Finite Group Databasebeta

← #28488222

Subgroups — #28488222

Order 15104.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837633C211yesmaximal, characteristic
2#21425082C211yesmaximal, characteristic, complemented
3#21424931C211yesmaximal, characteristic
4#21424123C211yesmaximal, characteristic, complemented
5#21423875C211yesmaximal, characteristic, complemented
6#21422665C211yesmaximal, characteristic
7C59×#1520C211yesmaximal, characteristic, complemented
8C22×#11745671C411yescharacteristic, complemented
9C23×C59⋊C8C2211yescharacteristic
10C2×#14273701C2211yescharacteristic
11#16837633—21no—
12#16837633—21no—
13#16837633—21no—
14#16837633—21no—
15C2×#14273660—21no—
16C2×#14273658C411yescharacteristic, complemented
17C2×#14273651C411yescharacteristic, complemented
18C2×#14273647C411yescharacteristic, complemented
19C2×#14273641—22no—
20C2×#14273637—21no—
21#16837262C2211yescharacteristic
22C4×C22⋊Dic59C2211yescharacteristic
23#16837169—22no—
24#16837164—21no—
25#16837156—21no—
26#16837137—22no—
27C2×C22⋊Q8×C59C2211yescharacteristic
28C2×C22⋊C8×C59C2211yescharacteristic
29C59×#380C2211yescharacteristic
30C59×#328—22no—
31C22×D236C2×C411yescharacteristic
32C22×C22⋊D59—21no—
33C2×#11745671—21no—
34C2×#11745671—21no—
35C2×#11745671—42no—
36C22×C59⋊C8—42no—
37C22×C59⋊C8C2×C411yescharacteristic
38C22×C59⋊C8C2×C411yescharacteristic
39C22×C59⋊C8D411yescharacteristic
40C22×C59⋊C8D411yescharacteristic
41C22×C59⋊C8D411yescharacteristic
42C22×C59⋊C8D411yescharacteristic
43#14273701—21no—
44#14273701—21no—
45#14273701—21no—
46#14273701—21no—
47C2×C4⋊Dic59—21no—
48C2×C4⋊Dic59C2×C411yescharacteristic
49C2×C22⋊Dic59—21no—
50C2×C22⋊Dic59D412yes—
51C2×(C2×D59)⋊C4—21no—
52C2×C4×Dic59—21no—
53#14273660—42no—
54#14273658—21no—
55#14273658—21no—
56#14273651—21no—
57#14273651—21no—
58#14273647—21no—
59#14273647—21no—
60#14273641—44no—
61#14273637—42no—
62C59⋊#95—21no—
63C59⋊#95—21no—
64C22×Q8×C59C2×C411yescharacteristic
65C23×C4×C59C2311yescharacteristic
66C22×C8×C59—21no—
67C22⋊Q8×C59—21no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59—21no—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59C2×C411yescharacteristic
74C2×C22⋊C4×C59—21no—
75C2×C22⋊C4×C59—21no—
76C22⋊C8×C59—21no—
77C22⋊C8×C59D412yes—
78#95×C59—21no—
79C24×C59C2×D411yescharacteristic
80C23×D59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59—21no—
83C2×Q8×C59—41no—
84C2×Q8×C59—41no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59C22×C411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59#4111yescharacteristic
95C22×C4×C59#4111yescharacteristic
96C2×C22⋊D59—41no—
97C2×C22⋊D59—41no—
98C2×C22⋊D59—42no—
99C2×C22⋊D59—42no—
100C22×Dic59—21no—
101C22×Dic59—22no—
102C22×Dic59—41no—
103C2×D236—21no—
104C2×D236—21no—
105C2×D236—41no—
106C2×D236—41no—
107C2×C8×C59—22no—
108C2×C8×C59—41no—
109C2×C8×C59—41no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—41no—
113C4⋊C4×C59—41no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59—41no—
118C22⋊C4×C59—42no—
119C22⋊Dic59—22no—
120C22⋊Dic59—22no—
121C22⋊Dic59—42no—
122#11745671—42no—
123#11745671—42no—
124(C2×D59)⋊C4—41no—
125(C2×D59)⋊C4—41no—
126C4⋊Dic59—21no—
