Finite Group Databasebeta

← #28499547

Subgroups — #28499547

Order 15104.

698 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 91 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#21424170C211yesmaximal, characteristic, complemented
2C2×#16837126C211yesmaximal, characteristic, complemented
3#21422117C212yesmaximal, complemented
4#21422093C212yesmaximal, complemented
5C59×#2404C211yesmaximal, characteristic
6C2×C22⋊(C59⋊C8)—21no—
7#16837258—21no—
8#16837169—21no—
9#16837169C2212yes—
10#16837126—21no—
11#16837126C2212yes—
12C22×#41×C59C411yescharacteristic, complemented
13C2×#120×C59—21no—
14C22×C22⋊C4×C59C2211yescharacteristic
15C59×#393C411yescharacteristic, complemented
16C59×#393C2211yescharacteristic
17C59×#392C411yescharacteristic, complemented
18C59×#392C2211yescharacteristic
19C59×#386—21no—
20C59×#386—21no—
21C59×#386—21no—
22C59×#381C411yescharacteristic, complemented
23C22×C59⋊C8—41no—
24C2×C22⋊Dic59—22no—
25C2×C22⋊Dic59—41no—
26C2×C22⋊Dic59—41no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—41no—
29C25×C59D411yescharacteristic
30C2×#41×C59—41no—
31C2×#41×C59—41no—
32C2×#41×C59—44no—
33C22×Q8×C59C2×C411yescharacteristic
34C22×D4×C59—21no—
35C22×D4×C59—21no—
36C22×D4×C59—21no—
37C22×D4×C59C2×C411yescharacteristic
38C23×C4×C59—21no—
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41#120×C59—44no—
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—41no—
48C2×C22⋊C4×C59—41no—
49C2×C22⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59C2×C411yescharacteristic
57#95×C59C2×C411yescharacteristic
58C24×C59—21no—
59C24×C59—21no—
60C24×C59—21no—
61C24×C59—22no—
62C24×C59—41no—
63C24×C59—42no—
64C24×C59—81no—
65C24×C59C22⋊C411yescharacteristic
66C24×C59C22⋊C411yescharacteristic
67C24×C59C2×D411yescharacteristic
68#41×C59—44no—
69#41×C59—44no—
70#41×C59—44no—
71#41×C59—44no—
72C2×Q8×C59—42no—
73C2×Q8×C59—41no—
74C2×D4×C59—24no—
75C2×D4×C59—41no—
76C2×D4×C59—42no—
77C2×D4×C59—41no—
78C2×D4×C59—41no—
79C2×D4×C59—42no—
80C2×D4×C59—42no—
81C2×D4×C59—42no—
82C2×D4×C59—42no—
83C2×D4×C59—42no—
84C2×D4×C59—81no—
85C2×D4×C59—81no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—21no—
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—42no—
94C22×C4×C59—42no—
95C22×C4×C59—42no—
96C22×C4×C59—42no—
97C22×C4×C59—81no—
98C22×C4×C59C22⋊C411yescharacteristic
99C22×C4×C59C22⋊C411yescharacteristic
100C22×C4×C59C22×C411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×Dic59—42no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—42no—
109C22⋊C4×C59—42no—
110C22⋊C4×C59—81no—
111C22⋊C4×C59—81no—
112C22⋊Dic59—42no—
113C22⋊Dic59—42no—
114C22⋊Dic59—82no—
115C22⋊Dic59—82no—
116C2×C59⋊C8—42no—
117C2×C59⋊C8—82no—
118C23×C59—22no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—22no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—41no—
128C23×C59—42no—
129C23×C59—42no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59—42no—
135C23×C59—41no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59—42no—
139C23×C59—41no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—81no—
143C23×C59—81no—
144C23×C59—84no—
145C23×C59—84no—
146C23×C59#9911yescharacteristic
147C23×C59#9911yescharacteristic
148C23×C59C2×C22⋊C411yescharacteristic
149C23×C59#13512yes—
150Q8×C59—42no—
151Q8×C59—42no—
152D4×C59—44no—
153D4×C59—42no—
154D4×C59—42no—
155D4×C59—44no—
156D4×C59—84no—
157D4×C59—84no—
158C2×C4×C59—22no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—42no—
164C2×C4×C59—41no—
165C2×C4×C59—42no—
166C2×C4×C59—48no—
167C2×C4×C59—42no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—41no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—81no—
180C2×Dic59—82no—
181C2×Dic59—82no—
182C2×Dic59—82no—
183C59⋊C8—82no—
184#11839—591nomaximal, Sylow
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—22no—
192C22×C59—21no—
193C22×C59—42no—
194C22×C59—42no—
195C22×C59—42no—
196C22×C59—42no—
197C22×C59—42no—
198C22×C59—42no—
199C22×C59—41no—
200C22×C59—41no—
201C22×C59—42no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—41no—
205C22×C59—81no—
206C22×C59—84no—
207C22×C59—81no—
208C22×C59—81no—
209C22×C59—81no—
210C22×C59—84no—
211C22×C59—81no—
212C22×C59—81no—
213C22×C59AΣL(1,𝔽16)12yes—
214C22×C59C2×#9911yescharacteristic
