Finite Group Databasebeta

← #33240474

Subgroups — #33240474

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989027C211yesmaximal, characteristic, complemented
2C22×#14556531C211yesmaximal, characteristic, complemented
3C2×#17988828C211yesmaximal, characteristic, complemented
4#22612042C212yesmaximal, complemented
5#22612037C212yesmaximal, complemented
6#22612025C212yesmaximal, complemented
7#22612020C212yesmaximal, complemented
8#22611720C211yesmaximal, characteristic, complemented
9#22611650C211yesmaximal, characteristic, complemented
10#22611643C211yesmaximal, characteristic, complemented
11C73×#3050C211yesmaximal, characteristic
12C22×Dic292C2211yescharacteristic
13C2×#14556570—21no—
14#17989027—24no—
15C2×#14556537C2212yescomplemented
16C2×#14556535C2212yescomplemented
17C2×#14556531—24no—
18C2×#14556531C2212yes—
19C2×#14556531C2211yescharacteristic
20C2×#14556531C2211yescharacteristic
21C22×#11924003C2212yes—
22#17988828—21no—
23#17988828—21no—
24#17988828C2212yes—
25#17988828C2212yes—
26C2×#14556525C2211yescharacteristic
27C2×#14556516—21no—
28#17988414C2212yescomplemented
29#17988412C2212yescomplemented
30#17988391C2212yescomplemented
31#17988390C2212yescomplemented
32#17988390C2212yescomplemented
33#17988388C2212yescomplemented
34C2×#135×C73C2211yescharacteristic
35C23×C8×C73C2211yescharacteristic
36#525×C73—21no—
37#467×C73C2211yescharacteristic
38#467×C73C2211yescharacteristic
39#466×C73C2211yescharacteristic
40#466×C73C2211yescharacteristic
41#465×C73—21no—
42#465×C73—21no—
43#427×C73C2211yescharacteristic
44#416×C73—21no—
45C22×Dic146—21no—
46C2×Dic292—22no—
47C2×Dic292—21no—
48C2×Dic292—22no—
49C2×Dic292C2312yes—
50#14556570—22no—
51#14556570—22no—
52#14556570—41no—
53#14556570—41no—
54C2×C4⋊Dic73—21no—
55C2×#11924007—41no—
56C2×C22⋊Dic73—41no—
57#14556537—24no—
58#14556535—24no—
59#14556531—24no—
60#14556531—22no—
61#14556531D414yescomplemented
62C2×#11924003—24no—
63C2×#11924003C2312yes—
64C2×#11924003C2312yes—
65#14556525—21no—
66#14556525C2312yes—
67#14556516—22no—
68#14556516—22no—
69C2×#41×C73—21no—
70C23×C4×C73D411yescharacteristic
71#135×C73—22no—
72#135×C73—22no—
73C2×Q16×C73C2311yescharacteristic
74C2×SD16×C73—21no—
75C2×D8×C73C2311yescharacteristic
76C22×C8×C73—22no—
77C22×C8×C73—22no—
78C22×C8×C73D411yescharacteristic
79C22×C8×C73D411yescharacteristic
80C22×C8×C73D411yescharacteristic
81C22×C8×C73C2311yescharacteristic
82C22×C8×C73C2311yescharacteristic
83C22×C8×C73C2311yescharacteristic
84#123×C73—41no—
85C22⋊Q8×C73—21no—
86C22⋊Q8×C73—21no—
87#121×C73—21no—
88#121×C73—21no—
89#120×C73—41no—
90C4×D4×C73—41no—
91C4×D4×C73—41no—
92#117×C73—21no—
93#107×C73C2311yescharacteristic
94#107×C73C2311yescharacteristic
95#106×C73—21no—
96#103×C73—21no—
97#103×C73—21no—
98#102×C73—21no—
99#102×C73—21no—
100C2×Dic146—22no—
101C2×Dic146—42no—
102C2×Dic146—42no—
103C2×Dic146—41no—
104C22×Dic73—41no—
105C22×Dic73—41no—
106Dic292—22no—
107Dic292—22no—
108Dic292—42no—
109C4⋊Dic73—22no—
110C4⋊Dic73—41no—
111#11924007—42no—
112#11924007—42no—
113C22⋊Dic73—42no—
114C22⋊Dic73—42no—
115#11924003—22no—
116#11924003C2×D414yes—
117C24×C73D811yescharacteristic
118#41×C73—42no—
119#41×C73—42no—
120C2×Q8×C73—21no—
121C2×D4×C73—21no—
122C2×D4×C73—41no—
123C2×D4×C73—41no—
124C2×D4×C73—41no—
125C22×C4×C73—22no—
126C22×C4×C73—22no—
127C22×C4×C73—41no—
128C22×C4×C73D811yescharacteristic
129C22×C4×C73D811yescharacteristic
130C22×C4×C73D811yescharacteristic
131C22×C4×C73C2×D411yescharacteristic
132C22×C4×C73C2×D411yescharacteristic
133C22×C4×C73C2×D411yescharacteristic
134Q16×C73—21no—
135Q16×C73—21no—
136SD16×C73—42no—
137D8×C73—21no—
138D8×C73—21no—
139C2×C8×C73—22no—
140C2×C8×C73—22no—
