Finite Group Databasebeta

← #28501260

Subgroups — #28501260

Order 15104.

472 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425320C211yesmaximal, characteristic
2#21423064C212yesmaximal
3#21422126C212yesmaximal, complemented
4#21422042C211yesmaximal, characteristic, complemented
5C59×#2551C211yesmaximal, characteristic, complemented
6#16837983C2211yescharacteristic
7#16837541—22no—
8C2×#14273655—21no—
9#16837401—21no—
10#16837171C2211yescharacteristic
11#16837135—22no—
12#16837128C2212yes—
13C2×C4⋊Q8×C59—22no—
14C59×#395—21no—
15C59×#395C2212yes—
16C59×#389—22no—
17C59×#384C2211yescharacteristic
18C4×C4⋊C4×C59—21no—
19C2×#11745685—21no—
20C2×#11745671—42no—
21C2×#11745664—21no—
22#14273707—41no—
23#14273702—41no—
24#14273701—22no—
25#14273655—42no—
26#14273651—42no—
27#14273650—41no—
28#14273647—41no—
29C22⋊(C59⋊C8)—22no—
30C22×Q8×C59—21no—
31C22×Q8×C59D412yes—
32C4⋊Q8×C59—48no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C4⋊C4×C59C2311yescharacteristic
40C2×C42×C59—22no—
41C2×C42×C59D411yescharacteristic
42#95×C59—21no—
43#95×C59—22no—
44#95×C59D412yes—
45C2×Q8×C59—44no—
46C2×Q8×C59—44no—
47C2×Q8×C59—42no—
48C2×Q8×C59—81no—
49C22×C4×C59—21no—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—21no—
53C22×C4×C59C2×D411yescharacteristic
54C22×C4×C59C2×D412yes—
55C2×C22⋊D59—41no—
56#11745685—41no—
57#11745685—41no—
58#11745685—41no—
59#11745685—41no—
60C2×D236—21no—
61C2×D236—22no—
62C2×C4×D59—41no—
63C2×Dic118—21no—
64C4⋊C4×C59—24no—
65C4⋊C4×C59—24no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—44no—
68C4⋊C4×C59—44no—
69C4⋊C4×C59—44no—
70C4⋊C4×C59—41no—
71C42×C59—21no—
72C42×C59—44no—
73C42×C59SD1612yescomplemented
74#11745671—84no—
75(C2×D59)⋊C4—42no—
76C4⋊Dic59—22no—
77#11745664—41no—
78#11745664—41no—
79C2×C59⋊C8—42no—
80C2×C59⋊C8—41no—
81C23×C59#12011yescharacteristic
82C22×D59—41no—
83Q8×C59—44no—
84Q8×C59—82no—
85Q8×C59—84no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—24no—
92C2×C4×C59—24no—
93C2×C4×C59—22no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—41no—
101C2×C4×C59C2×SD1611yescharacteristic
102C2×C4×C59#13711yescharacteristic
103C22⋊D59—81no—
104C22⋊D59—81no—
105C2×Dic59—41no—
106D236—41no—
107D236—41no—
108D236—82no—
109C4×D59—81no—
110C4×D59—81no—
111Dic118—41no—
112Dic118—41no—
113C59⋊C8—81no—
114C59⋊C8—82no—
115#12180—591nomaximal, Sylow
116C22×C59—21no—
117C22×C59—21no—
118C22×C59#44811yescharacteristic
119C22×C59#45411yescharacteristic
120C22×C59B(4,𝔽2)11yescharacteristic
121C2×D59—81no—
122C2×D59—81no—
123C2×D59—81no—
124C4×C59—24no—
125C4×C59—21no—
126C4×C59—21no—
127C4×C59—44no—
128C4×C59—42no—
129C4×C59—44no—
130C4×C59—81no—
131Dic59—81no—
132#2551D5911yescharacteristic, complemented
133#2039—591no—
134#1604—592no—
135#1354—592no—
136#1325—591no—
137C2×C59—21no—
138C2×C59—21no—
139C2×C59#159911yescharacteristic
140C2×C59#219611yescharacteristic
141C2×C59#220511yescharacteristic
142D59—161no—
143#537—591no—
144C2×C4⋊Q8—22no—
145#448—1182no—
146C2×#104—1181no—
147#417—1181no—
148#395—21no—
149#395C2×D5912yes—
150#389—22no—
151#384C2×D5911yescharacteristic
152C4×C4⋊C4—21no—
153#343—591no—
154#327—1182no—
155#324—592no—
156C59#1218011yescharacteristic, complemented, Sylow
157C2×#41—1181no—
158C22×Q8—21no—
159C22×Q8C22⋊D5912yes—
160C2×SD16—2362no—
161C2×M16—1181no—
162C4⋊Q8—48no—
163C22⋊Q8—2361no—
164#121—1182no—
165#121—2361no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22⋊D5911yescharacteristic
172C2×C4⋊C4C22×D5911yescharacteristic
173C2×C42—22no—
174C2×C42C22⋊D5911yescharacteristic
175#104—2362no—
176#103—2361no—
177#103—2362no—
178#102—2361no—
179C22⋊C8—1182no—
180#95—21no—
181#95—22no—
182#95C22⋊D5912yes—
183#41—2361no—
184#41—2361no—
185#41—2361no—
186#41—2361no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—42no—
190C2×Q8—81no—
191C2×Q8—1181no—
192C2×D4—1182no—
193C2×D4—1181no—
194C2×D4—2361no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—2361no—
200C22×C4C2×C22⋊D5911yescharacteristic
201C22×C4C2×C22⋊D5912yes—
202SD16—4724no—
203M16—2361no—
204M16—2361no—
205C2×C8—2362no—
206C2×C8—2361no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—1182no—
215C22⋊C4—2362no—
216C42—21no—
217C42—44no—
218C42#1174567112yescomplemented
219C23—2361no—
220C23#1427369611yescharacteristic
221Q8—44no—
222Q8—82no—
223Q8—84no—
224Q8—2361no—
225Q8—2361no—
226D4—4722no—
227D4—4721no—
228D4—4721no—
229D4—2361no—
230D4—2361no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—4721no—
247C2×C4—4721no—
248C2×C4—2361no—
249C2×C4C2×#1174567111yescharacteristic
250C2×C4#1427375811yescharacteristic
251C8—4721no—
252C8—4722no—
253C22—21no—
254C22—21no—
255C22—4721no—
256C22—4721no—
257C22—4721no—
258C22#1683754111yescharacteristic
259C22#1683757511yescharacteristic
260C22#1683758911yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—44no—
267C4—81no—
268C4—4721no—
269C2—21no—
270C2—21no—
271C2—9441no—
272C2#2142303611yescharacteristic
273C2#2142608011yescharacteristic
274C2#2142612611yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2551C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2551D59
hypocenterC59#12180
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
3. #21422126C2semidirect
4. #21422042C2semidirect
5. C59×#2551C2semidirect
73. C42×C59SD16semidirect
132. #2551D59semidirect
156. C59#12180semidirect
218. C42#11745671semidirect

Maximal subgroups

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

subgroupindex
#214253202
#214230642
#214221262
#214220422
C59×#25512
#1218059