacteristic
195C2×Dic73—22no—
196C2×Dic73—42no—
197D292—24no—
198D292—22no—
199D292#13514yes—
200C4×D73—42no—
201C4×D73—42no—
202C4×D73—44no—
203C8×C73—21no—
204C8×C73—21no—
205C8×C73—21no—
206C8×C73—21no—
207#10596831—42no—
208#10596831—42no—
209#10596831—42no—
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—81no—
213C22×C73—81no—
214C22×C73—81no—
215C22×C73#46312yes—
216C22×C73C2×#12111yescharacteristic
217C2×D73—24no—
218C2×D73—48no—
219C2×D73—42no—
220C2×D73—42no—
221C4×C73—21no—
222C4×C73—21no—
223C4×C73—21no—
224C4×C73—81no—
225C4×C73—81no—
226C4×C73—81no—
227C4×C73#46312yes—
228C4×C73C2×#13512yes—
229C4×C73C2×#13511yescharacteristic
230C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
231Dic73—42no—
232#19110—731nomaximal, Sylow
233C2×C73—22no—
234C2×C73—161no—
235C2×C73C2×#46311yescharacteristic
236C2×C73#305011yescharacteristic
237C2×C73#306711yescharacteristic
238D73—44no—
239#3131D7311yescharacteristic, complemented
240#3101—731no—
241#3100—731no—
242C2×C8×D4—731no—
243#1683—732no—
244#1681—732no—
245#1679—732no—
246#1677—732no—
247#1564—731no—
248#1542—731no—
249#1540—731no—
250C73#1911011yescharacteristic, complemented, Sylow
251C23×C8—1461no—
252#533C2×D7311yescharacteristic
253#532C2×D7311yescharacteristic
254C2×C4×D4—731no—
255#488—21no—
256#488—21no—
257#487C2×D7311yescharacteristic, complemented
258#487C2×D7311yescharacteristic, complemented
259#486C2×D7311yescharacteristic, complemented
260#486C2×D7311yescharacteristic, complemented
261#483—1461no—
262#480—1461no—
263#476—732no—
264#474—732no—
265#461—732no—
266#459—732no—
267#450—1461no—
268#447—1461no—
269C4×SD16—1462no—
270C4×D8—1462no—
271C8×D4—732no—
272C8×D4—732no—
273C8×D4—1461no—
274C8×D4—1461no—
275#431—731no—
276#431—731no—
277C2×C4⋊C8—731no—
278#416—21no—
279#416—21no—
280C2×C22⋊C8—1461no—
281C2×C4×C8C2×D7311yescharacteristic
282#335—732no—
283#334—732no—
284#333—732no—
285#332—732no—
286#332—732no—
287#331—732no—
288C2×#41—21no—
289C22×D4—731no—
290C23×C4—1461no—
291C2×SD16—1462no—
292C2×D8—1462no—
293C22×C8—2924no—
294C22×C8—1461no—
295C22×C8—1462no—
296C22×C8—1462no—
297C22×C8C22⋊D7311yescharacteristic
298C22×C8C22⋊D7311yescharacteristic
299C4⋊Q8C22×D7311yescharacteristic
300C4⋊Q8C22×D7311yescharacteristic
301C4⋊Q8C22×D7311yescharacteristic
302#127C22×D7311yescharacteristic
303#125—21no—
304#124—21no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307#121—21no—
308#121—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—732no—
314C4×D4—732no—
315C4×D4—1461no—
316C4×D4—1461no—
317#117—21no—
318C2×C4⋊C4—731no—
319C2×C22⋊C4—1461no—
320C2×C42C22×D7311yescharacteristic
321#107—1462no—
322#106—1462no—
323C4⋊C8—732no—
324C4⋊C8—732no—
325C4⋊C8—732no—
326C4⋊C8—1461no—
327#103—21no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—1462no—
332#102—21no—
333#102—21no—
334#102—21no—
335#102—21no—
336#102—1462no—
337#102—1462no—
338#102—1462no—
339C22⋊C8—1461no—
340C22⋊C8—1461no—
341C22⋊C8—1462no—
342C22⋊C8—1462no—
343C4×C8—1461no—
344C4×C8—1461no—
345C4×C8C22⋊D7311yescharacteristic
346C4×C8C22⋊D7311yescharacteristic
347C4×C8C22×D7311yescharacteristic
348C4×C8C22×D7311yescharacteristic
349C24—1461no—
350#41—42no—
351#41—42no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—732no—
