Finite Group Databasebeta

← #28485549

Subgroups — #28485549

Order 15104.

580 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433576C211yesmaximal, characteristic
2#21425022C211yesmaximal, characteristic, complemented
3#21424116C211yesmaximal, characteristic, complemented
4#21422763C211yesmaximal, characteristic, complemented
5#21421998C211yesmaximal, characteristic, complemented
6#21421971C211yesmaximal, characteristic, complemented
7C59×#2988C211yesmaximal, characteristic, complemented
8C2×#14273752—21no—
9C2×#14273724C2211yescharacteristic
10#16837602—21no—
11#16837602C412yescomplemented
12#16837597—21no—
13#16837597C412yescomplemented
14#16837426C2211yescharacteristic
15#16837426C2211yescharacteristic
16C2×#14273654—21no—
17#16837406—21no—
18C2×#14273649—22no—
19C4×C4⋊Dic59C2211yescharacteristic
20#16837161—21no—
21#16837145—21no—
22#16837145—21no—
23C4×#41×C59C2211yescharacteristic
24C8×Q8×C59—21no—
25C8×D4×C59—22no—
26C8×D4×C59—21no—
27C59×#431—21no—
28C59×#431C2211yescharacteristic
29C2×C4⋊C8×C59—21no—
30C2×C4⋊C8×C59C2211yescharacteristic
31C22×D236—21no—
32C22×D236—21no—
33C22×D236—21no—
34#14273752—42no—
35#14273752—42no—
36C2×#11745671—22no—
37C2×#11745669—22no—
38C2×#11745664—21no—
39C2×#11745664—21no—
40#14273724—21no—
41#14273724—21no—
42#14273724—21no—
43#14273724C2×C412yes—
44C2×C4⋊Dic59—21no—
45C2×C4×Dic59—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)C2×C412yes—
50#14273654—42no—
51#14273649—44no—
52#14273647—22no—
53#14273647—22no—
54C59⋊#95—21no—
55C2×#41×C59D411yescharacteristic
56C22×C8×C59—22no—
57C22×C8×C59D412yes—
58C4×Q8×C59C2×C411yescharacteristic
59C4×Q8×C59D411yescharacteristic
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59C2×C411yescharacteristic
63C4×D4×C59D411yescharacteristic
64#117×C59—21no—
65#117×C59D411yescharacteristic
66C2×C42×C59—21no—
67C2×C42×C59C2311yescharacteristic
68C4⋊C8×C59—22no—
69C4⋊C8×C59—21no—
70C4⋊C8×C59—21no—
71C4⋊C8×C59D411yescharacteristic
72C4⋊C8×C59D411yescharacteristic
73C22⋊C8×C59—22no—
74C22⋊C8×C59—21no—
75C4×C8×C59—22no—
76C4×C8×C59—21no—
77C23×D59—41no—
78#41×C59—21no—
79#41×C59—21no—
80#41×C59D812yes—
81#41×C59SD1612yes—
82C2×Q8×C59C22⋊C411yescharacteristic
83C2×D4×C59—21no—
84C2×D4×C59C22⋊C411yescharacteristic
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59#4111yescharacteristic
91C22×Dic59—41no—
92C2×D236—22no—
93C2×D236—22no—
94C2×D236—22no—
95C2×D236—42no—
96C2×D236—42no—
97C2×D236—41no—
98C2×D236—41no—
99C2×D236—41no—
100C2×D236—41no—
101C2×D236—41no—
102C2×D236—41no—
103C2×D236—41no—
104C2×D236—82no—
105C2×C8×C59—21no—
106C2×C8×C59—22no—
107C2×C8×C59—21no—
108C2×C8×C59—42no—
109C2×C8×C59—82no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59C22⋊C411yescharacteristic
113C4⋊C4×C59C22⋊C411yescharacteristic
114C22⋊C4×C59—21no—
115C22⋊C4×C59—21no—
116C22⋊C4×C59—21no—
117C42×C59—21no—
118C42×C59—21no—
119C42×C59—21no—
120C42×C59C22×C411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D411yescharacteristic
123C42×C59#4111yescharacteristic
124#11745671—44no—
125#11745669—44no—
126C4⋊Dic59—22no—
127C4⋊Dic59—41no—
128C4×Dic59—41no—
129C4×Dic59—41no—
130#11745664—41no—
131#11745664—41no—
132#11745664—41no—
133#11745664—41no—
134C2×C59⋊C8—22no—
135C2×C59⋊C8—41no—
136C23×C59—21no—
137C23×C59#12111yescharacteristic
138C22×D59—42no—
139C22×D59—82no—
140C22×D59—82no—
