Finite Group Databasebeta

← #28498302

Subgroups — #28498302

Order 15104.

748 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 95 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#21422434C211yesmaximal, characteristic, complemented
2#21422417C211yesmaximal, characteristic, complemented
3#21422024C214yesmaximal, complemented
4C59×#2408C211yesmaximal, characteristic
5#16837455—21no—
6#16837366—21no—
7#16837128—21no—
8#16837126—21no—
9#16837126C2212yes—
10#16837126C2212yes—
11C23×D4×C59C411yescharacteristic
12C2×#121×C59—21no—
13C2×#120×C59C412yes—
14C2×C4×D4×C59C2211yescharacteristic
15C59×#394—21no—
16C59×#392—21no—
17C59×#390—21no—
18C59×#380C411yescharacteristic
19C59×#379C2212yes—
20C2×#11745664—41no—
21C4⋊(C59⋊C8)—41no—
22C22⋊(C59⋊C8)—22no—
23C22⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—41no—
25C4×C59⋊C8—41no—
26C25×C59C2×C412yes—
27C22×D4×C59—22no—
28C22×D4×C59—21no—
29C22×D4×C59—21no—
30C22×D4×C59—41no—
31C22×D4×C59—41no—
32C22×D4×C59—44no—
33C22×D4×C59C2×C411yescharacteristic
34C23×C4×C59C2×C411yescharacteristic
35C23×C4×C59D411yescharacteristic
36#121×C59—44no—
37#120×C59—44no—
38C4×D4×C59—41no—
39C4×D4×C59—41no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59C2×C412yes—
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59C2311yescharacteristic
48C2×C42×C59D411yescharacteristic
49#95×C59—21no—
50C24×C59—22no—
51C24×C59—24no—
52C24×C59—42no—
53C24×C59—44no—
54C24×C59—42no—
55C24×C59—82no—
56C24×C59C22⋊C411yescharacteristic
57C24×C59C22×C411yescharacteristic
58C2×D4×C59—24no—
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C2×D4×C59—42no—
63C2×D4×C59—42no—
64C2×D4×C59—44no—
65C2×D4×C59—48no—
66C2×D4×C59—44no—
67C2×D4×C59—44no—
68C2×D4×C59—44no—
69C2×D4×C59—44no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C2×D4×C59—82no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—22no—
78C22×C4×C59—42no—
79C22×C4×C59—42no—
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C4⋊C4×C59—41no—
86C4⋊C4×C59—42no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—44no—
90C22⋊C4×C59—44no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—82no—
93C42×C59—21no—
94C42×C59—21no—
95#11745664—82no—
96C2×C59⋊C8—42no—
97C2×C59⋊C8—42no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—24no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—42no—
109C23×C59—42no—
110C23×C59—42no—
111C23×C59—44no—
112C23×C59—44no—
113C23×C59—42no—
114C23×C59—42no—
115C23×C59—42no—
116C23×C59—84no—
117C23×C59—88no—
118C23×C59—88no—
119C23×C59—84no—
120C23×C59C2×C22⋊C411yescharacteristic
121C23×C59#13112yes—
122D4×C59—44no—
123D4×C59—44no—
124D4×C59—48no—
125D4×C59—44no—
126D4×C59—88no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—22no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—42no—
134C2×C4×C59—42no—
135C2×C4×C59—42no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—81no—
143C2×C4×C59#9911yescharacteristic
144C2×C4×C59#9911yescharacteristic
145C2×C4×C59#10012yes—
146C59⋊C8—82no—
147C59⋊C8—82no—
148#11561—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—22no—
152C22×C59—42no—
153C22×C59—42no—
154C22×C59—42no—
155C22×C59—44no—
156C22×C59—44no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59—41no—
160C22×C59—44no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—81no—
164C22×C59—82no—
165C22×C59—82no—
166C22×C59—88no—
167C22×C59—88no—
168C22×C59—82no—
169C22×C59—82no—
170C22×C59#40811yescharacteristic
