Finite Group Databasebeta

← #28499011

Subgroups — #28499011

Order 15104.

648 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 87 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#21427311C211yesmaximal, characteristic
2#21422417C211yesmaximal, characteristic, complemented
3#21422116C212yesmaximal, complemented
4#21422092C212yesmaximal, complemented
5C4×#99×C59C211yesmaximal, characteristic, complemented
6C22×#11745685C411yescharacteristic, complemented
7C2×C4×C22⋊D59—21no—
8#16837455—21no—
9#16837322C411yescharacteristic, complemented
10#16837319C411yescharacteristic, complemented
11#16837317—21no—
12#16837314C2211yescharacteristic
13#16837311C2211yescharacteristic
14#16837310—21no—
15C4×(C2×D59)⋊C4—21no—
16C4×C22⋊Dic59C411yescharacteristic, complemented
17#16837126—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59—21no—
21C4×C22⋊C4×C59—21no—
22C59×#342C2212yes—
23C2×#11745685—44no—
24C2×#11745685—41no—
25C2×#11745685—41no—
26C22×Dic118C2×C411yescharacteristic
27C22×D236C2×C411yescharacteristic
28C22×C22⋊D59—21no—
29C22×C22⋊D59—21no—
30C22×C4×D59—21no—
31C4×C22⋊D59—44no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×C4×Dic59—21no—
39C4⋊(C59⋊C8)—41no—
40C22⋊(C59⋊C8)—22no—
41C4×C59⋊C8—41no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C22×D4×C59D411yescharacteristic
47C23×C4×C59C2×C411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C4×D4×C59—41no—
50C4×D4×C59—41no—
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59—22no—
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C2×C42×C59—41no—
56C2×C42×C59D411yescharacteristic
57#99×C59—24no—
58#95×C59—41no—
59C24×C59C22⋊C411yescharacteristic
60C24×C59C2×D411yescharacteristic
61C23×D59—21no—
62C2×D4×C59—22no—
63C2×D4×C59—22no—
64C2×D4×C59—41no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—42no—
70C22×C4×C59—42no—
71C22×C4×C59—42no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—42no—
80C2×C22⋊D59—42no—
81C2×C22⋊D59—42no—
82C2×C22⋊D59—42no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86#11745685—44no—
87#11745685—44no—
88#11745685—44no—
89#11745685—44no—
90C2×D236—41no—
91C2×D236—42no—
92C2×C4×D59—24no—
93C2×C4×D59—42no—
94C2×C4×D59—41no—
95C2×C4×D59—41no—
96C2×Dic118—41no—
97C2×Dic118—42no—
98C4⋊C4×C59—41no—
99C22⋊C4×C59—24no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—42no—
103C42×C59—21no—
104C42×C59—21no—
105C42×C59—82no—
106C22⋊Dic59—42no—
107C22⋊Dic59—42no—
108(C2×D59)⋊C4—42no—
109(C2×D59)⋊C4—42no—
110(C2×D59)⋊C4—42no—
111Dic59⋊C4—42no—
112C4×Dic59—42no—
113C2×C59⋊C8—42no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59C2×C22⋊C411yescharacteristic
124C23×C59#13512yes—
125C22×D59—22no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130C22×D59—41no—
131C22×D59—41no—
132D4×C59—44no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—22no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—44no—
146C2×C4×C59—81no—
147C2×C4×C59—82no—
148C2×C4×C59—82no—
149C2×C4×C59#9911yescharacteristic
150C2×C4×C59#9911yescharacteristic
151C2×C4×C59#10412yes—
152C22⋊D59—48no—
153C22⋊D59—48no—
154C2×Dic59—22no—
155C2×Dic59—22no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163D236—42no—
164D236—42no—
165C4×D59—44no—
166C4×D59—44no—
167Dic118—42no—
168Dic118—42no—
169C59⋊C8—82no—
170#11725—591nomaximal, Sylow
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—22no—
174C22×C59—42no—
175C22×C59—41no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59C2×#9911yescharacteristic
184C22×C59#41611yescharacteristic
185C22×C59C2×#10411yescharacteristic
186C2×D59—42no—
187C2×D59—42no—
188C2×D59—42no—
189C2×D59—81no—
190C2×D59—81no—
191C2×D59—81no—
192C2×D59—81no—
193C4×C59—21no—
194C4×C59—22no—
195C4×C59—41no—
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—84no—
199C4×C59AΣL(1,𝔽16)12yes—
200Dic59—42no—
201Dic59—42no—
202Dic59—81no—
203#2388—591no—
204C4×#99D5911yescharacteristic, complemented
205#1460—591no—
206#1351—592no—
207#1343—592no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59—42no—
