Finite Group Databasebeta

← #28491684

Subgroups — #28491684

Order 15104.

640 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 107 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#21430066C211yesmaximal, characteristic
2#21424031C212yesmaximal
3#21421975C212yesmaximal
4#21421939C211yesmaximal, characteristic
5C59×#1446C211yesmaximal, characteristic
6C2×#14273651—22no—
7C2×#14273651—22no—
8C2×C22⋊(C59⋊C8)C2212yes—
9#16837344—22no—
10#16837336—22no—
11#16837275—22no—
12#16837262C2212yes—
13C23×Q8×C59C2211yescharacteristic
14C2×#95×C59—22no—
15C2×#95×C59C2211yescharacteristic
16C2×#95×C59C2211yescharacteristic
17C22×C59⋊C8—22no—
18C2×C4⋊Dic59D412yes—
19C2×C4⋊Dic59D412yes—
20C2×Dic59⋊C4—22no—
21C2×Dic59⋊C4—22no—
22C2×C22⋊Dic59—22no—
23#14273651—44no—
24#14273651—44no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27C59⋊#95—22no—
28C59⋊#95—22no—
29C59⋊#95—22no—
30C22×Q8×C59—21no—
31C22×Q8×C59—22no—
32C22×Q8×C59—22no—
33C22×Q8×C59—44no—
34C22×Q8×C59Q812yes—
35C23×C4×C59—21no—
36C23×C4×C59—21no—
37C23×C4×C59D412yes—
38C23×C4×C59C2311yescharacteristic
39#95×C59—22no—
40#95×C59—22no—
41#95×C59—44no—
42C24×C59C2×D411yescharacteristic
43C2×Q8×C59—24no—
44C2×Q8×C59—24no—
45C2×Q8×C59—24no—
46C2×Q8×C59—44no—
47C2×Q8×C59—42no—
48C2×Q8×C59—42no—
49C2×Q8×C59—44no—
50C2×Q8×C59—48no—
51C2×Q8×C59—42no—
52C22×C4×C59—24no—
53C22×C4×C59—22no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—42no—
59C22×C4×C59—42no—
60C22×C4×C59—41no—
61C22×C4×C59—41no—
62C22×C4×C59—42no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59—42no—
68C22×C4×C59C2×D412yes—
69C22×C4×C59C2×Q811yescharacteristic
70C22×C4×C59#4112yes—
71C22×C4×C59#4111yescharacteristic
72C22×Dic59—22no—
73C22×Dic59—22no—
74C22⋊Dic59—42no—
75C22⋊Dic59—42no—
76C4⋊Dic59—42no—
77C4⋊Dic59—42no—
78Dic59⋊C4—82no—
79Dic59⋊C4—82no—
80C2×C59⋊C8—42no—
81C2×C59⋊C8—42no—
82C2×C59⋊C8—42no—
83C23×C59—21no—
84C23×C59—21no—
85C23×C59—22no—
86C23×C59#12011yescharacteristic
87C23×C59C22⋊Q811yescharacteristic
88C23×C59#12411yescharacteristic
89C23×C59#13512yes—
90C23×C59#13712yes—
91Q8×C59—28no—
92Q8×C59—44no—
93Q8×C59—44no—
94Q8×C59—44no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—24no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—44no—
103C2×C4×C59—42no—
104C2×C4×C59—44no—
105C2×C4×C59—44no—
106C2×C4×C59—42no—
107C2×C4×C59—44no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—82no—
112C2×C4×C59—81no—
113C2×C4×C59—81no—
114C2×C4×C59—82no—
115C2×C4×C59C22⋊Q812yes—
116C2×C4×C59#12412yes—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C59⋊C8—82no—
124#10345—591nomaximal, Sylow
125C22×C59—22no—
126C22×C59—22no—
127C22×C59—22no—
128C22×C59—22no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—21no—
133C22×C59#39311yescharacteristic
134C22×C59#45212yes—
135C22×C59#45511yescharacteristic
136C22×C59#45611yescharacteristic
137C22×C59B(4,𝔽2)12yes—
138C22×C59#47912yes—
139C22×C59#48811yescharacteristic
140C22×C59#49111yescharacteristic
141C4×C59—22no—
142C4×C59—22no—
143C4×C59—44no—
144C4×C59—82no—
145C4×C59—82no—
146Dic59—82no—
147Dic59—82no—
148#2646—591no—
149#1787—592no—
150#1446D5911yescharacteristic
151#1294—592no—
152#1280—591no—
153C2×C59—21no—
154C2×C59—22no—
155C2×C59—21no—
156C2×C59#160312yes—
157C2×C59#160612yes—
158C2×C59#202611yescharacteristic
159C2×C59#202811yescharacteristic
160C2×C59#202911yescharacteristic
161C23×Q8C2×D5911yescharacteristic
162C2×#103—1182no—
163C2×#103—1182no—
164C2×C22⋊C8—592no—
165#400—1182no—
166#398—1182no—
167#385—1182no—
168#380—592no—
169C2×#95—22no—
170C2×#95C2×D5911yescharacteristic
171C2×#95C2×D5911yescharacteristic
172C59#1034511yescharacteristic, complemented, Sylow
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8Dic11812yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C22⋊D5912yes—
181C23×C4C22×D5911yescharacteristic
182C22×C8—1182no—
183C2×C4⋊C4—592no—
184C2×C4⋊C4—592no—
185C2×C4⋊C4—1182no—
186C2×C4⋊C4—1182no—
187C2×C22⋊C4—1182no—
188#103—2364no—
189#103—2364no—
190C22⋊C8—1182no—
191C22⋊C8—1182no—
192#95—22no—
193#95—22no—
194#95—44no—
195#95—1182no—
196#95—1182no—
197#95—1182no—
198C24C2×C22⋊D5911yescharacteristic
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—44no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—48no—
207C2×Q8—42no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—1182no—
225C22×C4—1182no—
226C22×C4C2×Dic11811yescharacteristic
227C22×C4#1174568511yescharacteristic
228C22×C4D4×D5912yes—
229C22×C4#1174568712yes—
230C2×C8—2362no—
231C2×C8—2362no—
232C2×C8—2362no—
233C4⋊C4—4722no—
234C4⋊C4—4722no—
235C4⋊C4—2362no—
236C4⋊C4—2362no—
237C22⋊C4—2362no—
238C22⋊C4—2362no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1427369511yescharacteristic
243C23#1427370611yescharacteristic
244C23#1427371411yescharacteristic
245C23#1427374712yes—
246C23#1427375812yes—
247Q8—28no—
248Q8—44no—
249Q8—44no—
250Q8—44no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—2362no—
272C2×C4—2362no—
273C2×C4—2362no—
274C2×C4—2362no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4#1427370412yes—
278C2×C4#1427371312yes—
279C8—4722no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22#1683731411yescharacteristic
289C22#1683756212yes—
290C22#1683757911yescharacteristic
291C22#1683758411yescharacteristic
292C22#1683758812yes—
293C22#1683770912yes—
294C22#1683775611yescharacteristic
295C22#1683777011yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—82no—
300C4—82no—
301C4—4722no—
302C4—4722no—
303C2—21no—
304C2—22no—
305C2—21no—
306C2#2142305712yes—
307C2#2142307412yes—
308C2#2142524711yescharacteristic
309C2#2142525711yescharacteristic
310C2#2142526011yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1446C2
socleC23×C59#120
prime radicalC24×C59C2×D4
hypercenter#1446D59
hypocenterC59#10345
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
172. C59#10345semidirect

Maximal subgroups

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

subgroupindex
#214300662
#214240312
#214219752
#214219392
C59×#14462
#1034559