Finite Group Databasebeta

← #28498975

Subgroups — #28498975

Order 15104.

592 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 107 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#21429901C211yesmaximal, characteristic
2C2×#16837169C211yesmaximal, characteristic
3#21422121C212yesmaximal, complemented
4#21422118C212yesmaximal, complemented
5C2×C59×#323C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic, complemented
7C22×C22⋊Dic59—21no—
8#16837338C411yescharacteristic, complemented
9#16837338C2211yescharacteristic
10#16837325—21no—
11#16837325—21no—
12#16837314C411yescharacteristic, complemented
13#16837314C2211yescharacteristic
14#16837275—21no—
15#16837262C411yescharacteristic, complemented
16#16837258—21no—
17#16837169—21no—
18#16837169C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×Dic118C2×C411yescharacteristic, complemented
24C23×Dic59—41no—
25#14273706—41no—
26#14273706—41no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59C2×C411yescharacteristic, complemented
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—24no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—41no—
41C2×C22⋊Dic59—42no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic, complemented
45C59⋊#95C2×C411yescharacteristic, complemented
46C25×C59D411yescharacteristic
47C23×C4×C59C2×C411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C22×C8×C59—41no—
50C2×C22⋊C4×C59—41no—
51C2×C22⋊C4×C59—41no—
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59C2311yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59—21no—
59C24×C59—21no—
60C24×C59—22no—
61C24×C59—81no—
62C24×C59—81no—
63C24×C59C22⋊C411yescharacteristic
64C24×C59C22⋊C411yescharacteristic
65C24×C59SD1612yes—
66C24×C59Q1612yes—
67C24×C59C2×D411yescharacteristic
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—81no—
73C22×C4×C59—81no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×Dic59—21no—
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—44no—
83C22×Dic59—42no—
84C22×Dic59—82no—
85C22×Dic59—82no—
86C2×Dic118—41no—
87C2×Dic118—81no—
88C2×C8×C59—42no—
89C2×C8×C59—82no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊Dic59—24no—
95C22⋊Dic59—44no—
96C22⋊Dic59—41no—
97C22⋊Dic59—41no—
98C22⋊Dic59—42no—
99C22⋊Dic59—42no—
100C22⋊Dic59—81no—
101C22⋊Dic59—81no—
102C22⋊Dic59—81no—
103C22⋊Dic59—81no—
104C4⋊Dic59—41no—
105C4⋊Dic59—81no—
106Dic59⋊C4—42no—
107Dic59⋊C4—81no—
108Dic59⋊C4—81no—
109C23×C59—22no—
110C23×C59—22no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—22no—
114C23×C59—22no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—42no—
120C23×C59—42no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—81no—
126C23×C59—81no—
127C23×C59—82no—
128C23×C59—82no—
129C23×C59—84no—
130C23×C59#9911yescharacteristic
131C23×C59#9911yescharacteristic
132C23×C59#10312yes—
133C23×C59#10312yes—
134C23×C59#10412yes—
135C23×C59C2×C22⋊C411yescharacteristic
136C23×C59C2×SD1611yescharacteristic
137C23×C59C2×Q1611yescharacteristic
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—81no—
148C2×C4×C59—81no—
149C2×C4×C59—81no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—44no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—88no—
164C2×Dic59—82no—
165C2×Dic59—82no—
166Dic118—82no—
167C8×C59—82no—
168#11716—591nomaximal, Sylow
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—22no—
177C22×C59—21no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—42no—
181C22×C59—41no—
182C22×C59—41no—
183C22×C59—44no—
184C22×C59—44no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—41no—
188C22×C59—41no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—82no—
192C22×C59—82no—
193C22×C59#32812yes—
194C22×C59#32812yes—
195C22×C59C2×#9911yescharacteristic
196C22×C59C2×#10311yescharacteristic
197C22×C59C2×#10411yescharacteristic
198C4×C59—41no—
199C4×C59—41no—
200C4×C59—41no—
201C4×C59—41no—
202Dic59—81no—
203Dic59—81no—
204Dic59—84no—
205Dic59—81no—
206Dic59—81no—
207#2626—591no—
208C2×#342—591no—
