Finite Group Databasebeta

← #28475713

Subgroups — #28475713

Order 15104.

560 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 75 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#21422423C211yesmaximal, characteristic
2#21422333C211yesmaximal, characteristic
3C59×#1819C211yesmaximal, characteristic
4C22×C22⋊Dic59—21no—
5C22×C4×Dic59C411yescharacteristic, complemented
6#16837366—21no—
7C2×C59⋊#95—21no—
8C2×C59⋊#95C411yescharacteristic, complemented
9#16837126—21no—
10#16837126C412yescomplemented
11C22×C22⋊C4×C59C2211yescharacteristic
12C59×#379—21no—
13C59×#342—21no—
14C59×#342C412yescomplemented
15C23×Dic59—21no—
16C23×Dic59—21no—
17C2×#11745664—41no—
18C2×C22⋊Dic59—24no—
19C2×C22⋊Dic59—42no—
20C2×C22⋊Dic59—42no—
21C2×C4×Dic59—41no—
22C2×C4×Dic59—42no—
23C22⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—41no—
25C59⋊#95—42no—
26C59⋊#95—41no—
27C25×C59D411yescharacteristic
28C23×C4×C59C2×C411yescharacteristic
29C23×C4×C59D411yescharacteristic
30C2×C22⋊C4×C59—22no—
31C2×C22⋊C4×C59—41no—
32C2×C22⋊C4×C59—41no—
33C2×C22⋊C4×C59—41no—
34C2×C22⋊C4×C59—41no—
35C2×C22⋊C4×C59C2×C411yescharacteristic
36C2×C22⋊C4×C59C2×C411yescharacteristic
37C2×C22⋊C4×C59D411yescharacteristic
38C2×C22⋊C4×C59Q811yescharacteristic
39C24×C59—22no—
40C24×C59—21no—
41C24×C59—21no—
42C24×C59—41no—
43C24×C59—41no—
44C24×C59—41no—
45C24×C59—41no—
46C24×C59—41no—
47C24×C59C22⋊C411yescharacteristic
48C24×C59C22⋊C411yescharacteristic
49C24×C59C4⋊C411yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—41no—
55C22×C4×C59—41no—
56C22×C4×C59—41no—
57C22×C4×C59—41no—
58C22×C4×C59—42no—
59C22×C4×C59C4211yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C4⋊C411yescharacteristic
62C22×C4×C59C4⋊C411yescharacteristic
63C22×Dic59—22no—
64C22×Dic59—22no—
65C22×Dic59—22no—
66C22×Dic59—22no—
67C22×Dic59—41no—
68C22×Dic59—41no—
69C22×Dic59—41no—
70C22×Dic59—41no—
71C22×Dic59—41no—
72C22×Dic59—41no—
73C22×Dic59—41no—
74C22×Dic59—41no—
75C22×Dic59—41no—
76C22×Dic59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—42no—
82C22⋊C4×C59—42no—
83C22⋊C4×C59—81no—
84C22⋊C4×C59—81no—
85C22⋊C4×C59—81no—
86C22⋊C4×C59—81no—
87C22⋊Dic59—48no—
88C4×Dic59—42no—
89C4×Dic59—42no—
90#11745664—81no—
91#11745664—81no—
92C2×C59⋊C8—42no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—42no—
105C23×C59—42no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—42no—
111C23×C59—42no—
112C23×C59—42no—
113C23×C59—42no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—81no—
119C23×C59—81no—
120C23×C59—81no—
121C23×C59—81no—
122C23×C59—81no—
123C23×C59—81no—
124C23×C59#9511yescharacteristic
125C23×C59#9911yescharacteristic
126C23×C59#9911yescharacteristic
127C23×C59#10412yes—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—42no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—81no—
142C2×C4×C59—82no—
143C2×C4×C59—82no—
144C2×C4×C59—82no—
145C2×Dic59—24no—
146C2×Dic59—42no—
147C2×Dic59—42no—
148C2×Dic59—42no—
149C2×Dic59—42no—
150C2×Dic59—42no—
151C2×Dic59—42no—
152C2×Dic59—42no—
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—81no—
156C2×Dic59—81no—
157C2×Dic59—81no—
158C2×Dic59—81no—
159C59⋊C8—82no—
160#7213—591nomaximal, Sylow
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—22no—
176C22×C59—42no—
177C22×C59—42no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59—41no—
182C22×C59—41no—
183C22×C59—81no—
184C22×C59—82no—
185C22×C59—82no—
186C22×C59—82no—
187C22×C59—82no—
188C22×C59—81no—
189C22×C59H(3,ℤ/4)11yescharacteristic
190C22×C59#33911yescharacteristic
191C22×C59#34211yescharacteristic
192C22×C59AΣL(1,𝔽16)11yescharacteristic
193C22×C59#35211yescharacteristic
