Finite Group Databasebeta

← #28499633

Subgroups — #28499633

Order 15104.

334 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 89 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#21425892C211yesmaximal, characteristic
3#21424222C211yesmaximal, 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#16837271—21no—
19Dic59⋊C16—21no—
20C22⋊(C59⋊C16)C2211yescharacteristic
21C16×Dic59—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×Dic59⋊C4—41no—
41C2×Dic59⋊C4—41no—
42C2×C4×Dic59D411yescharacteristic
43C2×C59⋊C16—21no—
44#14273661C2×C411yescharacteristic
45#14273660—21no—
46#14273652—21no—
47#14273651—21no—
48#14273650—21no—
49C8×Dic59C2×C411yescharacteristic
50C8×Dic59C2×C411yescharacteristic
51C8×Dic59D411yescharacteristic
52C8×Dic59D411yescharacteristic
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—
88Dic59⋊C4—82no—
89Dic59⋊C4—82no—
90C4×Dic59—41no—
91C4×Dic59—41no—
92C4×Dic59C22⋊C411yescharacteristic
93C4×Dic59C22⋊C411yescharacteristic
94C4×Dic59SD1611yescharacteristic
95C4×Dic59SD1611yescharacteristic
96C2×C59⋊C8—41no—
97C2×C59⋊C8C22⋊C411yescharacteristic
98C2×C59⋊C8C22⋊C411yescharacteristic
99Dic236—41no—
100C16×C59—41no—
101C59⋊C16—41no—
102C23×C59C2×D811yescharacteristic
103Q8×C59—41no—
104Q8×C59—41no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—81no—
110C2×C4×C59—81no—
111C2×C4×C59—81no—
112C2×C4×C59C2×C22⋊C411yescharacteristic
113C2×C4×C59C2×SD1611yescharacteristic
114C2×Dic59—22no—
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
124C2×Dic59SD3212yes—
125C2×Dic59Q3212yes—
126Dic118—41no—
127Dic118—41no—
128Dic118—82no—
129C8×C59—21no—
130C8×C59—21no—
131C59⋊C8—21no—
132C59⋊C8—21no—
133#11857—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59C2×#10211yescharacteristic
137C22×C59C2×SD3211yescharacteristic
138C22×C59C2×Q3211yescharacteristic
139C4×C59—21no—
140C4×C59—81no—
141C4×C59—81no—
142C4×C59—161no—
143C4×C59#41011yescharacteristic
144C4×C59#42111yescharacteristic
145Dic59—21no—
146Dic59—81no—
147Dic59—81no—
148Dic59—161no—
149Dic59#35812yes—
150Dic59#35812yes—
151Dic59#36012yes—
152#3170—591no—
153#2162—591no—
154#1834—591no—
155#1348—591no—
156#1348—591no—
157#1348—591no—
158#1348D5911yescharacteristic
159C2×C59—21no—
160C2×C59—21no—
161C2×C59#151711yescharacteristic
162C2×C59C2×#35811yescharacteristic
163C2×C59C2×#36011yescharacteristic
164#533—1181no—
165#500—591no—
166#494—591no—
167#487—1181no—
168#467—591no—
169#467—1181no—
170#467C2×D5911yescharacteristic
171C4×Q16—591no—
172C4×Q16—591no—
173C4×Q16—1181no—
174C2×C4×C8—591no—
175#384—1181no—
176C4⋊C16—1181no—
177C22⋊C16—591no—
178C22⋊C16C2×D5911yescharacteristic
179C4×C16—1181no—
180#340—591no—
181#340—1181no—
182#340C2×D5911yescharacteristic
183C59#1185711yescharacteristic, complemented, Sylow
184C2×Q16—591no—
185C2×Q16—1181no—
186C2×Q16C4×D5911yescharacteristic
187C22×C8—591no—
188C22×C8C22×D5911yescharacteristic
189C4⋊Q8—1181no—
190C4⋊Q8—1181no—
191#125—2361no—
192C22⋊Q8—21no—
193C22⋊Q8—1181no—
194C22⋊Q8—2361no—
195C4×Q8—1181no—
196C4×Q8—1181no—
197C4×Q8—2361no—
198#117—2361no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—1181no—
201C2×C4⋊C4—2361no—
202C2×C4⋊C4—2361no—
203C2×C42—591no—
204C2×C16—21no—
205C2×C16—1181no—
206#107—591no—
207#107—1181no—
208#107C4×D5911yescharacteristic
209#103—21no—
210#103—1181no—
211#103—1181no—
212#103—1181no—
213C4×C8—591no—
214C4×C8—591no—
215C4×C8—591no—
216C4×C8—591no—
217C2×Q8—21no—
218C2×Q8—1181no—
219C2×Q8—2361no—
220C22×C4—41no—
221C22×C4—591no—
222C22×C4—591no—
223C22×C4—2361no—
224C22×C4D4×D5911yescharacteristic
225Q16—41no—
226Q16—2361no—
227Q16—2362no—
228C2×C8—41no—
229C2×C8—591no—
230C2×C8—591no—
231C2×C8—2361no—
232C2×C8C2×C4×D5911yescharacteristic
233C2×C8D4×D5911yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—1181no—
239C4⋊C4—4722no—
240C4⋊C4—4722no—
241C4⋊C4—2361no—
242C4⋊C4—2361no—
243C4⋊C4—2361no—
244C4⋊C4—2361no—
245C4⋊C4—2361no—
246C4⋊C4—2361no—
247C4⋊C4—2361no—
248C4⋊C4—2361no—
249C22⋊C4—41no—
250C22⋊C4—2361no—
251C42—591no—
252C42—591no—
253C42—591no—
254C42—591no—
255C42—2361no—
256C42—2361no—
257C16—41no—
258C16—2361no—
259C23D8×D5911yescharacteristic
260Q8—41no—
261Q8—41no—
262Q8—4722no—
263Q8—2361no—
264Q8—2361no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—591no—
273C2×C4—591no—
274C2×C4—591no—
275C2×C4—592no—
276C2×C4—592no—
277C2×C4—591no—
278C2×C4—1182no—
279C2×C4—4721no—
280C2×C4—4721no—
281C2×C4—4721no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4C22⋊C4×D5911yescharacteristic
285C2×C4SD16×D5911yescharacteristic
286C8—21no—
287C8—21no—
288C8—1181no—
289C8—1181no—
290C22—21no—
291C22—21no—
292C22#102×D5911yescharacteristic
293C22SD32×D5911yescharacteristic
294C22Q32×D5911yescharacteristic
295C4—21no—
296C4—81no—
297C4—81no—
298C4—161no—
299C4—592no—
300C4—592no—
301C4—592no—
302C4—1181no—
303C4—9441no—
304C4—4721no—
305C4—4721no—
306C4#1683737411yescharacteristic
307C4#1683741711yescharacteristic
308C2—21no—
309C2—21no—
310C2#2142265411yescharacteristic
311C2#358×D5911yescharacteristic
312C2#360×D5911yescharacteristic
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#11857
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
183. C59#11857semidirect

Maximal subgroups

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

subgroupindex
#214346772
#214258922
#214242222
#214221092
#214221082
#214221072
C59×#13482
#1185759