Finite Group Databasebeta

← #28482248

Subgroups — #28482248

Order 15104.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#21427038C211yesmaximal, characteristic
2#21424081C211yesmaximal, characteristic
3#21422412C211yesmaximal, characteristic
4#21421976C211yesmaximal, characteristic, complemented
5#21421974C211yesmaximal, characteristic, complemented
6#21421963C211yesmaximal, characteristic, complemented
7C59×#1289C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D59C2211yescharacteristic
9C2×#14273641—21no—
10C2×C22⋊(C59⋊C8)C2211yescharacteristic
11#16837292—21no—
12#16837283—21no—
13#16837279—21no—
14#16837269C411yescharacteristic, complemented
15#16837264C411yescharacteristic, complemented
16#16837262C2211yescharacteristic
17C22⋊C4×Dic59—21no—
18#16837249C411yescharacteristic, complemented
19#16837247C411yescharacteristic, complemented
20C2×C59⋊#95C2211yescharacteristic
21#16837168—21no—
22#16837127—21no—
23#16837125—21no—
24C2×C22⋊C8×C59C2211yescharacteristic
25C59×#380C2211yescharacteristic
26C59×#380C2211yescharacteristic
27C22×C22⋊D59D411yescharacteristic
28C23×Dic59—21no—
29C22×C4×D59C2×C411yescharacteristic
30C22×C59⋊C8—21no—
31C4×C22⋊D59—21no—
32C4×C22⋊D59—21no—
33C4×C22⋊D59—21no—
34C4×C22⋊D59—21no—
35C2×C4⋊Dic59—21no—
36C2×Dic59⋊C4D411yescharacteristic
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59D411yescharacteristic
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4D411yescharacteristic
42C2×C4×Dic59—21no—
43C2×C4×Dic59—21no—
44C2×C4×Dic59C2×C411yescharacteristic
45#14273641—42no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C59⋊#95C2×C411yescharacteristic
55C59⋊#95C2×C411yescharacteristic
56C23×C4×C59C2311yescharacteristic
57C22×C8×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C22⋊C4×C59—21no—
61C2×C22⋊C4×C59—21no—
62C22⋊C8×C59—21no—
63C22⋊C8×C59—21no—
64#95×C59—21no—
65#95×C59C2×C411yescharacteristic
66#95×C59C2×C411yescharacteristic
67C24×C59C2×D411yescharacteristic
68C23×D59C22⋊C411yescharacteristic
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—41no—
72C22×C4×C59—81no—
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59#4111yescharacteristic
76C22×C4×C59#4111yescharacteristic
77C22×C4×C59#4111yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C2×C22⋊D59—21no—
80C2×C22⋊D59—21no—
81C2×C22⋊D59—21no—
82C2×C22⋊D59—21no—
83C2×C22⋊D59—21no—
84C2×C22⋊D59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—42no—
88C22×Dic59—41no—
89C22×Dic59—41no—
90C22×Dic59—41no—
91C22×Dic59—81no—
92C22×Dic59C22⋊C411yescharacteristic
93C22×Dic59C22⋊C411yescharacteristic
94C22×Dic59C22⋊C411yescharacteristic
95C2×C4×D59—21no—
96C2×C4×D59—21no—
97C2×C4×D59—81no—
98C2×C8×C59—41no—
99C2×C8×C59—81no—
100C4⋊C4×C59—81no—
101C4⋊C4×C59—81no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊Dic59—21no—
107C22⋊Dic59—21no—
108C22⋊Dic59—41no—
109C22⋊Dic59—41no—
110(C2×D59)⋊C4—21no—
111(C2×D59)⋊C4—21no—
112(C2×D59)⋊C4—42no—
113(C2×D59)⋊C4—81no—
114C4⋊Dic59—81no—
115Dic59⋊C4—21no—
116Dic59⋊C4—21no—
117C4×Dic59—21no—
118C4×Dic59—21no—
119C4×Dic59—42no—
120C4×Dic59—81no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—81no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59C2×C22⋊C411yescharacteristic
