Finite Group Databasebeta

← #33069785

Subgroups — #33069785

Order 18688.

716 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#22620026C211yesmaximal, characteristic, complemented
2#22620018C211yesmaximal, characteristic
3C2×C8×D292C211yesmaximal, characteristic, complemented
4#22609379C212yesmaximal, complemented
5#22609365C212yesmaximal, complemented
6#22609303C212yesmaximal, complemented
7#22609269C212yesmaximal, complemented
8#22608694C211yesmaximal, characteristic, complemented
9#22608631C211yesmaximal, characteristic, complemented
10#22608626C211yesmaximal, characteristic, complemented
11C73×#3167C211yesmaximal, characteristic, complemented
12C22×C8×D73—21no—
13C2×C4×D292C2211yescharacteristic
14#17988920—21no—
15#17988903—21no—
16#17988874C2212yes—
17#17988864C2212yes—
18#17988789C2212yes—
19#17988779C2212yes—
20#17988737—21no—
21#17988720—21no—
22#17988697—22no—
23#17988685—22no—
24C8×D292—21no—
25C8×D292—21no—
26C8×D292C2212yes—
27C8×D292C2212yes—
28#17988644C2211yescharacteristic
29#17988644C2211yescharacteristic
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#498×C73C2211yescharacteristic, complemented
41#492×C73C2211yescharacteristic, complemented
42#487×C73C2211yescharacteristic, complemented
43#486×C73C2211yescharacteristic, complemented
44#465×C73—21no—
45#465×C73—21no—
46C4×SD16×C73—21no—
47C4×SD16×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—
77#14556502—21no—
78#14556502—21no—
79C2×#41×C73—21no—
80C2×SD16×C73—21no—
81C2×SD16×C73—21no—
82C22×C8×C73D411yescharacteristic
83C22×C8×C73D411yescharacteristic
84C4⋊Q8×C73C2311yescharacteristic
85C4⋊Q8×C73C2311yescharacteristic
86C4⋊Q8×C73C2311yescharacteristic
87#127×C73C2311yescharacteristic
88#125×C73—21no—
89#124×C73—21no—
90C22⋊Q8×C73—21no—
91C22⋊Q8×C73—21no—
92#121×C73—21no—
93#121×C73—21no—
94C4×Q8×C73—21no—
95C4×Q8×C73—21no—
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98#117×C73—21no—
99C2×C42×C73C2311yescharacteristic
100#106×C73—21no—
101#106×C73—21no—
102#103×C73—21no—
103#103×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—82no—
132C2×Q8×C73—21no—
133C2×Q8×C73—21no—
134C2×Q8×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—21no—
138C2×D4×C73—41no—
139C22×C4×C73—41no—
140C22×C4×C73C2×D411yescharacteristic
141C22×C4×C73C2×D411yescharacteristic
142C22×C4×C73#4111yescharacteristic
143SD16×C73—42no—
144SD16×C73—42no—
145C2×C8×C73—22no—
146C2×C8×C73—41no—
147C2×C8×C73C2×D411yescharacteristic
148C2×C8×C73C2×D411yescharacteristic
149C2×C8×C73C2×D411yescharacteristic
150C2×C8×C73C2×D411yescharacteristic
151C4⋊C4×C73—21no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—41no—
158C4⋊C4×C73—41no—
159C22⋊C4×C73—41no—
160C22⋊C4×C73—41no—
161C22⋊C4×C73—41no—
162C42×C73—41no—
163C42×C73C2×D411yescharacteristic
164C42×C73C2×D411yescharacteristic
165C42×C73#4111yescharacteristic
166C42×C73C2411yescharacteristic
167C23×C73—41no—
168C23×C73#12111yescharacteristic
169C22×D73—24no—
170C22×D73—22no—
171C22×D73—42no—
172C22×D73—42no—
173Q8×C73—42no—
174Q8×C73—41no—
175Q8×C73—41no—
176Q8×C73—81no—
177D4×C73—42no—
178D4×C73—41no—
179D4×C73—41no—
180D4×C73—81no—
181D4×C73—82no—
182C2×C4×C73—22no—
