Finite Group Databasebeta

← #28488196

Subgroups — #28488196

Order 15104.

564 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#21425080C211yesmaximal, characteristic, complemented
3#21424931C211yesmaximal, characteristic
4C8×C22⋊Dic59C211yesmaximal, characteristic, complemented
5#21422667C211yesmaximal, characteristic, complemented
6#21422663C211yesmaximal, characteristic
7C59×#1807C211yesmaximal, characteristic, complemented
8C22×#11745660C411yescharacteristic, complemented
9C2×#14273706C2211yescharacteristic
10C2×#14273701C2211yescharacteristic
11#16837631—21no—
12#16837631—21no—
13#16837631—21no—
14#16837631—21no—
15C2×#14273659C411yescharacteristic, complemented
16C2×#14273652C411yescharacteristic, complemented
17C2×#14273649C411yescharacteristic, complemented
18C2×#14273641—22no—
19C2×C22⋊(C59⋊C8)C2211yescharacteristic
20C2×C8×Dic59—21no—
21C4×C22⋊Dic59C2211yescharacteristic
22#16837168—22no—
23#16837156—21no—
24#16837138—22no—
25#16837135—22no—
26C23×C8×C59C2211yescharacteristic
27C2×#106×C59—21no—
28C59×#380C2211yescharacteristic
29C59×#380C2211yescharacteristic
30C59×#340—21no—
31C22×Dic118C2×C411yescharacteristic
32C22×D236C2×C411yescharacteristic
33C22×C22⋊D59—21no—
34C2×#11745660—21no—
35C2×#11745660—21no—
36C2×#11745660—42no—
37C22×C59⋊C8—21no—
38#14273706—21no—
39#14273706—21no—
40#14273706—21no—
41#14273706—21no—
42#14273701—21no—
43#14273701—21no—
44#14273701—21no—
45#14273701—21no—
46C2×C4⋊Dic59C2×C411yescharacteristic
47C2×C4⋊Dic59C2×C411yescharacteristic
48C2×Dic59⋊C4—21no—
49C2×C22⋊Dic59—21no—
50C2×C22⋊Dic59D412yes—
51C2×(C2×D59)⋊C4—21no—
52C2×C4×Dic59—21no—
53#14273659—21no—
54#14273659—21no—
55#14273652—21no—
56#14273652—21no—
57#14273649—21no—
58#14273649—21no—
59#14273641—44no—
60C22⋊(C59⋊C8)—21no—
61C22⋊(C59⋊C8)D412yes—
62C8×Dic59—42no—
63C59⋊#95—21no—
64C23×C4×C59C2311yescharacteristic
65C22×C8×C59—42no—
66C22×C8×C59C2×C411yescharacteristic
67C22×C8×C59C2×C411yescharacteristic
68C22×C8×C59D411yescharacteristic
69C22×C8×C59D411yescharacteristic
70C22×C8×C59D411yescharacteristic
71C22×C8×C59D411yescharacteristic
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59—21no—
74C2×C22⋊C4×C59—21no—
75C2×C22⋊C4×C59—21no—
76#106×C59—42no—
77#95×C59—21no—
78#95×C59—21no—
79C24×C59C2×D411yescharacteristic
80C23×D59—21no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59C22×C411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59#4111yescharacteristic
90C22×C4×C59#4111yescharacteristic
91C2×C22⋊D59—41no—
92C2×C22⋊D59—41no—
93C2×C22⋊D59—42no—
94C2×C22⋊D59—42no—
95C22×Dic59—22no—
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—21no—
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—41no—
111C2×C8×C59—41no—
112C2×C8×C59—41no—
113C2×C8×C59C22⋊C411yescharacteristic
114C2×C8×C59C22⋊C411yescharacteristic
115C2×C8×C59C22⋊C411yescharacteristic
116C2×C8×C59C22⋊C411yescharacteristic
117C4⋊C4×C59—41no—
118C4⋊C4×C59—41no—
119C4⋊C4×C59—41no—
120C4⋊C4×C59—41no—
121C22⋊C4×C59—42no—
122C22⋊C4×C59—42no—
123C22⋊Dic59—22no—
124C22⋊Dic59—41no—
125C22⋊Dic59—41no—
126C22⋊Dic59SD1614yescomplemented
127(C2×D59)⋊C4—41no—
128(C2×D59)⋊C4—41no—
129C4⋊Dic59—21no—
130C4⋊Dic59—21no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133Dic59⋊C4—41no—
134Dic59⋊C4—41no—
135C4×Dic59—41no—
136C4×Dic59—41no—
137C2×C59⋊C8—22no—
138C2×C59⋊C8—41no—
139C2×C59⋊C8—41no—
140#11745660—42no—
141#11745660—42no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59C4×D411yescharacteristic
145C23×C59#12111yescharacteristic
146C23×C59#12711yescharacteristic
147C23×C59C2×SD1612yes—
148C23×C59#13512yes—
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—81no—
163C2×C4×C59—81no—
164C2×C4×C59—81no—
165C2×C4×C59—81no—
166C2×C4×C59—81no—
167C2×C4×C59—81no—
168C2×C4×C59C2×C22⋊C411yescharacteristic
169C2×C4×C59C4×D411yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C2×C4×C59#12411yescharacteristic
172C22⋊D59—82no—
173C22⋊D59—81no—
174C22⋊D59—81no—
175C2×Dic59—24no—
176C2×Dic59—42no—
177C2×Dic59—42no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—41no—
181C2×Dic59—42no—
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#9912yescomplemented
194C59⋊C8—42no—
195#9781—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×C59C4×SD1612yes—
206C22×C59#46512yes—
207C22×C59#49211yescharacteristic
208C22×C59#49511yescharacteristic
209C2×D59—81no—
210C2×D59—81no—
211C2×D59—81no—
212C2×D59—162no—
