Finite Group Databasebeta

← #28484729

Subgroups — #28484729

Order 15104.

684 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#21429903C211yesmaximal, characteristic
2#21426709C211yesmaximal, characteristic
3#21422668C211yesmaximal, characteristic, complemented
4C2×#16837125C211yesmaximal, characteristic
5#21421964C211yesmaximal, characteristic, complemented
6#21421963C211yesmaximal, characteristic, complemented
7C59×#1520C211yesmaximal, characteristic, complemented
8C22×(C2×D59)⋊C4C2211yescharacteristic
9C2×C22⋊C4×D59—21no—
10C2×C22⋊(C59⋊C8)C2211yescharacteristic
11#16837337C412yescomplemented
12#16837324—22no—
13#16837313C412yescomplemented
14#16837276—22no—
15#16837274—21no—
16#16837273—22no—
17#16837262C2211yescharacteristic
18#16837262C2211yescharacteristic
19#16837260—21no—
20#16837257—21no—
21#16837139—22no—
22#16837125—22no—
23C2×C22⋊Q8×C59C2211yescharacteristic
24C2×C22⋊C8×C59C2211yescharacteristic
25C59×#380C2211yescharacteristic
26C59×#328—22no—
27C24×D59D411yescharacteristic
28C23×Dic59D411yescharacteristic
29C22×C4×D59—22no—
30C22×C59⋊C8—21no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59—21no—
33C2×Dic59⋊C4—22no—
34C2×Dic59⋊C4—22no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—22no—
42C2×(C2×D59)⋊C4—22no—
43C2×(C2×D59)⋊C4—22no—
44C2×(C2×D59)⋊C4C2×C412yes—
45C2×(C2×D59)⋊C4D412yes—
46C22⋊C4×D59—42no—
47C22⋊C4×D59—42no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C59⋊#95—22no—
51C59⋊#95—22no—
52C59⋊#95C2×C412yescomplemented
53C59⋊#95D412yes—
54C22×Q8×C59C2×C411yescharacteristic, complemented
55C23×C4×C59C2311yescharacteristic
56C22×C8×C59—21no—
57C22⋊Q8×C59—21no—
58C22⋊Q8×C59—21no—
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59C2×C411yescharacteristic, complemented
64C2×C22⋊C4×C59—21no—
65C2×C22⋊C4×C59—21no—
66C22⋊C8×C59—21no—
67C22⋊C8×C59—21no—
68#95×C59D412yes—
69C24×C59C2×D411yescharacteristic
70C23×D59—21no—
71C23×D59—21no—
72C23×D59—21no—
73C23×D59—21no—
74C23×D59—81no—
75C23×D59—81no—
76C23×D59C22⋊C412yes—
77C23×D59SD1612yes—
78C23×D59Q1612yes—
79C2×Q8×C59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—81no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—22no—
85C22×C4×C59—81no—
86C22×C4×C59C22×C411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59#4111yescharacteristic
91C22×C4×C59#4111yescharacteristic
92C22×Dic59—21no—
93C22×Dic59—21no—
94C22×Dic59—22no—
95C22×Dic59—22no—
96C22×Dic59—81no—
97C22×Dic59C22⋊C412yes—
98C2×C4×D59—44no—
99C2×C4×D59—82no—
100C2×C4×D59—82no—
101C2×C8×C59—41no—
102C2×C8×C59—81no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—81no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C22⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C22⋊Dic59—41no—
113C22⋊Dic59—41no—
114C22⋊Dic59—41no—
115C22⋊Dic59—41no—
116(C2×D59)⋊C4—22no—
117(C2×D59)⋊C4—22no—
118(C2×D59)⋊C4—22no—
119(C2×D59)⋊C4—22no—
120(C2×D59)⋊C4—44no—
121(C2×D59)⋊C4—82no—
122(C2×D59)⋊C4—82no—
123C4⋊Dic59—81no—
124C4⋊Dic59—81no—
125Dic59⋊C4—82no—
126Dic59⋊C4—82no—
127C2×C59⋊C8—41no—
128C2×C59⋊C8—81no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59C2×C22⋊C411yescharacteristic
132C23×C59#12011yescharacteristic
133C23×C59#12311yescharacteristic
134C23×C59C2×SD1611yescharacteristic
135C23×C59C2×Q1611yescharacteristic
136C23×C59AGL(1,ℤ/8)11yescharacteristic
