Finite Group Databasebeta

← #28501278

Subgroups — #28501278

Order 15104.

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

normal subgroups only

This group has 362 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#21425320C211yesmaximal, characteristic
2#21423132C211yesmaximal, characteristic
3#21423064C211yesmaximal, characteristic
4#21422126C211yesmaximal, characteristic, complemented
5#21422123C211yesmaximal, characteristic, complemented
6#21422042C211yesmaximal, characteristic, complemented
7C59×#2843C211yesmaximal, characteristic, complemented
8#16837983C2211yescharacteristic
9#16837720—21no—
10#16837541—21no—
11C2×#14273655—21no—
12#16837401—21no—
13#16837171C2211yescharacteristic
14#16837135—21no—
15#16837135—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#400—21no—
19C59×#400C2211yescharacteristic
20C59×#398—21no—
21C59×#398—21no—
22C59×#398—21no—
23C59×#395—21no—
24C59×#395C2211yescharacteristic
25C59×#384C2211yescharacteristic
26C59×#383—21no—
27C2×#11745685—21no—
28C2×#11745671—41no—
29C2×#11745664—21no—
30#14273707—41no—
31#14273702—41no—
32#14273701—21no—
33#14273701—21no—
34#14273658—41no—
35#14273655—42no—
36#14273651—41no—
37#14273650—41no—
38#14273647—41no—
39#14273647—41no—
40C22⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—21no—
42C22×Q8×C59D411yescharacteristic
43C2×C4⋊C4×C59—21no—
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×C59D411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59C2311yescharacteristic
52C2×C42×C59D411yescharacteristic
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59D411yescharacteristic
62#95×C59D411yescharacteristic
63C2×Q8×C59—41no—
64C2×Q8×C59—81no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C2×C22⋊D59—41no—
75#11745685—41no—
76#11745685—41no—
77#11745685—41no—
78#11745685—41no—
79C2×D236—21no—
80C2×D236—21no—
81C2×D236—21no—
82C2×C4×D59—41no—
83C2×Dic118—21no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—81no—
88C4⋊C4×C59—81no—
89C4⋊C4×C59—81no—
90C4⋊C4×C59—81no—
91C4⋊C4×C59—81no—
92C4⋊C4×C59—81no—
93C42×C59—21no—
94C42×C59SD1612yescomplemented
95#11745671—82no—
96(C2×D59)⋊C4—41no—
97(C2×D59)⋊C4—41no—
98C4⋊Dic59—21no—
99C4⋊Dic59—21no—
100#11745664—41no—
101#11745664—41no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8—41no—
105C23×C59#12011yescharacteristic
106C22×D59—41no—
107Q8×C59—42no—
108Q8×C59—81no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59C2×SD1611yescharacteristic
133C2×C4×C59#13711yescharacteristic
134C22⋊D59—81no—
135C22⋊D59—81no—
136C2×Dic59—41no—
137D236—41no—
138D236—41no—
139D236—81no—
140D236—81no—
141C4×D59—81no—
142C4×D59—81no—
143Dic118—41no—
144Dic118—41no—
145C59⋊C8—81no—
146C59⋊C8—81no—
147C59⋊C8—81no—
148#12183—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59#44811yescharacteristic
152C22×C59#45411yescharacteristic
153C22×C59B(4,𝔽2)11yescharacteristic
154C2×D59—81no—
155C2×D59—81no—
156C2×D59—81no—
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—42no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—81no—
163C4×C59—81no—
164C4×C59—81no—
165C4×C59—81no—
166C4×C59—81no—
167Dic59—81no—
168#2843D5911yescharacteristic, complemented
169#2039—591no—
170#1616—591no—
171#1604—591no—
172#1354—591no—
173#1353—591no—
174#1325—591no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59#159911yescharacteristic
178C2×C59#220011yescharacteristic
179C2×C59#220311yescharacteristic
180D59—161no—
181#537—591no—
182#481—1181no—
183#448—1181no—
184C2×#104—1181no—
185#417—1181no—
186#400—21no—
187#400C2×D5911yescharacteristic
188#398—21no—
189#398—21no—
190#398—21no—
191#395—21no—
192#395C2×D5911yescharacteristic
193#384C2×D5911yescharacteristic
194#383—21no—
195#343—591no—
196#327—1181no—
197#327—1181no—
198#324—591no—
199#324—591no—
200C59#1218311yescharacteristic, complemented, Sylow
201C2×#41—1181no—
202C22×Q8C22⋊D5911yescharacteristic
203C2×SD16—2361no—
204C2×M16—1181no—
205C22⋊Q8—2361no—
206#121—1181no—
207#121—1181no—
208#121—2361no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22⋊D5911yescharacteristic
216C2×C4⋊C4C22⋊D5911yescharacteristic
217C2×C4⋊C4C22×D5911yescharacteristic
218C2×C42C22⋊D5911yescharacteristic
219#106—2361no—
220#104—2362no—
221#103—2361no—
222#103—2361no—
223#102—2361no—
224#102—2361no—
225C22⋊C8—1181no—
226C22⋊C8—1181no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C22⋊D5911yescharacteristic
236#95C22⋊D5911yescharacteristic
237#41—2361no—
238#41—2361no—
239#41—2361no—
240#41—2361no—
241C2×Q8—41no—
242C2×Q8—81no—
243C2×Q8—1181no—
244C2×D4—1181no—
245C2×D4—1181no—
246C2×D4—1181no—
247C2×D4—2361no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—2361no—
255C22×C4C2×C22⋊D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257C22×C4C2×C22⋊D5911yescharacteristic
258SD16—4722no—
259M16—2361no—
260M16—2361no—
261C2×C8—2361no—
262C2×C8—2361no—
263C2×C8—2361no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—1181no—
274C4⋊C4—1181no—
275C22⋊C4—2361no—
276C22⋊C4—2361no—
277C42—21no—
278C42#1174567112yescomplemented
279C23—2361no—
280C23#1427369611yescharacteristic
281Q8—42no—
282Q8—81no—
283Q8—2361no—
284Q8—2361no—
285D4—4721no—
286D4—4721no—
287D4—4721no—
288D4—4721no—
289D4—2361no—
290D4—2361no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—4721no—
315C2×C4—4721no—
316C2×C4—2361no—
317C2×C4C2×#1174567111yescharacteristic
318C2×C4#1427375811yescharacteristic
319C8—4721no—
320C8—4721no—
321C8—4721no—
322C22—21no—
323C22—21no—
324C22—4721no—
325C22—4721no—
326C22—4721no—
327C22#1683754111yescharacteristic
328C22#1683757511yescharacteristic
329C22#1683758911yescharacteristic
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—4721no—
341C2—21no—
342C2—21no—
343C2—9441no—
344C2#2142303611yescharacteristic
345C2#2142610311yescharacteristic
346C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2843C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2843D59
hypocenterC59#12183
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2843

Complements

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

NH ≅ G/Ntype
4. #21422126C2semidirect
5. #21422123C2semidirect
6. #21422042C2semidirect
7. C59×#2843C2semidirect
94. C42×C59SD16semidirect
168. #2843D59semidirect
200. C59#12183semidirect
278. C42#11745671semidirect

Maximal subgroups

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

subgroupindex
#214253202
#214231322
#214230642
#214221262
#214221232
#214220422
C59×#28432
#1218359