Finite Group Databasebeta

← #28491009

Subgroups — #28491009

Order 15104.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837604C211yesmaximal, characteristic
2#21429014C211yesmaximal, characteristic
3#21425058C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
5C2×#16837129C211yesmaximal, characteristic
6#21421980C211yesmaximal, characteristic, complemented
7C2×C59×#475C211yesmaximal, characteristic, complemented
8C22×#11745671—21no—
9C2×#14273703—21no—
10C2×C4×D236C2211yescharacteristic
11#16837604—22no—
12#16837604—22no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273651—21no—
15C2×C4×C59⋊C8C2211yescharacteristic
16#16837324—22no—
17#16837304—21no—
18#16837300—21no—
19#16837272C2211yescharacteristic
20#16837259—21no—
21#16837164—22no—
22Q8⋊(C59⋊C8)—22no—
23#16837129—22no—
24C2×C4⋊Q8×C59C2211yescharacteristic
25C2×C4×Q8×C59C2211yescharacteristic
26C59×#475—22no—
27C59×#475—22no—
28C2×#106×C59—21no—
29C2×C4⋊C8×C59C2211yescharacteristic
30C2×#103×C59—21no—
31C22×D236D411yescharacteristic
32C22×C4×D59—21no—
33C2×#11745671—24no—
34C2×#11745671—42no—
35C2×#11745671—42no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38#14273703—42no—
39#14273703—42no—
40C4×D236—21no—
41C4×D236—21no—
42C4×D236—21no—
43C4×D236—21no—
44C2×C4⋊Dic59—21no—
45C2×C4⋊Dic59—21no—
46C2×C4⋊Dic59D411yescharacteristic
47C2×C4⋊Dic59D412yes—
48C2×Dic59⋊C4—22no—
49C2×(C2×D59)⋊C4—22no—
50C2×(C2×D59)⋊C4—21no—
51C2×(C2×D59)⋊C4—22no—
52C4⋊(C59⋊C8)—21no—
53C4⋊(C59⋊C8)D412yes—
54#14273651—24no—
55C4×C59⋊C8—21no—
56C4×C59⋊C8—21no—
57C59⋊#95—22no—
58C22×Q8×C59—21no—
59C22×Q8×C59D411yescharacteristic
60C22×C8×C59—21no—
61C4⋊Q8×C59—21no—
62C4⋊Q8×C59—21no—
63C4⋊Q8×C59—21no—
64C4⋊Q8×C59D412yes—
65C4×Q8×C59—21no—
66C4×Q8×C59—21no—
67C4×Q8×C59—21no—
68C4×Q8×C59—21no—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59D411yescharacteristic
73C2×C42×C59—21no—
74C2×C42×C59C2311yescharacteristic
75#106×C59—24no—
76C4⋊C8×C59—21no—
77C4⋊C8×C59D412yes—
78#103×C59—24no—
79C23×D59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59—22no—
83C2×Q8×C59—21no—
84C2×Q8×C59—21no—
85C2×Q8×C59—21no—
86C2×Q8×C59—41no—
87C2×Q8×C59—41no—
88C2×Q8×C59—41no—
89C2×Q8×C59—41no—
90C2×Q8×C59—41no—
91C2×Q8×C59SD1612yescomplemented
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—41no—
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×Dic59—21no—
99C22×Dic59—22no—
100C2×D236—21no—
101C2×D236—21no—
102C2×D236—21no—
103C2×D236—21no—
104C2×D236—21no—
105C2×D236SD1612yes—
106C2×C4×D59—41no—
107C2×C4×D59—41no—
108C2×C4×D59—41no—
109C2×C4×D59—41no—
110C2×C4×D59—41no—
111C2×C4×D59—41no—
112C2×C8×C59—22no—
113C2×C8×C59—41no—
114C2×C8×C59—41no—
115C4⋊C4×C59—22no—
116C4⋊C4×C59—21no—
117C4⋊C4×C59—22no—
118C4⋊C4×C59—21no—
119C4⋊C4×C59—41no—
120C4⋊C4×C59—41no—
121C4⋊C4×C59—41no—
122C4⋊C4×C59—41no—
123C42×C59—41no—
124C42×C59—41no—
125C42×C59C2×D411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128C42×C59#4111yescharacteristic
129#11745671—48no—
130(C2×D59)⋊C4—44no—
131(C2×D59)⋊C4—41no—
132(C2×D59)⋊C4—41no—
133(C2×D59)⋊C4—82no—
134C4⋊Dic59—21no—
135C4⋊Dic59—21no—
136C4⋊Dic59—41no—
137C4⋊Dic59—41no—
138C4⋊Dic59—41no—
139C4⋊Dic59—41no—
140C4⋊Dic59—42no—
141Dic59⋊C4—44no—
142C2×C59⋊C8—22no—
143C2×C59⋊C8—22no—
144C2×C59⋊C8—41no—
145C2×C59⋊C8—41no—
146C2×C59⋊C8—41no—
147C2×C59⋊C8—41no—
148C23×C59#12011yescharacteristic
149C22×D59—22no—
150C22×D59—41no—
151C22×D59—41no—
152C22×D59—41no—
153C22×D59—41no—
154C22×D59—41no—
155C22×D59—41no—
156Q8×C59—22no—
157Q8×C59—22no—
158Q8×C59—41no—
159Q8×C59—41no—
160Q8×C59—42no—
161Q8×C59—42no—
162Q8×C59—81no—
163Q8×C59—81no—
