Finite Group Databasebeta

← #28743090

Subgroups — #28743090

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437230C211yesmaximal, characteristic, complemented
2C2×#16837837C211yesmaximal, characteristic, complemented
3#21434582C211yesmaximal, characteristic, complemented
4#21426172C214yesmaximal, complemented
5#21426165C214yesmaximal, complemented
6#21425862C211yesmaximal, characteristic, complemented
7#21425843C211yesmaximal, characteristic, complemented
8C2×#16837214C211yesmaximal, characteristic, complemented
9C59×#2957C211yesmaximal, characteristic, complemented
10C2×#14273755—21no—
11C2×#14273749C2211yescharacteristic
12C22×D472C2211yescharacteristic
13#16837971—21no—
14#16837843—21no—
15#16837842C2212yescomplemented
16#16837838—21no—
17#16837837—22no—
18#16837837C2214yes—
19#16837837C2212yescomplemented
20#16837832—22no—
21C2×#14273670—21no—
22C2×#14273668—21no—
23C2×#14273667C2211yescharacteristic
24C2×#14273667C2211yescharacteristic
25#16837791—22no—
26#16837786C2212yescomplemented
27#16837779C2212yescomplemented
28#16837223C2214yescomplemented
29#16837217C2212yescomplemented
30#16837214C2214yescomplemented
31#16837214C2212yescomplemented
32C2×#135×C59C2211yescharacteristic
33C59×#568—21no—
34C59×#444—22no—
35C59×#443C2214yescomplemented
36C2×#108×C59C2211yescharacteristic
37C59×#421—21no—
38C59×#405C2211yescharacteristic
39C2×#11745685—21no—
40C22×D236—21no—
41#14273755—42no—
42#14273755—42no—
43#14273749—21no—
44#14273749—21no—
45#14273749—21no—
46#14273749—21no—
47C2×Dic236C2311yescharacteristic
48C2×#11745660—21no—
49C2×#11745660—22no—
50C2×D472—21no—
51C2×D472—21no—
52C2×D472—21no—
53C2×D472—21no—
54C2×D472—21no—
55C2×D472—22no—
56C2×D472C2312yes—
57C2×D472C2311yescharacteristic
58#14273724—22no—
59#14273715—22no—
60#14273702—42no—
61#14273697—42no—
62#14273673—22no—
63#14273671—22no—
64#14273670—24no—
65#14273670—22no—
66#14273668—24no—
67#14273668—22no—
68#14273667—21no—
69#14273667—21no—
70#14273667C2312yes—
71#14273667C2312yes—
72C2×C59⋊C16—21no—
73C2×C59⋊C16—21no—
74C2×#41×C59—21no—
75#135×C59—21no—
76#135×C59—21no—
77#135×C59D412yescomplemented
78#135×C59C2312yes—
79C2×Q16×C59D411yescharacteristic, complemented
80C2×SD16×C59—21no—
81C2×D8×C59D411yescharacteristic, complemented
82#131×C59—24no—
83#131×C59—42no—
84C2×M16×C59—21no—
85C2×M16×C59—42no—
86C2×M16×C59D411yescharacteristic
87C22×C8×C59C2311yescharacteristic
88#117×C59D411yescharacteristic
89#108×C59—21no—
90#108×C59C2312yes—
91#104×C59—24no—
92#97×C59D412yes—
93C4×C8×C59C2312yes—
94C23×D59—41no—
95#41×C59—22no—
96#41×C59—22no—
97#41×C59—41no—
98#41×C59—41no—
99C2×Q8×C59—21no—
100C2×D4×C59—21no—
101C2×D4×C59—41no—
102C22×C4×C59—41no—
103C22×C4×C59C2×D411yescharacteristic
104C2×C22⋊D59—41no—
105#11745685—41no—
106#11745685—41no—
107#11745685—41no—
108#11745685—41no—
109C2×D236—22no—
110C2×D236—21no—
111C2×D236—41no—
112C2×D236—41no—
113C2×D236—41no—
114C2×D236—41no—
115C2×D236—41no—
116C2×D236—44no—
117C2×C4×D59—41no—
118C2×Dic118—21no—
119Q16×C59—21no—
120Q16×C59C2×D412yes—
121SD16×C59—22no—
122SD16×C59—22no—
123D8×C59—21no—
124D8×C59C2×D412yes—
125M16×C59—22no—
126M16×C59—41no—
127M16×C59—44no—
128M16×C59D812yes—
129M16×C59D812yes—
130C2×C8×C59—41no—
131C2×C8×C59—44no—
132C2×C8×C59C2×D411yescharacteristic
133C2×C8×C59C2×D411yescharacteristic
134C2×C8×C59C2×D411yescharacteristic
135C2×C8×C59C2×D412yes—
136C2×C8×C59#4111yescharacteristic
137C2×C8×C59#4111yescharacteristic
138C2×C8×C59C2411yescharacteristic
139C4⋊C4×C59—21no—
140C22⋊C4×C59—21no—
141C42×C59C2×D412yes—
142(C2×D59)⋊C4—44no—
143Dic236—21no—
144Dic236—21no—
145D472—22no—
146D472—21no—