141C2×C8×C73—22no—
142C2×C8×C73—21no—
143C2×C8×C73—21no—
144C2×C8×C73—21no—
145C2×C8×C73—21no—
146C2×C8×C73C2×D412yes—
147C2×C8×C73C2×D411yescharacteristic
148C2×C8×C73C2×D411yescharacteristic
149C2×C8×C73C2×D411yescharacteristic
150C2×C8×C73#4112yes—
151C2×C8×C73C2411yescharacteristic
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—41no—
155C4⋊C4×C73—41no—
156C22⋊C4×C73—41no—
157C22⋊C4×C73—41no—
158C22⋊C4×C73—41no—
159C22⋊C4×C73—41no—
160C22⋊C4×C73—41no—
161C42×C73—41no—
162C23×C73—22no—
163C23×C73—22no—
164C23×C73—41no—
165C23×C73—41no—
166C23×C73C2×D811yescharacteristic
167C23×C73C2×D811yescharacteristic
168C23×C73C2×D811yescharacteristic
169Q8×C73—41no—
170Q8×C73—41no—
171D4×C73—41no—
172D4×C73—41no—
173D4×C73—82no—
174D4×C73—82no—
175D4×C73—82no—
176C2×C4×C73—22no—
177C2×C4×C73—21no—
178C2×C4×C73—21no—
179C2×C4×C73—21no—
180C2×C4×C73—21no—
181C2×C4×C73—22no—
182C2×C4×C73—22no—
183C2×C4×C73—41no—
184C2×C4×C73—41no—
185C2×C4×C73—41no—
186C2×C4×C73—82no—
187C2×C4×C73#12112yes—
188C2×C4×C73C2×D811yescharacteristic
189C2×C4×C73C2×D811yescharacteristic
190C2×C4×C73C2×D811yescharacteristic
191C2×C4×C73#13512yes—
192C2×C4×C73C22×D411yescharacteristic
193C2×Dic73—42no—
194C2×Dic73—42no—
195C2×Dic73—82no—
196C2×Dic73—82no—
197Dic146—42no—
198Dic146—42no—
199Dic146—82no—
200C8×C73—21no—
201C8×C73—21no—
202C8×C73#12112yes—
203C8×C73C22×D411yescharacteristic
204C8×C73C2×#4111yescharacteristic
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73—21no—
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—44no—
213C22×C73—81no—
214C22×C73—81no—
215C22×C73—81no—
216C22×C73#46612yes—
217C22×C73#51012yescomplemented
218C22×C73C22×D811yescharacteristic
219C4×C73—21no—
220C4×C73—21no—
221C4×C73—81no—
222C4×C73—81no—
223C4×C73—81no—
224C4×C73#46612yes—
225C4×C73C2×#12111yescharacteristic
226C4×C73C2×#13511yescharacteristic
227Dic73—82no—
228Dic73—82no—
229#33606—731nomaximal, Sylow
230C2×C73—22no—
231C2×C73—22no—
232C2×C73—41no—
233C2×C73—161no—
234C2×C73C2×#46611yescharacteristic
235C2×C73C2×#51011yescharacteristic
236C2×C73#341411yescharacteristic
237C2×#510—731no—
238C22×M32—731no—
239#3050D7311yescharacteristic
240C2×#467—731no—
241#2220—732no—
242#2219—732no—
243#2217—732no—
244#2216—732no—
245#2159—731no—
246#2141—731no—
247#2139—731no—
248C73#3360611yescharacteristic, complemented, Sylow
249C2×#135C2×D7311yescharacteristic
250C22×Q16—731no—
251C23×C8C2×D7311yescharacteristic
252#525—21no—
253C2×C22⋊Q8—1461no—
254#510—1464no—
255C2×Q32—732no—
256C2×SD32—732no—
257C2×M32—731no—
258C2×M32—731no—
259C2×M32—732no—
260C2×M32—1464no—
261C22×C16—732no—
262#467—732no—
263#467—732no—
264#467—1461no—
265#467—1461no—
266#467C2×D7311yescharacteristic
267#467C2×D7311yescharacteristic
268#466C2×D7311yescharacteristic
269#466C2×D7311yescharacteristic
270#465—21no—
271#465—21no—
272#427C2×D7311yescharacteristic
273C2×#107—731no—
274#416—21no—
275C2×#103—1461no—
276#367—732no—
277#366—732no—
278#358—732no—
279#358—732no—
280#358—732no—
281#357—732no—
282C2×#41—21no—
283C22×Q8—1461no—
284C23×C4C22⋊D7311yescharacteristic
285#135—22no—
286#135—22no—
287C2×Q16—732no—
288C2×Q16—1462no—
289C2×Q16—1461no—
290C2×Q16—1462no—
291C2×Q16C22×D7311yescharacteristic
292C2×SD16—21no—
293C2×D8C22×D7311yescharacteristic
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C22⋊D7311yescharacteristic
297C22×C8C22⋊D7311yescharacteristic
298C22×C8C22⋊D7311yescharacteristic
299C22×C8C22×D7311yescharacteristic
300C22×C8C22×D7311yescharacteristic
301C22×C8C22×D7311yescharacteristic
302#123—41no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—2921no—
306C22⋊Q8—2921no—
307C22⋊Q8—1462no—
308C22⋊Q8—1462no—
309#121—21no—
310#121—21no—
