Finite Group Databasebeta

← #28501849

Subgroups — #28501849

Order 15104.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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#21425252C211yesmaximal, characteristic
2#21423137C211yesmaximal, characteristic, complemented
3#21423103C211yesmaximal, characteristic, complemented
4#21422123C211yesmaximal, characteristic, complemented
5#21422102C211yesmaximal, characteristic, complemented
6#21422028C211yesmaximal, characteristic
7C59×#2593C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837737—21no—
10#16837732—21no—
11#16837407—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#400C2211yescharacteristic
19C59×#398C2211yescharacteristic
20C59×#394—21no—
21C59×#390C2211yescharacteristic
22C59×#388—21no—
23C59×#388—21no—
24C59×#388—21no—
25C59×#388—21no—
26C4×C4⋊C4×C59—21no—
27C2×#11745664—21no—
28#14273726—41no—
29#14273718—41no—
30#14273706—21no—
31#14273706—21no—
32#14273686—21no—
33#14273660—41no—
34#14273658—41no—
35#14273650—41no—
36#14273650—41no—
37#14273648—41no—
38#14273648—41no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22×D4×C59D411yescharacteristic
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59C2311yescharacteristic
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C42×C59—21no—
54C2×C42×C59—21no—
55C2×C42×C59D411yescharacteristic
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59D411yescharacteristic
61#95×C59D411yescharacteristic
62C24×C59—21no—
63C2×D4×C59—21no—
64C2×D4×C59—21no—
65C2×D4×C59—41no—
66C2×D4×C59—41no—
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—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—42no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—81no—
88C22⋊C4×C59—81no—
89C42×C59—42no—
90C42×C59—41no—
91C42×C59—81no—
92C22⋊Dic59—41no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95C4⋊Dic59—21no—
96C4⋊Dic59—21no—
97Dic59⋊C4—41no—
98Dic59⋊C4—41no—
99C4×Dic59—41no—
100#11745664—41no—
101#11745664—41no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59#12011yescharacteristic
113D4×C59—42no—
114D4×C59—81no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
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#13511yescharacteristic
137C2×C4×C59#13711yescharacteristic
138C2×Dic59—41no—
139C2×Dic59—41no—
140Dic118—81no—
141Dic118—81no—
142C59⋊C8—81no—
143C59⋊C8—81no—
144C59⋊C8—81no—
145#12272—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—81no—
149C22×C59—81no—
150C22×C59—81no—
151C22×C59—81no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59#45211yescharacteristic
156C22×C59#45511yescharacteristic
157C22×C59B(4,𝔽2)11yescharacteristic
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—42no—
161C4×C59—42no—
162C4×C59—41no—
163C4×C59—41no—
164C4×C59—81no—
165C4×C59—81no—
166C4×C59—81no—
167Dic59—81no—
168Dic59—81no—
169#2593D5911yescharacteristic
170#2027—591no—
171#1617—591no—
172#1611—591no—
173#1353—591no—
174#1346—591no—
175#1318—591no—
176C2×C59—21no—
177C2×C59—21no—
178C2×C59—161no—
179C2×C59#160611yescharacteristic
180C2×C59#219811yescharacteristic
181C2×C59#220411yescharacteristic
182#544—591no—
183#484—1181no—
184#483—1181no—
185#418—1181no—
186#400C2×D5911yescharacteristic
187#398C2×D5911yescharacteristic
188#394—21no—
189#390C2×D5911yescharacteristic
190#388—21no—
191#388—21no—
192#388—21no—
193#388—21no—
194C4×C4⋊C4—21no—
195#343—591no—
196#339—1181no—
197#328—1181no—
198#328—1181no—
199#324—591no—
200#324—591no—
201C59#1227211yescharacteristic, complemented, Sylow
202C22×D4C22⋊D5911yescharacteristic
203C2×M16—1181no—
204C4⋊Q8—2361no—
205#125—2361no—
206C22⋊Q8—1181no—
207C22⋊Q8—1181no—
208#117—1181no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22⋊D5911yescharacteristic
212C2×C4⋊C4C22⋊D5911yescharacteristic
213C2×C4⋊C4C22×D5911yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22⋊D5911yescharacteristic
223#107—2361no—
224#106—2361no—
225#103—2361no—
226#103—2361no—
227#102—2361no—
228#102—2361no—
229C22⋊C8—1181no—
230C22⋊C8—1181no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C22⋊D5911yescharacteristic
236#95C22⋊D5911yescharacteristic
237C24—21no—
238C2×Q8—1181no—
239C2×Q8—1181no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4C2×C22⋊D5911yescharacteristic
250C22×C4C2×C22⋊D5911yescharacteristic
251C22×C4C2×C22⋊D5911yescharacteristic
252M16—2361no—
253M16—2361no—
254C2×C8—2361no—
255C2×C8—2361no—
256C2×C8—2361no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—1181no—
264C4⋊C4—1181no—
265C4⋊C4—1181no—
266C4⋊C4—1181no—
267C4⋊C4—2361no—
268C4⋊C4—2361no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—2361no—
276C42—41no—
277C42—42no—
278C42—81no—
279C42—2361no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#1427369611yescharacteristic
288Q8—4721no—
289Q8—4721no—
290D4—42no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
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—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4#1427374711yescharacteristic
316C2×C4#1427375711yescharacteristic
317C8—4721no—
318C8—4721no—
319C8—4721no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#1683756411yescharacteristic
330C22#1683758011yescharacteristic
331C22#1683758911yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—4721no—
342C4—4721no—
343C2—21no—
344C2—21no—
345C2—161no—
346C2#2142307611yescharacteristic
347C2#2142609211yescharacteristic
348C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2593C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2593D59
hypocenterC59#12272
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2593

Complements

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

NH ≅ G/Ntype
2. #21423137C2semidirect
3. #21423103C2semidirect
4. #21422123C2semidirect
5. #21422102C2semidirect
201. C59#12272semidirect

Maximal subgroups

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

subgroupindex
#214252522
#214231372
#214231032
#214221232
#214221022
#214220282
C59×#25932
#1227259