209C2×#323D5911yescharacteristic
210#1352—592no—
211#1351—592no—
212C2×C59—21no—
213C2×C59—22no—
214C2×C59—21no—
215C2×C59—22no—
216C2×C59—41no—
217C2×C59—42no—
218C2×C59—41no—
219C2×C59C2×#32811yescharacteristic
220C2×C59#212111yescharacteristic
221C2×C59#212711yescharacteristic
222C2×C22⋊Q8—591no—
223C22×C22⋊C4—1181no—
224C22×C22⋊C4C2×D5911yescharacteristic
225C2×C22⋊C8—21no—
226#399—591no—
227#399—591no—
228#396—1181no—
229#396—1181no—
230#393—591no—
231#393—591no—
232#385—1181no—
233#380—591no—
234#379—1181no—
235#342—592no—
236#342—1181no—
237#323—21no—
238#323C2×D5912yes—
239C59#1171611yescharacteristic, complemented, Sylow
240C25D23611yescharacteristic
241C22×Q8—591no—
242C23×C4—2361no—
243C23×C4C4×D5911yescharacteristic
244C23×C4C22⋊D5911yescharacteristic
245C22×C8—41no—
246C22⋊Q8—2361no—
247C22⋊Q8—2361no—
248C2×C4⋊C4—591no—
249C2×C4⋊C4—1181no—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1181no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—1184no—
256C2×C22⋊C4—1182no—
257C2×C22⋊C4—1181no—
258C2×C22⋊C4—1181no—
259C2×C22⋊C4—1181no—
260C2×C22⋊C4—1181no—
261C2×C22⋊C4—1181no—
262C2×C22⋊C4—1181no—
263C2×C22⋊C4—2361no—
264C2×C22⋊C4—2362no—
265C2×C22⋊C4C4×D5911yescharacteristic
266C2×C22⋊C4D23611yescharacteristic
267C2×C22⋊C4C22⋊D5911yescharacteristic
268C2×C22⋊C4C22×D5911yescharacteristic
269C22⋊C8—22no—
270C22⋊C8—41no—
271#95—591no—
272#95—591no—
273#95—1181no—
274#95—1181no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24#1174566012yes—
281C24Dic23612yes—
282C24(C2×D59)⋊C411yescharacteristic
283C24(C2×D59)⋊C411yescharacteristic
284C24C2×D23611yescharacteristic
285C2×Q8—4721no—
286C2×Q8—2361no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4—1181no—
297C22×C4—4722no—
298C22×C4—4722no—
299C22×C4—2364no—
300C22×C4—2362no—
301C22×C4(C2×D59)⋊C411yescharacteristic
302C22×C4(C2×D59)⋊C411yescharacteristic
303C22×C4C2×C4×D5911yescharacteristic
304C22×C4C2×C22⋊D5911yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—4721no—
308C4⋊C4—4721no—
309C4⋊C4—4721no—
310C4⋊C4—2361no—
311C4⋊C4—2362no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—1184no—
317C22⋊C4—4721no—
318C22⋊C4—4721no—
319C22⋊C4—4721no—
320C22⋊C4—4721no—
321C22⋊C4—2364no—
322C22⋊C4—2361no—
323C22⋊C4—2361no—
324C22⋊C4—2362no—
325C22⋊C4—2362no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1427364111yescharacteristic
348C23#1427364111yescharacteristic
349C23#1427365212yes—
350C23#1427365212yes—
351C23#1427365312yes—
352C23C2×(C2×D59)⋊C411yescharacteristic
353C23C2×#1174566011yescharacteristic
354C23C2×Dic23611yescharacteristic
355Q8—4722no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—4728no—
369C2×C4—4722no—
370C2×C4—4722no—
371C2×C4—2364no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4—2361no—
378C2×C4—2361no—
379C2×C4—2361no—
380C2×C4—2361no—
381C2×C4—2361no—
382C2×C4—2361no—
383C2×C4—2361no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1683714012yes—
410C22#1683714012yes—
411C22C2×#1427364111yescharacteristic
412C22C2×#1427365211yescharacteristic
413C22C2×#1427365311yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—4721no—
419C4—4721no—
420C4—4724no—
421C4—4721no—
422C4—4721no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×#1683714011yescharacteristic
431C2#2142574311yescharacteristic
432C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC2×C59×#323C2
socleC22×C59C2×#99
prime radicalC25×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC2×#323D59
hypocenterC59#11716
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #21422121C2semidirect
4. #21422118C2semidirect
6. C2×#14273706C4semidirect
8. #16837338C4semidirect
12. #16837314C4semidirect
15. #16837262C4semidirect
23. C22×Dic118C2×C4semidirect
28. C2×C4⋊Dic59C2×C4semidirect
44. C59⋊#95C2×C4semidirect
45. C59⋊#95C2×C4semidirect
239. C59#11716semidirect

Maximal subgroups

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

subgroupindex
#214299012
C2×#168371692
#214221212
#214221182
C2×C59×#3232
#1171659