311#120—41no—
312C4×D4—41no—
313C4×D4—41no—
314#117—21no—
315C2×C4⋊C4—2921no—
316C2×C4⋊C4—1461no—
317C2×C22⋊C4—2921no—
318Q32—1464no—
319SD32—1464no—
320M32—734no—
321M32—1464no—
322M32—1462no—
323C2×C16—732no—
324C2×C16—732no—
325C2×C16—1464no—
326#107—732no—
327#107—1461no—
328#107C22×D7311yescharacteristic
329#107C22×D7311yescharacteristic
330#106—21no—
331#103—21no—
332#103—21no—
333#103—1462no—
334#103—1462no—
335#102—21no—
336#102—21no—
337C24#1192401311yescharacteristic
338#41—42no—
339#41—42no—
340C2×Q8—21no—
341C2×Q8—2921no—
342C2×Q8—2922no—
343C2×Q8—2922no—
344C2×Q8—1462no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—2921no—
353C22×C4—2921no—
354C22×C4#1192401311yescharacteristic
355C22×C4#1192401311yescharacteristic
356C22×C4#1192401311yescharacteristic
357C22×C4C2×C22⋊D7311yescharacteristic
358C22×C4C2×C22⋊D7311yescharacteristic
359C22×C4C2×C22⋊D7311yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—2922no—
363Q16—1462no—
364Q16—1462no—
365SD16—42no—
366D8—21no—
367D8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8D4×D7312yes—
376C2×C8C2×C22⋊D7311yescharacteristic
377C2×C8C2×C22⋊D7311yescharacteristic
378C2×C8C2×C22⋊D7311yescharacteristic
379C2×C8#1192402712yes—
380C2×C8C23×D7311yescharacteristic
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—2922no—
386C4⋊C4—2922no—
387C4⋊C4—2921no—
388C4⋊C4—1462no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—2922no—
395C22⋊C4—2922no—
396C42—41no—
397C16—734no—
398C16—1462no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23C2×#1192401311yescharacteristic
404C23C2×#1192401311yescharacteristic
405C23C2×#1192401311yescharacteristic
406Q8—41no—
407Q8—41no—
408Q8—2922no—
409Q8—2922no—
410Q8—5842no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—82no—
415D4—82no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—82no—
427C2×C4—2922no—
428C2×C4—2922no—
429C2×C4—5842no—
430C2×C4—5842no—
431C2×C4#1455656312yes—
432C2×C4C2×#1192401311yescharacteristic
433C2×C4C2×#1192401311yescharacteristic
434C2×C4C2×#1192401311yescharacteristic
435C2×C4#1455661112yes—
436C2×C4C22×C22⋊D7311yescharacteristic
437C8—21no—
438C8—21no—
439C8#1455656312yes—
440C8C2×D4×D7311yescharacteristic
441C8C2×#1192402711yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22#1798882112yes—
454C22#1798902712yescomplemented
455C22C22×#1192401311yescharacteristic
456C4—21no—
457C4—21no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—5842no—
462C4—5842no—
463C4#1798882112yes—
464C4C2×#1455656311yescharacteristic
465C4C2×#1455661111yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—161no—
470C2C2×#1798882111yescharacteristic
471C2C2×#1798902711yescharacteristic
472C2#2262306511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3050C2
socleC22×C73C22×D8
hypercenter#3050D73
hypocenterC73#33606
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#17989027C2semidirect
2. C22×#14556531C2semidirect
3. C2×#17988828C2semidirect
4. #22612042C2semidirect
5. #22612037C2semidirect
6. #22612025C2semidirect
7. #22612020C2semidirect
8. #22611720C2semidirect
9. #22611650C2semidirect
10. #22611643C2semidirect
15. C2×#14556537C22semidirect
16. C2×#14556535C22semidirect
28. #17988414C22semidirect
29. #17988412C22semidirect
30. #17988391C22semidirect
31. #17988390C22semidirect
32. #17988390C22semidirect
33. #17988388C22semidirect
61. #14556531D4semidirect
217. C22×C73#510semidirect
248. C73#33606semidirect
454. C22#17989027semidirect

Maximal subgroups

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

subgroupindex
C2×#179890272
C22×#145565312
C2×#179888282
#226120422
#226120372
#226120252
#226120202
#226117202
#226116502
#226116432
C73×#30502
#3360673