Finite Group Databasebeta

← #28485155

Subgroups — #28485155

Order 15104.

750 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837541C211yesmaximal, characteristic
2Q8×C22⋊Dic59C211yesmaximal, characteristic, complemented
3#21424931C211yesmaximal, characteristic
4#21422665C211yesmaximal, characteristic
5#21421963C211yesmaximal, characteristic, complemented
6#21421942C211yesmaximal, characteristic, complemented
7C59×#1787C211yesmaximal, characteristic, complemented
8C22×#11745671C412yescomplemented
9C2×#14273701C2211yescharacteristic
10C2×Q8×Dic59—21no—
11#16837541—21no—
12#16837541—21no—
13#16837541—22no—
14C2×#14273651C412yescomplemented
15C2×#14273641—21no—
16C2×#14273641—21no—
17C2×C22⋊(C59⋊C8)C2211yescharacteristic
18#16837309—22no—
19#16837309—21no—
20#16837262—21no—
21#16837262C2211yescharacteristic
22C4×C22⋊Dic59—21no—
23C4×C22⋊Dic59C2211yescharacteristic
24#16837137—22no—
25C23×Q8×C59C2211yescharacteristic
26C2×#103×C59—22no—
27C2×C22⋊C8×C59C2211yescharacteristic
28C59×#380C2211yescharacteristic
29C22×D236C2×C411yescharacteristic
30C22×C22⋊D59—21no—
31C2×#11745671—22no—
32C2×#11745671—22no—
33C2×#11745671—44no—
34C22×C59⋊C8C2×C412yes—
35#14273701—21no—
36#14273701—21no—
37#14273701—21no—
38#14273701—21no—
39Q8×Dic59—44no—
40C2×C4⋊Dic59—22no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59C2×C411yescharacteristic
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59D411yescharacteristic
45C2×C22⋊Dic59D411yescharacteristic
46C2×(C2×D59)⋊C4—21no—
47C2×C4×Dic59—22no—
48C2×C4×Dic59—21no—
49#14273651—22no—
50#14273651—22no—
51#14273641—42no—
52#14273641—42no—
53C22⋊(C59⋊C8)—21no—
54C22⋊(C59⋊C8)—21no—
55C59⋊#95—22no—
56C59⋊#95—22no—
57C59⋊#95—21no—
58C59⋊#95D412yes—
59C22×Q8×C59—21no—
60C22×Q8×C59—21no—
61C22×Q8×C59—21no—
62C22×Q8×C59—44no—
63C22×Q8×C59C2×C412yes—
64C22×Q8×C59D411yescharacteristic
65C22×Q8×C59D411yescharacteristic
66C22×Q8×C59D412yes—
67C23×C4×C59—21no—
68C23×C4×C59C2311yescharacteristic
69C22×C8×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72#103×C59—44no—
73C22⋊C8×C59—21no—
74C22⋊C8×C59—21no—
75#95×C59—21no—
76C24×C59C2×D411yescharacteristic
77C23×D59—21no—
78C2×Q8×C59—24no—
79C2×Q8×C59—22no—
80C2×Q8×C59—22no—
81C2×Q8×C59—21no—
82C2×Q8×C59—21no—
83C2×Q8×C59—24no—
84C2×Q8×C59—24no—
85C2×Q8×C59—42no—
86C2×Q8×C59—42no—
87C2×Q8×C59—42no—
88C2×Q8×C59—44no—
89C2×Q8×C59—44no—
90C2×Q8×C59—44no—
91C2×Q8×C59C22⋊C412yes—
92C2×Q8×C59C22⋊C412yes—
93C22×C4×C59—21no—
94C22×C4×C59—22no—
95C22×C4×C59—21no—
96C22×C4×C59—22no—
97C22×C4×C59—42no—
98C22×C4×C59—44no—
99C22×C4×C59—41no—
100C22×C4×C59C22×C411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59#4111yescharacteristic
105C22×C4×C59#4111yescharacteristic
106C2×C22⋊D59—41no—
107C2×C22⋊D59—41no—
108C2×C22⋊D59—41no—
109C2×C22⋊D59—41no—
110C2×C22⋊D59—41no—
111C2×C22⋊D59—41no—
112C22×Dic59—21no—
113C22×Dic59—21no—
114C22×Dic59—21no—
115C22×Dic59—22no—
116C2×D236—21no—
117C2×D236—21no—
118C2×D236—42no—
119C2×C8×C59—42no—
120C2×C8×C59—41no—
121C4⋊C4×C59—42no—
122C22⋊C4×C59—41no—
123C22⋊C4×C59—41no—
124C22⋊Dic59—21no—
125C22⋊Dic59—21no—
126C22⋊Dic59—21no—
127C22⋊Dic59—41no—
128C22⋊Dic59—41no—
129C22⋊Dic59SD1612yescomplemented
130#11745671—44no—
131#11745671—44no—
132(C2×D59)⋊C4—41no—
133(C2×D59)⋊C4—41no—
134C4⋊Dic59—21no—
135C4⋊Dic59—21no—
136C4⋊Dic59—44no—
137C4⋊Dic59—42no—
138C4×Dic59—44no—
139C4×Dic59—42no—
140C2×C59⋊C8—22no—
141C2×C59⋊C8—42no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59C4×D411yescharacteristic
145C23×C59#12011yescharacteristic