147D472—22no—
148D472—21no—
149D472—41no—
150D472—41no—
151D472—44no—
152#11745660—41no—
153#11745660—41no—
154#11745660—44no—
155C59⋊C16—22no—
156C59⋊C16—22no—
157C23×C59—41no—
158C23×C59AGL(1,ℤ/8)11yescharacteristic
159C22×D59—42no—
160C22×D59—41no—
161C22×D59—81no—
162C22×D59—81no—
163C22×D59—162no—
164Q8×C59—22no—
165Q8×C59—41no—
166D4×C59—22no—
167D4×C59—42no—
168D4×C59—42no—
169D4×C59—41no—
170C2×C4×C59—22no—
171C2×C4×C59—41no—
172C2×C4×C59—42no—
173C2×C4×C59—42no—
174C2×C4×C59#12111yescharacteristic
175C2×C4×C59#12111yescharacteristic
176C2×C4×C59C2×D811yescharacteristic
177C2×C4×C59C2×D811yescharacteristic
178C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C59C22×D411yescharacteristic
180C22⋊D59—81no—
181C22⋊D59—81no—
182C2×Dic59—41no—
183D236—41no—
184D236—42no—
185D236—42no—
186D236—41no—
187D236—81no—
188D236—81no—
189D236—164no—
190C4×D59—81no—
191C4×D59—81no—
192Dic118—41no—
193Dic118—41no—
194C8×C59—42no—
195C8×C59#12111yescharacteristic
196C8×C59#12111yescharacteristic
197C8×C59C2×D814yes—
198C8×C59C22×D411yescharacteristic
199C8×C59C2×#4111yescharacteristic
200#33595—591nomaximal, Sylow
201C22×C59—41no—
202C22×C59—42no—
203C22×C59—161no—
204C22×C59#45911yescharacteristic
205C22×C59#45911yescharacteristic
206C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
207C2×D59—82no—
208C2×D59—84no—
209C2×D59—81no—
210C2×D59—82no—
211C2×D59—162no—
212C2×D59—162no—
213C4×C59—42no—
214C4×C59—42no—
215C4×C59#45911yescharacteristic
216C4×C59#45911yescharacteristic
217C4×C59C2×#12111yescharacteristic
218C4×C59C22×D811yescharacteristic
219Dic59—81no—
220#3414—591no—
221C2×#509—591no—
222#3155—591no—
223#2957D5911yescharacteristic, complemented
224#2213—594no—
225#2212—594no—
226#2153—591no—
227#2148—591no—
228C2×#361—591no—
229C2×C59—21no—
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—82no—
233C2×C59C2×#45911yescharacteristic
234D59—161no—
235D59—162no—
236C2×AGL(1,ℤ/8)—1181no—
237C2×#135—591no—
238C2×#135C2×D5911yescharacteristic
239C22×D8—591no—
240#568—21no—
241#535—1181no—
242#510—592no—
243#510—1181no—
244#509—594no—
245#509—592no—
246#509—1182no—
247#509—1181no—
248#508—1182no—
249C2×SD32—1181no—
250C2×D16—1181no—
251C2×M32—591no—
252C2×M32—591no—
253#496—1182no—
254#495—592no—
255#493—592no—
256#444—22no—
257#443C2×D5914yescomplemented
258C2×#108C2×D5911yescharacteristic
259#421—21no—
260#405C2×D5911yescharacteristic
261#364—594no—
262#362—592no—
263#361—594no—
264#361—592no—
265C59#3359511yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C2×#41—1181no—
268C22×D4—1181no—
269AGL(1,ℤ/8)—2362no—
270AGL(1,ℤ/8)—2362no—
271#135—21no—
272#135—21no—
273#135—1181no—
274#135—1181no—
275#135—1181no—
276#135—1181no—
277#135D23612yescomplemented
278#135C22×D5912yes—
279C2×Q16—591no—
280C2×Q16D23611yescharacteristic, complemented
281C2×SD16—21no—
282C2×SD16—1181no—
283C2×SD16—1182no—
284C2×D8—592no—
285C2×D8—591no—
286C2×D8—1181no—
287C2×D8—1181no—
288C2×D8—1181no—
289C2×D8—1181no—
290C2×D8—1181no—
291C2×D8—1182no—
292C2×D8D23611yescharacteristic, complemented
293#131—24no—
294#131—42no—
295C2×M16—21no—
296C2×M16—42no—
297C2×M16C22⋊D5911yescharacteristic
298C22×C8C22×D5911yescharacteristic
299#127—1182no—
300#124—1182no—
301#121—2362no—
302#120—2362no—
303#117C22⋊D5911yescharacteristic
304Q32—1182no—
305SD32—1184no—
306SD32—1182no—
307SD32—1182no—
308D16—1184no—
309D16—1182no—
310M32—592no—
311M32—592no—
312M32—1181no—
313M32—1181no—
314C2×C16—1181no—
315C2×C16—1181no—
316#108—21no—
317#108C22×D5912yes—
318#104—24no—
319#97C22⋊D5912yes—
320C4×C8C22×D5912yes—
321C24—2361no—
322#41—22no—
323#41—22no—
324#41—41no—
