Finite Group Databasebeta

← #28499530

Subgroups — #28499530

Order 15104.

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

normal subgroups only

This group has 286 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#21425908C211yesmaximal, characteristic, complemented
2#21424245C211yesmaximal, characteristic, complemented
3#21422090C211yesmaximal, characteristic, complemented
4#21422087C211yesmaximal, characteristic
5#21422084C211yesmaximal, characteristic, complemented
6#21422081C211yesmaximal, characteristic
7C59×#3183C211yesmaximal, characteristic
8#16837223—21no—
9#16837184C2211yescharacteristic
10#16837184C2211yescharacteristic
11#16837180—21no—
12#16837173—21no—
13#16837162—21no—
14C59⋊#338C2211yescharacteristic
15#16837159C2211yescharacteristic
16C59×#529—21no—
17C59×#499C411yescharacteristic, complemented
18C59×#495C411yescharacteristic, complemented
19C59×#488—21no—
20C59×#470C2211yescharacteristic
21C59×#469C2211yescharacteristic
22C59×#468—21no—
23C59×#442C411yescharacteristic, complemented
24C59×#441C411yescharacteristic, complemented
25C59×#440—21no—
26C4×M16×C59C2211yescharacteristic
27C2×#11745664—21no—
28#14273686—21no—
29#14273667—21no—
30#14273667—21no—
31C2×Q16×C59C2×C411yescharacteristic
32C2×SD16×C59—21no—
33C2×D8×C59C2×C411yescharacteristic
34C2×M16×C59D411yescharacteristic
35C2×M16×C59C2311yescharacteristic
36#126×C59—41no—
37#125×C59—21no—
38#124×C59—21no—
39#123×C59—41no—
40C22⋊Q8×C59—21no—
41#121×C59—21no—
42C4×Q8×C59—21no—
43C4×D4×C59—21no—
44C4×D4×C59—41no—
45#117×C59—41no—
46C2×C42×C59D411yescharacteristic
47#107×C59—21no—
48#106×C59C2×C411yescharacteristic
49#106×C59C2×C411yescharacteristic
50#103×C59—21no—
51#103×C59—21no—
52#102×C59—21no—
53#102×C59—21no—
54#97×C59C2×C411yescharacteristic
55#97×C59D411yescharacteristic
56C4×C8×C59C2×C411yescharacteristic
57C4×C8×C59D411yescharacteristic
58C2×Q8×C59—21no—
59C2×D4×C59—21no—
60C2×D4×C59—41no—
61C22×C4×C59—21no—
62C22×C4×C59—41no—
63C22×C4×C59C2×D411yescharacteristic
64Q16×C59—41no—
65SD16×C59—42no—
66D8×C59—41no—
67M16×C59—41no—
68M16×C59—41no—
69C2×C8×C59C22⋊C411yescharacteristic
70C2×C8×C59C22⋊C411yescharacteristic
71C2×C8×C59C22×C411yescharacteristic
72C2×C8×C59C2×D411yescharacteristic
73C4⋊C4×C59—21no—
74C4⋊C4×C59—21no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C42×C59—21no—
85C42×C59—41no—
86C42×C59C22⋊C411yescharacteristic
87C42×C59C22⋊C411yescharacteristic
88C22⋊Dic59—41no—
89C4⋊Dic59—41no—
90C4×Dic59—41no—
91#11745664—41no—
92#11745664—41no—
93C2×C59⋊C8—41no—
94C59⋊C16—41no—
95C59⋊C16—41no—
96C23×C59—41no—
97C23×C59AGL(1,ℤ/8)11yescharacteristic
98Q8×C59—41no—
99Q8×C59—41no—
100D4×C59—41no—
101D4×C59—41no—
102D4×C59—82no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—81no—
111C2×C4×C59—82no—
112C2×C4×C59C2×C22⋊C411yescharacteristic
113C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
114C2×Dic59—81no—
115C8×C59—21no—
116C8×C59—21no—
117C8×C59—21no—
118C8×C59—21no—
119C59⋊C8—81no—
120#11836—591nomaximal, Sylow
121C22×C59—21no—
122C22×C59—41no—
123C22×C59—81no—
124C22×C59—81no—
125C22×C59—81no—
126C22×C59#41811yescharacteristic
127C4×C59—21no—
128C4×C59—41no—
129C4×C59—41no—
130C4×C59—81no—
131C4×C59—81no—
132C4×C59—81no—
133C4×C59#41011yescharacteristic
134C4×C59#42111yescharacteristic
135Dic59—161no—
136#3183D5911yescharacteristic
137#2165—591no—
138#1841—591no—
139#1342—591no—
140#1341—591no—
141#1340—591no—
142#1339—591no—
