Finite Group Databasebeta

← #28501693

Subgroups — #28501693

Order 15104.

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

normal subgroups only

This group has 428 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#21425446C211yesmaximal, characteristic, complemented
2#21423103C211yesmaximal, characteristic, complemented
3#21423035C211yesmaximal, characteristic
4#21422102C211yesmaximal, characteristic, complemented
5#21422099C211yesmaximal, characteristic
6#21422040C211yesmaximal, characteristic, complemented
7C59×#2609C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837732—21no—
10#16837553—21no—
11#16837442—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C2×#117×C59—21no—
19C59×#398C2211yescharacteristic
20C59×#396—21no—
21C59×#394C2211yescharacteristic
22C59×#388—21no—
23C59×#387—21no—
24C59×#386—21no—
25C59×#385—21no—
26C59×#382C2211yescharacteristic
27C2×#11745670—41no—
28C2×#11745664—21no—
29#14273726—41no—
30#14273718—41no—
31#14273706—21no—
32#14273706—21no—
33#14273686—21no—
34#14273660—41no—
35#14273658—41no—
36#14273658—41no—
37#14273650—41no—
38#14273648—41no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22×D4×C59D411yescharacteristic
42C23×C4×C59—21no—
43#117×C59—42no—
44#117×C59—42no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C42×C59—21no—
57C2×C42×C59D411yescharacteristic
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62#95×C59D411yescharacteristic
63#95×C59D411yescharacteristic
64C24×C59—21no—
65C2×D4×C59—41no—
66C2×D4×C59—42no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—42no—
73C22×C4×C59—42no—
74C22×C4×C59—42no—
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C2×Dic118—21no—
79C2×Dic118—21no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—42no—
82C4⋊C4×C59—42no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C22⋊C4×C59—42no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—42no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—81no—
93C42×C59—21no—
94C42×C59—42no—
95C42×C59SD1612yescomplemented
96C22⋊Dic59—41no—
97#11745670—82no—
98C4⋊Dic59—21no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102Dic59⋊C4—41no—
103Dic59⋊C4—41no—
104C4×Dic59—41no—
105#11745664—22no—
106#11745664—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C23×C59—22no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59#12011yescharacteristic
118D4×C59—42no—
119D4×C59—81no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—48no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59C2×SD1611yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×Dic59—41no—
145C2×Dic59—41no—
146Dic118—81no—
147Dic118—81no—
148C59⋊C8—42no—
149C59⋊C8—81no—
150C59⋊C8—81no—
151#12248—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—42no—
155C22×C59—42no—
156C22×C59—42no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59#45011yescharacteristic
162C22×C59#45611yescharacteristic
163C22×C59B(4,𝔽2)11yescharacteristic
164C4×C59—21no—
165C4×C59—21no—
166C4×C59—42no—
167C4×C59—42no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—81no—
171C4×C59—81no—
172C4×C59—81no—
173Dic59—81no—
174Dic59—81no—
175#2609D5911yescharacteristic
176#2061—591no—
177#1611—591no—
178#1599—591no—
179#1346—591no—
180#1345—591no—
181#1324—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—82no—
185C2×C59#160411yescharacteristic
186C2×C59#219711yescharacteristic
187C2×C59#220411yescharacteristic
188#544—591no—
189C2×#117—21no—
190#483—1181no—
191#450—1181no—
192#428—1181no—
193#398C2×D5911yescharacteristic
194#396—21no—
195#394C2×D5911yescharacteristic
196#388—21no—
197#387—21no—
198#386—21no—
199#385—21no—
200#382C2×D5911yescharacteristic
201#343—591no—
202#339—1181no—
203#328—1181no—
204#328—1181no—
205#324—591no—
206#324—591no—
207C59#1224811yescharacteristic, complemented, Sylow
208C22×D4C22⋊D5911yescharacteristic
209C23×C4—21no—
210C2×SD16—2361no—
211C2×M16—1181no—
212C4⋊Q8—2361no—
213#125—2361no—
214C22⋊Q8—1181no—
215C22⋊Q8—1181no—
216#117—42no—
217#117—42no—
218#117—1181no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22⋊D5911yescharacteristic
223C2×C4⋊C4C22⋊D5911yescharacteristic
224C2×C4⋊C4C22×D5911yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42C22⋊D5911yescharacteristic
232#107—2361no—
233#106—2361no—
234#106—2361no—
235#103—2361no—
236#102—2361no—
237C22⋊C8—1181no—
238C22⋊C8—1181no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95C22⋊D5911yescharacteristic
244#95C22⋊D5911yescharacteristic
245C24—21no—
246C2×Q8—1181no—
247C2×Q8—1181no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×C22⋊D5911yescharacteristic
259C22×C4C2×C22⋊D5911yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261SD16—4722no—
262M16—1182no—
263M16—2361no—
264C2×C8—2361no—
265C2×C8—2361no—
266C2×C8—2361no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—1181no—
275C4⋊C4—1181no—
276C4⋊C4—1181no—
277C4⋊C4—1181no—
278C4⋊C4—2361no—
279C4⋊C4—2361no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—81no—
286C22⋊C4—2361no—
287C42—21no—
288C42—42no—
289C42—2361no—
290C42#1174567012yescomplemented
291C23—22no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#1427369611yescharacteristic
299Q8—4721no—
300Q8—4721no—
301D4—42no—
302D4—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4C2×#1174567011yescharacteristic
328C2×C4#1427375311yescharacteristic
329C8—4721no—
330C8—4721no—
331C8—2362no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1683755311yescharacteristic
342C22#1683758511yescharacteristic
343C22#1683758911yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—4721no—
354C4—4721no—
355C2—21no—
356C2—21no—
357C2—82no—
358C2#2142306511yescharacteristic
359C2#2142608611yescharacteristic
360C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2609C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2609D59
hypocenterC59#12248
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2609

Complements

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

NH ≅ G/Ntype
1. #21425446C2semidirect
2. #21423103C2semidirect
4. #21422102C2semidirect
6. #21422040C2semidirect
95. C42×C59SD16semidirect
207. C59#12248semidirect
290. C42#11745670semidirect

Maximal subgroups

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

subgroupindex
#214254462
#214231032
#214230352
#214221022
#214220992
#214220402
C59×#26092
#1224859