126C23×C59#12311yescharacteristic
127C23×C59#12311yescharacteristic
128C23×C59#13511yescharacteristic
129C23×C59#13511yescharacteristic
130C23×C59AGL(1,ℤ/8)11yescharacteristic
131C23×C59#13711yescharacteristic
132C22×D59—21no—
133C22×D59—21no—
134C22×D59—21no—
135C22×D59—41no—
136C22×D59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—81no—
150C2×C4×C59—81no—
151C2×C4×C59—81no—
152C2×C4×C59#11711yescharacteristic
153C2×C4×C59#11711yescharacteristic
154C2×C4×C59#12311yescharacteristic
155C2×C4×C59#12311yescharacteristic
156C22⋊D59—41no—
157C22⋊D59—41no—
158C22⋊D59—41no—
159C22⋊D59—41no—
160C2×Dic59—22no—
161C2×Dic59—21no—
162C2×Dic59—21no—
163C2×Dic59—21no—
164C2×Dic59—21no—
165C2×Dic59—21no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—82no—
174C2×Dic59—82no—
175C2×Dic59—81no—
176C2×Dic59—81no—
177C2×Dic59—81no—
178C4×D59—81no—
179C4×D59—81no—
180C8×C59—81no—
181C59⋊C8—81no—
182#8763—591nomaximal, Sylow
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59#38111yescharacteristic
194C22×C59#41611yescharacteristic
195C22×C59#41711yescharacteristic
196C22×C59#48211yescharacteristic
197C22×C59#48211yescharacteristic
198C22×C59#48511yescharacteristic
199C22×C59#48511yescharacteristic
200C2×D59—41no—
201C2×D59—41no—
202C2×D59—41no—
203C2×D59—81no—
204C2×D59—81no—
205C4×C59—21no—
206C4×C59—21no—
207C4×C59—21no—
208C4×C59—21no—
209C4×C59—81no—
210C4×C59—81no—
211C4×C59—161no—
212Dic59—42no—
213Dic59—41no—
214Dic59—41no—
215Dic59—41no—
216Dic59—81no—
217Dic59—161no—
218#2353—591no—
219#1799—591no—
220#1459—591no—
221#1294—591no—
222#1294—591no—
223#1289—591no—
224#1289D5911yescharacteristic, complemented
225C2×C59—21no—
226C2×C59—21no—
227C2×C59—21no—
228C2×C59—21no—
229C2×C59—21no—
230C2×C59—21no—
231C2×C59#182511yescharacteristic
232C2×C59#182711yescharacteristic
233C2×C59#182811yescharacteristic
234D59—81no—
235C2×C4×D4—591no—
236C2×#99—1181no—
237C2×C22⋊C8—591no—
238C2×C22⋊C8C2×D5911yescharacteristic
239#388—1181no—
240#387—1181no—
241#386—1181no—
242#383—591no—
243#381—591no—
244#380—591no—
245#380C2×D5911yescharacteristic
246#380C2×D5911yescharacteristic
247C4×C22⋊C4—591no—
248C4×C22⋊C4—1181no—
249#376—591no—
250C2×#95—591no—
251#342—1181no—
252#324—1181no—
253#323—1181no—
254C59#876311yescharacteristic, complemented, Sylow
255C22×D4—591no—
256C23×C4—591no—
257C23×C4—1181no—
258C23×C4C22×D5911yescharacteristic
259C22×C8—21no—
260C22×C8—1181no—
261C4×D4—1181no—
262C4×D4—1181no—
263C4×D4—1181no—
264C4×D4—1181no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—591no—
268C2×C4⋊C4—1181no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—591no—
272C2×C22⋊C4—591no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C22⋊C4—1181no—
276C2×C42—591no—
277C2×C42—1181no—
278C2×C42—1181no—
279#99—2362no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—1181no—
283C22⋊C8—1181no—
284#95—21no—
285#95—591no—
286#95—591no—
287#95—1181no—
288#95—1181no—
289#95—1181no—
290#95—1181no—
291#95—1181no—
292#95—1181no—
293#95C4×D5911yescharacteristic
294#95C4×D5911yescharacteristic
295C24—591no—
296C24D4×D5911yescharacteristic
297C2×D4—1181no—
298C2×D4—1181no—
299C2×D4—1181no—
300C2×D4—1181no—
301C2×D4—1181no—
