Finite Group Databasebeta

← #28499641

Subgroups — #28499641

Order 15104.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#21434677C211yesmaximal, characteristic
2#21425896C211yesmaximal, characteristic
3#21424225C211yesmaximal, characteristic
4#21422109C211yesmaximal, characteristic
5#21422108C211yesmaximal, characteristic
6#21422107C211yesmaximal, characteristic
7C59×#1348C211yesmaximal, characteristic
8#16837954—21no—
9#16837806C411yescharacteristic, complemented
10#16837780C411yescharacteristic, complemented
11#16837751—21no—
12#16837643C2211yescharacteristic
13#16837641—21no—
14Q16×Dic59C411yescharacteristic, complemented
15#16837496C411yescharacteristic, complemented
16#16837494—21no—
17C2×C8×Dic59C2211yescharacteristic
18#16837267—21no—
19Dic59⋊C16—21no—
20C22⋊(C59⋊C16)C2211yescharacteristic
21#16837177—21no—
22#16837165C2211yescharacteristic
23#16837164—21no—
24C59×#467C2211yescharacteristic
25C59×C22⋊C16C2211yescharacteristic
26C59×#340C2211yescharacteristic
27C2×Dic236C2×C411yescharacteristic
28C2×#11745672—21no—
29C22×C59⋊C8D411yescharacteristic
30#14273726—21no—
31#14273725—21no—
32#14273717—41no—
33#14273706—21no—
34#14273704—41no—
35Q8×Dic59—21no—
36#14273692—21no—
37#14273692—41no—
38#14273683—41no—
39C2×C4⋊Dic59—21no—
40C2×C4×Dic59D411yescharacteristic
41C2×C59⋊C16—21no—
42#14273661C2×C411yescharacteristic
43#14273660—21no—
44#14273652—21no—
45#14273651—21no—
46#14273650—21no—
47C8×Dic59C2×C411yescharacteristic
48C8×Dic59C2×C411yescharacteristic
49C8×Dic59D411yescharacteristic
50C8×Dic59D411yescharacteristic
51C59⋊#95—41no—
52C59⋊#95—41no—
53C2×Q16×C59C2×C411yescharacteristic
54C22×C8×C59C2311yescharacteristic
55C22⋊Q8×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C16×C59—21no—
58#107×C59C2×C411yescharacteristic
59#103×C59—21no—
60C2×Q8×C59—21no—
61C22×C4×C59—41no—
62C22×C4×C59C2×D411yescharacteristic
63C22×Dic59—41no—
64C22×Dic59D811yescharacteristic
65C22×Dic59D811yescharacteristic
66C2×Dic118—21no—
67C2×Dic118—41no—
68Q16×C59—41no—
69C2×C8×C59—41no—
70C2×C8×C59C22×C411yescharacteristic
71C2×C8×C59C2×D411yescharacteristic
72C4⋊C4×C59—21no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊Dic59—41no—
78#11745672—42no—
79C4⋊Dic59—21no—
80C4⋊Dic59—41no—
81C4⋊Dic59—41no—
82C4⋊Dic59—41no—
83Dic59⋊C4—41no—
84Dic59⋊C4—41no—
85Dic59⋊C4—41no—
86Dic59⋊C4—41no—
87Dic59⋊C4—41no—
88C4×Dic59—41no—
89C4×Dic59—41no—
90C4×Dic59C22⋊C411yescharacteristic
91C4×Dic59C22⋊C411yescharacteristic
92C4×Dic59SD1611yescharacteristic
93C4×Dic59SD1611yescharacteristic
94C2×C59⋊C8—41no—
95C2×C59⋊C8C22⋊C411yescharacteristic
96C2×C59⋊C8C22⋊C411yescharacteristic
97Dic236—41no—
98C16×C59—41no—
99C59⋊C16—41no—
100C23×C59C2×D811yescharacteristic
101Q8×C59—41no—
102Q8×C59—41no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—81no—
108C2×C4×C59—81no—
109C2×C4×C59—81no—
110C2×C4×C59C2×C22⋊C411yescharacteristic
111C2×C4×C59C2×SD1611yescharacteristic
112C2×Dic59—21no—
113C2×Dic59—21no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—41no—
117C2×Dic59—81no—
118C2×Dic59—81no—
119C2×Dic59—81no—
120C2×Dic59#10211yescharacteristic
121C2×Dic59#10211yescharacteristic
122C2×Dic59#10211yescharacteristic
123C2×Dic59#10211yescharacteristic
124Dic118—41no—
125Dic118—41no—
126Dic118—82no—
127C8×C59—21no—
128C8×C59—21no—
129C59⋊C8—21no—
130C59⋊C8—21no—
131#11859—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59C2×#10211yescharacteristic
135C22×C59#51011yescharacteristic
136C22×C59#51011yescharacteristic
137C4×C59—21no—
138C4×C59—81no—
139C4×C59—81no—
140C4×C59—161no—
141C4×C59#41011yescharacteristic