164C2×C4×C59—21no—
165C2×C4×C59—21no—
166C2×C4×C59—22no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—42no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—81no—
174C2×C4×C59—81no—
175C2×C4×C59#12011yescharacteristic
176C2×C4×C59#12011yescharacteristic
177C2×C4×C59#12111yescharacteristic
178C2×C4×C59#12111yescharacteristic
179C2×C4×C59#12111yescharacteristic
180C2×C4×C59#12711yescharacteristic
181C2×C4×C59C2×SD1611yescharacteristic
182C2×C4×C59C2×SD1611yescharacteristic
183C2×C4×C59#13511yescharacteristic
184C2×C4×C59#13511yescharacteristic
185C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C59#13711yescharacteristic
188C2×C4×C59#13711yescharacteristic
189C2×Dic59—41no—
190C2×Dic59—42no—
191C2×Dic59—41no—
192C2×Dic59—42no—
193C2×Dic59—41no—
194C2×Dic59—42no—
195D236—22no—
196D236—22no—
197D236—41no—
198D236—41no—
199C4×D59—81no—
200C4×D59—81no—
201C4×D59—81no—
202C4×D59—81no—
203C8×C59—42no—
204C59⋊C8—42no—
205C59⋊C8—42no—
206#10232—591nomaximal, Sylow
207C22×C59#39211yescharacteristic
208C22×C59#44811yescharacteristic
209C22×C59#44911yescharacteristic
210C22×C59#45011yescharacteristic
211C22×C59#45211yescharacteristic
212C22×C59#45511yescharacteristic
213C22×C59#45611yescharacteristic
214C2×D59—42no—
215C2×D59—42no—
216C2×D59—42no—
217C2×D59—81no—
218C2×D59—81no—
219C2×D59—81no—
220C2×D59—81no—
221C4×C59—21no—
222C4×C59—21no—
223C4×C59—42no—
224C4×C59—82no—
225C4×C59—81no—
226C4×C59#44811yescharacteristic
227C4×C59#44911yescharacteristic
228C4×C59#45011yescharacteristic
229C4×C59#45211yescharacteristic
230C4×C59#46511yescharacteristic
231C4×C59#46511yescharacteristic
232C4×C59#46811yescharacteristic
233C4×C59#46811yescharacteristic
234Dic59—82no—
235Dic59—81no—
236C2×#475D5911yescharacteristic, complemented
237C2×#461—591no—
238#2538—591no—
239#1990—591no—
240C2×Q8⋊C8—591no—
241C2×D4⋊C8—591no—
242#1296—591no—
243C2×C59#164211yescharacteristic
244C2×C59#164311yescharacteristic
245C2×C59#165011yescharacteristic
246C2×C59#165111yescharacteristic
247C2×C59#200011yescharacteristic
248C2×C59#200011yescharacteristic
249C2×C59#200711yescharacteristic
250D59—82no—
251C22×SD16—1181no—
252C2×C4⋊Q8C2×D5911yescharacteristic
253C2×C22⋊Q8—1181no—
254C2×C4×Q8C2×D5911yescharacteristic
255C2×C4×D4—591no—
256#475—22no—
257#475—22no—
258#461—1182no—
259#461—1182no—
260C2×#106—21no—
261C2×C4⋊C8—591no—
262C2×C4⋊C8C2×D5911yescharacteristic
263C2×#103—21no—
264C2×#103—1181no—
265C2×C4×C8—591no—
266#396—1182no—
267#390—1181no—
268#389—1181no—
269#384—591no—
270#380—1181no—
271#340—1182no—
272Q8⋊C8—1182no—
273D4⋊C8—1182no—
274C59#1023211yescharacteristic, complemented, Sylow
275C22×Q8—21no—
276C22×Q8C22⋊D5911yescharacteristic
277C22×D4—591no—
278C23×C4—1181no—
279C2×SD16—1184no—
280C2×SD16—2362no—
281C2×SD16—2362no—
282C22×C8—21no—
283C22×C8—1181no—
284C22×C8—1181no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8C22⋊D5912yes—
289C22⋊Q8—2362no—
290C22⋊Q8—2362no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×D4—1181no—
296C4×D4—1181no—
297C4×D4—1181no—
298C4×D4—1181no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—591no—
303C2×C4⋊C4—592no—
304C2×C4⋊C4—1182no—
305C2×C4⋊C4—1181no—
306C2×C4⋊C4—1181no—
307C2×C4⋊C4C22⋊D5911yescharacteristic
308C2×C22⋊C4—1182no—
309C2×C22⋊C4—1182no—
310C2×C22⋊C4—1181no—
311C2×C42—21no—
312C2×C42C22×D5911yescharacteristic
313#106—24no—
314C4⋊C8—21no—
315C4⋊C8—592no—
316C4⋊C8—1181no—
317C4⋊C8C22⋊D5912yes—
318#103—24no—
319#103—1184no—
320C4×C8—1181no—
321C4×C8—1181no—
322#95—1182no—
323C24—1181no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8#1174567112yescomplemented