137C23×C59#13711yescharacteristic
138C22×D59—22no—
139C22×D59—22no—
140C22×D59—22no—
141C22×D59—22no—
142C22×D59—22no—
143C22×D59—22no—
144C22×D59—42no—
145C22×D59—42no—
146C22×D59—42no—
147C22×D59—42no—
148C22×D59—82no—
149C22×D59—82no—
150C22×D59—81no—
151C22×D59—81no—
152C22×D59—82no—
153C22×D59—82no—
154C22×D59#10314yes—
155Q8×C59—81no—
156Q8×C59—81no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—42no—
164C2×C4×C59—42no—
165C2×C4×C59—42no—
166C2×C4×C59—42no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—42no—
170C2×C4×C59C4×D411yescharacteristic
171C2×C4×C59C4×D411yescharacteristic
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59#12111yescharacteristic
174C2×Dic59—22no—
175C2×Dic59—42no—
176C2×Dic59—42no—
177C2×Dic59—42no—
178C2×Dic59—42no—
179C2×Dic59—42no—
180C2×Dic59—42no—
181C2×Dic59—42no—
182C2×Dic59—42no—
183C2×Dic59—81no—
184C4×D59—88no—
185C8×C59—81no—
186C59⋊C8—81no—
187#9199—591nomaximal, Sylow
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—41no—
195C22×C59—41no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59#38611yescharacteristic
199C22×C59C2×#10311yescharacteristic
200C22×C59#41711yescharacteristic
201C22×C59#45011yescharacteristic
202C22×C59#45111yescharacteristic
203C22×C59#48311yescharacteristic
204C22×C59#48411yescharacteristic
205C2×D59—24no—
206C2×D59—44no—
207C2×D59—44no—
208C2×D59—42no—
209C2×D59—42no—
210C2×D59—44no—
211C2×D59—82no—
212C2×D59—82no—
213C2×D59—82no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—21no—
217C4×C59—21no—
218C4×C59—82no—
219C4×C59—81no—
220C4×C59—81no—
221Dic59—42no—
222Dic59—82no—
223Dic59—82no—
224#2626—591no—
225#2315—591no—
226#1520—591no—
227#1520D5911yescharacteristic, complemented
228C2×#323—591no—
229#1289—591no—
230#1289—591no—
231C2×C59—21no—
232C2×C59—21no—
233C2×C59—21no—
234C2×C59—21no—
235C2×C59—21no—
236C2×C59—21no—
237C2×C59#187211yescharacteristic
238C2×C59#191111yescharacteristic
239C2×C59#191311yescharacteristic
240D59—44no—
241C2×C22⋊Q8C2×D5911yescharacteristic
242C22×C22⋊C4—591no—
243C22×C22⋊C4—1181no—
244C2×C22⋊C8—591no—
245C2×C22⋊C8C2×D5911yescharacteristic
246#399—592no—
247#396—1182no—
248#393—592no—
249#385—1182no—
250#385—1181no—
251#385—1182no—
252#380—591no—
253#380—591no—
254#380—1181no—
255#380C2×D5911yescharacteristic
256#379—1181no—
257#328—22no—
258#328—1182no—
259#323—1182no—
260C59#919911yescharacteristic, complemented, Sylow
261C25—591no—
262C22×Q8C4×D5911yescharacteristic, complemented
263C23×C4—591no—
264C23×C4—1182no—
265C23×C4C22×D5911yescharacteristic
266C22×C8—21no—
267C22×C8—1181no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—1182no—
275C2×C4⋊C4—1182no—
276C2×C4⋊C4—1181no—
277C2×C4⋊C4—1181no—
278C2×C4⋊C4C4×D5911yescharacteristic, complemented
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—592no—
282C2×C22⋊C4—592no—
283C2×C22⋊C4—1181no—
284C2×C22⋊C4—1181no—
285C2×C22⋊C4—1181no—
286C2×C22⋊C4—1181no—
287C2×C22⋊C4—1182no—
288C2×C22⋊C4—1182no—
289C2×C22⋊C4—1182no—
290C2×C22⋊C4—1181no—
291C2×C22⋊C4—1181no—
292C2×C22⋊C4—2362no—
293C2×C22⋊C4—2362no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—1181no—
297C22⋊C8—1181no—
298#95—592no—
299#95—592no—
300#95—1182no—
301#95—1182no—
302#95C22⋊D5912yes—
303C24—592no—
304C24—592no—
305C24—592no—
306C24—1181no—
307C24—1181no—