142C4×C59#42111yescharacteristic
143Dic59—21no—
144Dic59—21no—
145Dic59—21no—
146Dic59—21no—
147Dic59—81no—
148Dic59—81no—
149Dic59—161no—
150#3170—591no—
151#2163—591no—
152#1835—591no—
153#1348—591no—
154#1348—591no—
155#1348—591no—
156#1348D5911yescharacteristic
157C2×C59—21no—
158C2×C59—21no—
159C2×C59#151711yescharacteristic
160C2×C59#214111yescharacteristic
161C2×C59#214511yescharacteristic
162#533—1181no—
163#500—591no—
164#494—591no—
165#487—1181no—
166#467—591no—
167#467—1181no—
168#467C2×D5911yescharacteristic
169C4×Q16—591no—
170C4×Q16—591no—
171C4×Q16—1181no—
172C2×C4×C8—591no—
173#382—1181no—
174C4⋊C16—1181no—
175C22⋊C16—591no—
176C22⋊C16C2×D5911yescharacteristic
177#346—1181no—
178#340—591no—
179#340—1181no—
180#340C2×D5911yescharacteristic
181C59#1185911yescharacteristic, complemented, Sylow
182C2×Q16—591no—
183C2×Q16—1181no—
184C2×Q16C4×D5911yescharacteristic
185C22×C8—591no—
186C22×C8C22×D5911yescharacteristic
187C4⋊Q8—1181no—
188C4⋊Q8—1181no—
189#125—2361no—
190C22⋊Q8—21no—
191C22⋊Q8—1181no—
192C22⋊Q8—2361no—
193C4×Q8—1181no—
194C4×Q8—1181no—
195C4×Q8—2361no—
196#117—2361no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—1181no—
199C2×C42—591no—
200C2×C16—21no—
201C2×C16—1181no—
202#107—591no—
203#107—1181no—
204#107C4×D5911yescharacteristic
205#103—21no—
206#103—1181no—
207#103—1181no—
208#103—1181no—
209C4×C8—591no—
210C4×C8—591no—
211C4×C8—591no—
212C4×C8—591no—
213#95—2361no—
214#95—2361no—
215C2×Q8—21no—
216C2×Q8—1181no—
217C2×Q8—2361no—
218C22×C4—41no—
219C22×C4—591no—
220C22×C4—591no—
221C22×C4—2361no—
222C22×C4D4×D5911yescharacteristic
223Q16—41no—
224Q16—2361no—
225Q16—2362no—
226C2×C8—41no—
227C2×C8—591no—
228C2×C8—591no—
229C2×C8—2361no—
230C2×C8C2×C4×D5911yescharacteristic
231C2×C8D4×D5911yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—1181no—
237C4⋊C4—2361no—
238C4⋊C4—2361no—
239C4⋊C4—2361no—
240C4⋊C4—2361no—
241C4⋊C4—2361no—
242C4⋊C4—2361no—
243C4⋊C4—2361no—
244C4⋊C4—2361no—
245C22⋊C4—41no—
246C22⋊C4—2361no—
247C42—591no—
248C42—591no—
249C42—591no—
250C42—591no—
251C42—2361no—
252C42—2361no—
253C16—41no—
254C16—2361no—
255C23D8×D5911yescharacteristic
256Q8—41no—
257Q8—41no—
258Q8—4722no—
259Q8—2361no—
260Q8—2361no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—591no—
269C2×C4—591no—
270C2×C4—591no—
271C2×C4—591no—
272C2×C4—1181no—
273C2×C4—1181no—
274C2×C4—4721no—
275C2×C4—4721no—
276C2×C4—4721no—
277C2×C4—2361no—
278C2×C4—2361no—
279C2×C4—2361no—
280C2×C4C22⋊C4×D5911yescharacteristic
281C2×C4SD16×D5911yescharacteristic
282C8—21no—
283C8—21no—
284C8—1181no—
285C8—1181no—
286C22—21no—
287C22—21no—
288C22#102×D5911yescharacteristic
289C22#1683784111yescharacteristic
290C22#1683784411yescharacteristic
291C4—21no—
292C4—81no—
293C4—81no—
294C4—161no—
295C4—1181no—
296C4—1181no—
297C4—1181no—
298C4—1181no—
299C4—9441no—
300C4—4721no—
301C4—4721no—
302C4#1683737411yescharacteristic
303C4#1683741711yescharacteristic
304C2—21no—
305C2—21no—
306C2#2142265411yescharacteristic
307C2#2142581411yescharacteristic
308C2#2142583011yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1348C2
socleC22×C59C2×#102
prime radicalC23×C59C2×D8
cosocleC22×C8×C59C23
hypercenter#1348D59
hypocenterC59#11859
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
9. #16837806C4semidirect
10. #16837780C4semidirect
14. Q16×Dic59C4semidirect
15. #16837496C4semidirect
181. C59#11859semidirect

Maximal subgroups

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

subgroupindex
#214346772
#214258962
#214242252
#214221092
#214221082
#214221072
C59×#13482
#1185959