Finite Group Databasebeta

← #28485591

Subgroups — #28485591

Order 15104.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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#21434226C211yesmaximal, characteristic
2#21425039C211yesmaximal, characteristic
3#21422607C211yesmaximal, characteristic
4#21422543C211yesmaximal, characteristic
5#21421989C211yesmaximal, characteristic
6#21421952C211yesmaximal, characteristic
7C59×#2276C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837732—21no—
10#16837687C412yescomplemented
11#16837612C412yescomplemented
12#16837558—21no—
13#16837507—22no—
14#16837459C2211yescharacteristic
15#16837459C2211yescharacteristic
16C4×#11745664C2211yescharacteristic
17#16837162—22no—
18#16837138—21no—
19#16837138—21no—
20Q8⋊(C59⋊C8)—22no—
21C2×C4×Q8×C59—21no—
22C2×C4×Q8×C59C2211yescharacteristic
23C4×C4⋊C4×C59—21no—
24C4×C4⋊C4×C59—22no—
25C4×C4⋊C4×C59—21no—
26C4×C4⋊C4×C59C2211yescharacteristic
27C59×#376—21no—
28C59×#376C2211yescharacteristic
29C2×#11745672—22no—
30C2×#11745664—21no—
31#14273727C2×C411yescharacteristic
32#14273727C2×C411yescharacteristic
33#14273719—21no—
34#14273706—21no—
35#14273706—21no—
36C4×Dic118—21no—
37C4×Dic118—21no—
38#14273686—21no—
39#14273658—22no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)C2×C412yes—
42#14273651—22no—
43#14273650—22no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46#14273638—21no—
47#14273638—21no—
48#14273638—21no—
49C4×C59⋊C8—21no—
50C22×Q8×C59D411yescharacteristic
51C4×Q8×C59—28no—
52C4×Q8×C59—21no—
53C4×Q8×C59—21no—
54C4×Q8×C59C2×C412yes—
55C4×Q8×C59D412yes—
56C2×C4⋊C4×C59—22no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59D411yescharacteristic
60C2×C42×C59—22no—
61C2×C42×C59—21no—
62C2×C42×C59—21no—
63C2×C42×C59—22no—
64C2×C42×C59D412yes—
65C2×C42×C59C2311yescharacteristic
66#95×C59—22no—
67#95×C59—21no—
68#95×C59—21no—
69#95×C59D411yescharacteristic
70#95×C59D411yescharacteristic
71C2×Q8×C59—24no—
72C2×Q8×C59—21no—
73C2×Q8×C59C22⋊C412yes—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—22no—
78C22×C4×C59—21no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C2×Dic118—21no—
83C2×Dic118—21no—
84C4⋊C4×C59—28no—
85C4⋊C4×C59—24no—
86C4⋊C4×C59—22no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59C22⋊C412yes—
90C42×C59—28no—
91C42×C59—24no—
92C42×C59—22no—
93C42×C59—22no—
94C42×C59—42no—
95C42×C59—82no—
96C42×C59C22×C411yescharacteristic
97C42×C59C2×D411yescharacteristic
98C42×C59#4111yescharacteristic
99C42×C59#4111yescharacteristic
100C22⋊Dic59—41no—
101#11745672—44no—
102C4⋊Dic59—21no—
103C4⋊Dic59—21no—
104C4⋊Dic59—21no—
105C4⋊Dic59—21no—
106Dic59⋊C4—41no—
107Dic59⋊C4—41no—
108C4×Dic59—41no—
109#11745664—22no—
110#11745664—41no—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—41no—
114C23×C59#12011yescharacteristic
115Q8×C59—24no—
116Q8×C59—22no—
117Q8×C59#10414yes—
118C2×C4×C59—24no—
119C2×C4×C59—22no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—24no—
125C2×C4×C59—22no—
126C2×C4×C59—22no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—42no—
130C2×C4×C59—41no—
131C2×C4×C59—42no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59C2×C22⋊C411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59#12111yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12311yescharacteristic
142C2×Dic59—41no—
143C2×Dic59—41no—
144Dic118—42no—
145Dic118—81no—
146C59⋊C8—42no—
147C59⋊C8—42no—
148C59⋊C8—81no—
149#9324—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59#38611yescharacteristic
