Finite Group Databasebeta

← #28484750

Subgroups — #28484750

Order 15104.

762 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 127 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#21429830C211yesmaximal, characteristic
2#21426710C211yesmaximal, characteristic, complemented
3#21422665C212yesmaximal, complemented
4#21421964C212yesmaximal, complemented
5C2×C59×#323C211yesmaximal, characteristic, complemented
6C2×#14273701C2211yescharacteristic
7C2×#14273696—21no—
8C22×C22⋊Dic59—21no—
9#16837327C412yescomplemented
10#16837318—22no—
11#16837311C412yescomplemented
12#16837281—21no—
13#16837277—24no—
14#16837262—21no—
15#16837262C2211yescharacteristic
16#16837262C2212yes—
17#16837137—24no—
18C22×C22⋊C4×C59C2211yescharacteristic
19C2×C22⋊C8×C59C2212yes—
20C59×#323—22no—
21C22×D236C2×C411yescharacteristic, complemented
22C22×C22⋊D59—21no—
23C22×C22⋊D59—22no—
24C23×Dic59—22no—
25#14273701—22no—
26#14273701—22no—
27#14273696—42no—
28#14273696—42no—
29C2×C4⋊Dic59—22no—
30C2×C4⋊Dic59—22no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59C2×C411yescharacteristic, complemented
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—22no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—22no—
37C2×C22⋊Dic59—42no—
38C2×C22⋊Dic59—42no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—22no—
41C59⋊#95—22no—
42C59⋊#95—22no—
43C59⋊#95C2×C412yescomplemented
44C59⋊#95D414yescomplemented
45C25×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C23×C4×C59C2311yescharacteristic
48C22×C8×C59—22no—
49C2×C22⋊C4×C59—22no—
50C2×C22⋊C4×C59—22no—
51C2×C22⋊C4×C59C2×C412yes—
52C2×C22⋊C4×C59D412yes—
53C22⋊C8×C59—22no—
54C22⋊C8×C59—22no—
55C24×C59—21no—
56C24×C59—21no—
57C24×C59—21no—
58C24×C59—21no—
59C24×C59—81no—
60C24×C59—81no—
61C24×C59C22⋊C412yes—
62C24×C59D812yes—
63C24×C59SD1612yes—
64C24×C59C2×D411yescharacteristic
65C23×D59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—81no—
69C22×C4×C59—81no—
70C22×C4×C59C22⋊C412yes—
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D412yes—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59#4112yes—
75C2×C22⋊D59—44no—
76C2×C22⋊D59—41no—
77C2×C22⋊D59—41no—
78C2×C22⋊D59—41no—
79C2×C22⋊D59—41no—
80C2×C22⋊D59—41no—
81C2×C22⋊D59—41no—
82C2×C22⋊D59—82no—
83C2×C22⋊D59—82no—
84C22×Dic59—21no—
85C22×Dic59—22no—
86C22×Dic59—24no—
87C22×Dic59—44no—
88C22×Dic59—82no—
89C22×Dic59—82no—
90C2×D236—22no—
91C2×D236—81no—
92C2×C8×C59—42no—
93C2×C8×C59—82no—
94C22⋊C4×C59—24no—
95C22⋊C4×C59—24no—
96C22⋊Dic59—44no—
97C22⋊Dic59—44no—
98C22⋊Dic59—42no—
99C22⋊Dic59—41no—
100C22⋊Dic59—41no—
101C22⋊Dic59—42no—
102(C2×D59)⋊C4—42no—
103(C2×D59)⋊C4—82no—
104C4⋊Dic59—22no—
105C4⋊Dic59—82no—
106C4⋊Dic59—82no—
107C4⋊Dic59—81no—
108C23×C59—22no—
109C23×C59—22no—
110C23×C59—22no—
111C23×C59—22no—
112C23×C59—22no—
113C23×C59—22no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—42no—
117C23×C59—42no—
118C23×C59—42no—
119C23×C59—42no—
120C23×C59—81no—
121C23×C59—81no—
122C23×C59—82no—