127C4⋊Dic59—21no—
128C4⋊Dic59—41no—
129C4⋊Dic59—41no—
130C4×Dic59—41no—
131C4×Dic59—41no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—22no—
134C2×C59⋊C8—22no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C2×C59⋊C8—41no—
138C2×C59⋊C8C22⋊C411yescharacteristic
139C2×C59⋊C8C22⋊C411yescharacteristic
140C2×C59⋊C8C22⋊C411yescharacteristic
141C2×C59⋊C8C22⋊C411yescharacteristic
142C23×C59—41no—
143C23×C59—41no—
144C23×C59C4×D411yescharacteristic
145C23×C59#12111yescharacteristic
146C23×C59#12711yescharacteristic
147C23×C59AGL(1,ℤ/8)12yes—
148C23×C59#13712yes—
149C22×D59—41no—
150C22×D59—42no—
151C22×D59—81no—
152C22×D59—81no—
153Q8×C59—41no—
154Q8×C59—41no—
155Q8×C59—81no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—81no—
168C2×C4×C59—81no—
169C2×C4×C59—81no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59C2×C22⋊C411yescharacteristic
173C2×C4×C59C4×D411yescharacteristic
174C2×C4×C59#12111yescharacteristic
175C2×C4×C59#12411yescharacteristic
176C22⋊D59—82no—
177C22⋊D59—81no—
178C22⋊D59—81no—
179C2×Dic59—42no—
180C2×Dic59—42no—
181C2×Dic59—41no—
182C2×Dic59—42no—
183C2×Dic59—42no—
184C2×Dic59—81no—
185C2×Dic59—81no—
186C2×Dic59—81no—
187D236—41no—
188D236—41no—
189D236—81no—
190C8×C59—42no—
191C59⋊C8—21no—
192C59⋊C8—21no—
193C59⋊C8—21no—
194C59⋊C8#9912yescomplemented
195#9786—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×C59#44012yes—
206C22×C59#46812yes—
207C22×C59#49611yescharacteristic
208C22×C59#49711yescharacteristic
209C2×D59—81no—
210C2×D59—81no—
211C2×D59—81no—
212C2×D59—162no—
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—81no—
217C4×C59—81no—
218C4×C59—161no—
219C4×C59C2×#9911yescharacteristic
220C4×C59#41011yescharacteristic
221Dic59—82no—
222Dic59—81no—
223Dic59—161no—
224C2×#465—591no—
225#1996—591no—
226#1964—591no—
227#1808—591no—
228#1752—591no—
229#1520D5911yescharacteristic, complemented
230#1519—591no—
231C2×C59—22no—
232C2×C59—21no—
233C2×C59—22no—
234C2×C59—21no—
235C2×C59#196411yescharacteristic
236C2×C59#197311yescharacteristic
237C2×C59#197611yescharacteristic
238D59—161no—
239C22×SD16—591no—
240C23×C8—591no—
241C2×C22⋊Q8C2×D5911yescharacteristic
242C2×#121—591no—
243#465—1181no—
244#465—1181no—
245#465—1181no—
246#465—1181no—
247C2×#107—1181no—
248C2×#106—591no—
249C2×#103—591no—
250C2×#102—591no—
251C2×#99—1182no—
252C2×C22⋊C8C2×D5911yescharacteristic
253C2×#97—1181no—
254#380—591no—
255#380C2×D5911yescharacteristic
256C4×C22⋊C4—591no—
257#342—1182no—
258#340—1181no—
259#336—1181no—
260#328—22no—
261#327—1182no—
262C59#978611yescharacteristic, complemented, Sylow
263C22×Q8C4×D5911yescharacteristic
264C22×D4—591no—
265C22×D4—1181no—
266C23×C4C22×D5911yescharacteristic
267C2×SD16—1181no—
268C2×SD16—1181no—
269C2×SD16—2362no—
270C22×C8—21no—
271C22×C8—591no—
272C22×C8—591no—
273C22×C8—591no—
274C22×C8—591no—
275C22×C8—591no—
276C22×C8—591no—
277C22×C8—2362no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
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—1181no—
290C2×C4⋊C4C4×D5911yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—592no—
294C2×C22⋊C4—1181no—
295C2×C22⋊C4—1181no—
296C2×C42—1181no—
297#107—2362no—
