Finite Group Databasebeta

← #28501445

Subgroups — #28501445

Order 15104.

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

normal subgroups only

This group has 404 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#21425252C211yesmaximal, characteristic
2#21423103C212yesmaximal, complemented
3#21422102C212yesmaximal, complemented
4#21422028C211yesmaximal, characteristic
5C59×#2724C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837732—22no—
8#16837407—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C2×#123×C59—21no—
14C59×#398C2212yes—
15C59×#397—22no—
16C59×#390C2211yescharacteristic
17C59×#389—21no—
18C59×#385—22no—
19C2×#11745664—21no—
20#14273726—41no—
21#14273718—41no—
22#14273706—22no—
23#14273686—21no—
24#14273658—42no—
25#14273650—42no—
26#14273648—41no—
27#14273648—41no—
28C22⋊(C59⋊C8)—22no—
29C22×D4×C59D411yescharacteristic
30C23×C4×C59—21no—
31#123×C59—42no—
32#123×C59—42no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59D412yes—
37C2×C4⋊C4×C59C2311yescharacteristic
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—21no—
40C2×C22⋊C4×C59—22no—
41C2×C42×C59D411yescharacteristic
42#95×C59—22no—
43#95×C59—22no—
44#95×C59D412yes—
45C24×C59—21no—
46C2×D4×C59—21no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×D4×C59SD1612yes—
50C22×C4×C59—21no—
51C22×C4×C59—22no—
52C22×C4×C59—22no—
53C22×C4×C59—42no—
54C22×C4×C59—81no—
55C22×C4×C59—81no—
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59C2×D412yes—
58C2×Dic118—22no—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—42no—
63C4⋊C4×C59—42no—
64C4⋊C4×C59—81no—
65C22⋊C4×C59—44no—
66C22⋊C4×C59—42no—
67C22⋊C4×C59—44no—
68C42×C59—41no—
69C22⋊Dic59—41no—
70C4⋊Dic59—21no—
71C4⋊Dic59—21no—
72C4⋊Dic59—22no—
73Dic59⋊C4—42no—
74C4×Dic59—41no—
75#11745664—41no—
76#11745664—41no—
77C2×C59⋊C8—42no—
78C2×C59⋊C8—41no—
79C23×C59—22no—
80C23×C59—42no—
81C23×C59—42no—
82C23×C59—41no—
83C23×C59—41no—
84C23×C59#12011yescharacteristic
85D4×C59—42no—
86D4×C59—81no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—42no—
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—84no—
101C2×C4×C59C2×SD1611yescharacteristic
102C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
103C2×Dic59—41no—
104C2×Dic59—41no—
105Dic118—82no—
106C59⋊C8—81no—
107C59⋊C8—82no—
108#12209—591nomaximal, Sylow
109C22×C59—21no—
110C22×C59—21no—
111C22×C59—44no—
112C22×C59—42no—
113C22×C59—82no—
114C22×C59—81no—
115C22×C59—81no—
116C22×C59#45011yescharacteristic
117C22×C59#45511yescharacteristic
118C22×C59B(4,𝔽2)11yescharacteristic
119C4×C59—21no—
120C4×C59—21no—
121C4×C59—42no—
122C4×C59—42no—
123C4×C59—82no—
124C4×C59—82no—
125Dic59—81no—
126Dic59—81no—
127#2724D5911yescharacteristic
128#2027—591no—
129#1611—592no—
130#1346—592no—
131#1318—591no—
132C2×C59—21no—
133C2×C59—21no—
134C2×C59—82no—
135C2×C59#160111yescharacteristic
136C2×C59#219811yescharacteristic
137C2×C59#220311yescharacteristic
138#544—591no—
139C2×#123—21no—
140#483—1182no—
141#418—1181no—
142#398C2×D5912yes—
143#397—22no—
144#390C2×D5911yescharacteristic
145#389—21no—
146#385—22no—
147#343—591no—
148#339—1181no—
149#328—1182no—
150#324—592no—
151C59#1220911yescharacteristic, complemented, Sylow
152C22×D4C22⋊D5911yescharacteristic
153C23×C4—21no—
154C2×M16—1181no—
155C4⋊Q8—2361no—
156#125—2361no—
157#123—42no—
158#123—42no—
159C22⋊Q8—1182no—
160#117—1181no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22⋊D5912yes—
165C2×C4⋊C4C22×D5911yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42C22⋊D5911yescharacteristic
170#106—2362no—
171#103—2362no—
172#102—2361no—
173#102—2361no—
174C22⋊C8—1182no—
175#95—22no—
176#95—22no—
177#95C22⋊D5912yes—
178C24—21no—
179C2×Q8—1182no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4#1174567012yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×C22⋊D5911yescharacteristic
191C22×C4C2×C22⋊D5912yes—
192M16—2361no—
193M16—2361no—
194C2×C8—2362no—
195C2×C8—2361no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—1181no—
203C4⋊C4—1181no—
204C4⋊C4—1182no—
205C4⋊C4—2362no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—2361no—
210C42—41no—
211C42—2361no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#1427369611yescharacteristic
218Q8—4722no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—2361no—
236C2×C4—2361no—
237C2×C4C2×#1174567011yescharacteristic
238C2×C4#1427375311yescharacteristic
239C8—4721no—
240C8—4722no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1683755311yescharacteristic
249C22#1683758011yescharacteristic
250C22#1683758911yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—4721no—
258C4—4721no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2142304811yescharacteristic
263C2#2142609211yescharacteristic
264C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2724C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2724D59
hypocenterC59#12209
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #21423103C2semidirect
3. #21422102C2semidirect
151. C59#12209semidirect

Maximal subgroups

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

subgroupindex
#214252522
#214231032
#214221022
#214220282
C59×#27242
#1220959