Finite Group Databasebeta

← #28492234

Subgroups — #28492234

Order 15104.

530 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429019C211yesmaximal, characteristic
2#21424223C211yesmaximal, characteristic, complemented
3#21424102C211yesmaximal, characteristic, complemented
4#21424097C211yesmaximal, characteristic
5C2×#16837132C211yesmaximal, characteristic, complemented
6C2×#16837129C211yesmaximal, characteristic
7C59×#1296C211yesmaximal, characteristic, complemented
8C2×#14273707—21no—
9C2×C4×Dic118C2211yescharacteristic
10C2×C4×D236C2211yescharacteristic
11C2×#14273652—21no—
12C2×#14273649—21no—
13C2×C4×C59⋊C8C2211yescharacteristic
14#16837324—22no—
15#16837303—21no—
16#16837301—21no—
17#16837272C2211yescharacteristic
18#16837259—21no—
19#16837165—21no—
20#16837165—21no—
21#16837164—22no—
22#16837132—22no—
23#16837129—22no—
24C2×C4⋊C8×C59C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C59×#384C2211yescharacteristic
27C22×Dic118D411yescharacteristic
28C22×D236D411yescharacteristic
29C22×C4×D59—21no—
30C22×C59⋊C8—21no—
31#14273707—42no—
32#14273707—42no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35C4×Dic118—21no—
36C4×Dic118—21no—
37C4×D236—21no—
38C4×D236—21no—
39C4×D236—21no—
40C4×D236—21no—
41C2×C4⋊Dic59—22no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—21no—
44C2×C4⋊Dic59—21no—
45C2×C4⋊Dic59D411yescharacteristic
46C2×C4⋊Dic59D411yescharacteristic
47C2×Dic59⋊C4—21no—
48C2×(C2×D59)⋊C4—22no—
49C2×(C2×D59)⋊C4—21no—
50C2×(C2×D59)⋊C4—22no—
51C2×C4×Dic59—21no—
52#14273652—42no—
53#14273649—42no—
54C4×C59⋊C8—21no—
55C4×C59⋊C8D412yes—
56C59⋊#95—22no—
57C22×C8×C59—21no—
58C22×C8×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59D412yes—
62C2×C42×C59C2311yescharacteristic
63C4⋊C8×C59—21no—
64C4⋊C8×C59—21no—
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67C23×D59—21no—
68C22×C4×C59—22no—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—41no—
76C2×D236—21no—
77C2×D236—21no—
78C2×D236—21no—
79C2×D236—21no—
80C2×D236—21no—
81C2×D236Q1612yes—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C4×D59—41no—
85C2×C4×D59—41no—
86C2×C4×D59—41no—
87C2×C4×D59—41no—
88C2×Dic118—21no—
89C2×Dic118—21no—
90C2×Dic118—21no—
91C2×Dic118—21no—
92C2×Dic118—21no—
93C2×Dic118SD1612yes—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C2×C8×C59—41no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C4⋊C4×C59—42no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C42×C59C2×D411yescharacteristic
106C42×C59#4111yescharacteristic
107C42×C59#4111yescharacteristic
108C42×C59#4111yescharacteristic
109(C2×D59)⋊C4—44no—
110(C2×D59)⋊C4—41no—
111(C2×D59)⋊C4—41no—
112(C2×D59)⋊C4—82no—
113C4⋊Dic59—21no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117C4⋊Dic59—44no—
118C4⋊Dic59—42no—
119C4⋊Dic59—41no—
120C4⋊Dic59—41no—
121C4⋊Dic59—41no—
122C4⋊Dic59—41no—
123Dic59⋊C4—41no—
124Dic59⋊C4—41no—
125C4×Dic59—41no—
126C4×Dic59—41no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—41no—
130C23×C59#12011yescharacteristic
131C22×D59—22no—
132C22×D59—41no—
133C22×D59—41no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59#12311yescharacteristic
147C2×C4×C59#12311yescharacteristic
148C2×C4×C59#12411yescharacteristic
149C2×C4×C59C2×SD1611yescharacteristic
150C2×C4×C59C2×Q1611yescharacteristic