123C23×C59—82no—
124C23×C59—82no—
125C23×C59—82no—
126C23×C59#10214yes—
127C23×C59C2×C22⋊C411yescharacteristic
128C23×C59#12011yescharacteristic
129C23×C59#12311yescharacteristic
130C23×C59C2×D811yescharacteristic
131C23×C59C2×SD1611yescharacteristic
132C23×C59AGL(1,ℤ/8)11yescharacteristic
133C23×C59#13711yescharacteristic
134C22×D59—42no—
135C22×D59—42no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139C2×C4×C59—22no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—81no—
147C2×C4×C59C4×D412yes—
148C2×C4×C59#12112yes—
149C22⋊D59—88no—
150C22⋊D59—81no—
151C22⋊D59—81no—
152C22⋊D59—81no—
153C22⋊D59—81no—
154C2×Dic59—44no—
155C2×Dic59—44no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158C2×Dic59—41no—
159C2×Dic59—42no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—44no—
163C2×Dic59—88no—
164D236—82no—
165C8×C59—82no—
166#9203—591nomaximal, Sylow
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—24no—
174C22×C59—41no—
175C22×C59—41no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59—44no—
179C22×C59—44no—
180C22×C59—44no—
181C22×C59—42no—
182C22×C59—42no—
183C22×C59—82no—
184C22×C59—82no—
185C22×C59—82no—
186C22×C59#38611yescharacteristic
187C22×C59C2×#10211yescharacteristic
188C22×C59#41711yescharacteristic
189C22×C59#44711yescharacteristic
190C22×C59#44811yescharacteristic
191C22×C59#48011yescharacteristic
192C22×C59#48111yescharacteristic
193C2×D59—82no—
194C2×D59—82no—
195C2×D59—81no—
196C2×D59—81no—
197C2×D59—81no—
198C4×C59—22no—
199C4×C59—22no—
200C4×C59—42no—
201Dic59—84no—
202Dic59—82no—
203Dic59—81no—
204#2617—591no—
205#2315—591no—
206#1519—592no—
207C2×#323D5911yescharacteristic, complemented
208#1289—592no—
209C2×C59—21no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—21no—
213C2×C59—21no—
214C2×C59—21no—
215C2×C59—44no—
216C2×C59#187411yescharacteristic
217C2×C59#191111yescharacteristic
218C2×C59#191311yescharacteristic
219D59—161no—
220C2×#121—591no—
221C2×#120—1181no—
222C22×C22⋊C4—1181no—
223C22×C22⋊C4C2×D5911yescharacteristic
224C2×C22⋊C8C2×D5912yes—
225#397—592no—
226#394—1182no—
227#392—592no—
228#386—1181no—
229#385—1184no—
230#380—591no—
231#380—592no—
232#380—1181no—
233#327—1184no—
234#323—22no—
235C59#920311yescharacteristic, complemented, Sylow
236C25D23611yescharacteristic
237C22×D4—591no—
238C22×D4—1182no—
239C22×D4—1181no—
240C23×C4—1182no—
241C23×C4D23611yescharacteristic
242C23×C4C22×D5911yescharacteristic
243C22×C8—22no—
244#121—1182no—
245#121—1182no—
246#120—2362no—
247#120—2362no—
248C2×C4⋊C4—591no—
249C2×C4⋊C4—1182no—
250C2×C4⋊C4—1182no—
251C2×C4⋊C4—1181no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—1182no—
255C2×C22⋊C4—1182no—
256C2×C22⋊C4—1182no—
257C2×C22⋊C4—1182no—
258C2×C22⋊C4—1181no—
259C2×C22⋊C4—1181no—
260C2×C22⋊C4—2362no—
261C2×C22⋊C4—2362no—
262C2×C22⋊C4C4×D5912yes—
263C2×C22⋊C4C22⋊D5912yes—
264C22⋊C8—22no—
265C22⋊C8—22no—
266#95—592no—
267#95—594no—
268#95—1182no—
269#95—1182no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—81no—
