Finite Group Databasebeta

← #28501508

Subgroups — #28501508

Order 15104.

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

normal subgroups only

This group has 460 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#21423069C212yesmaximal, complemented
3#21422126C212yesmaximal, complemented
4#21422028C211yesmaximal, characteristic
5C59×#2683C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837558—22no—
8#16837407—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C2×C22⋊Q8×C59—22no—
14C59×#396—22no—
15C59×#395C2212yes—
16C59×#390C2211yescharacteristic
17C59×#389—21no—
18C59×#386—21no—
19C2×#11745672—42no—
20C2×#11745664—21no—
21#14273726—41no—
22#14273718—41no—
23#14273706—22no—
24#14273686—21no—
25#14273651—42no—
26#14273648—41no—
27#14273648—41no—
28C22⋊(C59⋊C8)—22no—
29C22×Q8×C59D412yes—
30C22×D4×C59D411yescharacteristic
31C23×C4×C59—21no—
32C22⋊Q8×C59—48no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59C2311yescharacteristic
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—21no—
41C2×C42×C59D411yescharacteristic
42#95×C59—22no—
43#95×C59D412yes—
44C24×C59—21no—
45C2×Q8×C59—44no—
46C2×Q8×C59—42no—
47C2×D4×C59—21no—
48C2×D4×C59—41no—
49C2×D4×C59—41no—
50C2×D4×C59Q1612yescomplemented
51C22×C4×C59—22no—
52C22×C4×C59—21no—
53C22×C4×C59—22no—
54C22×C4×C59—44no—
55C22×C4×C59—81no—
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59C2×D412yes—
58C2×Dic118—22no—
59C4⋊C4×C59—24no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—44no—
62C4⋊C4×C59—44no—
63C4⋊C4×C59—81no—
64C22⋊C4×C59—44no—
65C22⋊C4×C59—44no—
66C22⋊C4×C59—81no—
67C42×C59—41no—
68C22⋊Dic59—41no—
69#11745672—84no—
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
85Q8×C59—44no—
86Q8×C59—82no—
87D4×C59—42no—
88D4×C59—81no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59—24no—
93C2×C4×C59—22no—
94C2×C4×C59—42no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—84no—
103C2×C4×C59C2×Q1611yescharacteristic
104C2×C4×C59#13711yescharacteristic
105C2×Dic59—41no—
106C2×Dic59—41no—
107Dic118—82no—
108C59⋊C8—81no—
109C59⋊C8—82no—
110#12220—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59—44no—
114C22×C59—42no—
115C22×C59—82no—
116C22×C59—81no—
117C22×C59—81no—
118C22×C59#45111yescharacteristic
119C22×C59#45511yescharacteristic
120C22×C59B(4,𝔽2)11yescharacteristic
121C4×C59—21no—
122C4×C59—21no—
123C4×C59—42no—
124C4×C59—42no—
125C4×C59—44no—
126C4×C59—82no—
127Dic59—81no—
128Dic59—81no—
129#2683D5911yescharacteristic
130#2027—591no—
131#1605—592no—
132#1354—592no—
133#1318—591no—
134C2×C59—21no—
135C2×C59—21no—
136C2×C59—82no—
137C2×C59#160211yescharacteristic
138C2×C59#219611yescharacteristic
139C2×C59#220311yescharacteristic
140#544—591no—
141C2×C22⋊Q8—22no—
142#451—1182no—
143#418—1181no—
144#396—22no—
145#395C2×D5912yes—
146#390C2×D5911yescharacteristic
147#389—21no—
148#386—21no—
149#343—591no—
150#339—1181no—
151#328—1182no—
152#324—592no—
153C59#1222011yescharacteristic, complemented, Sylow
154C22×Q8C22⋊D5912yes—
155C22×D4C22⋊D5911yescharacteristic
156C23×C4—21no—
157C2×Q16—2362no—
158C2×M16—1181no—
159C4⋊Q8—2361no—
160#125—2361no—
161C22⋊Q8—48no—
162C22⋊Q8—1182no—
163#117—1181no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22×D5911yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C42C22⋊D5911yescharacteristic
173#103—2362no—
174#102—2361no—
175#102—2361no—
176C22⋊C8—1182no—
177#95—22no—
178#95C22⋊D5912yes—
179C24—21no—
180C2×Q8—44no—
181C2×Q8—42no—
182C2×Q8—1182no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#1174567212yescomplemented
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4C2×C22⋊D5911yescharacteristic
193C22×C4C2×C22⋊D5912yes—
194Q16—4724no—
195M16—2361no—
196M16—2361no—
197C2×C8—2362no—
198C2×C8—2361no—
199C4⋊C4—24no—
200C4⋊C4—41no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—1181no—
205C4⋊C4—1181no—
206C4⋊C4—1182no—
207C4⋊C4—2362no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—81no—
211C22⋊C4—2361no—
212C42—41no—
213C42—2361no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#1427369611yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—4722no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—84no—
239C2×C4—2361no—
240C2×C4—2361no—
241C2×C4C2×#1174567211yescharacteristic
242C2×C4#1427375811yescharacteristic
243C8—4721no—
244C8—4722no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1683755811yescharacteristic
253C22#1683758011yescharacteristic
254C22#1683758911yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—4721no—
262C4—4721no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2142305311yescharacteristic
267C2#2142608011yescharacteristic
268C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2683C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2683D59
hypocenterC59#12220
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2683

Complements

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

NH ≅ G/Ntype
2. #21423069C2semidirect
3. #21422126C2semidirect
50. C2×D4×C59Q16semidirect
153. C59#12220semidirect
186. C2×D4#11745672semidirect

Maximal subgroups

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

subgroupindex
#214252522
#214230692
#214221262
#214220282
C59×#26832
#1222059