151C2×C4×C59#13511yescharacteristic
152C2×C4×C59#13511yescharacteristic
153C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C59#13711yescharacteristic
157C2×Dic59—22no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—81no—
167C2×Dic59—81no—
168C2×Dic59—81no—
169D236—22no—
170D236—22no—
171D236—41no—
172D236—41no—
173C4×D59—81no—
174C4×D59—81no—
175C4×D59—81no—
176C4×D59—81no—
177Dic118—22no—
178Dic118—22no—
179Dic118—41no—
180Dic118—41no—
181C8×C59—81no—
182C8×C59—81no—
183C59⋊C8—42no—
184#10436—591nomaximal, Sylow
185C22×C59#39411yescharacteristic
186C22×C59#44911yescharacteristic
187C22×C59#44911yescharacteristic
188C22×C59#45011yescharacteristic
189C22×C59#45111yescharacteristic
190C22×C59#45411yescharacteristic
191C22×C59#45511yescharacteristic
192C2×D59—42no—
193C2×D59—42no—
194C2×D59—42no—
195C2×D59—81no—
196C2×D59—81no—
197C2×D59—81no—
198C2×D59—81no—
199C4×C59—21no—
200C4×C59—21no—
201C4×C59—82no—
202C4×C59#46511yescharacteristic
203C4×C59#46711yescharacteristic
204C4×C59#46811yescharacteristic
205C4×C59#46911yescharacteristic
206C4×C59#48111yescharacteristic
207C4×C59#48211yescharacteristic
208C4×C59#48411yescharacteristic
209C4×C59#48511yescharacteristic
210Dic59—42no—
211Dic59—81no—
212Dic59—81no—
213Dic59—161no—
214#2538—591no—
215#1834—591no—
216#1803—591no—
217#1802—591no—
218C2×Q8⋊C8—591no—
219C2×D4⋊C8—591no—
220#1296D5911yescharacteristic, complemented
221C2×C59#162811yescharacteristic
222C2×C59#163711yescharacteristic
223C2×C59#164211yescharacteristic
224C2×C59#164411yescharacteristic
225C2×C59#203411yescharacteristic
226C2×C59#203611yescharacteristic
227C2×C59#204011yescharacteristic
228D59—82no—
229C2×C22⋊Q8—1181no—
230C2×C4×Q8—591no—
231C2×C4×D4—591no—
232C2×C4⋊C8C2×D5911yescharacteristic
233C2×C4⋊C8C2×D5911yescharacteristic
234C2×#103—1181no—
235C2×#102—1181no—
236C2×C4×C8—591no—
237#396—1182no—
238#390—1181no—
239#389—1181no—
240#384—591no—
241#384C2×D5911yescharacteristic
242#380—1181no—
243#340—1181no—
244#340—1181no—
245#340—1182no—
246Q8⋊C8—1182no—
247D4⋊C8—1182no—
248C59#1043611yescharacteristic, complemented, Sylow
249C22×Q8—591no—
250C22×D4—591no—
251C23×C4—1181no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—1181no—
255C22⋊Q8—2362no—
256C22⋊Q8—2362no—
257C4×Q8—1181no—
258C4×Q8—1181no—
259C4×Q8—1181no—
260C4×Q8—1181no—
261C4×D4—1181no—
262C4×D4—1181no—
263C4×D4—1181no—
264C4×D4—1181no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—591no—
268C2×C4⋊C4—591no—
269C2×C4⋊C4—1182no—
270C2×C4⋊C4—1181no—
271C2×C4⋊C4—1181no—
272C2×C4⋊C4—1181no—
273C2×C4⋊C4—1181no—
274C2×C4⋊C4D23612yes—
275C2×C22⋊C4—1182no—
276C2×C22⋊C4—1182no—
277C2×C22⋊C4—1181no—
278C2×C42—1181no—
279C2×C42C22×D5911yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284#103—2362no—
285#102—2362no—
286C4×C8—592no—
287C4×C8—1181no—
288#95—1182no—
289C24—1181no—
290C2×Q8—592no—
291C2×Q8—1181no—
292C2×Q8—1181no—
293C2×Q8—1181no—
294C2×Q8—1181no—
295C2×Q8—1181no—
296C2×D4—592no—
297C2×D4—1181no—
298C2×D4—1181no—
299C2×D4—1181no—
300C2×D4—1181no—
301C2×D4—1181no—
302C22×C4—22no—
303C22×C4—1181no—
304C22×C4—1181no—
305C22×C4—1181no—
306C22×C4—2361no—
307C22×C4—2361no—
308C22×C4—2361no—
309C22×C4—2361no—
310C22×C4—2361no—
311C22×C4—2361no—
312C22×C4—2361no—
313C22×C4C2×D23611yescharacteristic