141C22×D59—81no—
142C22×D59—81no—
143Q8×C59—21no—
144Q8×C59#10212yescomplemented
145D4×C59—22no—
146D4×C59—22no—
147D4×C59—21no—
148D4×C59#10212yescomplemented
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—22no—
153C2×C4×C59—22no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—81no—
160C2×C4×C59C2×C22⋊C411yescharacteristic
161C2×C4×C59C4×D411yescharacteristic
162C2×C4×C59C4×D411yescharacteristic
163C2×C4×C59#12011yescharacteristic
164C2×C4×C59#12111yescharacteristic
165C2×C4×C59#12311yescharacteristic
166C2×C4×C59C2×D811yescharacteristic
167C2×C4×C59C2×SD1611yescharacteristic
168C2×C4×C59#13511yescharacteristic
169C2×C4×C59#13511yescharacteristic
170C2×Dic59—42no—
171C2×Dic59—81no—
172C2×Dic59—81no—
173D236—44no—
174D236—42no—
175D236—42no—
176D236—82no—
177D236—82no—
178D236—81no—
179D236—81no—
180D236—81no—
181D236—81no—
182C8×C59—42no—
183C8×C59—41no—
184C8×C59—41no—
185C59⋊C8—42no—
186C59⋊C8—81no—
187#9318—591nomaximal, Sylow
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—22no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59#38611yescharacteristic
194C22×C59#46511yescharacteristic
195C22×C59#46611yescharacteristic
196C2×D59—82no—
197C2×D59—82no—
198C2×D59—82no—
199C2×D59—162no—
200C2×D59—162no—
201C4×C59—21no—
202C4×C59—22no—
203C4×C59—21no—
204C4×C59—21no—
205C4×C59—41no—
206C4×C59—41no—
207C4×C59C2×#10211yescharacteristic
208C4×C59#41611yescharacteristic
209C4×C59#45311yescharacteristic
210C4×C59#45411yescharacteristic
211C4×C59#46511yescharacteristic
212C4×C59#46611yescharacteristic
213Dic59—82no—
214#3038—591no—
215#2988D5911yescharacteristic, complemented
216#1982—591no—
217#1806—591no—
218#1545—591no—
219#1303—591no—
220#1292—591no—
221C2×C59—21no—
222C2×C59—21no—
223C2×C59—42no—
224C2×C59#188311yescharacteristic
225C2×C59#189311yescharacteristic
226C2×C59#190311yescharacteristic
227D59—162no—
228C2×AGL(1,ℤ/8)—1181no—
229C2×#127—591no—
230C4×#41C2×D5911yescharacteristic
231#460—592no—
232#460—1181no—
233#459—592no—
234#459—1181no—
235C8×Q8—21no—
236C8×D4—22no—
237C8×D4—21no—
238#431—21no—
239#431C2×D5911yescharacteristic
240#423—591no—
241#423—591no—
242C2×C4⋊C8—21no—
243C2×C4⋊C8C2×D5911yescharacteristic
244C2×#104—1181no—
245#418—1181no—
246C2×#102—1182no—
247C4×C4⋊C4—591no—
248#339—1181no—
249#331—1181no—
250#331—1181no—
251C59#931811yescharacteristic, complemented, Sylow
252C2×#41D23611yescharacteristic
253C22×D4—1181no—
254C22×D4—1181no—
255C22×D4—1181no—
256AGL(1,ℤ/8)—2362no—
257AGL(1,ℤ/8)—2362no—
258C2×SD16—1182no—
259C2×D8—1182no—
260C2×M16—1181no—
261C2×M16—1181no—
262C22×C8—22no—
263C22×C8C22⋊D5912yes—
264#127—592no—
265#127—1181no—
266#127—1181no—
267#127—1181no—
268C4×Q8C4×D5911yescharacteristic
269C4×Q8C22⋊D5911yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D5911yescharacteristic
273C4×D4C22⋊D5911yescharacteristic
274#117—21no—
275#117D23611yescharacteristic
276C2×C4⋊C4—1181no—
277C2×C42—21no—
278C2×C42—1181no—
279C2×C42C22×D5911yescharacteristic
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—592no—
284C4⋊C8—1181no—
285C4⋊C8—1181no—
286C4⋊C8—1181no—
287C4⋊C8C22⋊D5911yescharacteristic
288C4⋊C8C22⋊D5911yescharacteristic
289#104—2362no—
290#102—1182no—
291#102—1182no—
292#102—2364no—
293C22⋊C8—22no—
294C22⋊C8—21no—
295C4×C8—22no—
296C4×C8—21no—
297#95—1181no—
298C24—2361no—
299#41—21no—
300#41—21no—
301#41#1174566012yes—
