Finite Group Databasebeta

← #28499113

Subgroups — #28499113

Order 15104.

352 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 81 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#21425896C211yesmaximal, characteristic, complemented
3#21423988C211yesmaximal, 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×#14273660—21no—
18C2×C8×Dic59C2211yescharacteristic
19C4⋊C4×Dic59—21no—
20Dic59⋊C16—21no—
21C22⋊(C59⋊C16)C2211yescharacteristic
22#16837177—21no—
23#16837165C2211yescharacteristic
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×Dic59—41no—
41C2×C4×Dic59D411yescharacteristic
42C2×C59⋊C16—21no—
43#14273661C2×C411yescharacteristic
44#14273660—24no—
45#14273658—21no—
46#14273651—21no—
47#14273649—21no—
48#14273647—21no—
49C8×Dic59C2×C411yescharacteristic
50C8×Dic59C2×C411yescharacteristic
51C8×Dic59D411yescharacteristic
52C8×Dic59D411yescharacteristic
53C59⋊#95—41no—
54C2×Q16×C59C2×C411yescharacteristic
55C22×C8×C59C2311yescharacteristic
56C22⋊Q8×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C16×C59—21no—
59#107×C59C2×C411yescharacteristic
60#103×C59—21no—
61C2×Q8×C59—21no—
62C22×C4×C59—41no—
63C22×C4×C59C2×D411yescharacteristic
64C2×C22⋊D59—41no—
65C22×Dic59—41no—
66C22×Dic59D811yescharacteristic
67C22×Dic59D811yescharacteristic
68C2×D236—21no—
69C2×C4×D59—41no—
70Q16×C59—41no—
71C2×C8×C59—41no—
72C2×C8×C59C22×C411yescharacteristic
73C2×C8×C59C2×D411yescharacteristic
74C4⋊C4×C59—22no—
75C4⋊C4×C59—21no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79#11745671—42no—
80(C2×D59)⋊C4—41no—
81(C2×D59)⋊C4—41no—
82(C2×D59)⋊C4—41no—
83(C2×D59)⋊C4—41no—
84C4⋊Dic59—22no—
85C4⋊Dic59—21no—
86C4⋊Dic59—41no—
87C4⋊Dic59—41no—
88C4⋊Dic59—41no—
89Dic59⋊C4—41no—
90C4×Dic59—41no—
91C4×Dic59—82no—
92C4×Dic59C22⋊C411yescharacteristic
93C4×Dic59C22⋊C411yescharacteristic
94C4×Dic59SD1611yescharacteristic
95C4×Dic59SD1611yescharacteristic
96C2×C59⋊C8—22no—
97C2×C59⋊C8C22⋊C411yescharacteristic
98C2×C59⋊C8C22⋊C411yescharacteristic
99D472—41no—
100C16×C59—41no—
101C59⋊C16—41no—
102C23×C59C2×D811yescharacteristic
103C22×D59—41no—
104Q8×C59—41no—
105Q8×C59—41no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—42no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—81no—
112C2×C4×C59—81no—
113C2×C4×C59C2×C22⋊C411yescharacteristic
114C2×C4×C59C2×SD1611yescharacteristic
115C22⋊D59—81no—
116C22⋊D59—81no—
117C2×Dic59—21no—
118C2×Dic59—21no—
119C2×Dic59—41no—
120C2×Dic59—42no—
121C2×Dic59—41no—
122C2×Dic59—81no—
123C2×Dic59—81no—
124C2×Dic59#10211yescharacteristic
125C2×Dic59#10211yescharacteristic
126C2×Dic59#10211yescharacteristic
127C2×Dic59#10211yescharacteristic
128D236—41no—
129D236—41no—
130C4×D59—82no—
131C8×C59—21no—
132C8×C59—21no—
133C59⋊C8—21no—
134C59⋊C8#10412yes—
135#11746—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59C2×#10211yescharacteristic
139C22×C59#50911yescharacteristic
140C22×C59#51011yescharacteristic
141C2×D59—81no—
142C2×D59—81no—
143C2×D59—81no—
144C4×C59—21no—
145C4×C59—82no—
146C4×C59—81no—
147C4×C59—81no—
148C4×C59#41011yescharacteristic
149C4×C59C2×#10411yescharacteristic
150Dic59—21no—
151Dic59—21no—
152Dic59—21no—
153Dic59—21no—
154Dic59—82no—
155Dic59—81no—
156#3168—591no—
157#2163—591no—
158C4×#107—591no—
159#1348—591no—
160#1348D5911yescharacteristic, complemented
161#1347—591no—
162#1347—591no—
163C2×C59—21no—
164C2×C59—21no—
165C2×C59#150211yescharacteristic