314C22×C4D4×D5911yescharacteristic
315C22×C4D4×D5911yescharacteristic
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—1182no—
323C2×C8—2361no—
324C2×C8—2361no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1181no—
331C4⋊C4—1181no—
332C4⋊C4—1181no—
333C4⋊C4—1181no—
334C4⋊C4—2362no—
335C4⋊C4—2364no—
336C4⋊C4—2361no—
337C4⋊C4—2361no—
338C4⋊C4—2361no—
339C4⋊C4—2361no—
340C4⋊C4—2361no—
341C4⋊C4—2361no—
342C22⋊C4—4722no—
343C22⋊C4—2364no—
344C22⋊C4—2361no—
345C22⋊C4—2361no—
346C42—2361no—
347C42—2361no—
348C42#1174568511yescharacteristic
349C42#1174568511yescharacteristic
350C42D4×D5911yescharacteristic
351C42#1174568911yescharacteristic
352C23—1182no—
353C23—2361no—
354C23—2361no—
355C23—2361no—
356C23—2361no—
357C23—2361no—
358C23—2361no—
359C23#1427369711yescharacteristic
360Q8—1182no—
361Q8—1182no—
362Q8—2361no—
363Q8—2361no—
364D4—1182no—
365D4—1182no—
366D4—2361no—
367D4—2361no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—1182no—
374C2×C4—4721no—
375C2×C4—4721no—
376C2×C4—4721no—
377C2×C4—4721no—
378C2×C4—4721no—
379C2×C4—4721no—
380C2×C4—4721no—
381C2×C4—2361no—
382C2×C4—2361no—
383C2×C4—2361no—
384C2×C4—2361no—
385C2×C4—2361no—
386C2×C4—2361no—
387C2×C4—2361no—
388C2×C4—2361no—
389C2×C4#1427370011yescharacteristic
390C2×C4#1427370011yescharacteristic
391C2×C4#1427370211yescharacteristic
392C2×C4#1427371211yescharacteristic
393C2×C4#1427371211yescharacteristic
394C2×C4#1427371511yescharacteristic
395C2×C4SD16×D5911yescharacteristic
396C2×C4Q16×D5911yescharacteristic
397C2×C4#1427374811yescharacteristic
398C2×C4#1427375011yescharacteristic
399C2×C4#1427375111yescharacteristic
400C2×C4#1427375111yescharacteristic
401C2×C4#1427375411yescharacteristic
402C2×C4#1427375611yescharacteristic
403C8—81no—
404C8—81no—
405C8—2362no—
406C22—4721no—
407C22—4721no—
408C22—4721no—
409C22—4721no—
410C22—2362no—
411C22—2362no—
412C22—2362no—
413C22#1683732011yescharacteristic
414C22#1683754811yescharacteristic
415C22#1683755011yescharacteristic
416C22#1683755411yescharacteristic
417C22#1683755911yescharacteristic
418C22#1683757711yescharacteristic
419C22#1683758211yescharacteristic
420C4—21no—
421C4—21no—
422C4—82no—
423C4—9441no—
424C4—4721no—
425C4—4721no—
426C4—2362no—
427C4#1683763411yescharacteristic
428C4#1683764211yescharacteristic
429C4#1683764711yescharacteristic
430C4#1683765411yescharacteristic
431C4#1683772211yescharacteristic
432C4#1683772911yescharacteristic
433C4#1683773911yescharacteristic
434C4#1683774611yescharacteristic
435C2—4722no—
436C2#2142320711yescharacteristic
437C2#2142326211yescharacteristic
438C2#2142329711yescharacteristic
439C2#2142331111yescharacteristic
440C2#2142528811yescharacteristic
441C2#2142530211yescharacteristic
442C2#2142532611yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1296C2
socleC23×C59#120
prime radicalC22×D236D4
hypercenter#1296D59
hypocenterC59#10436
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #21424223C2semidirect
3. #21424102C2semidirect
5. C2×#16837132C2semidirect
7. C59×#1296C2semidirect
220. #1296D59semidirect
248. C59#10436semidirect

Maximal subgroups

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

subgroupindex
#214290192
#214242232
#214241022
#214240972
C2×#168371322
C2×#168371292
C59×#12962
#1043659