215C22×C59#41611yescharacteristic
216C22×C59#42111yescharacteristic
217C4×C59—41no—
218C4×C59—42no—
219C4×C59—41no—
220C4×C59—41no—
221C4×C59—41no—
222C4×C59—81no—
223C4×C59—81no—
224Dic59—162no—
225#2404D5911yescharacteristic
226#1819—591no—
227C2×#323—591no—
228#1351—592no—
229#1343—592no—
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—22no—
233C2×C59—42no—
234C2×C59—41no—
235C2×C59—81no—
236C2×C59—82no—
237C2×C59#151611yescharacteristic
238C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
239C2×C59#212711yescharacteristic
240C22×#41Dic5911yescharacteristic, complemented
241C2×#120—21no—
242C22×C22⋊C4C2×D5911yescharacteristic
243C2×C22⋊C8—1181no—
244#393Dic5911yescharacteristic, complemented
245#393C2×D5911yescharacteristic
246#392Dic5911yescharacteristic, complemented
247#392C2×D5911yescharacteristic
248#386—21no—
249#386—21no—
250#386—21no—
251#381Dic5911yescharacteristic, complemented
252#379—1181no—
253#342—592no—
254#342—1181no—
255#323—592no—
256#323—1181no—
257C59#1183911yescharacteristic, complemented, Sylow
258C25C22⋊D5911yescharacteristic
259C2×#41—41no—
260C2×#41—41no—
261C2×#41—44no—
262C22×Q8C2×Dic5911yescharacteristic
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4C2×Dic5911yescharacteristic
267C23×C4—21no—
268C23×C4C2×Dic5911yescharacteristic
269C23×C4C22⋊D5911yescharacteristic
270C22×C8—2361no—
271#120—44no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—1182no—
280C2×C22⋊C4—2361no—
281C2×C22⋊C4—2361no—
282C2×C22⋊C4C2×Dic5911yescharacteristic
283C2×C22⋊C4C22⋊D5911yescharacteristic
284C2×C22⋊C4C22⋊D5911yescharacteristic
285C2×C22⋊C4C22×D5911yescharacteristic
286C22⋊C8—1182no—
287C22⋊C8—2361no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95C2×Dic5911yescharacteristic
292#95C2×Dic5911yescharacteristic
293C24—21no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—41no—
298C24—42no—
299C24—81no—
300C24C22⋊Dic5911yescharacteristic
301C24C22⋊Dic5911yescharacteristic
302C24C2×C22⋊D5911yescharacteristic
303#41—44no—
304#41—44no—
305#41—44no—
306#41—44no—
307C2×Q8—42no—
308C2×Q8—41no—
309C2×D4—24no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—81no—
320C2×D4—81no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—81no—
333C22×C4—2362no—
334C22×C4C22⋊Dic5911yescharacteristic
335C22×C4C22⋊Dic5911yescharacteristic
336C22×C4C22×Dic5911yescharacteristic
337C22×C4C2×C22⋊D5911yescharacteristic
338C2×C8—4722no—
339C2×C8—2362no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—4722no—
350C22⋊C4—4722no—
351C22⋊C4—2362no—
352C22⋊C4—2362no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—41no—
377C23—81no—
378C23—81no—
379C23—84no—
380C23—84no—
381C23#1427364211yescharacteristic
382C23#1427364211yescharacteristic
383C23C2×C22⋊Dic5911yescharacteristic
384C23#1427374712yes—
385Q8—42no—
386Q8—42no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—44no—
391D4—84no—
392D4—84no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—4722no—
416C2×C4—4722no—
417C2×C4—4722no—
418C8—4722no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—81no—
440C22—84no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—84no—
445C22—81no—
446C22—81no—
447C22#1683718912yes—
448C22C2×#1427364211yescharacteristic
449C22#1683739811yescharacteristic
450C22#1683741911yescharacteristic
451C4—41no—
452C4—42no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—81no—
457C4—81no—
458C4—9442no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—42no—
463C2—41no—
464C2—81no—
465C2—82no—
466C2#2142265211yescharacteristic
467C2C2×#1683718911yescharacteristic
468C2#2142576511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2404C2
socleC22×C59C2×#99
cosocleC2×C22⋊C4×C59C23
hypercenter#2404D59
hypocenterC59#11839
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. #21424170C2semidirect
2. C2×#16837126C2semidirect
3. #21422117C2semidirect
4. #21422093C2semidirect
12. C22×#41×C59C4semidirect
15. C59×#393C4semidirect
17. C59×#392C4semidirect
22. C59×#381C4semidirect
240. C22×#41Dic59semidirect
244. #393Dic59semidirect
246. #392Dic59semidirect
251. #381Dic59semidirect
257. C59#11839semidirect

Maximal subgroups

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

subgroupindex
#214241702
C2×#168371262
#214221172
#214220932
C59×#24042
#1183959