Finite Group Databasebeta

← #28501265

Subgroups — #28501265

Order 15104.

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

normal subgroups only

This group has 372 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#21423132C212yesmaximal
3#21422123C212yesmaximal, complemented
4#21422042C211yesmaximal, characteristic, complemented
5C59×#2571C211yesmaximal, characteristic, complemented
6#16837983C2211yescharacteristic
7#16837720—22no—
8C2×#14273655—21no—
9#16837401—21no—
10#16837171C2211yescharacteristic
11#16837135—22no—
12#16837128C2212yes—
13C59×#400—21no—
14C59×#400C2212yes—
15C59×#389—22no—
16C59×#384C2211yescharacteristic
17C59×#382—22no—
18C4×C4⋊C4×C59—21no—
19C2×#11745685—21no—
20C2×#11745664—21no—
21#14273707—41no—
22#14273702—41no—
23#14273701—22no—
24#14273658—42no—
25#14273655—42no—
26#14273650—41no—
27#14273647—41no—
28#14273647—42no—
29C22⋊(C59⋊C8)—22no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59—22no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59D411yescharacteristic
35C2×C4⋊C4×C59D412yes—
36C2×C4⋊C4×C59C2311yescharacteristic
37C2×C42×C59—22no—
38C2×C42×C59D411yescharacteristic
39#95×C59—21no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59—22no—
43#95×C59D412yes—
44C22×C4×C59—21no—
45C22×C4×C59—22no—
46C22×C4×C59—22no—
47C22×C4×C59—21no—
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59C2×D412yes—
50C2×C22⋊D59—41no—
51#11745685—41no—
52#11745685—41no—
53#11745685—41no—
54#11745685—41no—
55C2×D236—21no—
56C2×D236—22no—
57C2×C4×D59—41no—
58C2×Dic118—21no—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—44no—
62C4⋊C4×C59—42no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—81no—
66C42×C59—21no—
67C42×C59—44no—
68C42×C59SD1612yes—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—22no—
71#11745664—41no—
72#11745664—41no—
73C2×C59⋊C8—42no—
74C2×C59⋊C8—41no—
75C23×C59#12011yescharacteristic
76C22×D59—41no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—22no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—41no—
91C2×C4×C59—42no—
92C2×C4×C59C2×SD1611yescharacteristic
93C2×C4×C59#13711yescharacteristic
94C22⋊D59—81no—
95C22⋊D59—81no—
96C2×Dic59—41no—
97D236—41no—
98D236—41no—
99D236—82no—
100C4×D59—81no—
101C4×D59—81no—
102Dic118—41no—
103Dic118—41no—
104C59⋊C8—81no—
105C59⋊C8—82no—
106#12181—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—21no—
109C22×C59#44811yescharacteristic
110C22×C59#45411yescharacteristic
111C22×C59B(4,𝔽2)11yescharacteristic
112C2×D59—81no—
113C2×D59—81no—
114C2×D59—81no—
115C4×C59—24no—
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—42no—
119C4×C59—82no—
120C4×C59—81no—
121C4×C59—82no—
122Dic59—81no—
123#2571D5911yescharacteristic, complemented
124#2039—591no—
125#1616—592no—
126#1353—592no—
127#1325—591no—
128C2×C59—21no—
129C2×C59—21no—
130C2×C59#159911yescharacteristic
131C2×C59#219811yescharacteristic
132C2×C59#220411yescharacteristic
133D59—161no—
134#537—591no—
135#481—1182no—
136C2×#104—1181no—
137#417—1181no—
138#400—21no—
139#400C2×D5912yes—
140#389—22no—
141#384C2×D5911yescharacteristic
142#382—22no—
143C4×C4⋊C4—21no—
144#343—591no—
145#327—1182no—
146#324—592no—
147C59#1218111yescharacteristic, complemented, Sylow
148C2×#41—1181no—
149C2×M16—1181no—
150C22⋊Q8—2361no—
151#121—1182no—
152#121—2361no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D5911yescharacteristic
158C2×C4⋊C4C22⋊D5912yes—
159C2×C4⋊C4C22×D5911yescharacteristic
160C2×C42—22no—
161C2×C42C22⋊D5911yescharacteristic
162#106—2362no—
163#104—2362no—
164#103—2361no—
165#102—2361no—
166#102—2362no—
167C22⋊C8—1182no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95C22⋊D5912yes—
173#41—2361no—
174#41—2361no—
175#41—2361no—
176#41—2361no—
177C2×Q8—1181no—
178C2×D4—1182no—
179C2×D4—1181no—
180C2×D4—2361no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—2361no—
186C22×C4C2×C22⋊D5911yescharacteristic
187C22×C4C2×C22⋊D5912yes—
188M16—2361no—
189M16—2361no—
190C2×C8—2362no—
191C2×C8—2361no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—1182no—
200C22⋊C4—2362no—
201C42—21no—
202C42—44no—
203C42#1174567112yes—
204C23—2361no—
205C23#1427369611yescharacteristic
206Q8—2361no—
207Q8—2361no—
208D4—4722no—
209D4—4721no—
210D4—4721no—
211D4—2361no—
212D4—2361no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—4721no—
229C2×C4—4721no—
230C2×C4—2361no—
231C2×C4C2×#1174567111yescharacteristic
232C2×C4#1427375811yescharacteristic
233C8—4721no—
234C8—4722no—
235C22—21no—
236C22—21no—
237C22—4721no—
238C22—4721no—
239C22—4721no—
240C22#1683754111yescharacteristic
241C22#1683757511yescharacteristic
242C22#1683758911yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4—4721no—
251C2—21no—
252C2—21no—
253C2—9441no—
254C2#2142303611yescharacteristic
255C2#2142609211yescharacteristic
256C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2571C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2571D59
hypocenterC59#12181
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #21422123C2semidirect
4. #21422042C2semidirect
5. C59×#2571C2semidirect
123. #2571D59semidirect
147. C59#12181semidirect

Maximal subgroups

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

subgroupindex
#214253202
#214231322
#214221232
#214220422
C59×#25712
#1218159