Finite Group Databasebeta

← #28488225

Subgroups — #28488225

Order 15104.

552 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837631C211yesmaximal, characteristic
2#21425081C211yesmaximal, characteristic, complemented
3#21424932C211yesmaximal, characteristic
4#21424125C211yesmaximal, characteristic, complemented
5#21422669C211yesmaximal, characteristic, complemented
6#21422665C211yesmaximal, characteristic
7C59×#1752C211yesmaximal, characteristic, complemented
8C22×#11745660C411yescharacteristic, complemented
9C2×#14273706C2211yescharacteristic
10C2×#14273701C2211yescharacteristic
11#16837631—21no—
12#16837631—21no—
13#16837631—21no—
14#16837631—21no—
15C2×#14273661—21no—
16C2×#14273659C411yescharacteristic, complemented
17C2×#14273652C411yescharacteristic, complemented
18C2×#14273649C411yescharacteristic, complemented
19C2×#14273641—22no—
20#16837262C2211yescharacteristic
21#16837262C2211yescharacteristic
22#16837168—22no—
23#16837165—21no—
24#16837140—22no—
25#16837137—22no—
26C23×C8×C59C2211yescharacteristic
27C2×C22⋊C8×C59C2211yescharacteristic
28C2×#97×C59—21no—
29C4×C22⋊C4×C59C2211yescharacteristic
30C59×#336—21no—
31C22×Dic118C2×C411yescharacteristic
32C22×D236C2×C411yescharacteristic
33C22×C22⋊D59—21no—
34C2×#11745660—21no—
35C2×#11745660—21no—
36C2×#11745660—42no—
37#14273706—21no—
38#14273706—21no—
39#14273706—21no—
40#14273706—21no—
41#14273701—21no—
42#14273701—21no—
43#14273701—21no—
44#14273701—21no—
45C2×C4⋊Dic59—21no—
46C2×C4⋊Dic59—21no—
47C2×C4⋊Dic59C2×C411yescharacteristic
48C2×C4⋊Dic59C2×C411yescharacteristic
49C2×Dic59⋊C4—21no—
50C2×C22⋊Dic59—21no—
51C2×C22⋊Dic59—21no—
52C2×C22⋊Dic59—21no—
53C2×(C2×D59)⋊C4—21no—
54#14273661—42no—
55#14273659—21no—
56#14273659—21no—
57#14273652—21no—
58#14273652—21no—
59#14273649—21no—
60#14273649—21no—
61#14273641—44no—
62C59⋊#95—21no—
63C59⋊#95—21no—
64C23×C4×C59C2311yescharacteristic
65C22×C8×C59—21no—
66C22×C8×C59—42no—
67C22×C8×C59C2×C411yescharacteristic
68C22×C8×C59C2×C411yescharacteristic
69C22×C8×C59D411yescharacteristic
70C22×C8×C59D411yescharacteristic
71C22×C8×C59D411yescharacteristic
72C22×C8×C59D411yescharacteristic
73C2×C22⋊C4×C59D412yes—
74C2×C42×C59—21no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59D412yes—
77#97×C59—42no—
78#95×C59—21no—
79C24×C59C2×D411yescharacteristic
80C23×D59—21no—
81C22×C4×C59—22no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59C22×C411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×C4×C59#4111yescharacteristic
90C2×C22⋊D59—41no—
91C2×C22⋊D59—41no—
92C2×C22⋊D59—42no—
93C2×C22⋊D59—42no—
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—41no—
98C22×Dic59—41no—
99C2×D236—21no—
100C2×D236—21no—
101C2×D236—41no—
102C2×D236—41no—
103C2×Dic118—21no—
104C2×Dic118—21no—
105C2×Dic118—41no—
106C2×Dic118—41no—
107C2×C8×C59—22no—
108C2×C8×C59—22no—
109C2×C8×C59—22no—
110C2×C8×C59—22no—
111C2×C8×C59—41no—
112C2×C8×C59—41no—
113C2×C8×C59—41no—
114C2×C8×C59—41no—
115C2×C8×C59—41no—
116C2×C8×C59C22⋊C411yescharacteristic
117C2×C8×C59C22⋊C411yescharacteristic
118C2×C8×C59C22⋊C411yescharacteristic
119C2×C8×C59C22⋊C411yescharacteristic
120C22⋊C4×C59—22no—
121C22⋊C4×C59—22no—
122C42×C59—41no—
123C42×C59—41no—
124C22⋊Dic59—41no—
125C22⋊Dic59—41no—
126C22⋊Dic59—42no—
127C22⋊Dic59—42no—
128(C2×D59)⋊C4—41no—
129(C2×D59)⋊C4—41no—
130C4⋊Dic59—21no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133C4⋊Dic59—21no—
134C4⋊Dic59—41no—