166C2×C59#213911yescharacteristic
167C2×C59#214511yescharacteristic
168D59—161no—
169#529—1181no—
170#499—591no—
171#495—591no—
172#488—1181no—
173#467C2×D5911yescharacteristic
174#466—591no—
175#465—1181no—
176C4×Q16—591no—
177C4×SD16—1181no—
178C4×D8—591no—
179C2×#107—1181no—
180C2×C4×C8—591no—
181C4×C4⋊C4—1181no—
182C4⋊C16—1181no—
183C22⋊C16—591no—
184C22⋊C16C2×D5911yescharacteristic
185#346—1181no—
186#340—591no—
187#340C2×D5911yescharacteristic
188C59#1174611yescharacteristic, complemented, Sylow
189C2×Q16C4×D5911yescharacteristic
190C2×SD16—1181no—
191C2×D8—591no—
192C22×C8—591no—
193C22×C8C22×D5911yescharacteristic
194#126—2361no—
195#125—1181no—
196#124—1181no—
197#123—2361no—
198C22⋊Q8—21no—
199#121—1181no—
200C4×Q8—1181no—
201C4×D4—1181no—
202C4×D4—2361no—
203#117—2361no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—1181no—
206C2×C42—591no—
207C2×C42—2361no—
208C2×C16—21no—
209C2×C16—1181no—
210#107—591no—
211#107—1184no—
212#107C4×D5911yescharacteristic
213#106—1181no—
214#103—21no—
215#103—1181no—
216#102—1181no—
217#102—1181no—
218C4×C8—591no—
219C4×C8—591no—
220C4×C8—591no—
221C4×C8—591no—
222#95—2361no—
223C2×Q8—21no—
224C2×D4—1181no—
225C2×D4—2361no—
226C22×C4—41no—
227C22×C4—591no—
228C22×C4—591no—
229C22×C4—2361no—
230C22×C4—2361no—
231C22×C4D4×D5911yescharacteristic
232Q16—41no—
233SD16—2362no—
234D8—2361no—
235C2×C8—41no—
236C2×C8—591no—
237C2×C8—591no—
238C2×C8—1182no—
239C2×C8C2×C4×D5911yescharacteristic
240C2×C8D4×D5911yescharacteristic
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—1182no—
246C4⋊C4—1181no—
247C4⋊C4—2361no—
248C4⋊C4—2361no—
249C4⋊C4—2361no—
250C4⋊C4—2361no—
251C22⋊C4—41no—
252C22⋊C4—2361no—
253C22⋊C4—2361no—
254C22⋊C4—2361no—
255C22⋊C4—2361no—
256C42—591no—
257C42—591no—
258C42—591no—
259C42—591no—
260C42—4722no—
261C42—2361no—
262C16—41no—
263C16—2361no—
264C23—2361no—
265C23D8×D5911yescharacteristic
266Q8—41no—
267Q8—41no—
268D4—4721no—
269D4—4721no—
270D4—2361no—
271D4—2361no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—591no—
280C2×C4—591no—
281C2×C4—591no—
282C2×C4—591no—
283C2×C4—1181no—
284C2×C4—1181no—
285C2×C4—4722no—
286C2×C4—4721no—
287C2×C4—4721no—
288C2×C4—2361no—
289C2×C4—2362no—
290C2×C4—2361no—
291C2×C4C22⋊C4×D5911yescharacteristic
292C2×C4SD16×D5911yescharacteristic
293C8—21no—
294C8—21no—
295C8—592no—
296C8—1181no—
297C22—21no—
298C22—21no—
299C22—4721no—
300C22—4721no—
301C22—4721no—
302C22#102×D5911yescharacteristic
303C22#1683783911yescharacteristic
304C22#1683784411yescharacteristic
305C4—21no—
306C4—82no—
307C4—81no—
308C4—81no—
309C4—1181no—
310C4—1181no—
311C4—1181no—
312C4—1181no—
313C4—4722no—
314C4—4721no—
315C4#1683737411yescharacteristic
316C4#104×D5911yescharacteristic
317C2—21no—
318C2—21no—
319C2—9441no—
320C2#2142258911yescharacteristic
321C2#2142580711yescharacteristic
322C2#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#11746
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. #21425896C2semidirect
3. #21423988C2semidirect
4. #21422108C2semidirect
7. C59×#1348C2semidirect
9. #16837803C4semidirect
10. #16837787C4semidirect
14. Q16×Dic59C4semidirect
16. #16837484C4semidirect
160. #1348D59semidirect
188. C59#11746semidirect

Maximal subgroups

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

subgroupindex
#214346672
#214258962
#214239882
#214221082
#214221062
#214221042
C59×#13482
#1174659