360C2×D4—1464no—
361C2×D4—1461no—
362C22×C4—41no—
363C22×C4—2922no—
364C22×C4—2922no—
365C22×C4—1461no—
366C22×C4—1462no—
367C22×C4—1462no—
368C22×C4D4×D7311yescharacteristic
369C22×C4C2×C22⋊D7311yescharacteristic
370C22×C4#1192402711yescharacteristic
371SD16—2924no—
372D8—2924no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—2924no—
376C2×C8—2924no—
377C2×C8—2922no—
378C2×C8—1462no—
379C2×C8—1462no—
380C2×C8—1462no—
381C2×C8C2×C22⋊D7311yescharacteristic
382C2×C8C2×C22⋊D7311yescharacteristic
383C2×C8C2×C22⋊D7311yescharacteristic
384C2×C8C2×C22⋊D7311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—732no—
394C4⋊C4—1461no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—1462no—
399C22⋊C4—1462no—
400C42—41no—
401C42D4×D7311yescharacteristic
402C42C2×C22⋊D7311yescharacteristic
403C42#1192402711yescharacteristic
404C42C23×D7311yescharacteristic
405C23—41no—
406C23—2922no—
407C23—2922no—
408C23—1464no—
409C23—1462no—
410C23#1455656211yescharacteristic
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415D4—42no—
416D4—41no—
417D4—41no—
418D4—81no—
419D4—82no—
420D4—734no—
421D4—1464no—
422D4—1462no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—82no—
429C2×C4—2922no—
430C2×C4—2924no—
431C2×C4—2922no—
432C2×C4—2922no—
433C2×C4—1462no—
434C2×C4#1455656211yescharacteristic
435C2×C4#1455656211yescharacteristic
436C2×C4#1455656211yescharacteristic
437C2×C4#1455661112yes—
438C2×C4#1455661612yes—
439C2×C4C2×D4×D7311yescharacteristic
440C2×C4C22×C22⋊D7311yescharacteristic
441C2×C4C2×#1192402711yescharacteristic
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—2922no—
447C8—2922no—
448C8—2922no—
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—2928no—
455C22—2922no—
456C22—2922no—
457C22—1464no—
458C22#1798879912yes—
459C22C2×#1455656211yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—2922no—
467C4#1798879912yes—
468C4#135×D7312yes—
469C4C2×#1455661111yescharacteristic
470C4C2×#1455661611yescharacteristic
471C2—22no—
472C2—161no—
473C2—2924no—
474C2C2×#1798879911yescharacteristic
475C2#2261950811yescharacteristic
476C2#2261964711yescharacteristic
subgroupG/H
centerC2×C4#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3131C2
socleC22×C73C2×#121
prime radical#22619927C2
hypercenter#3131D73
hypocenterC73#19110
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #22619935C2semidirect
3. C2×C8×D292C2semidirect
4. #22609379C2semidirect
5. #22609365C2semidirect
6. #22609351C2semidirect
7. #22609337C2semidirect
8. #22608679C2semidirect
9. #22608588C2semidirect
10. #22608580C2semidirect
11. C73×#3131C2semidirect
32. #17988340C22semidirect
33. #17988338C22semidirect
34. #17988336C22semidirect
35. #17988333C22semidirect
36. #17988333C22semidirect
37. #17988331C22semidirect
42. #487×C73C22semidirect
43. #487×C73C22semidirect
44. #486×C73C22semidirect
45. #486×C73C22semidirect
239. #3131D73semidirect
250. C73#19110semidirect
257. #487C2×D73semidirect
258. #487C2×D73semidirect
259. #486C2×D73semidirect
260. #486C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199352
#226199272
C2×C8×D2922
#226093792
#226093652
#226093512
#226093372
#226086792
#226085882
#226085802
C73×#31312
#1911073