135C4⋊Dic59—41no—
136C4⋊Dic59—41no—
137C4⋊Dic59—41no—
138Dic59⋊C4—41no—
139Dic59⋊C4—41no—
140#11745660—42no—
141#11745660—42no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59C4×D411yescharacteristic
145C23×C59#12111yescharacteristic
146C23×C59#12711yescharacteristic
147C23×C59AGL(1,ℤ/8)12yes—
148C23×C59#13712yes—
149C22×D59—41no—
150C22×D59—42no—
151C22×D59—81no—
152C22×D59—81no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—42no—
163C2×C4×C59—42no—
164C2×C4×C59—42no—
165C2×C4×C59C2×C22⋊C411yescharacteristic
166C2×C4×C59C4×D411yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#12411yescharacteristic
169C22⋊D59—82no—
170C22⋊D59—81no—
171C22⋊D59—81no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—42no—
176C2×Dic59—81no—
177C2×Dic59—81no—
178C2×Dic59—81no—
179C2×Dic59—81no—
180C2×Dic59—81no—
181C2×Dic59—81no—
182C2×Dic59—81no—
183C2×Dic59—81no—
184D236—41no—
185D236—41no—
186D236—81no—
187Dic118—41no—
188Dic118—41no—
189Dic118—81no—
190C8×C59—21no—
191C8×C59—21no—
192C8×C59—21no—
193C8×C59—42no—
194C8×C59#9912yescomplemented
195#9786—591nomaximal, Sylow
196C22×C59—21no—
197C22×C59—21no—
198C22×C59—21no—
199C22×C59—21no—
200C22×C59—21no—
201C22×C59—21no—
202C22×C59—42no—
203C22×C59—42no—
204C22×C59#39011yescharacteristic
205C22×C59#44012yes—
206C22×C59#46812yes—
207C22×C59#49611yescharacteristic
208C22×C59#49711yescharacteristic
209C2×D59—81no—
210C2×D59—81no—
211C2×D59—81no—
212C2×D59—162no—
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—82no—
217C4×C59C2×#9911yescharacteristic
218C4×C59#41011yescharacteristic
219Dic59—81no—
220Dic59—81no—
221Dic59—81no—
222Dic59—161no—
223Dic59—161no—
224C2×#465—591no—
225#1996—591no—
226#1964—591no—
227#1808—591no—
228#1752D5911yescharacteristic, complemented
229#1520—591no—
230#1519—591no—
231C2×C59—21no—
232C2×C59—22no—
233C2×C59—22no—
234C2×C59—21no—
235C2×C59#196411yescharacteristic
236C2×C59#197311yescharacteristic
237C2×C59#197611yescharacteristic
238D59—161no—
239C22×SD16—591no—
240C23×C8C2×D5911yescharacteristic
241C2×C22⋊Q8—591no—
242C2×#121—591no—
243#465—1181no—
244#465—1181no—
245#465—1181no—
246#465—1181no—
247C2×#107—1181no—
248C2×#106—591no—
249C2×#103—591no—
250C2×#102—591no—
251C2×#99—1182no—
252C2×C22⋊C8C2×D5911yescharacteristic
253C2×#97—21no—
254#380—591no—
255#380—591no—
256C4×C22⋊C4C2×D5911yescharacteristic
257#342—1182no—
258#340—1181no—
259#336—21no—
260#328—1182no—
261#327—1182no—
262C59#978611yescharacteristic, complemented, Sylow
263C22×Q8—591no—
264C22×D4—591no—
265C22×D4—1181no—
266C23×C4C22×D5911yescharacteristic
267C2×SD16—1181no—
268C2×SD16—1181no—
269C2×SD16—2362no—
270C22×C8—21no—
271C22×C8—42no—
272C22×C8C4×D5911yescharacteristic
273C22×C8C4×D5911yescharacteristic
274C22×C8D23611yescharacteristic
275C22×C8D23611yescharacteristic
276C22×C8C22⋊D5911yescharacteristic
277C22×C8C22⋊D5911yescharacteristic
278C22⋊Q8—1181no—
279C22⋊Q8—1181no—
280C22⋊Q8—1181no—
281C22⋊Q8—1181no—
282#121—1181no—
283#121—1181no—
284#121—1181no—
285#121—1181no—
286C2×C4⋊C4—591no—
287C2×C4⋊C4—591no—
288C2×C4⋊C4—1181no—
289C2×C4⋊C4—1181no—
290C2×C4⋊C4—1181no—
291C2×C22⋊C4—1181no—
292C2×C22⋊C4—1181no—
293C2×C22⋊C4—1181no—
294C2×C22⋊C4—1181no—
295C2×C22⋊C4D23612yes—
296C2×C42—21no—
297#107—2362no—
298#106—1181no—
299#106—1181no—
300#103—1181no—
301#103—1181no—
302#102—1181no—
303#102—1181no—
304#99—2364no—
305C22⋊C8—21no—
306C22⋊C8D23612yes—
307#97—42no—
308#95—21no—