171C22×C59C2×#9911yescharacteristic
172C22×C59C2×#10011yescharacteristic
173C4×C59—21no—
174C4×C59—22no—
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—41no—
178C4×C59—82no—
179C4×C59AΣL(1,𝔽16)12yes—
180#2408D5911yescharacteristic
181#1465—591no—
182#1460—591no—
183#1316—594no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—42no—
187C2×C59—81no—
188C2×C59—84no—
189C2×C59#147111yescharacteristic
190C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
191C2×C59#212211yescharacteristic
192C23×D4Dic5911yescharacteristic
193C2×#121—21no—
194C2×#120Dic5912yes—
195C2×C4×D4C2×D5911yescharacteristic
196#431—1181no—
197#407—1181no—
198#394—21no—
199#392—21no—
200#390—21no—
201#380Dic5911yescharacteristic
202#379C2×D5912yes—
203#324—1181no—
204#323—592no—
205#323—592no—
206#323—1181no—
207C59#1156111yescharacteristic, complemented, Sylow
208C25C2×Dic5912yes—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—41no—
213C22×D4—41no—
214C22×D4—44no—
215C22×D4C2×Dic5911yescharacteristic
216C23×C4C2×Dic5911yescharacteristic
217C23×C4C22⋊D5911yescharacteristic
218C2×M16—2361no—
219#121—44no—
220#120—44no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊D5911yescharacteristic
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4C2×Dic5912yes—
229C2×C22⋊C4C22⋊D5911yescharacteristic
230C2×C22⋊C4C22×D5911yescharacteristic
231C2×C42C22⋊D5911yescharacteristic
232C4⋊C8—2361no—
233C22⋊C8—1182no—
234C22⋊C8—1182no—
235C22⋊C8—2361no—
236C4×C8—2361no—
237#95—21no—
238C24—22no—
239C24—24no—
240C24—42no—
241C24—44no—
242C24—42no—
243C24—82no—
244C24C22⋊Dic5911yescharacteristic
245C24C22×Dic5911yescharacteristic
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—82no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4C22⋊Dic5911yescharacteristic
269C22×C4C22⋊Dic5911yescharacteristic
270C22×C4C22⋊Dic5911yescharacteristic
271C22×C4C2×C22⋊D5911yescharacteristic
272C22×C4C2×C22⋊D5911yescharacteristic
273M16—4722no—
274C2×C8—2362no—
275C2×C8—2362no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—21no—
285C42—21no—
286C23—21no—
287C23—21no—
288C23—24no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—84no—
305C23—88no—
306C23—88no—
307C23—84no—
308C23C2×C22⋊Dic5911yescharacteristic
309C23#1427373412yes—
310D4—44no—
311D4—44no—
312D4—48no—
313D4—44no—
314D4—88no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4#1427364211yescharacteristic
332C2×C4#1427364211yescharacteristic
333C2×C4#1427364412yes—
334C8—4722no—
335C8—4722no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—44no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22—88no—
354C22—88no—
355C22—82no—
356C22—82no—
357C22#1683736911yescharacteristic
358C22C2×#1427364211yescharacteristic
359C22C2×#1427364411yescharacteristic
360C4—21no—
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4#1683718912yes—
367C2—21no—
368C2—21no—
369C2—42no—
370C2—81no—
371C2—84no—
372C2#2142245611yescharacteristic
373C2C2×#1683718911yescharacteristic
374C2#2142574611yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2408C2
socleC22×C59C2×#99
cosocleC2×C22⋊C4×C59C23
hypercenter#2408D59
hypocenterC59#11561
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #21422434C2semidirect
2. #21422417C2semidirect
3. #21422024C2semidirect
207. C59#11561semidirect

Maximal subgroups

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

subgroupindex
#214224342
#214224172
#214220242
C59×#24082
#1156159