275C24—81no—
276C24—1181no—
277C24#1174566012yes—
278C24D47212yes—
279C24(C2×D59)⋊C412yes—
280C24C2×D23611yescharacteristic
281C2×D4—1182no—
282C2×D4—4721no—
283C2×D4—4722no—
284C2×D4—4722no—
285C2×D4—2364no—
286C2×D4—2361no—
287C2×D4—2361no—
288C2×D4—2361no—
289C2×D4—2361no—
290C2×D4—2361no—
291C2×D4—2361no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—1182no—
297C22×C4—1181no—
298C22×C4—1184no—
299C22×C4—4722no—
300C22×C4—4722no—
301C22×C4—2364no—
302C22×C4(C2×D59)⋊C412yes—
303C22×C4C2×C4×D5911yescharacteristic
304C22×C4D4×D5912yes—
305C22×C4#1174568712yes—
306C22×C4C2×C22⋊D5911yescharacteristic
307C2×C8—42no—
308C2×C8—82no—
309C4⋊C4—1182no—
310C4⋊C4—4722no—
311C4⋊C4—4721no—
312C4⋊C4—4722no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—4722no—
316C22⋊C4—2364no—
317C22⋊C4—2364no—
318C22⋊C4—2362no—
319C22⋊C4—2361no—
320C22⋊C4—2361no—
321C22⋊C4—2362no—
322C22⋊C4—2362no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23—82no—
341C23—2362no—
342C23—2362no—
343C23—2361no—
344C23—2361no—
345C23—2361no—
346C23#1427364914yes—
347C23C2×(C2×D59)⋊C411yescharacteristic
348C23#1427369711yescharacteristic
349C23#1427371111yescharacteristic
350C23C2×D47211yescharacteristic
351C23C2×#1174566011yescharacteristic
352C23#1427375511yescharacteristic
353C23#1427376011yescharacteristic
354D4—4722no—
355D4—4728no—
356D4—4721no—
357D4—4721no—
358D4—4721no—
359D4—4721no—
360C2×C4—22no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—4728no—
369C2×C4—2362no—
370C2×C4—2364no—
371C2×C4—2364no—
372C2×C4—2362no—
373C2×C4—2362no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4—2364no—
378C2×C4(C22⋊D59)⋊C412yes—
379C2×C4#1427369912yes—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—4722no—
401C22—4722no—
402C22—4721no—
403C22—4721no—
404C22—4721no—
405C22#1683727911yescharacteristic
406C22C2×#1427364911yescharacteristic
407C22#1683740411yescharacteristic
408C22#1683753811yescharacteristic
409C22#1683754311yescharacteristic
410C22#1683771611yescharacteristic
411C22#1683772111yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—4724no—
416C4—4722no—
417C4—4721no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—44no—
425C2—9441no—
426C2#2142441111yescharacteristic
427C2#2142462611yescharacteristic
428C2#2142463611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#323C2
socleC22×C59#386
prime radical#21429830C2
cosocleC23×C4×C59C23
hypercenterC2×#323D59
hypocenterC59#9203
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #21426710C2semidirect
3. #21422665C2semidirect
4. #21421964C2semidirect
5. C2×C59×#323C2semidirect
9. #16837327C4semidirect
11. #16837311C4semidirect
21. C22×D236C2×C4semidirect
32. C2×C4⋊Dic59C2×C4semidirect
43. C59⋊#95C2×C4semidirect
44. C59⋊#95D4semidirect
207. C2×#323D59semidirect
235. C59#9203semidirect

Maximal subgroups

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

subgroupindex
#214298302
#214267102
#214226652
#214219642
C2×C59×#3232
#920359