309#95—1181no—
310#95—1181no—
311C24—1181no—
312C24C2×D23611yescharacteristic
313C2×Q8—1181no—
314C2×Q8—1181no—
315C2×Q8—2361no—
316C2×Q8—2361no—
317C2×D4—1181no—
318C2×D4—1181no—
319C2×D4—2361no—
320C2×D4—2361no—
321C2×D4—2361no—
322C2×D4—2361no—
323C2×D4—2362no—
324C2×D4—2362no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—1181no—
329C22×C4—1181no—
330C22×C4—1181no—
331C22×C4—2361no—
332C22×C4—2361no—
333C22×C4C2×C4×D5911yescharacteristic
334C22×C4C2×D23611yescharacteristic
335C22×C4C2×D23611yescharacteristic
336C22×C4#1174568511yescharacteristic
337C22×C4#1174568511yescharacteristic
338C22×C4C2×C22⋊D5911yescharacteristic
339SD16—2362no—
340SD16—2362no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8(C2×D59)⋊C411yescharacteristic
351C2×C8(C2×D59)⋊C411yescharacteristic
352C2×C8(C2×D59)⋊C411yescharacteristic
353C2×C8(C2×D59)⋊C411yescharacteristic
354C4⋊C4—1181no—
355C4⋊C4—1181no—
356C4⋊C4—1181no—
357C4⋊C4—1181no—
358C4⋊C4—2361no—
359C4⋊C4—2361no—
360C4⋊C4—2361no—
361C4⋊C4—2361no—
362C4⋊C4—2361no—
363C4⋊C4—2361no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—2362no—
367C22⋊C4—2362no—
368C22⋊C4—2361no—
369C22⋊C4—2361no—
370C22⋊C4—2361no—
371C22⋊C4—2361no—
372C42—41no—
373C42—41no—
374C23—41no—
375C23—41no—
376C23—4721no—
377C23—4721no—
378C23—2362no—
379C23—2361no—
380C23C4×D23611yescharacteristic
381C23#1427370111yescharacteristic
382C23#1427372411yescharacteristic
383C23#1427375512yes—
384C23#1427376012yes—
385Q8—4721no—
386Q8—2361no—
387Q8—2361no—
388D4—4721no—
389D4—4722no—
390D4—4721no—
391D4—4721no—
392D4—2361no—
393D4—2361no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—4721no—
407C2×C4—4721no—
408C2×C4—4721no—
409C2×C4—4721no—
410C2×C4—4721no—
411C2×C4—4721no—
412C2×C4—4721no—
413C2×C4—4721no—
414C2×C4—2362no—
415C2×C4—2361no—
416C2×C4—2361no—
417C2×C4—2361no—
418C2×C4C2×(C2×D59)⋊C411yescharacteristic
419C2×C4C4×D23611yescharacteristic
420C2×C4#1427370111yescharacteristic
421C2×C4#1427371511yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—42no—
426C8#1427364112yescomplemented
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—9442no—
436C22—4721no—
437C22—4721no—
438C22—4721no—
439C22#1683730411yescharacteristic
440C22#1683750412yes—
441C22#1683764812yes—
442C22#1683779111yescharacteristic
443C22#1683779611yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—82no—
448C4—9441no—
449C4—9441no—
450C4—4721no—
451C4—4721no—
452C4—4721no—
453C4C2×#1427364111yescharacteristic
454C4#1683737511yescharacteristic
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—9441no—
460C2#2142493211yescharacteristic
461C2#2142497711yescharacteristic
462C2#2142499411yescharacteristic
subgroupG/H
centerC22#16837304
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1752C2
socleC22×C59#390
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1752D59
hypocenterC59#9786
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
2. #21425081C2semidirect
4. #21424125C2semidirect
5. #21422669C2semidirect
7. C59×#1752C2semidirect
8. C22×#11745660C4semidirect
16. C2×#14273659C4semidirect
17. C2×#14273652C4semidirect
18. C2×#14273649C4semidirect
194. C8×C59#99semidirect
228. #1752D59semidirect
262. C59#9786semidirect
426. C8#14273641semidirect

Maximal subgroups

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

subgroupindex
C2×#168376312
#214250812
#214249322
#214241252
#214226692
#214226652
C59×#17522
#978659