298#106—1181no—
299#106—1181no—
300#103—1181no—
301#103—1181no—
302#102—1181no—
303#102—1181no—
304#99—2364no—
305C22⋊C8—21no—
306C22⋊C8C22⋊D5912yes—
307#97—2362no—
308#95—21no—
309#95—1181no—
310#95—1181no—
311C24—1181no—
312C24D4×D5911yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
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—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—1181no—
331C22×C4—1182no—
332C22×C4—2361no—
333C22×C4C2×C4×D5911yescharacteristic
334C22×C4#1174568511yescharacteristic
335C22×C4D4×D5911yescharacteristic
336C22×C4D4×D5911yescharacteristic
337C22×C4#1174568711yescharacteristic
338C22×C4C2×C22⋊D5911yescharacteristic
339SD16—2362no—
340SD16—2362no—
341C2×C8—22no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—591no—
345C2×C8—591no—
346C2×C8—591no—
347C2×C8—591no—
348C2×C8—1182no—
349C2×C8—1182no—
350C2×C8—1182no—
351C2×C8—2361no—
352C2×C8—2361no—
353C2×C8—2361no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—1181no—
361C4⋊C4—1181no—
362C4⋊C4—2361no—
363C4⋊C4—2361no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—1182no—
368C22⋊C4—1182no—
369C22⋊C4—2362no—
370C22⋊C4—2361no—
371C22⋊C4—2361no—
372C42—2361no—
373C42—2361no—
374C23—41no—
375C23—41no—
376C23—4721no—
377C23—4721no—
378C23—2362no—
379C23—2361no—
380C23(C22⋊D59)⋊C411yescharacteristic
381C23#1427370011yescharacteristic
382C23#1427372311yescharacteristic
383C23#1427375412yes—
384C23#1427375912yes—
385Q8—41no—
386Q8—41no—
387Q8—81no—
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—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—4721no—
411C2×C4—4721no—
412C2×C4—4721no—
413C2×C4—2362no—
414C2×C4—2362no—
415C2×C4—2362no—
416C2×C4—2362no—
417C2×C4—2361no—
418C2×C4C22⋊C4×D5911yescharacteristic
419C2×C4C4×C22⋊D5911yescharacteristic
420C2×C4#1427369911yescharacteristic
421C2×C4#1427371311yescharacteristic
422C8—42no—
423C8—592no—
424C8—1181no—
425C8—1181no—
426C8—1181no—
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#1683730211yescharacteristic
440C22#1683750512yes—
441C22#1683765112yes—
442C22#1683778811yescharacteristic
443C22#1683779511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—161no—
450C4—9441no—
451C4—4722no—
452C4—4721no—
453C4#99×D5911yescharacteristic
454C4#1683737411yescharacteristic
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—9441no—
460C2#2142493011yescharacteristic
461C2#2142497411yescharacteristic
462C2#2142499311yescharacteristic
subgroupG/H
centerC22#16837302
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1520C2
socleC22×C59#390
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1520D59
hypocenterC59#9786
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
2. #21425082C2semidirect
4. #21424123C2semidirect
5. #21423875C2semidirect
7. C59×#1520C2semidirect
8. C22×#11745671C4semidirect
16. C2×#14273658C4semidirect
17. C2×#14273651C4semidirect
18. C2×#14273647C4semidirect
194. C59⋊C8#99semidirect
229. #1520D59semidirect
262. C59#9786semidirect

Maximal subgroups

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

subgroupindex
C2×#168376332
#214250822
#214249312
#214241232
#214238752
#214226652
C59×#15202
#978659