Finite Group Databasebeta

← #28499771

Subgroups — #28499771

Order 15104.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#467×D59C211yesmaximal, characteristic
2#21425705C211yesmaximal, characteristic
3#21424120C211yesmaximal, characteristic
4#21422109C211yesmaximal, characteristic, complemented
5#21422108C211yesmaximal, characteristic, complemented
6#21422107C211yesmaximal, characteristic, complemented
7C59×#1348C211yesmaximal, characteristic, complemented
8C2×Q16×D59C411yescharacteristic, complemented
9C22×C8×D59C2211yescharacteristic
10C22⋊Q8×D59—21no—
11C2×#14273666—21no—
12#16837644C411yescharacteristic, complemented
13#16837643C2211yescharacteristic
14#16837642—21no—
15#16837641—21no—
16#16837640C411yescharacteristic, complemented
17#107×D59C411yescharacteristic, complemented
18C2×#14273658—21no—
19#103×D59—21no—
20#16837259—21no—
21C22⋊(C59⋊C16)C2211yescharacteristic
22(C2×D59)⋊C16—21no—
23#16837165C2211yescharacteristic
24C59×#467C2211yescharacteristic
25C59×C22⋊C16C2211yescharacteristic
26C59×#340C2211yescharacteristic
27C2×Q8×D59—21no—
28C22×C4×D59D411yescharacteristic
29C2×Dic236C2×C411yescharacteristic
30C2×#11745672—21no—
31Q16×D59—42no—
32C22×C59⋊C8D411yescharacteristic
33C2×C8×D59—42no—
34C2×C8×D59C2×C411yescharacteristic
35C2×C8×D59C2×C411yescharacteristic
36C2×C8×D59D411yescharacteristic
37C2×C8×D59D411yescharacteristic
38#14273707—21no—
39#14273706—21no—
40#14273705—41no—
41#14273704—41no—
42#14273703—21no—
43C2×C4⋊Dic59—21no—
44C4⋊C4×D59—21no—
45C4⋊C4×D59—41no—
46C2×(C2×D59)⋊C4—41no—
47C22⋊C4×D59—41no—
48#14273666—42no—
49C2×C59⋊C16—21no—
50#14273661C2×C411yescharacteristic
51#14273660—21no—
52#14273658—42no—
53#14273652—21no—
54#14273651—21no—
55#14273650—21no—
56C59⋊#95—41no—
57C2×Q16×C59C2×C411yescharacteristic
58C22×C8×C59C2311yescharacteristic
59C22⋊Q8×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C16×C59—21no—
62#107×C59C2×C411yescharacteristic
63#103×C59—21no—
64C23×D59D811yescharacteristic
65C2×Q8×C59—21no—
66C22×C4×C59—41no—
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—41no—
69C22×Dic59D811yescharacteristic
70Q8×D59—42no—
71Q8×D59—42no—
72C2×C4×D59—41no—
73C2×C4×D59—41no—
74C2×C4×D59—41no—
75C2×C4×D59—41no—
76C2×C4×D59C22⋊C411yescharacteristic
77C2×C4×D59C22⋊C411yescharacteristic
78C2×C4×D59SD1611yescharacteristic
79C2×C4×D59SD1611yescharacteristic
80C2×Dic118—21no—
81C2×Dic118—41no—
82Q16×C59—41no—
83C2×C8×C59—41no—
84C2×C8×C59C22×C411yescharacteristic
85C2×C8×C59C2×D411yescharacteristic
86C4⋊C4×C59—21no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊Dic59—41no—
92#11745672—42no—
93(C2×D59)⋊C4—41no—
94(C2×D59)⋊C4—41no—
95(C2×D59)⋊C4—82no—
96C4⋊Dic59—21no—
97C4⋊Dic59—41no—
98C4⋊Dic59—41no—
99C4⋊Dic59—41no—
100Dic59⋊C4—41no—
101Dic59⋊C4—41no—
102Dic59⋊C4—41no—
103C2×C59⋊C8—22no—