146C23×C59#12111yescharacteristic
147C23×C59C2×SD1611yescharacteristic
148C23×C59#13511yescharacteristic
149C23×C59AGL(1,ℤ/8)11yescharacteristic
150C23×C59#13711yescharacteristic
151C22×D59—41no—
152C22×D59—41no—
153C22×D59—41no—
154C22×D59—81no—
155C22×D59—81no—
156Q8×C59—24no—
157Q8×C59—22no—
158Q8×C59—22no—
159Q8×C59—22no—
160Q8×C59—44no—
161Q8×C59—44no—
162Q8×C59—42no—
163Q8×C59#9914yescomplemented
164C2×C4×C59—24no—
165C2×C4×C59—22no—
166C2×C4×C59—22no—
167C2×C4×C59—22no—
168C2×C4×C59—22no—
169C2×C4×C59—22no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—44no—
174C2×C4×C59—44no—
175C2×C4×C59—42no—
176C2×C4×C59—81no—
177C2×C4×C59—81no—
178C2×C4×C59—81no—
179C2×C4×C59C2×C22⋊C411yescharacteristic
180C2×C4×C59C4×D411yescharacteristic
181C2×C4×C59#12111yescharacteristic
182C2×C4×C59#12311yescharacteristic
183C22⋊D59—81no—
184C22⋊D59—81no—
185C22⋊D59—81no—
186C22⋊D59—81no—
187C2×Dic59—22no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—41no—
191C2×Dic59—42no—
192C2×Dic59—41no—
193C2×Dic59—42no—
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—41no—
198C2×Dic59—42no—
199D236—42no—
200D236—81no—
201C8×C59—81no—
202C59⋊C8—42no—
203#9262—591nomaximal, Sylow
204C22×C59—21no—
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—41no—
211C22×C59—41no—
212C22×C59—41no—
213C22×C59—41no—
214C22×C59#38611yescharacteristic
215C22×C59C4×SD1611yescharacteristic
216C22×C59#44011yescharacteristic
217C22×C59#44811yescharacteristic
218C22×C59#44911yescharacteristic
219C22×C59#46011yescharacteristic
220C22×C59#46411yescharacteristic
221C2×D59—81no—
222C2×D59—81no—
223C2×D59—81no—
224C2×D59—161no—
225C2×D59—161no—
226C4×C59—21no—
227C4×C59—21no—
228C4×C59—24no—
229C4×C59—21no—
230C4×C59—42no—
231C4×C59—161no—
232C4×C59C2×#9912yes—
233Dic59—42no—
234Dic59—81no—
235Dic59—82no—
236Dic59—81no—
237C2×#448—591no—
238Q8×C22⋊C4—591no—
239#1964—591no—
240#1787D5911yescharacteristic, complemented
241#1519—591no—
242#1289—591no—
243#1282—591no—
244C2×C59—21no—
245C2×C59—21no—
246C2×C59—21no—
247C2×C59—21no—
248C2×C59—21no—
249C2×C59—21no—
250C2×C59#188011yescharacteristic
251C2×C59#189811yescharacteristic
252C2×C59#190511yescharacteristic
253D59—161no—
254C23×Q8C2×D5911yescharacteristic
255C22×SD16—592no—
256C2×#121—591no—
257C2×C4×Q8—1181no—
258#448—1181no—
259#448—1181no—
260#448—1182no—
261C2×#103—22no—
262C2×#103—592no—
263C2×#99—1181no—
264C2×#99—1181no—
265C2×C22⋊C8—591no—
266C2×C22⋊C8C2×D5911yescharacteristic
267#391—1182no—
268#391—1181no—
269#380—591no—
270#380—1181no—
271#380C2×D5911yescharacteristic
272C4×C22⋊C4—591no—
273C4×C22⋊C4—1181no—
274#327—1182no—
275C59#926211yescharacteristic, complemented, Sylow
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—44no—
280C22×Q8C4×D5912yes—
281C22×Q8D23611yescharacteristic
282C22×Q8D23611yescharacteristic
283C22×Q8C22⋊D5912yes—
284C22×D4—591no—
285C22×D4—1181no—
286C23×C4—21no—
287C23×C4C22×D5911yescharacteristic
288C2×SD16—1182no—
289C2×SD16—1182no—
290C2×SD16—2364no—
291C22×C8—21no—
292C22×C8—592no—
293#121—1181no—
294#121—1181no—
295#121—1181no—
296#121—1181no—
297C4×Q8—2364no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—591no—
300C2×C4⋊C4—1182no—
301C2×C4⋊C4—1181no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—591no—
304C2×C22⋊C4—591no—
305C2×C22⋊C4—1181no—
306C2×C22⋊C4—1181no—
307C2×C42—1182no—
308C2×C42—1181no—
309#103—44no—
310#103—1182no—
311#103—1182no—
312#99—2362no—
313#99—2362no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—1181no—