143C2×C59—21no—
144C2×C59—41no—
145C2×C59—161no—
146C2×C59#151411yescharacteristic
147#529—21no—
148#499Dic5911yescharacteristic, complemented
149#495Dic5911yescharacteristic, complemented
150#488—21no—
151#470C2×D5911yescharacteristic
152#469C2×D5911yescharacteristic
153#468—21no—
154#442Dic5911yescharacteristic, complemented
155#441Dic5911yescharacteristic, complemented
156#440—21no—
157C4×M16C2×D5911yescharacteristic
158#364—1181no—
159#349—591no—
160#349—591no—
161#347—1181no—
162#344—1181no—
163#339—1181no—
164#338—591no—
165H(3,ℤ/4)—591no—
166C59#1183611yescharacteristic, complemented, Sylow
167C2×Q16C2×Dic5911yescharacteristic
168C2×SD16—21no—
169C2×D8C2×Dic5911yescharacteristic
170C2×M16—1181no—
171C2×M16C22⋊D5911yescharacteristic
172C2×M16C22×D5911yescharacteristic
173#126—41no—
174#125—21no—
175#124—21no—
176#123—41no—
177C22⋊Q8—21no—
178#121—21no—
179C4×Q8—21no—
180C4×D4—21no—
181C4×D4—41no—
182#117—41no—
183#117—1181no—
184C2×C42C22⋊D5911yescharacteristic
185M32—1181no—
186M32—1181no—
187#107—21no—
188#106C2×Dic5911yescharacteristic
189#106C2×Dic5911yescharacteristic
190#103—21no—
191#103—21no—
192#102—21no—
193#102—21no—
194#97C2×Dic5911yescharacteristic
195#97C22⋊D5911yescharacteristic
196C4×C8C2×Dic5911yescharacteristic
197C4×C8C22⋊D5911yescharacteristic
198C2×Q8—21no—
199C2×D4—21no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4C2×C22⋊D5911yescharacteristic
204Q16—41no—
205SD16—42no—
206D8—41no—
207M16—41no—
208M16—41no—
209M16—2361no—
210M16—2361no—
211C2×C8—2361no—
212C2×C8C22⋊Dic5911yescharacteristic
213C2×C8C22⋊Dic5911yescharacteristic
214C2×C8C22×Dic5911yescharacteristic
215C2×C8C2×C22⋊D5911yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—2361no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—2361no—
229C42—21no—
230C42—41no—
231C42—2361no—
232C42C22⋊Dic5911yescharacteristic
233C42C22⋊Dic5911yescharacteristic
234C16—2361no—
235C16—2361no—
236C23—41no—
237C23#1427375311yescharacteristic
238Q8—41no—
239Q8—41no—
240D4—41no—
241D4—41no—
242D4—82no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—4721no—
253C2×C4C2×C22⋊Dic5911yescharacteristic
254C2×C4#1427375311yescharacteristic
255C8—21no—
256C8—21no—
257C8—21no—
258C8—21no—
259C8—4721no—
260C22—21no—
261C22—41no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22#1683740711yescharacteristic
266C4—21no—
267C4—41no—
268C4—41no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—9441no—
273C4#1683737611yescharacteristic
274C4#1683741911yescharacteristic
275C2—21no—
276C2—41no—
277C2—161no—
278C2#2142264311yescharacteristic
subgroupG/H
centerC2#21422643
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#3183C2
socleC2×C59#1514
prime radicalC59×#469C22
cosocleC2×M16×C59C23
hypercenter#3183D59
hypocenterC59#11836
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3183

Complements

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

NH ≅ G/Ntype
1. #21425908C2semidirect
2. #21424245C2semidirect
3. #21422090C2semidirect
5. #21422084C2semidirect
17. C59×#499C4semidirect
18. C59×#495C4semidirect
23. C59×#442C4semidirect
24. C59×#441C4semidirect
148. #499Dic59semidirect
149. #495Dic59semidirect
154. #442Dic59semidirect
155. #441Dic59semidirect
166. C59#11836semidirect

Maximal subgroups

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

subgroupindex
#214259082
#214242452
#214220902
#214220872
#214220842
#214220812
C59×#31832
#1183659