336C2×D4—592no—
337C2×D4—1181no—
338C2×D4—1181no—
339C2×D4—1181no—
340C2×D4—1181no—
341C2×D4—1181no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4—1181no—
346C22×C4—1182no—
347C22×C4—2361no—
348C22×C4—2361no—
349C22×C4—2361no—
350C22×C4—2361no—
351C22×C4—2361no—
352C22×C4—2361no—
353C22×C4D4×D5911yescharacteristic
354C22×C4D4×D5911yescharacteristic
355C22×C4C2×C22⋊D5911yescharacteristic
356SD16—2368no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—1182no—
361C2×C8—1182no—
362C2×C8—2361no—
363C2×C8—2361no—
364C2×C8—2361no—
365C2×C8—2361no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1181no—
375C4⋊C4—1181no—
376C4⋊C4—2362no—
377C4⋊C4—2364no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4—2361no—
381C4⋊C4—2361no—
382C22⋊C4—4722no—
383C22⋊C4—2364no—
384C22⋊C4—2361no—
385C22⋊C4—2361no—
386C42—41no—
387C42—41no—
388C42D4×D5911yescharacteristic
389C42#1174568711yescharacteristic
390C42C2×C22⋊D5911yescharacteristic
391C42C2×C22⋊D5911yescharacteristic
392C23—1182no—
393C23—2361no—
394C23—2361no—
395C23—2361no—
396C23—2361no—
397C23—2361no—
398C23—2361no—
399C23#1427369511yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—42no—
405Q8—42no—
406Q8—81no—
407Q8—81no—
408D4—1182no—
409D4—1182no—
410D4—2361no—
411D4—2361no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—4721no—
424C2×C4—4721no—
425C2×C4—4721no—
426C2×C4—4721no—
427C2×C4—2362no—
428C2×C4—2362no—
429C2×C4—2361no—
430C2×C4—2361no—
431C2×C4—2361no—
432C2×C4—2362no—
433C2×C4#1427369511yescharacteristic
434C2×C4#1427369611yescharacteristic
435C2×C4#1427369811yescharacteristic
436C2×C4#1427369911yescharacteristic
437C2×C4#1427369911yescharacteristic
438C2×C4#1427372311yescharacteristic
439C2×C4SD16×D5911yescharacteristic
440C2×C4C2×#1174567111yescharacteristic
441C2×C4#1427374711yescharacteristic
442C2×C4#1427374811yescharacteristic
443C2×C4#1427375211yescharacteristic
444C2×C4#1427375411yescharacteristic
445C2×C4#1427375611yescharacteristic
446C2×C4#1427375811yescharacteristic
447C8—42no—
448C8—2362no—
449C8—2362no—
450C22—4721no—
451C22—4721no—
452C22—4721no—
453C22—4721no—
454C22—2362no—
455C22—2362no—
456C22—2362no—
457C22#1683731011yescharacteristic
458C22#1683753911yescharacteristic
459C22#1683754611yescharacteristic
460C22#1683755211yescharacteristic
461C22#1683756211yescharacteristic
462C22#1683757811yescharacteristic
463C22#1683758311yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—82no—
468C4—81no—
469C4—4722no—
470C4—4721no—
471C4#1683754111yescharacteristic
472C4#1683754711yescharacteristic
473C4#1683755111yescharacteristic
474C4#1683756111yescharacteristic
475C4#1683762811yescharacteristic
476C4#1683763311yescharacteristic
477C4#1683764511yescharacteristic
478C4#1683765011yescharacteristic
479C2—4722no—
480C2#2142329511yescharacteristic
481C2#2142330211yescharacteristic
482C2#2142335111yescharacteristic
483C2#2142335811yescharacteristic
484C2#2142509611yescharacteristic
485C2#2142509711yescharacteristic
486C2#2142514111yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#475C2
socleC23×C59#120
prime radicalC22×D236D4
hypercenterC2×#475D59
hypocenterC59#10232
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #21425058C2semidirect
4. C2×Q8⋊(C59⋊C8)C2semidirect
6. #21421980C2semidirect
7. C2×C59×#475C2semidirect
91. C2×Q8×C59SD16semidirect
236. C2×#475D59semidirect
274. C59#10232semidirect
335. C2×Q8#11745671semidirect

Maximal subgroups

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

subgroupindex
C2×#168376042
#214290142
#214250582
C2×Q8⋊(C59⋊C8)2
C2×#168371292
#214219802
C2×C59×#4752
#1023259