183C2×C4×C73—22no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—41no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—82no—
190C2×C4×C73#12111yescharacteristic
191C2×C4×C73#12111yescharacteristic
192C2×C4×C73#12111yescharacteristic
193C2×C4×C73C22×D411yescharacteristic
194C2×C4×C73C22×D411yescharacteristic
195C2×C4×C73C2×#4111yescharacteristic
196C2×Dic73—22no—
197C2×Dic73—42no—
198D292—24no—
199D292—22no—
200D292#13514yes—
201C4×D73—44no—
202C4×D73—42no—
203C4×D73—42no—
204C8×C73—21no—
205C8×C73—21no—
206C8×C73—21no—
207C8×C73AGL(1,ℤ/8)12yes—
208#10596831—42no—
209#10596831—42no—
210#10596831—42no—
211C22×C73—21no—
212C22×C73—21no—
213C22×C73—81no—
214C22×C73—81no—
215C22×C73—81no—
216C22×C73#47112yes—
217C22×C73C2×#12111yescharacteristic
218C2×D73—24no—
219C2×D73—48no—
220C2×D73—42no—
221C2×D73—42no—
222C4×C73—21no—
223C4×C73—21no—
224C4×C73—21no—
225C4×C73—21no—
226C4×C73—81no—
227C4×C73—81no—
228C4×C73—81no—
229C4×C73C2×#13512yes—
230C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C73#57511yescharacteristic
232Dic73—42no—
233#19257—731nomaximal, Sylow
234C2×C73—22no—
235C2×C73—161no—
236C2×C73#303611yescharacteristic
237C2×C73#305011yescharacteristic
238C2×C73C2×#47111yescharacteristic
239D73—44no—
240#3167D7311yescharacteristic, complemented
241#3110—731no—
242#3109—731no—
243C2×C8×D4—731no—
244#1683—732no—
245#1681—732no—
246#1671—732no—
247#1665—732no—
248#1568—731no—
249#1552—731no—
250#1551—731no—
251C73#1925711yescharacteristic, complemented, Sylow
252C23×C8—1461no—
253#533C2×D7311yescharacteristic
254#532C2×D7311yescharacteristic
255C2×C4×D4—731no—
256#498C2×D7311yescharacteristic, complemented
257#492C2×D7311yescharacteristic, complemented
258#487C2×D7311yescharacteristic, complemented
259#486C2×D7311yescharacteristic, complemented
260#484—1461no—
261#481—1461no—
262#476—732no—
263#474—732no—
264#465—21no—
265#465—21no—
266#461—732no—
267#459—732no—
268#450—1461no—
269#447—1461no—
270#442—1462no—
271#440—1462no—
272C4×SD16—21no—
273C4×SD16—21no—
274C8×D4—732no—
275C8×D4—732no—
276C8×D4—1461no—
277C8×D4—1461no—
278#432—731no—
279#432—731no—
280C2×C4⋊C8—731no—
281C2×C22⋊C8—1461no—
282C2×C4×C8C2×D7311yescharacteristic
283#335—732no—
284#334—732no—
285#333—732no—
286#332—732no—
287#332—732no—
288#331—732no—
289C2×#41—21no—
290C22×D4—731no—
291C23×C4—1461no—
292C2×SD16—21no—
293C2×SD16—21no—
294C2×SD16—1462no—
295C2×D8—1462no—
296C22×C8—2924no—
297C22×C8—1461no—
298C22×C8—1462no—
299C22×C8—1462no—
300C22×C8C22⋊D7311yescharacteristic
301C22×C8C22⋊D7311yescharacteristic
302C4⋊Q8C22×D7311yescharacteristic
303C4⋊Q8C22×D7311yescharacteristic
304C4⋊Q8C22×D7311yescharacteristic
305#127C22×D7311yescharacteristic
306#125—21no—
307#124—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—21no—
311#121—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—732no—
317C4×D4—732no—
318C4×D4—1461no—
319C4×D4—1461no—
320#117—21no—
321C2×C4⋊C4—731no—
322C2×C22⋊C4—1461no—
323C2×C42C22×D7311yescharacteristic
324#107—1462no—
325#106—21no—
326#106—21no—
327#106—1462no—
328C4⋊C8—732no—
329C4⋊C8—732no—
330C4⋊C8—732no—
331C4⋊C8—1461no—
332#103—21no—
333#103—21no—
334#103—1462no—
335#102—21no—
336#102—21no—
337#102—1462no—
338#102—1462no—