317C22⋊C8—1181no—
318#95—21no—
319#95—592no—
320#95—1182no—
321#95—1182no—
322#95—1181no—
323C24—1181no—
324C24D4×D5911yescharacteristic
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8(C2×D59)⋊C412yes—
339C2×Q8(C2×D59)⋊C412yes—
340C2×D4—1181no—
341C2×D4—1181no—
342C2×D4—2362no—
343C2×D4—2361no—
344C2×D4—2361no—
345C2×D4—2361no—
346C2×D4—2361no—
347C2×D4—2361no—
348C2×D4—2361no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—41no—
356C22×C4—1181no—
357C22×C4—1181no—
358C22×C4—1182no—
359C22×C4—1181no—
360C22×C4C2×C4×D5911yescharacteristic
361C22×C4C2×D23611yescharacteristic
362C22×C4D4×D5911yescharacteristic
363C22×C4#1174568711yescharacteristic
364C22×C4#1174568711yescharacteristic
365C22×C4C2×C22⋊D5911yescharacteristic
366SD16—2364no—
367SD16—2364no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—1182no—
371C2×C8—2362no—
372C4⋊C4—42no—
373C4⋊C4—1181no—
374C4⋊C4—1181no—
375C4⋊C4—2364no—
376C4⋊C4—2362no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—592no—
380C22⋊C4—1181no—
381C22⋊C4—1181no—
382C22⋊C4—1181no—
383C22⋊C4—2361no—
384C22⋊C4—2361no—
385C22⋊C4—2361no—
386C22⋊C4—2361no—
387C42—2364no—
388C42—2362no—
389C23—41no—
390C23—41no—
391C23—4721no—
392C23—4721no—
393C23—2361no—
394C23—2361no—
395C23—2361no—
396C23(C22⋊D59)⋊C411yescharacteristic
397C23#1427369711yescharacteristic
398C23#1427369911yescharacteristic
399C23SD16×D5911yescharacteristic
400C23#1427375011yescharacteristic
401C23#1427375411yescharacteristic
402C23#1427375911yescharacteristic
403Q8—24no—
404Q8—22no—
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—44no—
409Q8—42no—
410Q8#1427364114yescomplemented
411D4—4721no—
412D4—4721no—
413D4—4721no—
414D4—4721no—
415D4—4721no—
416D4—2362no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—1182no—
433C2×C4—2362no—
434C2×C4—2361no—
435C2×C4—2362no—
436C2×C4—2361no—
437C2×C4—2361no—
438C2×C4—2361no—
439C2×C4—2361no—
440C2×C4—2361no—
441C2×C4—2362no—
442C2×C4—2361no—
443C2×C4—2361no—
444C2×C4C2×(C2×D59)⋊C411yescharacteristic
445C2×C4(C22⋊D59)⋊C411yescharacteristic
446C2×C4#1427369911yescharacteristic
447C2×C4#1427371111yescharacteristic
448C8—81no—
449C8—2362no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—9441no—
461C22—9441no—
462C22—4721no—
463C22—4721no—
464C22—4721no—
465C22#1683727911yescharacteristic
466C22#1683748711yescharacteristic
467C22#1683750511yescharacteristic
468C22#1683754211yescharacteristic
469C22#1683755011yescharacteristic
470C22#1683760011yescharacteristic
471C22#1683762611yescharacteristic
472C4—21no—
473C4—24no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—161no—
478C4—4721no—
479C4—4722no—
480C4—4721no—
481C4—2362no—
482C4C2×#1427364112yes—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—9441no—
490C2#2142444911yescharacteristic
491C2#2142454711yescharacteristic
492C2#2142458811yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1787C2
socleC22×C59#386
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1787D59
hypocenterC59#9262
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. Q8×C22⋊Dic59C2semidirect
5. #21421963C2semidirect
6. #21421942C2semidirect
7. C59×#1787C2semidirect
8. C22×#11745671C4semidirect
14. C2×#14273651C4semidirect
129. C22⋊Dic59SD16semidirect
163. Q8×C59#99semidirect
240. #1787D59semidirect
275. C59#9262semidirect
410. Q8#14273641semidirect

Maximal subgroups

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

subgroupindex
C2×#168375412
Q8×C22⋊Dic592
#214249312
#214226652
#214219632
#214219422
C59×#17872
#926259