211C2×C59—81no—
212C2×C59#150111yescharacteristic
213C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
214C2×C59#212611yescharacteristic
215D59—82no—
216C22×#41—591no—
217C2×C4×D4—1181no—
218C2×C4×D4C2×D5911yescharacteristic
219#431—1181no—
220C2×#99—21no—
221#395—591no—
222#394—591no—
223#394—1181no—
224#393—591no—
225#392—591no—
226#392—1181no—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—591no—
229C4×C22⋊C4—1181no—
230#342C2×D5912yes—
231#323—592no—
232#323—1181no—
233C59#1172511yescharacteristic, complemented, Sylow
234C2×#41—2361no—
235C2×#41—2361no—
236C2×#41—2364no—
237C22×Q8—591no—
238C22×D4—591no—
239C22×D4—1181no—
240C22×D4—1181no—
241C22×D4C22⋊D5911yescharacteristic
242C23×C4—1181no—
243C23×C4C4×D5911yescharacteristic
244C23×C4C22⋊D5911yescharacteristic
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—2364no—
248C2×C4⋊C4—1181no—
249C2×C4⋊C4C4×D5911yescharacteristic
250C2×C22⋊C4—22no—
251C2×C22⋊C4—1181no—
252C2×C22⋊C4—1181no—
253C2×C22⋊C4—1181no—
254C2×C22⋊C4—1181no—
255C2×C22⋊C4—1181no—
256C2×C22⋊C4D23611yescharacteristic
257C2×C22⋊C4C22×D5911yescharacteristic
258C2×C42—41no—
259C2×C42—1181no—
260C2×C42D23611yescharacteristic
261C4⋊C8—2361no—
262#99—24no—
263C22⋊C8—1182no—
264C4×C8—2361no—
265#95—41no—
266#95—591no—
267#95—591no—
268#95—1181no—
269#95—1181no—
270C24—1181no—
271C24(C2×D59)⋊C411yescharacteristic
272C24C2×C22⋊D5911yescharacteristic
273#41—2364no—
274#41—2364no—
275#41—2364no—
276#41—2364no—
277C2×Q8—2361no—
278C2×Q8—2362no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—2361no—
283C2×D4—2362no—
284C2×D4—2362no—
285C2×D4—2362no—
286C2×D4—2362no—
287C2×D4—2362no—
288C2×D4—2362no—
289C2×D4—2362no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—1184no—
298C22×C4—1181no—
299C22×C4—1181no—
300C22×C4—1181no—
301C22×C4—2362no—
302C22×C4—2361no—
303C22×C4—2361no—
304C22×C4(C2×D59)⋊C411yescharacteristic
305C22×C4(C2×D59)⋊C411yescharacteristic
306C22×C4(C2×D59)⋊C411yescharacteristic
307C22×C4C2×C4×D5911yescharacteristic
308C22×C4C2×D23611yescharacteristic
309C2×C8—2362no—
310C4⋊C4—41no—
311C4⋊C4—2362no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—2362no—
317C22⋊C4—2362no—
318C22⋊C4—2362no—
319C22⋊C4—2362no—
320C22⋊C4—2362no—
321C42—21no—
322C42—21no—
323C42—82no—
324C42—2362no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—1182no—
335C23—2361no—
336C23—2361no—
337C23—2361no—
338C23—2361no—
339C23—2361no—
340C23—2361no—
341C23C2×(C2×D59)⋊C411yescharacteristic
342C23#1427374712yes—
343Q8—2362no—
344Q8—2362no—
345D4—44no—
346D4—2362no—
347D4—2362no—
348D4—2368no—
349D4—2368no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—1182no—
367C2×C4—1182no—
368C2×C4—2364no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2364no—
377C2×C4#1427364111yescharacteristic
378C2×C4#1427364111yescharacteristic
379C2×C4#1427365512yes—
380C8—4722no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—4721no—
394C22—4721no—
395C22—4721no—
396C22—4721no—
397C22—2362no—
398C22—2362no—
399C22—2362no—
400C22C2×#1427364111yescharacteristic
401C22#1683739611yescharacteristic
402C22C2×#1427365511yescharacteristic
403C4—21no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—84no—
409C4—4721no—
410C4—2362no—
411C4—2362no—
412C4#1683718812yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—81no—
417C2—4722no—
418C2#2142258511yescharacteristic
419C2C2×#1683718811yescharacteristic
420C2#2142575911yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radical#21427311C2
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11725
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #21422417C2semidirect
3. #21422116C2semidirect
4. #21422092C2semidirect
5. C4×#99×C59C2semidirect
6. C22×#11745685C4semidirect
9. #16837322C4semidirect
10. #16837319C4semidirect
16. C4×C22⋊Dic59C4semidirect
204. C4×#99D59semidirect
233. C59#11725semidirect

Maximal subgroups

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

subgroupindex
#214273112
#214224172
#214221162
#214220922
C4×#99×C592
#1172559