Finite Group Databasebeta

← #28501430

Subgroups — #28501430

Order 15104.

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

normal subgroups only

This group has 500 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#21425263C211yesmaximal, characteristic, complemented
2#21423018C212yesmaximal, complemented
3#21422093C212yesmaximal, complemented
4#21422026C211yesmaximal, characteristic, complemented
5C59×#2815C211yesmaximal, characteristic
6#16838028C2211yescharacteristic
7#16837553—22no—
8#16837265—21no—
9#16837169C2211yescharacteristic
10#16837168—21no—
11#16837138—22no—
12#16837126C2212yes—
13C2×#121×C59—21no—
14C59×#397—22no—
15C59×#392—22no—
16C59×#392—21no—
17C59×#392C2212yes—
18C59×#381C2211yescharacteristic
19C2×#11745670—42no—
20#14273706—22no—
21#14273704—42no—
22C2×C22⋊Dic59—21no—
23C2×C22⋊Dic59—21no—
24#14273648—42no—
25C22⋊(C59⋊C8)—22no—
26C59⋊#95—42no—
27C22×D4×C59—21no—
28C22×D4×C59—22no—
29C22×D4×C59—21no—
30C22×D4×C59D412yes—
31C23×C4×C59D411yescharacteristic
32#121×C59—42no—
33#121×C59—42no—
34C2×C4⋊C4×C59—21no—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59D411yescharacteristic
39C2×C22⋊C4×C59C2311yescharacteristic
40#95×C59—21no—
41#95×C59—22no—
42#95×C59D412yes—
43C24×C59—22no—
44C24×C59C2×D411yescharacteristic
45C2×D4×C59—42no—
46C2×D4×C59—42no—
47C2×D4×C59—42no—
48C2×D4×C59—42no—
49C2×D4×C59—82no—
50C2×D4×C59—81no—
51C2×D4×C59—81no—
52C2×D4×C59—82no—
53C2×D4×C59—82no—
54C2×D4×C59—82no—
55C2×D4×C59—81no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—41no—
61C22×C4×C59—41no—
62C22×C4×C59SD1612yescomplemented
63C22×C4×C59C2×D412yes—
64C22×Dic59—22no—
65C22×Dic59—41no—
66C2×Dic118—22no—
67C4⋊C4×C59—42no—
68C22⋊C4×C59—41no—
69C22⋊C4×C59—44no—
70C22⋊C4×C59—41no—
71C22⋊C4×C59—82no—
72C22⋊C4×C59—82no—
73C22⋊Dic59—41no—
74C22⋊Dic59—41no—
75C22⋊Dic59—41no—
76C22⋊Dic59—41no—
77#11745670—84no—
78C4⋊Dic59—22no—
79Dic59⋊C4—42no—
80C2×C59⋊C8—42no—
81C23×C59—21no—
82C23×C59—21no—
83C23×C59—42no—
84C23×C59—42no—
85C23×C59—42no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59#12011yescharacteristic
92C23×C59C2×SD1611yescharacteristic
93C23×C59AGL(1,ℤ/8)11yescharacteristic
94D4×C59—44no—
95D4×C59—82no—
96D4×C59—84no—
97D4×C59—84no—
98D4×C59—84no—
99D4×C59—84no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—41no—
111C2×C4×C59—81no—
112C2×Dic59—42no—
113C2×Dic59—82no—
114C2×Dic59—81no—
115C2×Dic59—82no—
116Dic118—82no—
117C59⋊C8—82no—
118#12206—591nomaximal, Sylow
119C22×C59—22no—
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—42no—
125C22×C59—81no—
126C22×C59—82no—
127C22×C59—82no—
128C22×C59—82no—
129C22×C59—82no—
130C22×C59—82no—
131C22×C59—82no—
132C22×C59—82no—
133C22×C59#45011yescharacteristic
134C22×C59#45511yescharacteristic
135C22×C59B(4,𝔽2)11yescharacteristic
136C4×C59—42no—
137C4×C59—42no—
138C4×C59—82no—
139C4×C59—81no—
140Dic59—82no—
141Dic59—161no—
142#2815D5911yescharacteristic
143#2030—591no—
144#1596—592no—
145#1343—592no—
146#1317—591no—
147C2×C59—21no—
148C2×C59—21no—
149C2×C59—41no—
150C2×C59—41no—
151C2×C59—162no—
152C2×C59#160111yescharacteristic
153C2×C59#220012yes—
154#543—591no—
155C2×#121—21no—
156#450—1182no—
157#397—22no—
158#392—22no—
159#392—21no—
160#392C2×D5912yes—
161#381—1181no—
162#381C2×D5911yescharacteristic
163#342—591no—
164#342—1181no—
165#328—1182no—
166#323—592no—
167C59#1220611yescharacteristic, complemented, Sylow
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—21no—
171C22×D4C22⋊D5912yes—
172C23×C4C22⋊D5911yescharacteristic
173C2×SD16—2362no—
174C22⋊Q8—1182no—
175C22⋊Q8—2362no—
176#121—42no—
177#121—42no—
178C2×C4⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1181no—
183C2×C22⋊C4—1181no—
184C2×C22⋊C4C22⋊D5911yescharacteristic
185C2×C22⋊C4C22×D5911yescharacteristic
186#102—2362no—
187C22⋊C8—1182no—
188#95—21no—
189#95—22no—
190#95—2362no—
191#95C22⋊D5912yes—
192C24—22no—
193C24C2×C22⋊D5911yescharacteristic
194C2×Q8—1182no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—81no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—1182no—
213C22×C4—2361no—
214C22×C4#1174567012yescomplemented
215C22×C4C2×C22⋊D5912yes—
216SD16—4724no—
217C2×C8—2362no—
218C4⋊C4—42no—
219C4⋊C4—1182no—
220C4⋊C4—2362no—
221C22⋊C4—41no—
222C22⋊C4—44no—
223C22⋊C4—41no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C22⋊C4—2361no—
227C22⋊C4—2361no—
228C22⋊C4—2361no—
229C22⋊C4—2361no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23#1427369611yescharacteristic
241C23C2×#1174567011yescharacteristic
242C23#1427375311yescharacteristic
243Q8—4722no—
244D4—44no—
245D4—82no—
246D4—84no—
247D4—84no—
248D4—84no—
249D4—84no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—4721no—
263C2×C4—4722no—
264C2×C4—4722no—
265C2×C4—2362no—
266C8—4722no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22#1683755311yescharacteristic
282C22#1683758011yescharacteristic
283C22#1683758911yescharacteristic
284C4—42no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4—9441no—
289C4—4722no—
290C2—21no—
291C2—21no—
292C2—41no—
293C2—41no—
294C2—162no—
295C2#2142304811yescharacteristic
296C2#2142610312yes—
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2815C2
socleC22×C59B(4,𝔽2)
hypercenter#2815D59
hypocenterC59#12206
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2815

Complements

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

NH ≅ G/Ntype
1. #21425263C2semidirect
2. #21423018C2semidirect
3. #21422093C2semidirect
4. #21422026C2semidirect
62. C22×C4×C59SD16semidirect
167. C59#12206semidirect
214. C22×C4#11745670semidirect

Maximal subgroups

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

subgroupindex
#214252632
#214230182
#214220932
#214220262
C59×#28152
#1220659