194C22×C59#35311yescharacteristic
195C22×C59#35411yescharacteristic
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—41no—
200C4×C59—162no—
201Dic59—44no—
202Dic59—82no—
203#1819D5911yescharacteristic
204#1462—591no—
205#1437—591no—
206C2×C59—21no—
207C2×C59—21no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—42no—
213C2×C59—81no—
214C2×C59#128411yescharacteristic
215C2×C59#139011yescharacteristic
216C2×C59#139311yescharacteristic
217C22×C22⋊C4—1181no—
218C22×C22⋊C4C2×D5911yescharacteristic
219C22×C42—591no—
220#407—1181no—
221#379—21no—
222C2×#95—591no—
223C2×#95—1181no—
224#342—21no—
225#342Dic5912yescomplemented
226#323—592no—
227#323—1181no—
228C59#721311yescharacteristic, complemented, Sylow
229C25C22⋊D5911yescharacteristic
230C23×C4—1181no—
231C23×C4—1181no—
232C23×C4C4×D5911yescharacteristic
233C23×C4D23611yescharacteristic
234C2×M16—2361no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—1184no—
241C2×C22⋊C4—2362no—
242C2×C22⋊C4—2362no—
243C2×C22⋊C4Dic11811yescharacteristic
244C2×C22⋊C4C4×D5911yescharacteristic
245C2×C22⋊C4C2×Dic5911yescharacteristic
246C2×C22⋊C4C22⋊D5911yescharacteristic
247C2×C42—2361no—
248C2×C42—2362no—
249C22⋊C8—1182no—
250C22⋊C8—2361no—
251#95—2361no—
252#95—2362no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24Dic59⋊C411yescharacteristic
262C24(C2×D59)⋊C411yescharacteristic
263C24C22⋊Dic5911yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—1182no—
274C22×C4—1182no—
275C22×C4—1182no—
276C22×C4—1182no—
277C22×C4—2361no—
278C22×C4—2361no—
279C22×C4—2361no—
280C22×C4—2361no—
281C22×C4—2361no—
282C22×C4—2361no—
283C22×C4—2361no—
284C22×C4—2361no—
285C22×C4—2361no—
286C22×C4—2361no—
287C22×C4C4×Dic5911yescharacteristic
288C22×C4Dic59⋊C411yescharacteristic
289C22×C4C4⋊Dic5911yescharacteristic
290C22×C4(C2×D59)⋊C411yescharacteristic
291M16—4721no—
292M16—4721no—
293C2×C8—2362no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C22⋊C4—2368no—
305C42—2362no—
306C42—2362no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23C59⋊#9511yescharacteristic
339C23#1427364111yescharacteristic
340C23#1427364111yescharacteristic
341C23#1427365412yes—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—1184no—
360C2×C4—4721no—
361C2×C4—4721no—
362C2×C4—4721no—
363C2×C4—4721no—
364C2×C4—2362no—
365C2×C4—2362no—
366C2×C4—2362no—
367C2×C4—2362no—
368C2×C4—2362no—
369C2×C4—2362no—
370C2×C4—2362no—
371C2×C4—2362no—
372C2×C4—2362no—
373C8—4722no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—81no—
402C22#1683715811yescharacteristic
403C22#1683716111yescharacteristic
404C22#1683716811yescharacteristic
405C22#1683718811yescharacteristic
406C22#1683719111yescharacteristic
407C22#1683719411yescharacteristic
408C22#1683719611yescharacteristic
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—162no—
414C4—4722no—
415C4—2364no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—81no—
424C2#2142194811yescharacteristic
425C2#2142221311yescharacteristic
426C2#2142222011yescharacteristic
subgroupG/H
centerC22#16837168
commutatorC22×C4×C59C42
FrattiniC22×C22⋊C4C2×D59
FittingC59×#1819C2
socleC22×C59#342
prime radicalC25×C59D4
cosocleC22×C22⋊C4×C59C22
hypercenter#1819D59
hypocenterC59#7213
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1819

Complements

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

NH ≅ G/Ntype
5. C22×C4×Dic59C4semidirect
8. C2×C59⋊#95C4semidirect
10. #16837126C4semidirect
14. C59×#342C4semidirect
225. #342Dic59semidirect
228. C59#7213semidirect

Maximal subgroups

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

subgroupindex
#214224232
#214223332
C59×#18192
#721359