Finite Group Databasebeta

← #28499626

Subgroups — #28499626

Order 15104.

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

normal subgroups only

This group has 324 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#21434667C211yesmaximal, characteristic
2#21425892C211yesmaximal, characteristic, complemented
3#21424225C211yesmaximal, characteristic, complemented
4#21422108C211yesmaximal, characteristic, complemented
5#21422106C211yesmaximal, characteristic
6#21422104C211yesmaximal, characteristic
7C59×#1348C211yesmaximal, characteristic, complemented
8#16837938—21no—
9#16837803C411yescharacteristic, complemented
10#16837787C411yescharacteristic, complemented
11#16837758—21no—
12#16837638C2211yescharacteristic
13#16837633—21no—
14Q16×Dic59C411yescharacteristic, complemented
15#16837487—21no—
16#16837484C411yescharacteristic, complemented
17C2×C8×Dic59C2211yescharacteristic
18#16837267—21no—
19Dic59⋊C16—21no—
20C22⋊(C59⋊C16)C2211yescharacteristic
21C16×Dic59—21no—
22#16837165C2211yescharacteristic
23#16837164—21no—
24C59×#467C2211yescharacteristic
25C59×C22⋊C16C2211yescharacteristic
26C59×#340C2211yescharacteristic
27C2×#11745671—21no—
28C2×D472C2×C411yescharacteristic
29C22×C59⋊C8D411yescharacteristic
30#14273722—41no—
31#14273718—21no—
32#14273716—21no—
33#14273711—41no—
34#14273701—21no—
35Q8×Dic59—21no—
36#14273689—21no—
37(C22⋊D59)⋊C4—41no—
38#14273685—41no—
39C2×C4⋊Dic59—21no—
40C2×C4×Dic59D411yescharacteristic
41C2×C59⋊C16—21no—
42#14273661C2×C411yescharacteristic
43#14273658—21no—
44#14273651—21no—
45#14273649—21no—
46#14273647—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
63C2×C22⋊D59—41no—
64C22×Dic59—41no—
65C22×Dic59D811yescharacteristic
66C22×Dic59D811yescharacteristic
67C2×D236—21no—
68C2×C4×D59—41no—
69Q16×C59—41no—
70C2×C8×C59—41no—
71C2×C8×C59C22×C411yescharacteristic
72C2×C8×C59C2×D411yescharacteristic
73C4⋊C4×C59—21no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78#11745671—42no—
79(C2×D59)⋊C4—41no—
80(C2×D59)⋊C4—41no—
81(C2×D59)⋊C4—41no—
82(C2×D59)⋊C4—41no—
83C4⋊Dic59—21no—
84C4⋊Dic59—41no—
85C4⋊Dic59—41no—
86C4⋊Dic59—41no—
87C4⋊Dic59—41no—
88Dic59⋊C4—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
97D472—41no—
98C16×C59—41no—
99C59⋊C16—41no—
100C23×C59C2×D811yescharacteristic
101C22×D59—41no—
102Q8×C59—41no—
103Q8×C59—41no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—81no—
109C2×C4×C59—81no—
110C2×C4×C59—81no—
111C2×C4×C59C2×C22⋊C411yescharacteristic
112C2×C4×C59C2×SD1611yescharacteristic
113C22⋊D59—81no—
114C22⋊D59—81no—
115C2×Dic59—21no—
116C2×Dic59—21no—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—81no—
120C2×Dic59—81no—
121C2×Dic59—81no—
122C2×Dic59#10211yescharacteristic
123C2×Dic59#10211yescharacteristic
124C2×Dic59#10211yescharacteristic
125C2×Dic59#10211yescharacteristic
126D236—41no—
127D236—41no—
128C4×D59—82no—
129C8×C59—21no—
130C8×C59—21no—
131C59⋊C8—21no—
132C59⋊C8—21no—
133#11855—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59C2×#10211yescharacteristic
137C22×C59#50811yescharacteristic
138C22×C59#50811yescharacteristic
139C2×D59—81no—
140C2×D59—81no—
141C2×D59—81no—
142C4×C59—21no—
143C4×C59—81no—
144C4×C59—81no—
145C4×C59—161no—
146C4×C59#41011yescharacteristic
147C4×C59#42111yescharacteristic
148Dic59—21no—
149Dic59—21no—
150Dic59—21no—
151Dic59—21no—
152Dic59—81no—
153Dic59—161no—
154#3168—591no—
155#2162—591no—
156#1835—591no—
157#1348—591no—
158#1348D5911yescharacteristic, complemented
159#1347—591no—
160#1347—591no—
161C2×C59—21no—