325#41—41no—
326#41—2361no—
327#41—2361no—
328#41—2361no—
329#41—2361no—
330C2×Q8—21no—
331C2×Q8—1181no—
332C2×D4—21no—
333C2×D4—41no—
334C2×D4—1182no—
335C2×D4—1181no—
336C2×D4—2364no—
337C2×D4—2361no—
338C2×D4—2361no—
339C2×D4—2361no—
340C2×D4—2361no—
341C2×D4—2361no—
342C2×D4—2361no—
343C22×C4—41no—
344C22×C4—2361no—
345C22×C4C2×C22⋊D5911yescharacteristic
346Q16—21no—
347Q16—1181no—
348Q16—1181no—
349Q16C2×D23612yes—
350SD16—22no—
351SD16—22no—
352SD16—2361no—
353SD16—2361no—
354SD16—2364no—
355D8—21no—
356D8—1182no—
357D8—1181no—
358D8—1182no—
359D8—1181no—
360D8—2361no—
361D8—2361no—
362D8—2364no—
363D8C2×D23612yes—
364M16—22no—
365M16—41no—
366M16—44no—
367M16#1174566912yes—
368M16#1174566912yes—
369C2×C8—41no—
370C2×C8—44no—
371C2×C8C2×D23611yescharacteristic
372C2×C8C2×D23611yescharacteristic
373C2×C8#1174568511yescharacteristic
374C2×C8#1174568511yescharacteristic
375C2×C8C2×C22⋊D5911yescharacteristic
376C2×C8C2×C22⋊D5912yes—
377C2×C8C23×D5911yescharacteristic
378C4⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—2364no—
381C42C2×C22⋊D5912yes—
382C16—1182no—
383C16—1182no—
384C23—41no—
385C23—9442no—
386C23—4721no—
387C23—4721no—
388C23—2362no—
389C23—2361no—
390C23#1427375211yescharacteristic
391Q8—22no—
392Q8—41no—
393Q8—2361no—
394Q8—2361no—
395D4—22no—
396D4—42no—
397D4—42no—
398D4—41no—
399D4—9444no—
400D4—4721no—
401D4—4721no—
402D4—4721no—
403D4—4721no—
404D4—2362no—
405D4—2361no—
406D4—2361no—
407D4—2362no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—4721no—
413C2×C4—4721no—
414C2×C4—2361no—
415C2×C4#1427370111yescharacteristic
416C2×C4#1427370111yescharacteristic
417C2×C4C2×#1174566911yescharacteristic
418C2×C4C2×#1174566911yescharacteristic
419C2×C4#1427375211yescharacteristic
420C2×C4C22×C22⋊D5911yescharacteristic
421C8—42no—
422C8#1427370111yescharacteristic
423C8#1427370111yescharacteristic
424C8C2×#1174566914yes—
425C8C22×D23611yescharacteristic
426C8C2×#1174568511yescharacteristic
427C22—41no—
428C22—42no—
429C22—161no—
430C22—9442no—
431C22—9442no—
432C22—4722no—
433C22—4724no—
434C22—4721no—
435C22—4722no—
436C22#1683759711yescharacteristic
437C22#1683759711yescharacteristic
438C22C2×#1427375211yescharacteristic
439C4—42no—
440C4—42no—
441C4—4721no—
442C4#1683759711yescharacteristic
443C4#1683759711yescharacteristic
444C4C2×#1427370111yescharacteristic
445C4C22×#1174566911yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—82no—
450C2—9442no—
451C2—9441no—
452C2C2×#1683759711yescharacteristic
subgroupG/H
centerC2C2×#16837597
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2957C2
socleC2×C59C2×#459
hypercenter#2957D59
hypocenterC59#33595
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
1. #21437230C2semidirect
2. C2×#16837837C2semidirect
3. #21434582C2semidirect
4. #21426172C2semidirect
5. #21426165C2semidirect
6. #21425862C2semidirect
7. #21425843C2semidirect
8. C2×#16837214C2semidirect
9. C59×#2957C2semidirect
15. #16837842C22semidirect
19. #16837837C22semidirect
26. #16837786C22semidirect
27. #16837779C22semidirect
28. #16837223C22semidirect
29. #16837217C22semidirect
30. #16837214C22semidirect
31. #16837214C22semidirect
35. C59×#443C22semidirect
77. #135×C59D4semidirect
79. C2×Q16×C59D4semidirect
81. C2×D8×C59D4semidirect
223. #2957D59semidirect
257. #443C2×D59semidirect
265. C59#33595semidirect
277. #135D236semidirect
280. C2×Q16D236semidirect
292. C2×D8D236semidirect

Maximal subgroups

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

subgroupindex
#214372302
C2×#168378372
#214345822
#214261722
#214261652
#214258622
#214258432
C2×#168372142
C59×#29572
#3359559