Finite Group Databasebeta

← #28501700

Subgroups — #28501700

Order 15104.

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

normal subgroups only

This group has 386 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#21425441C211yesmaximal, characteristic, complemented
2#21423103C211yesmaximal, characteristic, complemented
3#21423035C211yesmaximal, characteristic
4#21422102C211yesmaximal, characteristic, complemented
5#21422099C211yesmaximal, characteristic
6#21422042C211yesmaximal, characteristic, complemented
7C59×#2823C211yesmaximal, 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×#126×C59—21no—
19C59×#398—21no—
20C59×#398C2211yescharacteristic
21C59×#396—21no—
22C59×#396—21no—
23C59×#396—21no—
24C59×#394—21no—
25C59×#394C2211yescharacteristic
26C59×#384C2211yescharacteristic
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
42#126×C59—42no—
43#126×C59—42no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59C2311yescharacteristic
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59—21no—
59C2×C42×C59D411yescharacteristic
60#95×C59—21no—
61#95×C59—21no—
62#95×C59D411yescharacteristic
63#95×C59D411yescharacteristic
64C24×C59—21no—
65C2×D4×C59—41no—
66C2×D4×C59—81no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×Dic118—21no—
76C2×Dic118—21no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—81no—
85C4⋊C4×C59—81no—
86C4⋊C4×C59—81no—
87C22⋊C4×C59—42no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—81no—
92C22⋊C4×C59—81no—
93C22⋊C4×C59—81no—
94C42×C59—21no—
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—41no—
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—21no—
124C2×C4×C59—21no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
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×C59C2×SD1611yescharacteristic
142C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
143C2×Dic59—41no—
144C2×Dic59—41no—
145Dic118—81no—
146Dic118—81no—
147C59⋊C8—42no—
148C59⋊C8—81no—
149C59⋊C8—81no—
150#12249—591nomaximal, Sylow
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59#45011yescharacteristic
161C22×C59#45611yescharacteristic
162C22×C59B(4,𝔽2)11yescharacteristic
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—42no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—81no—
169C4×C59—81no—
170C4×C59—81no—
171C4×C59—81no—
172Dic59—81no—
173Dic59—81no—
174#2823D5911yescharacteristic
175#2060—591no—
176#1611—591no—
177#1599—591no—
178#1346—591no—
179#1345—591no—
180#1325—591no—
181C2×C59—21no—
182C2×C59—21no—
183C2×C59—161no—
184C2×C59#160411yescharacteristic
185C2×C59#220011yescharacteristic
186C2×C59#220311yescharacteristic
187#544—591no—
188C2×#126—21no—
189#483—1181no—
190#450—1181no—
191#428—1181no—
192#398—21no—
193#398C2×D5911yescharacteristic
194#396—21no—
195#396—21no—
196#396—21no—
197#394—21no—
198#394C2×D5911yescharacteristic
199#384C2×D5911yescharacteristic
200#343—591no—
201#339—1181no—
202#328—1181no—
203#328—1181no—
204#324—591no—
205#324—591no—
206C59#1224911yescharacteristic, complemented, Sylow
207C22×D4C22⋊D5911yescharacteristic
208C2×SD16—2361no—
209C2×M16—1181no—
210C4⋊Q8—2361no—
211#126—42no—
212#126—42no—
213#125—2361no—
214C22⋊Q8—1181no—
215C22⋊Q8—1181no—
216#117—1181no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D5911yescharacteristic
224C2×C4⋊C4C22⋊D5911yescharacteristic
225C2×C4⋊C4C22×D5911yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42C22⋊D5911yescharacteristic
233#107—2361no—
234#106—2361no—
235#106—2361no—
236#103—2361no—
237#102—2361no—
238C22⋊C8—1181no—
239C22⋊C8—1181no—
240#95—21no—
241#95—21no—
242#95C22⋊D5911yescharacteristic
243#95C22⋊D5911yescharacteristic
244C24—21no—
245C2×Q8—1181no—
246C2×Q8—1181no—
247C2×D4—41no—
248C2×D4—81no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C2×C22⋊D5911yescharacteristic
255C22×C4C2×C22⋊D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257SD16—4722no—
258M16—1182no—
259M16—2361no—
260C2×C8—2361no—
261C2×C8—2361no—
262C2×C8—2361no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—1181no—
274C4⋊C4—1181no—
275C4⋊C4—1181no—
276C4⋊C4—1181no—
277C4⋊C4—2361no—
278C4⋊C4—2361no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—2361no—
287C42—21no—
288C42—2361no—
289C42#1174567012yescomplemented
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1427369611yescharacteristic
298Q8—4721no—
299Q8—4721no—
300D4—42no—
301D4—81no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
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—2361no—
324C2×C4—2361no—
325C2×C4C2×#1174567011yescharacteristic
326C2×C4#1427375311yescharacteristic
327C8—4721no—
328C8—4721no—
329C8—2362no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1683755311yescharacteristic
340C22#1683758511yescharacteristic
341C22#1683758911yescharacteristic
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—4721no—
352C4—4721no—
353C2—21no—
354C2—21no—
355C2—161no—
356C2#2142306511yescharacteristic
357C2#2142610311yescharacteristic
358C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2823C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2823D59
hypocenterC59#12249
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2823

Complements

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

NH ≅ G/Ntype
1. #21425441C2semidirect
2. #21423103C2semidirect
4. #21422102C2semidirect
6. #21422042C2semidirect
95. C42×C59SD16semidirect
206. C59#12249semidirect
289. C42#11745670semidirect

Maximal subgroups

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

subgroupindex
#214254412
#214231032
#214230352
#214221022
#214220992
#214220422
C59×#28232
#1224959