162C2×C59—21no—
163C2×C59#151711yescharacteristic
164C2×C59#213811yescharacteristic
165C2×C59#214511yescharacteristic
166D59—161no—
167#529—1181no—
168#499—591no—
169#495—591no—
170#488—1181no—
171#467C2×D5911yescharacteristic
172#466—591no—
173#465—1181no—
174C4×Q16—591no—
175C4×SD16—1181no—
176C4×D8—591no—
177C2×C4×C8—591no—
178#382—1181no—
179C4⋊C16—1181no—
180C22⋊C16—591no—
181C22⋊C16C2×D5911yescharacteristic
182C4×C16—1181no—
183#340—591no—
184#340—1181no—
185#340C2×D5911yescharacteristic
186C59#1185511yescharacteristic, complemented, Sylow
187C2×Q16C4×D5911yescharacteristic
188C2×SD16—1181no—
189C2×D8—591no—
190C22×C8—591no—
191C22×C8C22×D5911yescharacteristic
192#126—2361no—
193#125—1181no—
194#124—1181no—
195#123—2361no—
196C22⋊Q8—21no—
197#121—1181no—
198C4×Q8—1181no—
199C4×D4—1181no—
200C4×D4—2361no—
201#117—2361no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1181no—
204C2×C42—591no—
205C2×C16—21no—
206C2×C16—1181no—
207#107—591no—
208#107C4×D5911yescharacteristic
209#106—1181no—
210#103—21no—
211#103—1181no—
212#102—1181no—
213#102—1181no—
214C4×C8—591no—
215C4×C8—591no—
216C4×C8—591no—
217C4×C8—591no—
218#95—2361no—
219#95—2361no—
220C2×Q8—21no—
221C2×D4—1181no—
222C2×D4—2361no—
223C22×C4—41no—
224C22×C4—591no—
225C22×C4—591no—
226C22×C4—2361no—
227C22×C4—2361no—
228C22×C4D4×D5911yescharacteristic
229Q16—41no—
230SD16—2362no—
231D8—2361no—
232C2×C8—41no—
233C2×C8—591no—
234C2×C8—591no—
235C2×C8—2361no—
236C2×C8C2×C4×D5911yescharacteristic
237C2×C8D4×D5911yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—1181no—
243C4⋊C4—2361no—
244C4⋊C4—2361no—
245C4⋊C4—2361no—
246C4⋊C4—2361no—
247C4⋊C4—2361no—
248C22⋊C4—41no—
249C22⋊C4—2361no—
250C22⋊C4—2361no—
251C22⋊C4—2361no—
252C22⋊C4—2361no—
253C42—591no—
254C42—591no—
255C42—591no—
256C42—591no—
257C42—2361no—
258C16—41no—
259C16—2361no—
260C23—2361no—
261C23D8×D5911yescharacteristic
262Q8—41no—
263Q8—41no—
264D4—4721no—
265D4—4721no—
266D4—2361no—
267D4—2361no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—591no—
276C2×C4—591no—
277C2×C4—591no—
278C2×C4—591no—
279C2×C4—1181no—
280C2×C4—1181no—
281C2×C4—4722no—
282C2×C4—4721no—
283C2×C4—4721no—
284C2×C4—4721no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4C22⋊C4×D5911yescharacteristic
288C2×C4SD16×D5911yescharacteristic
289C8—21no—
290C8—21no—
291C8—1181no—
292C8—1181no—
293C22—21no—
294C22—21no—
295C22—4721no—
296C22—4721no—
297C22—4721no—
298C22#102×D5911yescharacteristic
299C22#1683783311yescharacteristic
300C22#1683783511yescharacteristic
301C4—21no—
302C4—81no—
303C4—81no—
304C4—161no—
305C4—1181no—
306C4—1181no—
307C4—1181no—
308C4—1181no—
309C4—9441no—
310C4—4721no—
311C4#1683737411yescharacteristic
312C4#1683741711yescharacteristic
313C2—21no—
314C2—21no—
315C2—9441no—
316C2#2142265411yescharacteristic
317C2#2142580211yescharacteristic
318C2#2142583011yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1348C2
socleC22×C59C2×#102
prime radical#16837638C22
cosocleC22×C8×C59C23
hypercenter#1348D59
hypocenterC59#11855
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
2. #21425892C2semidirect
3. #21424225C2semidirect
4. #21422108C2semidirect
7. C59×#1348C2semidirect
9. #16837803C4semidirect
10. #16837787C4semidirect
14. Q16×Dic59C4semidirect
16. #16837484C4semidirect
158. #1348D59semidirect
186. C59#11855semidirect

Maximal subgroups

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

subgroupindex
#214346672
#214258922
#214242252
#214221082
#214221062
#214221042
C59×#13482
#1185559