Finite Group Databasebeta

← #28491869

Subgroups — #28491869

Order 15104.

360 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 77 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#21425039C211yesmaximal, characteristic
2#21422646C211yesmaximal, characteristic
3#21422607C211yesmaximal, characteristic
4#21421989C211yesmaximal, characteristic
5#21421960C211yesmaximal, characteristic
6#21421952C211yesmaximal, characteristic
7C59×#2605C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837459C2211yescharacteristic
10#16837459C2211yescharacteristic
11C4×#11745664C2211yescharacteristic
12#16837162—21no—
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837138—21no—
17#16837138—21no—
18C59×#400—21no—
19C59×#395—21no—
20C59×#391—21no—
21C59×#391—21no—
22C59×#387—21no—
23C59×#387—21no—
24C59×#383C2211yescharacteristic
25C4×C4⋊C4×C59C2211yescharacteristic
26C59×#376C2211yescharacteristic
27C2×#11745664—21no—
28#14273727Q811yescharacteristic, complemented
29#14273727Q811yescharacteristic, complemented
30#14273719—21no—
31#14273706—21no—
32#14273706—21no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35#14273686—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40#14273638—21no—
41#14273638—21no—
42#14273638—21no—
43C4×C59⋊C8—21no—
44C22×Q8×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C42×C59—21no—
50C2×C42×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C2×C42×C59C2311yescharacteristic
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59D411yescharacteristic
62#95×C59D411yescharacteristic
63#95×C59D411yescharacteristic
64#95×C59D411yescharacteristic
65C2×Q8×C59—42no—
66C2×Q8×C59—42no—
67C2×Q8×C59—81no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C2×Dic118—21no—
78C2×Dic118—21no—
79C4⋊C4×C59—42no—
80C4⋊C4×C59—42no—
81C4⋊C4×C59—42no—
82C4⋊C4×C59—81no—
83C42×C59—42no—
84C42×C59—41no—
85C42×C59—41no—
86C42×C59C2×Q811yescharacteristic
87C42×C59#4111yescharacteristic
88C42×C59#4111yescharacteristic
89C42×C59#4111yescharacteristic
90C22⋊Dic59—41no—
91C4⋊Dic59—21no—
92C4⋊Dic59—21no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95Dic59⋊C4—41no—
96Dic59⋊C4—41no—
97C4×Dic59—41no—
98#11745664—22no—
99#11745664—41no—
100C2×C59⋊C8—41no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8—41no—
103C23×C59#12011yescharacteristic
104Q8×C59—84no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59C22⋊Q811yescharacteristic
128C2×C4×C59C22⋊Q811yescharacteristic
129C2×C4×C59C22⋊Q811yescharacteristic
130C2×C4×C59#12411yescharacteristic
131C2×C4×C59#12411yescharacteristic
132C2×C4×C59#12411yescharacteristic
133C2×Dic59—41no—
134C2×Dic59—41no—
135Dic118—81no—
136Dic118—81no—
137C59⋊C8—42no—
138C59⋊C8—81no—
139C59⋊C8—81no—
140#10379—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59#39311yescharacteristic
144C22×C59#45511yescharacteristic
145C22×C59#45611yescharacteristic
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59—21no—
151C4×C59—21no—
152C4×C59—42no—
153C4×C59—42no—
154C4×C59—81no—
155C4×C59—81no—
156C4×C59—81no—
157C4×C59—81no—
158Dic59—81no—
159Dic59—81no—
160#2605D5911yescharacteristic
161#1986—591no—
162#1515—591no—
163#1506—591no—
164#1300—591no—
165#1288—591no—
166#1285—591no—
167C2×C59—21no—
168C2×C59—21no—
169C2×C59#202811yescharacteristic
170C2×C59#203111yescharacteristic
171C2×C59#203111yescharacteristic
172#533—591no—
173#432—591no—
174#432—591no—
175C4×M16—591no—
176#400—21no—
177#395—21no—
178#391—21no—
179#391—21no—
180#387—21no—
181#387—21no—
182#383C2×D5911yescharacteristic
183C4×C4⋊C4C2×D5911yescharacteristic
184#376C2×D5911yescharacteristic
185#339—1181no—
186#339—1181no—
187#328—1181no—
188#328—1181no—
189#328—1181no—
190#328—1181no—
191C59#1037911yescharacteristic, complemented, Sylow
192C22×Q8—21no—
193C2×M16—1181no—
194C4⋊Q8—591no—
195C4⋊Q8—591no—
196#125—1181no—
197C22⋊Q8—1181no—
198C22⋊Q8—1181no—
199C4×Q8—1181no—
200C4×Q8—1181no—
201#117—1181no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C42—21no—
207C2×C42C22⋊D5911yescharacteristic
208C2×C42C22⋊D5911yescharacteristic
209C2×C42C22×D5911yescharacteristic
210C4⋊C8—1181no—
211C4⋊C8—1181no—
212C22⋊C8—1181no—
213C22⋊C8—1181no—
214#97—1181no—
215#97—1181no—
216#97—1181no—
217C4×C8—1181no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C22⋊D5911yescharacteristic
227#95C22⋊D5911yescharacteristic
228#95C22⋊D5911yescharacteristic
229#95C22⋊D5911yescharacteristic
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—81no—
233C2×Q8—1181no—
234C2×Q8—1181no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C2×C22⋊D5911yescharacteristic
242C22×C4C2×C22⋊D5911yescharacteristic
243C22×C4C2×C22⋊D5911yescharacteristic
244M16—1182no—
245M16—2361no—
246C2×C8—2361no—
247C2×C8—2361no—
248C2×C8—2361no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—81no—
253C4⋊C4—1181no—
254C4⋊C4—1181no—
255C4⋊C4—1181no—
256C4⋊C4—1181no—
257C4⋊C4—2361no—
258C4⋊C4—2361no—
259C22⋊C4—2361no—
260C42—41no—
261C42—41no—
262C42—42no—
263C42—2361no—
264C42Q8×D5911yescharacteristic
265C42#1174568911yescharacteristic
266C42#1174568911yescharacteristic
267C42#1174568911yescharacteristic
268C23#1427369611yescharacteristic
269Q8—84no—
270Q8—4721no—
271Q8—4721no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4#1427370311yescharacteristic
297C2×C4#1427370311yescharacteristic
298C2×C4#1427370311yescharacteristic
299C2×C4#1427371611yescharacteristic
300C2×C4#1427371611yescharacteristic
301C2×C4#1427371611yescharacteristic
302C8—4721no—
303C8—4721no—
304C8—2362no—
305C22—21no—
306C22—21no—
307C22#1683731311yescharacteristic
308C22#1683758011yescharacteristic
309C22#1683758511yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—4721no—
323C4—4721no—
324C2—21no—
325C2—21no—
326C2#2142525411yescharacteristic
327C2#2142526511yescharacteristic
328C2#2142526511yescharacteristic
subgroupG/H
centerC22#16837313
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2605C2
socleC22×C59#393
prime radicalC23×C59#120
hypercenter#2605D59
hypocenterC59#10379
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2605

Complements

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

NH ≅ G/Ntype
28. #14273727Q8semidirect
29. #14273727Q8semidirect
191. C59#10379semidirect

Maximal subgroups

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

subgroupindex
#214250392
#214226462
#214226072
#214219892
#214219602
#214219522
C59×#26052
#1037959