308C24—1181no—
309C24—1181no—
310C24—4721no—
311C24—4721no—
312C24D4×D5911yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—81no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—81no—
320C22×C4—592no—
321C22×C4—1181no—
322C22×C4—1181no—
323C22×C4—1182no—
324C22×C4—1182no—
325C22×C4—4722no—
326C22×C4—4722no—
327C22×C4—4721no—
328C22×C4—2364no—
329C22×C4C2×C4×D5911yescharacteristic
330C22×C4#1174568511yescharacteristic
331C22×C4D4×D5911yescharacteristic
332C22×C4D4×D5911yescharacteristic
333C22×C4#1174568711yescharacteristic
334C22×C4C2×C22⋊D5911yescharacteristic
335C2×C8—41no—
336C2×C8—81no—
337C2×C8—4721no—
338C2×C8—2361no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—81no—
344C4⋊C4—4722no—
345C4⋊C4—4722no—
346C4⋊C4—4721no—
347C4⋊C4—4721no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—1182no—
353C22⋊C4—1182no—
354C22⋊C4—1182no—
355C22⋊C4—1182no—
356C22⋊C4—4722no—
357C22⋊C4—4722no—
358C22⋊C4—2364no—
359C22⋊C4—2361no—
360C22⋊C4—2361no—
361C22⋊C4—2361no—
362C22⋊C4—2361no—
363C23—41no—
364C23—41no—
365C23—594no—
366C23—1182no—
367C23—1182no—
368C23—1182no—
369C23—1182no—
370C23—1182no—
371C23—1182no—
372C23—4721no—
373C23—4721no—
374C23—4722no—
375C23—4722no—
376C23—4722no—
377C23—4722no—
378C23—2362no—
379C23—2362no—
380C23—2362no—
381C23—2362no—
382C23C22⋊C4×D5911yescharacteristic
383C23#1427369511yescharacteristic
384C23#1427370911yescharacteristic
385C23SD16×D5911yescharacteristic
386C23Q16×D5911yescharacteristic
387C23#1427375411yescharacteristic
388C23#1427375911yescharacteristic
389Q8—81no—
390Q8—81no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—1182no—
405C2×C4—4728no—
406C2×C4—4721no—
407C2×C4—2362no—
408C2×C4—2362no—
409C2×C4—2362no—
410C2×C4—2362no—
411C2×C4—2362no—
412C2×C4—2362no—
413C2×C4—2362no—
414C2×C4—2362no—
415C2×C4(C22⋊D59)⋊C411yescharacteristic
416C2×C4C4×C22⋊D5911yescharacteristic
417C2×C4#1427369811yescharacteristic
418C2×C4#1427370011yescharacteristic
419C8—81no—
420C8—4721no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—1184no—
432C22—4722no—
433C22—4722no—
434C22—4722no—
435C22—2364no—
436C22—2364no—
437C22—2364no—
438C22—2362no—
439C22—2362no—
440C22#1683727811yescharacteristic
441C22#103×D5911yescharacteristic
442C22#1683740211yescharacteristic
443C22#1683755111yescharacteristic
444C22#1683755611yescharacteristic
445C22#1683773111yescharacteristic
446C22#1683773611yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4—4722no—
455C4—4722no—
456C4—2362no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—2364no—
464C2#2142440011yescharacteristic
465C2#2142462511yescharacteristic
466C2#2142463511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1520C2
socleC22×C59#386
prime radicalC24×D59D4
cosocleC23×C4×C59C23
hypercenter#1520D59
hypocenterC59#9199
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
3. #21422668C2semidirect
5. #21421964C2semidirect
6. #21421963C2semidirect
7. C59×#1520C2semidirect
11. #16837337C4semidirect
13. #16837313C4semidirect
52. C59⋊#95C2×C4semidirect
54. C22×Q8×C59C2×C4semidirect
63. C2×C4⋊C4×C59C2×C4semidirect
227. #1520D59semidirect
260. C59#9199semidirect
262. C22×Q8C4×D59semidirect
278. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
#214299032
#214267092
#214226682
C2×#168371252
#214219642
#214219632
C59×#15202
#919959