Finite Group Databasebeta

← #28501613

Subgroups — #28501613

Order 15104.

324 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 59 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#21425441C211yesmaximal, characteristic
2#21423137C211yesmaximal, characteristic
3#21423103C211yesmaximal, characteristic
4#21422123C211yesmaximal, characteristic
5#21422102C211yesmaximal, characteristic
6#21422042C211yesmaximal, characteristic
7C59×#2794C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837737—21no—
10#16837732—21no—
11#16837442—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#400—21no—
19C59×#400—21no—
20C59×#400C2211yescharacteristic
21C59×#398C2211yescharacteristic
22C59×#387—21no—
23C59×#387—21no—
24C59×#387—21no—
25C59×#387—21no—
26C59×#384C2211yescharacteristic
27C2×#11745664—21no—
28#14273726—41no—
29#14273718—41no—
30#14273706—21no—
31#14273706—21no—
32#14273686—21no—
33#14273660—41no—
34#14273660—41no—
35#14273658—41no—
36#14273658—41no—
37#14273650—41no—
38#14273650—41no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59C2311yescharacteristic
49C2×C42×C59—21no—
50C2×C42×C59D411yescharacteristic
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59D411yescharacteristic
61#95×C59D411yescharacteristic
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C2×Dic118—21no—
72C2×Dic118—21no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—42no—
75C4⋊C4×C59—42no—
76C4⋊C4×C59—42no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—81no—
85C22⋊Dic59—41no—
86C4⋊Dic59—21no—
87C4⋊Dic59—21no—
88C4⋊Dic59—21no—
89C4⋊Dic59—21no—
90Dic59⋊C4—41no—
91Dic59⋊C4—41no—
92C4×Dic59—41no—
93#11745664—22no—
94#11745664—41no—
95C2×C59⋊C8—41no—
96C2×C59⋊C8—41no—
97C2×C59⋊C8—41no—
98C23×C59#12011yescharacteristic
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59#13511yescharacteristic
123C2×C4×C59#13711yescharacteristic
124C2×Dic59—41no—
125C2×Dic59—41no—
126Dic118—81no—
127Dic118—81no—
128C59⋊C8—42no—
129C59⋊C8—81no—
130C59⋊C8—81no—
131#12236—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59#45211yescharacteristic
135C22×C59#45611yescharacteristic
136C22×C59B(4,𝔽2)11yescharacteristic
137C4×C59—21no—
138C4×C59—21no—
139C4×C59—42no—
140C4×C59—41no—
141C4×C59—41no—
142C4×C59—81no—
143C4×C59—81no—
144C4×C59—81no—
145C4×C59—81no—
146C4×C59—81no—
147Dic59—81no—
148Dic59—81no—
149#2794D5911yescharacteristic
150#2060—591no—
151#1617—591no—
152#1611—591no—
153#1353—591no—
154#1346—591no—
155#1325—591no—
156C2×C59—21no—
157C2×C59—21no—
158C2×C59#160311yescharacteristic
159C2×C59#219811yescharacteristic
160C2×C59#220311yescharacteristic
161#544—591no—
162#484—1181no—
163#483—1181no—
164#428—1181no—
165#400—21no—
166#400—21no—
167#400C2×D5911yescharacteristic
168#398C2×D5911yescharacteristic
169#387—21no—
170#387—21no—
171#387—21no—
172#387—21no—
173#384C2×D5911yescharacteristic
174#343—591no—
175#339—1181no—
176#328—1181no—
177#328—1181no—
178#324—591no—
179#324—591no—
180C59#1223611yescharacteristic, complemented, Sylow
181C2×M16—1181no—
182C4⋊Q8—2361no—
183#125—2361no—
184C22⋊Q8—1181no—
185C22⋊Q8—1181no—
186#117—1181no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22⋊D5911yescharacteristic
192C2×C4⋊C4C22⋊D5911yescharacteristic
193C2×C4⋊C4C22⋊D5911yescharacteristic
194C2×C4⋊C4C22×D5911yescharacteristic
195C2×C42—21no—
196C2×C42C22⋊D5911yescharacteristic
197#107—2361no—
198#107—2361no—
199#106—2361no—
200#106—2361no—
201#103—2361no—
202#103—2361no—
203C22⋊C8—1181no—
204C22⋊C8—1181no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C22⋊D5911yescharacteristic
215#95C22⋊D5911yescharacteristic
216C2×Q8—1181no—
217C2×Q8—1181no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4C2×C22⋊D5911yescharacteristic
225C22×C4C2×C22⋊D5911yescharacteristic
226C22×C4C2×C22⋊D5911yescharacteristic
227M16—1182no—
228M16—2361no—
229C2×C8—2361no—
230C2×C8—2361no—
231C2×C8—2361no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—1181no—
242C4⋊C4—1181no—
243C4⋊C4—1181no—
244C4⋊C4—1181no—
245C4⋊C4—2361no—
246C4⋊C4—2361no—
247C22⋊C4—2361no—
248C42—21no—
249C42—21no—
250C42—81no—
251C42—2361no—
252C23#1427369611yescharacteristic
253Q8—4721no—
254Q8—4721no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—2361no—
279C2×C4—2361no—
280C2×C4#1427374711yescharacteristic
281C2×C4#1427375711yescharacteristic
282C8—4721no—
283C8—4721no—
284C8—2362no—
285C22—21no—
286C22—21no—
287C22#1683756411yescharacteristic
288C22#1683758511yescharacteristic
289C22#1683758911yescharacteristic
290C4—21no—
291C4—21no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—4721no—
301C4—4721no—
302C2—21no—
303C2—21no—
304C2#2142305911yescharacteristic
305C2#2142609211yescharacteristic
306C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2794C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2794D59
hypocenterC59#12236
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2794

Complements

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

NH ≅ G/Ntype
180. C59#12236semidirect

Maximal subgroups

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

subgroupindex
#214254412
#214231372
#214231032
#214221232
#214221022
#214220422
C59×#27942
#1223659