104C2×C59⋊C8C22⋊C411yescharacteristic
105C2×C59⋊C8C22⋊C411yescharacteristic
106Dic236—41no—
107C8×D59—22no—
108C8×D59—22no—
109C8×D59—41no—
110C8×D59—41no—
111C16×C59—41no—
112C59⋊C16—41no—
113C23×C59C2×D811yescharacteristic
114C22×D59—21no—
115C22×D59—21no—
116C22×D59—22no—
117C22×D59—41no—
118C22×D59#10211yescharacteristic
119C22×D59#10211yescharacteristic
120Q8×C59—41no—
121Q8×C59—41no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—81no—
127C2×C4×C59—81no—
128C2×C4×C59—81no—
129C2×C4×C59C2×C22⋊C411yescharacteristic
130C2×C4×C59C2×SD1611yescharacteristic
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—81no—
135C2×Dic59—81no—
136C2×Dic59—81no—
137C2×Dic59#10211yescharacteristic
138C2×Dic59#10211yescharacteristic
139C4×D59—21no—
140C4×D59—21no—
141C4×D59—21no—
142C4×D59—21no—
143C4×D59—41no—
144C4×D59—41no—
145C4×D59—82no—
146C4×D59—82no—
147Dic118—41no—
148Dic118—41no—
149Dic118—82no—
150C8×C59—21no—
151C8×C59—21no—
152C59⋊C8—21no—
153C59⋊C8#9912yes—
154#11889—591nomaximal, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59C2×#10211yescharacteristic
158C22×C59#51011yescharacteristic
159C22×C59#51011yescharacteristic
160C2×D59—21no—
161C2×D59—21no—
162C2×D59—21no—
163C2×D59—21no—
164C2×D59—21no—
165C2×D59—42no—
166C2×D59—42no—
167C2×D59#36012yes—
168C4×C59—21no—
169C4×C59—81no—
170C4×C59—81no—
171C4×C59—161no—
172C4×C59C2×#9911yescharacteristic
173C4×C59#42111yescharacteristic
174Dic59—21no—
175Dic59—21no—
176Dic59—81no—
177Dic59—81no—
178Dic59—161no—
179C2×#467—591no—
180#2112—591no—
181#1807—591no—
182#1348—591no—
183#1348—591no—
184#1348—591no—
185#1348D5911yescharacteristic, complemented
186C2×C59—21no—
187C2×C59—21no—
188C2×C59#151911yescharacteristic
189C2×C59#214111yescharacteristic
190C2×C59C2×#36011yescharacteristic
191D59—22no—
192D59—41no—
193C22×Q16—591no—
194C23×C8—591no—
195C2×C22⋊Q8—1181no—
196C2×M32—1181no—
197#467—591no—
198#467—591no—
199#467—591no—
200#467—1181no—
201#467—1181no—
202#467C2×D5911yescharacteristic
203C2×#107—591no—
204C2×#106—1181no—
205C2×#103—1181no—
206#380—1181no—
207C22⋊C16—591no—
208C22⋊C16—1181no—
209C22⋊C16C2×D5911yescharacteristic
210#340—591no—
211#340C2×D5911yescharacteristic
212C59#1188911yescharacteristic, complemented, Sylow
213C22×Q8—1181no—
214C23×C4—591no—
215C2×Q16—591no—
216C2×Q16—1181no—
217C2×Q16—2362no—
218C2×Q16C4×D5911yescharacteristic
219C22×C8—591no—
220C22×C8—591no—
221C22×C8—591no—
222C22×C8—591no—
223C22×C8—591no—
224C22×C8—2362no—
225C22×C8C22×D5911yescharacteristic
226C22⋊Q8—21no—
227C22⋊Q8—1181no—
228C22⋊Q8—1181no—
229C22⋊Q8—1181no—
230C22⋊Q8—2361no—
231C22⋊Q8—2361no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4—1181no—
235C2×C4⋊C4—2361no—
236C2×C22⋊C4—2361no—
237C2×C22⋊C4—2361no—
238M32—2362no—
239C2×C16—21no—
240C2×C16—1181no—