339#102—1462no—
340C22⋊C8—1461no—
341C22⋊C8—1461no—
342C22⋊C8—1462no—
343C22⋊C8—1462no—
344#97—1461no—
345#97—1461no—
346C4×C8C22⋊D7311yescharacteristic
347C4×C8C22⋊D7311yescharacteristic
348C4×C8C22×D7311yescharacteristic
349C4×C8C22×D7311yescharacteristic
350C24—1461no—
351#41—82no—
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—42no—
372SD16—42no—
373SD16—2924no—
374D8—2924no—
375C2×C8—22no—
376C2×C8—41no—
377C2×C8—2924no—
378C2×C8—2922no—
379C2×C8—2924no—
380C2×C8—1462no—
381C2×C8—1462no—
382C2×C8—1462no—
383C2×C8C2×C22⋊D7311yescharacteristic
384C2×C8C2×C22⋊D7311yescharacteristic
385C2×C8C2×C22⋊D7311yescharacteristic
386C2×C8C2×C22⋊D7311yescharacteristic
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—732no—
396C4⋊C4—1461no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—1462no—
401C22⋊C4—1462no—
402C42—41no—
403C42D4×D7311yescharacteristic
404C42C2×C22⋊D7311yescharacteristic
405C42#1192402711yescharacteristic
406C42C23×D7311yescharacteristic
407C23—41no—
408C23—2922no—
409C23—2922no—
410C23—1464no—
411C23—1462no—
412C23#1455656211yescharacteristic
413Q8—41no—
414Q8—42no—
415Q8—41no—
416Q8—81no—
417D4—41no—
418D4—42no—
419D4—41no—
420D4—81no—
421D4—82no—
422D4—734no—
423D4—1464no—
424D4—1462no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4—2924no—
434C2×C4—2922no—
435C2×C4—2922no—
436C2×C4—2922no—
437C2×C4—1462no—
438C2×C4#1455656211yescharacteristic
439C2×C4#1455656211yescharacteristic
440C2×C4#1455656211yescharacteristic
441C2×C4C2×D4×D7311yescharacteristic
442C2×C4C22×C22⋊D7311yescharacteristic
443C2×C4C2×#1192402711yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8—2922no—
448C8—2922no—
449C8—2922no—
450C8#1455661612yes—
451C22—21no—
452C22—21no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—2928no—
457C22—2922no—
458C22—2922no—
459C22—1464no—
460C22#1798884712yes—
461C22C2×#1455656211yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—2922no—
470C4#135×D7312yes—
471C4C2×#1455661611yescharacteristic
472C4#1798938511yescharacteristic
473C2—22no—
474C2—161no—
475C2—2924no—
476C2#2261939111yescharacteristic
477C2#2261950811yescharacteristic
478C2C2×#1798884711yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3167C2
socleC22×C73C2×#121
prime radical#22620018C2
hypercenter#3167D73
hypocenterC73#19257
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #22620026C2semidirect
3. C2×C8×D292C2semidirect
4. #22609379C2semidirect
5. #22609365C2semidirect
6. #22609303C2semidirect
7. #22609269C2semidirect
8. #22608694C2semidirect
9. #22608631C2semidirect
10. #22608626C2semidirect
11. C73×#3167C2semidirect
32. #17988340C22semidirect
33. #17988338C22semidirect
34. #17988336C22semidirect
35. #17988333C22semidirect
36. #17988333C22semidirect
37. #17988331C22semidirect
40. #498×C73C22semidirect
41. #492×C73C22semidirect
42. #487×C73C22semidirect
43. #486×C73C22semidirect
240. #3167D73semidirect
251. C73#19257semidirect
256. #498C2×D73semidirect
257. #492C2×D73semidirect
258. #487C2×D73semidirect
259. #486C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226200262
#226200182
C2×C8×D2922
#226093792
#226093652
#226093032
#226092692
#226086942
#226086312
#226086262
C73×#31672
#1925773