302#41D47212yes—
303C2×Q8(C2×D59)⋊C411yescharacteristic
304C2×D4—21no—
305C2×D4—1182no—
306C2×D4—1182no—
307C2×D4—1182no—
308C2×D4—4722no—
309C2×D4—2362no—
310C2×D4—2362no—
311C2×D4—2361no—
312C2×D4—2361no—
313C2×D4—2361no—
314C2×D4—2361no—
315C2×D4—2361no—
316C2×D4—2361no—
317C2×D4—2361no—
318C2×D4(C2×D59)⋊C411yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—2361no—
323C22×C4C2×D23611yescharacteristic
324C22×C4#1174568511yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326SD16—2364no—
327D8—2364no—
328M16—2361no—
329M16—2361no—
330M16—2361no—
331M16—2361no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—42no—
336C2×C8—82no—
337C2×C8—1182no—
338C2×C8—2361no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—1182no—
342C4⋊C4—2361no—
343C4⋊C4(C2×D59)⋊C411yescharacteristic
344C4⋊C4(C2×D59)⋊C411yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—2361no—
352C42—2361no—
353C42C2×C4×D5911yescharacteristic
354C42#1174568511yescharacteristic
355C42C2×C22⋊D5911yescharacteristic
356C42C2×C22⋊D5911yescharacteristic
357C23—21no—
358C23—4722no—
359C23—4722no—
360C23—4721no—
361C23—4721no—
362C23—2362no—
363C23#1427370111yescharacteristic
364Q8—21no—
365Q8#1427364912yescomplemented
366D4—22no—
367D4—22no—
368D4—21no—
369D4—4722no—
370D4—4722no—
371D4—4721no—
372D4—4721no—
373D4—4721no—
374D4—4721no—
375D4—2364no—
376D4—2362no—
377D4—2362no—
378D4#1427364912yescomplemented
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—4721no—
391C2×C4—4721no—
392C2×C4—2362no—
393C2×C4C2×(C2×D59)⋊C411yescharacteristic
394C2×C4C4×C22⋊D5911yescharacteristic
395C2×C4C4×C22⋊D5911yescharacteristic
396C2×C4#1427369611yescharacteristic
397C2×C4#1427370111yescharacteristic
398C2×C4#1427371011yescharacteristic
399C2×C4C2×D47211yescharacteristic
400C2×C4C2×#1174566011yescharacteristic
401C2×C4#1427374911yescharacteristic
402C2×C4#1427374911yescharacteristic
403C8—42no—
404C8—41no—
405C8—41no—
406C8—4721no—
407C8—2362no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—9442no—
414C22—9442no—
415C22—4722no—
416C22—4722no—
417C22—4722no—
418C22#1683728111yescharacteristic
419C22#1683763111yescharacteristic
420C22#1683763811yescharacteristic
421C4—21no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—4722no—
428C4C2×#1427364911yescharacteristic
429C4#1683739911yescharacteristic
430C4#1683757011yescharacteristic
431C4#1683757511yescharacteristic
432C4#1683763111yescharacteristic
433C4#1683763811yescharacteristic
434C2—21no—
435C2—21no—
436C2—42no—
437C2—9442no—
438C2#2142447211yescharacteristic
439C2#2142452411yescharacteristic
440C2#2142457611yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2988C2
socleC22×C59#386
prime radical#21433576C2
cosocleC2×C42×C59C23
hypercenter#2988D59
hypocenterC59#9318
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. #21425022C2semidirect
3. #21424116C2semidirect
4. #21422763C2semidirect
5. #21421998C2semidirect
6. #21421971C2semidirect
7. C59×#2988C2semidirect
11. #16837602C4semidirect
13. #16837597C4semidirect
144. Q8×C59#102semidirect
148. D4×C59#102semidirect
215. #2988D59semidirect
251. C59#9318semidirect
365. Q8#14273649semidirect
378. D4#14273649semidirect

Maximal subgroups

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

subgroupindex
#214335762
#214250222
#214241162
#214227632
#214219982
#214219712
C59×#29882
#931859