Finite Group Databasebeta

← #28503639

Subgroups — #28503639

Order 15104.

810 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837386C211yesmaximal, characteristic
2#21426688C211yesmaximal, characteristic
3#21422213C214yesmaximal, complemented
4C2×C59×#410C211yesmaximal, characteristic
5C22×#11745664—21no—
6C2×#14273686C411yescharacteristic, complemented
7C22×C4⋊Dic59C411yescharacteristic, complemented
8#16837386—22no—
9#16837386C414yescomplemented
10C2×#14273646—21no—
11C2×#14273644C2212yes—
12#16837309—21no—
13#16837305—21no—
14#16837282C412yescomplemented
15#16837282C2212yes—
16#16837262—21no—
17#16837262—21no—
18C22×#41×C59C2211yescharacteristic
19C2×#117×C59—21no—
20C59×#410—22no—
21C59×#410C414yescomplemented
22C2×#99×C59—21no—
23C2×#99×C59C2212yes—
24C23×Dic59C2×C412yes—
25C2×#11745664—24no—
26C2×#11745664—22no—
27C2×#11745664—41no—
28C2×#11745664—42no—
29C22×C59⋊C8—41no—
30#14273686—42no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—41no—
35C2×C4⋊Dic59—41no—
36C2×C22⋊Dic59—22no—
37C2×C22⋊Dic59—22no—
38C2×C22⋊Dic59C2×C412yes—
39C2×C4×Dic59—21no—
40#14273646—24no—
41#14273644—22no—
42#14273644C2×C414yes—
43C59⋊#95—22no—
44C59⋊#95—22no—
45C2×#41×C59—41no—
46C2×#41×C59—41no—
47C2×#41×C59—42no—
48C2×#41×C59C2×C412yes—
49C2×#41×C59C2×C412yes—
50C2×#41×C59D412yes—
51C2×#41×C59Q812yes—
52C22×Q8×C59D411yescharacteristic
53C22×D4×C59C2×C411yescharacteristic
54C22×D4×C59D411yescharacteristic
55C22×D4×C59C2311yescharacteristic
56C23×C4×C59C2×C411yescharacteristic
57C23×C4×C59D411yescharacteristic
58C23×C4×C59D411yescharacteristic
59#117×C59—24no—
60#117×C59—42no—
61C2×C4⋊C4×C59—41no—
62C2×C22⋊C4×C59—22no—
63C2×C42×C59—41no—
64#99×C59—24no—
65#99×C59—22no—
66#99×C59C2×C414yes—
67C24×C59C22⋊C411yescharacteristic
68C24×C59C22×C411yescharacteristic
69C24×C59C2×D411yescharacteristic
70#41×C59—42no—
71#41×C59—42no—
72#41×C59—42no—
73#41×C59—42no—
74#41×C59—42no—
75#41×C59—42no—
76#41×C59—42no—
77#41×C59—42no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59C22⋊C412yes—
81C2×Q8×C59C4⋊C412yes—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59C4212yes—
89C2×D4×C59C22⋊C412yes—
90C2×D4×C59C4⋊C412yes—
91C2×D4×C59C4⋊C412yes—
92C2×D4×C59C22×C411yescharacteristic
93C2×D4×C59C22×C411yescharacteristic
94C2×D4×C59C2×D411yescharacteristic
95C2×D4×C59C2×Q811yescharacteristic
96C22×C4×C59—21no—
97C22×C4×C59—21no—
98C22×C4×C59—22no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59—22no—
102C22×C4×C59—21no—
103C22×C4×C59—21no—
104C22×C4×C59—22no—
105C22×C4×C59—42no—
106C22×C4×C59C4212yes—
107C22×C4×C59C22⋊C411yescharacteristic
108C22×C4×C59C22⋊C411yescharacteristic
109C22×C4×C59C22⋊C412yes—
110C22×C4×C59C22⋊C412yes—
111C22×C4×C59C22⋊C411yescharacteristic
112C22×C4×C59C4⋊C412yes—
113C22×C4×C59C4⋊C412yes—
114C22×C4×C59C4⋊C412yes—
115C22×C4×C59C2×D411yescharacteristic
116C22×Dic59—22no—
117C22×Dic59—22no—
118C22×Dic59—42no—
119C22×Dic59—82no—
120C4⋊C4×C59—44no—
121C22⋊C4×C59—24no—
122C22⋊C4×C59—42no—
123C42×C59—44no—
124C22⋊Dic59—42no—
125C22⋊Dic59—82no—
126C22⋊Dic59—82no—
127C4⋊Dic59—42no—
128C4⋊Dic59—41no—
129C4⋊Dic59—41no—
130C4⋊Dic59—42no—
131C4×Dic59—42no—
132#11745664—24no—
133#11745664—42no—
134#11745664—44no—
135C2×C59⋊C8—42no—
136C2×C59⋊C8—44no—
137C23×C59—21no—
138C23×C59—21no—
139C23×C59—21no—
140C23×C59—21no—
141C23×C59—21no—
142C23×C59—21no—
143C23×C59—41no—