241#107—591no—
242#107—1181no—
243#107C4×D5911yescharacteristic
244#106—2362no—
245#103—21no—
246#103—1181no—
247#103—1181no—
248#103—1181no—
249#95—2361no—
250C24—591no—
251C2×Q8—21no—
252C2×Q8—1181no—
253C2×Q8—2361no—
254C2×Q8—2362no—
255C2×Q8—2362no—
256C22×C4—41no—
257C22×C4—591no—
258C22×C4—591no—
259C22×C4—591no—
260C22×C4—591no—
261C22×C4—591no—
262C22×C4—2361no—
263C22×C4—2361no—
264C22×C4—2361no—
265C22×C4—2361no—
266C22×C4—2361no—
267C22×C4D4×D5911yescharacteristic
268Q16—41no—
269Q16—2361no—
270Q16—2362no—
271C2×C8—41no—
272C2×C8—591no—
273C2×C8—591no—
274C2×C8—1182no—
275C2×C8—1182no—
276C2×C8—1182no—
277C2×C8—2361no—
278C2×C8—2361no—
279C2×C8C2×C4×D5911yescharacteristic
280C2×C8D4×D5911yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—1181no—
286C4⋊C4—2361no—
287C4⋊C4—2361no—
288C4⋊C4—2361no—
289C4⋊C4—2361no—
290C4⋊C4—2361no—
291C4⋊C4—2361no—
292C22⋊C4—41no—
293C22⋊C4—4722no—
294C22⋊C4—2361no—
295C22⋊C4—2361no—
296C22⋊C4—2361no—
297C16—41no—
298C16—2361no—
299C23—591no—
300C23—591no—
301C23—1182no—
302C23—1181no—
303C23—1181no—
304C23—2361no—
305C23D8×D5911yescharacteristic
306Q8—41no—
307Q8—41no—
308Q8—4722no—
309Q8—2361no—
310Q8—2361no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—591no—
319C2×C4—591no—
320C2×C4—1181no—
321C2×C4—1181no—
322C2×C4—1181no—
323C2×C4—1181no—
324C2×C4—4722no—
325C2×C4—4722no—
326C2×C4—4721no—
327C2×C4—4721no—
328C2×C4—4721no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4C22⋊C4×D5911yescharacteristic
335C2×C4SD16×D5911yescharacteristic
336C8—21no—
337C8—21no—
338C8—592no—
339C8—1181no—
340C22—21no—
341C22—21no—
342C22—592no—
343C22—1181no—
344C22—1181no—
345C22—1181no—
346C22—1181no—
347C22—1181no—
348C22—2362no—
349C22—2362no—
350C22#102×D5911yescharacteristic
351C22#1683784111yescharacteristic
352C22#1683784411yescharacteristic
353C4—21no—
354C4—81no—
355C4—81no—
356C4—161no—
357C4—1181no—
358C4—1181no—
359C4—9441no—
360C4—4721no—
361C4—4721no—
362C4#99×D5911yescharacteristic
363C4#1683741711yescharacteristic
364C2—21no—
365C2—21no—
366C2—1182no—
367C2—2361no—
368C2#2142266211yescharacteristic
369C2#2142581411yescharacteristic
370C2#360×D5911yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1348C2
socleC22×C59C2×#102
prime radicalC23×D59D8
cosocleC22×C8×C59C23
hypercenter#1348D59
hypocenterC59#11889
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
4. #21422109C2semidirect
5. #21422108C2semidirect
6. #21422107C2semidirect
7. C59×#1348C2semidirect
8. C2×Q16×D59C4semidirect
12. #16837644C4semidirect
16. #16837640C4semidirect
17. #107×D59C4semidirect
185. #1348D59semidirect
212. C59#11889semidirect

Maximal subgroups

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

subgroupindex
#467×D592
#214257052
#214241202
#214221092
#214221082
#214221072
C59×#13482
#1188959