302C2×D4—1181no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—81no—
307C22×C4—591no—
308C22×C4—591no—
309C22×C4—591no—
310C22×C4—1181no—
311C22×C4—1181no—
312C22×C4—1181no—
313C22×C4—1181no—
314C22×C4—4721no—
315C22×C4—4721no—
316C22×C4—2362no—
317C22×C4—2361no—
318C22×C4—2361no—
319C22×C4—2361no—
320C22×C4C2×C4×D5911yescharacteristic
321C22×C4#1174568511yescharacteristic
322C22×C4#1174568511yescharacteristic
323C22×C4D4×D5911yescharacteristic
324C22×C4#1174568711yescharacteristic
325C22×C4#1174568911yescharacteristic
326C2×C8—41no—
327C2×C8—81no—
328C2×C8—4721no—
329C2×C8—2361no—
330C4⋊C4—81no—
331C4⋊C4—81no—
332C4⋊C4—1181no—
333C4⋊C4—1181no—
334C4⋊C4—4721no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—1181no—
340C22⋊C4—1181no—
341C22⋊C4—1181no—
342C22⋊C4—1181no—
343C22⋊C4—4721no—
344C22⋊C4—2362no—
345C22⋊C4—2361no—
346C22⋊C4—2361no—
347C42—1181no—
348C42—1181no—
349C42—4721no—
350C42—2362no—
351C23—41no—
352C23—41no—
353C23—1181no—
354C23—1181no—
355C23—1181no—
356C23—2361no—
357C23—2361no—
358C23C22⋊C4×D5911yescharacteristic
359C23#1427370911yescharacteristic
360C23#1427371211yescharacteristic
361C23#1427374811yescharacteristic
362C23#1427375011yescharacteristic
363C23#1427375111yescharacteristic
364C23#1427375611yescharacteristic
365D4—2361no—
366D4—2361no—
367D4—2361no—
368D4—2361no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—1182no—
385C2×C4—1181no—
386C2×C4—1181no—
387C2×C4—1181no—
388C2×C4—1181no—
389C2×C4—1181no—
390C2×C4—4721no—
391C2×C4—4721no—
392C2×C4—4721no—
393C2×C4—4721no—
394C2×C4—4721no—
395C2×C4—4722no—
396C2×C4—4722no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4—2361no—
402C2×C4—2361no—
403C2×C4—2361no—
404C2×C4#1427368411yescharacteristic
405C2×C4#1427368511yescharacteristic
406C2×C4#1427370911yescharacteristic
407C2×C4#1427371211yescharacteristic
408C8—81no—
409C8—4721no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—4721no—
421C22—4721no—
422C22—2361no—
423C22—2361no—
424C22—2361no—
425C22#1683726411yescharacteristic
426C22#1683739711yescharacteristic
427C22#1683740011yescharacteristic
428C22#1683772511yescharacteristic
429C22#1683772911yescharacteristic
430C22#1683774211yescharacteristic
431C22#1683774611yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—81no—
437C4—81no—
438C4—161no—
439C4—9441no—
440C4—4721no—
441C4—2362no—
442C4—2361no—
443C4—2361no—
444C4—2361no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—4721no—
452C2#2142419011yescharacteristic
453C2#2142419811yescharacteristic
454C2#2142420311yescharacteristic
subgroupG/H
centerC22#16837264
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1289C2
socleC22×C59#381
prime radicalC22×C22⋊D59D4
cosocleC23×C4×C59C23
hypercenter#1289D59
hypocenterC59#8763
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
4. #21421976C2semidirect
5. #21421974C2semidirect
6. #21421963C2semidirect
7. C59×#1289C2semidirect
14. #16837269C4semidirect
15. #16837264C4semidirect
18. #16837249C4semidirect
19. #16837247C4semidirect
224. #1289D59semidirect
254. C59#8763semidirect

Maximal subgroups

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

subgroupindex
#214270382
#214240812
#214224122
#214219762
#214219742
#214219632
C59×#12892
#876359