144C23×C59—41no—
145C23×C59—41no—
146C23×C59—41no—
147C23×C59—41no—
148C23×C59—41no—
149C23×C59#9512yes—
150C23×C59C2×C4211yescharacteristic
151C23×C59C2×C22⋊C411yescharacteristic
152C23×C59C2×C22⋊C411yescharacteristic
153C23×C59C2×C4⋊C411yescharacteristic
154C23×C59C2×C4⋊C411yescharacteristic
155Q8×C59—42no—
156Q8×C59—42no—
157D4×C59—42no—
158D4×C59—42no—
159D4×C59—42no—
160D4×C59—42no—
161D4×C59—41no—
162D4×C59—41no—
163D4×C59—41no—
164D4×C59—41no—
165C2×C4×C59—21no—
166C2×C4×C59—21no—
167C2×C4×C59—21no—
168C2×C4×C59—21no—
169C2×C4×C59—21no—
170C2×C4×C59—21no—
171C2×C4×C59—22no—
172C2×C4×C59—22no—
173C2×C4×C59—22no—
174C2×C4×C59—22no—
175C2×C4×C59—22no—
176C2×C4×C59—22no—
177C2×C4×C59—22no—
178C2×C4×C59—22no—
179C2×C4×C59—22no—
180C2×C4×C59—22no—
181C2×C4×C59—22no—
182C2×C4×C59—22no—
183C2×C4×C59—44no—
184C2×C4×C59—82no—
185C2×C4×C59—82no—
186C2×C4×C59#9512yes—
187C2×C4×C59#9512yes—
188C2×C4×C59#9512yes—
189C2×C4×C59#9912yes—
190C2×C4×C59#9911yescharacteristic
191C2×C4×C59#9911yescharacteristic
192C2×C4×C59C2×C22⋊C411yescharacteristic
193C2×C4×C59C2×C4⋊C411yescharacteristic
194C2×Dic59—42no—
195C2×Dic59—42no—
196C2×Dic59—42no—
197C2×Dic59—42no—
198C2×Dic59—42no—
199C2×Dic59—42no—
200C2×Dic59—84no—
201C59⋊C8—44no—
202#12575—591nomaximal, Sylow
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—21no—
206C22×C59—22no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—42no—
211C22×C59—41no—
212C22×C59—41no—
213C22×C59—41no—
214C22×C59—41no—
215C22×C59—42no—
216C22×C59—41no—
217C22×C59—41no—
218C22×C59—41no—
219C22×C59—41no—
220C22×C59—41no—
221C22×C59—41no—
222C22×C59—81no—
223C22×C59—81no—
224C22×C59C2×#9511yescharacteristic
225C22×C59C2×#9911yescharacteristic
226C22×C59C2×#9911yescharacteristic
227C4×C59—21no—
228C4×C59—22no—
229C4×C59—22no—
230C4×C59—21no—
231C4×C59—84no—
232C4×C59#34212yes—
233C4×C59#34212yes—
234Dic59—82no—
235Dic59—82no—
236C2×#413—591no—
237C2×#410D5911yescharacteristic
238#2311—591no—
239#1390—594no—
240C2×C59—21no—
241C2×C59—21no—
242C2×C59—41no—
243C2×C59—41no—
244C2×C59—41no—
245C2×C59—42no—
246C2×C59—41no—
247C2×C59C2×#34211yescharacteristic
248C2×C59#212211yescharacteristic
249C2×C59#212311yescharacteristic
250C22×#41C2×D5911yescharacteristic
251C22×M16—1181no—
252C2×#117—21no—
253C2×#117—591no—
254C22×C4⋊C4—591no—
255#413—594no—
256#413—1182no—
257C2×#101—1181no—
258C2×#100—592no—
259#410—22no—
260#410Dic5914yescomplemented
261C2×#99—21no—
262C2×#99C2×D5912yes—
263#391—1181no—
264#390—1181no—
265#386—592no—
266#386—592no—
267#380—1181no—
268#380—1181no—
269C59#1257511yescharacteristic, complemented, Sylow
270C2×#41—41no—
271C2×#41—41no—
272C2×#41—42no—
273C2×#41Dic11812yes—
274C2×#41C4×D5912yes—
275C2×#41C2×Dic5912yes—
276C2×#41C22⋊D5912yes—
277C22×Q8D23611yescharacteristic
278C22×D4C4×D5911yescharacteristic
279C22×D4C22⋊D5911yescharacteristic
280C22×D4C22×D5911yescharacteristic
281C23×C4—592no—
282C23×C4C4×D5911yescharacteristic
283C23×C4D23611yescharacteristic
284C23×C4C22⋊D5911yescharacteristic
285C2×M16—1184no—
286C2×M16—1182no—
287C2×M16—2361no—
288C2×M16—2362no—
289C22×C8—2361no—
290#117—24no—
291#117—42no—
292#117—2362no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—1181no—
295C2×C4⋊C4—1181no—
296C2×C4⋊C4—1181no—
297C2×C4⋊C4—2361no—
298C2×C4⋊C4—2361no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—592no—