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—161no—
217C4×C59—161no—
218C4×C59C2×#9911yescharacteristic
219C4×C59#41011yescharacteristic
220Dic59—44no—
221Dic59—81no—
222Dic59—81no—
223Dic59—81no—
224C2×#465—591no—
225#1996—591no—
226#1964—591no—
227#1807D5911yescharacteristic, complemented
228C8×C22⋊C4—591no—
229#1520—591no—
230#1519—591no—
231C2×C59—22no—
232C2×C59—21no—
233C2×C59—21no—
234C2×C59—22no—
235C2×C59#196411yescharacteristic
236C2×C59#196811yescharacteristic
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×#106—21no—
248C2×#106—591no—
249C2×#103—591no—
250C2×#102—591no—
251C2×#99—1182no—
252C2×C22⋊C8—591no—
253C2×C4×C8—1181no—
254#380C2×D5911yescharacteristic
255#380C2×D5911yescharacteristic
256C4×C22⋊C4—591no—
257#342—1182no—
258#340—21no—
259#336—1181no—
260#328—1182no—
261#327—1182no—
262C59#978111yescharacteristic, 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—42no—
271C22×C8—1181no—
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—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—591no—
289C2×C4⋊C4—591no—
290C2×C4⋊C4—1181no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—592no—
294C2×C22⋊C4—1181no—
295C2×C22⋊C4—1181no—
296C2×C42—1181no—
297#106—42no—
298#106—1181no—
299#106—1181no—
300#103—1181no—
301#103—1181no—
302#102—1181no—
303#102—1181no—
304#99—2364no—
305C22⋊C8—592no—
306C22⋊C8—1181no—
307C4×C8—2362no—
308#95—21no—
309#95—21no—
310#95—1181no—
311C24—1181no—
312C24D4×D5911yescharacteristic
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—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—1182no—
330C22×C4—1181no—
331C22×C4—1181no—
332C22×C4—1181no—
333C22×C4C2×C4×D5911yescharacteristic
334C22×C4C2×D23611yescharacteristic
335C22×C4D4×D5911yescharacteristic
336C22×C4#1174568911yescharacteristic
337C22×C4#1174568911yescharacteristic
338C22×C4C2×C22⋊D5911yescharacteristic
339SD16—2362no—
340SD16—2362no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—1182no—
348C2×C8—2361no—
349C2×C8—2361no—
350C2×C8(C2×D59)⋊C411yescharacteristic
351C2×C8(C2×D59)⋊C411yescharacteristic
352C2×C8(C2×D59)⋊C411yescharacteristic
353C2×C8(C2×D59)⋊C411yescharacteristic
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—1181no—
359C4⋊C4—1181no—
360C4⋊C4—1181no—
361C4⋊C4—1181no—
362C4⋊C4—2361no—
363C4⋊C4—2361no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—594no—
367C22⋊C4—1182no—
368C22⋊C4—2361no—
369C22⋊C4—2361no—
370C22⋊C4—2361no—
371C22⋊C4—2361no—
372C42—2361no—
373C42—2361no—
374C23—41no—
375C23—41no—
376C23—4721no—
377C23—4721no—
378C23—2362no—
379C23—2361no—
380C23#1427368911yescharacteristic
381C23#1427370211yescharacteristic
382C23#1427372311yescharacteristic
383C23SD16×D5912yes—
384C23#1427374812yes—
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—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—1184no—
410C2×C4—4721no—
411C2×C4—4721no—
412C2×C4—2362no—
413C2×C4—2361no—
414C2×C4—2361no—
415C2×C4—2361no—
416C2×C4—2362no—
417C2×C4—2362no—
418C2×C4C2×(C2×D59)⋊C411yescharacteristic
419C2×C4#1427368911yescharacteristic
420C2×C4#1427370211yescharacteristic
421C2×C4#1427371611yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—2362no—
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#1683730311yescharacteristic
440C22#1683748912yes—
441C22#1683763212yes—
442C22#1683777311yescharacteristic
443C22#1683778311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—161no—
448C4—161no—
449C4—4721no—
450C4—4721no—
451C4—4721no—
452C4—2364no—
453C4C2×#1427364111yescharacteristic
454C4#1683737511yescharacteristic
455C2—21no—
456C2—21no—
457C2—22no—
458C2—22no—
459C2—9441no—
460C2#2142493111yescharacteristic
461C2#2142494811yescharacteristic
462C2#2142499211yescharacteristic
subgroupG/H
centerC22#16837303
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1807C2
socleC22×C59#390
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1807D59
hypocenterC59#9781
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. #21425080C2semidirect
4. C8×C22⋊Dic59C2semidirect
5. #21422667C2semidirect
7. C59×#1807C2semidirect
8. C22×#11745660C4semidirect
15. C2×#14273659C4semidirect
16. C2×#14273652C4semidirect
17. C2×#14273649C4semidirect
126. C22⋊Dic59SD16semidirect
193. C8×C59#99semidirect
227. #1807D59semidirect
262. C59#9781semidirect
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
#214250802
#214249312
C8×C22⋊Dic592
#214226672
#214226632
C59×#18072
#978159