301C2×C22⋊C4—1182no—
302C2×C22⋊C4—1182no—
303C2×C42—41no—
304C2×C42—1181no—
305#101—1184no—
306#100—594no—
307#100—1182no—
308#99—24no—
309#99—22no—
310#99C2×Dic5914yes—
311#95—1182no—
312#95—1182no—
313C24(C2×D59)⋊C411yescharacteristic
314C24C2×C4×D5911yescharacteristic
315C24C2×C22⋊D5911yescharacteristic
316#41—42no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8C4⋊Dic5912yes—
327C2×Q8(C2×D59)⋊C412yes—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C4×Dic5912yes—
335C2×D4Dic59⋊C412yes—
336C2×D4Dic59⋊C412yes—
337C2×D4C22⋊Dic5912yes—
338C2×D4C2×Dic11811yescharacteristic
339C2×D4C2×C4×D5911yescharacteristic
340C2×D4C22×Dic5911yescharacteristic
341C2×D4C2×C22⋊D5911yescharacteristic
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—1182no—
353C22×C4—1182no—
354C22×C4—4722no—
355C22×C4—2362no—
356C22×C4C4×Dic5912yes—
357C22×C4Dic59⋊C412yes—
358C22×C4Dic59⋊C412yes—
359C22×C4C4⋊Dic5912yes—
360C22×C4(C2×D59)⋊C411yescharacteristic
361C22×C4(C2×D59)⋊C411yescharacteristic
362C22×C4(C2×D59)⋊C412yes—
363C22×C4(C2×D59)⋊C411yescharacteristic
364C22×C4C22⋊Dic5912yes—
365C22×C4C2×D23611yescharacteristic
366M16—1184no—
367M16—2362no—
368M16—2364no—
369C2×C8—2362no—
370C2×C8—2364no—
371C4⋊C4—44no—
372C4⋊C4—2362no—
373C4⋊C4—2361no—
374C4⋊C4—2361no—
375C4⋊C4—2362no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—4722no—
379C22⋊C4—4722no—
380C22⋊C4—2362no—
381C42—44no—
382C42—2362no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C59⋊#9512yes—
396C23C2×C4×Dic5911yescharacteristic
397C23C2×(C2×D59)⋊C411yescharacteristic
398C23C2×C22⋊Dic5911yescharacteristic
399C23C2×Dic59⋊C411yescharacteristic
400C23C2×Dic59⋊C411yescharacteristic
401Q8—42no—
402Q8—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—42no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—44no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—4724no—
433C2×C4—2362no—
434C2×C4—2362no—
435C2×C4—2362no—
436C2×C4—2362no—
437C2×C4—2362no—
438C2×C4—2362no—
439C2×C4C59⋊#9512yes—
440C2×C4C59⋊#9512yes—
441C2×C4C59⋊#9512yes—
442C2×C4#1427364112yes—
443C2×C4#1427364111yescharacteristic
444C2×C4#1427364111yescharacteristic
445C2×C4C2×(C2×D59)⋊C411yescharacteristic
446C2×C4C2×C4⋊Dic5911yescharacteristic
447C8—2364no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—81no—
468C22—81no—
469C22C2×C59⋊#9511yescharacteristic
470C22C2×#1427364111yescharacteristic
471C22C2×#1427364111yescharacteristic
472C4—21no—
473C4—22no—
474C4—22no—
475C4—21no—
476C4—84no—
477C4—4722no—
478C4—4722no—
479C4#1683716812yes—
480C4#1683716812yes—
481C2—21no—
482C2—21no—
483C2—41no—
484C2—41no—
485C2—41no—
486C2—42no—
487C2—41no—
488C2C2×#1683716811yescharacteristic
489C2#2142574511yescharacteristic
490C2#2142574911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC23×C59C2×C42
FrattiniC22×D4C22×D59
FittingC2×C59×#410C2
socleC22×C59C2×#99
prime radicalC22×#41×C59C22
cosocleC22×D4×C59C23
hypercenterC2×#410D59
hypocenterC59#12575
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #21422213C2semidirect
6. C2×#14273686C4semidirect
7. C22×C4⋊Dic59C4semidirect
9. #16837386C4semidirect
14. #16837282C4semidirect
21. C59×#410C4semidirect
260. #410Dic59semidirect
269. C59#12575semidirect

Maximal subgroups

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

subgroupindex
C2×#168373862
#214266882
#214222132
C2×C59×#4102
#1257559