153C22×C59#45511yescharacteristic
154C22×C59#45611yescharacteristic
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—24no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—44no—
162C4×C59—42no—
163C4×C59—42no—
164C4×C59—42no—
165C4×C59C2×#10412yes—
166Dic59—81no—
167Dic59—81no—
168#3113—591no—
169#2276D5911yescharacteristic
170#1986—591no—
171#1506—591no—
172#1491—591no—
173#1300—591no—
174#1285—591no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59#188211yescharacteristic
178C2×C59#189611yescharacteristic
179C2×C59#190511yescharacteristic
180#533—591no—
181C2×C4×Q8—21no—
182C2×C4×Q8C2×D5911yescharacteristic
183#483—1181no—
184#475—592no—
185#462—592no—
186#451—1181no—
187#441—1182no—
188#432—591no—
189#432—591no—
190C4×M16—591no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2×D5911yescharacteristic
195#376—21no—
196#376C2×D5911yescharacteristic
197#339—1182no—
198#328—1181no—
199#328—1181no—
200Q8⋊C8—1182no—
201C59#932411yescharacteristic, complemented, Sylow
202C22×Q8C22⋊D5911yescharacteristic
203C2×Q16—1182no—
204C2×M16—1181no—
205C4⋊Q8—591no—
206C4⋊Q8—591no—
207#125—1181no—
208C22⋊Q8—1181no—
209C22⋊Q8—1181no—
210C4×Q8—28no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—1181no—
214C4×Q8—1181no—
215C4×Q8C4×D5912yes—
216C4×Q8D23612yes—
217#117—1181no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22⋊D5911yescharacteristic
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42C22⋊D5912yes—
227C2×C42C22×D5911yescharacteristic
228#106—1182no—
229C4⋊C8—592no—
230C4⋊C8—1181no—
231#103—1182no—
232#103—1182no—
233C22⋊C8—1181no—
234C22⋊C8—1181no—
235#97—1181no—
236#97—1181no—
237#97—1181no—
238C4×C8—1181no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22⋊D5911yescharacteristic
243#95C22⋊D5911yescharacteristic
244C2×Q8—24no—
245C2×Q8—21no—
246C2×Q8—1181no—
247C2×Q8—1181no—
248C2×Q8(C2×D59)⋊C412yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4C2×C22⋊D5911yescharacteristic
255C22×C4C2×C22⋊D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257Q16—2364no—
258M16—1182no—
259M16—2361no—
260C2×C8—1182no—
261C2×C8—2361no—
262C2×C8—2361no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—1181no—
269C4⋊C4—1181no—
270C4⋊C4—1181no—
271C4⋊C4—1181no—
272C4⋊C4—2361no—
273C4⋊C4—2361no—
274C4⋊C4(C2×D59)⋊C412yes—
275C22⋊C4—2361no—
276C42—28no—
277C42—24no—
278C42—22no—
279C42—22no—
280C42—42no—
281C42—82no—
282C42—2361no—
283C42C2×C4×D5911yescharacteristic
284C42C2×D23611yescharacteristic
285C42#1174568511yescharacteristic
286C42#1174568511yescharacteristic
287C23#1427369611yescharacteristic
288Q8—24no—
289Q8—22no—
290Q8—4721no—
291Q8—2362no—
292Q8#1427365514yes—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4C2×(C2×D59)⋊C411yescharacteristic
314C2×C4C4×C22⋊D5911yescharacteristic
315C2×C4C4×C22⋊D5911yescharacteristic
316C2×C4#1427370111yescharacteristic
317C2×C4#1427370111yescharacteristic
318C2×C4#1427371011yescharacteristic
319C8—4721no—
320C8—2362no—
321C8—2362no—
322C22—21no—
323C22—21no—
324C22#1683728111yescharacteristic
325C22#1683758011yescharacteristic
326C22#1683758511yescharacteristic
327C4—21no—
328C4—21no—
329C4—24no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—4721no—
338C4—4721no—
339C4C2×#1427365512yes—
340C2—21no—
341C2—21no—
342C2#2142446511yescharacteristic
343C2#2142453911yescharacteristic
344C2#2142459011yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2276C2
socleC22×C59#386
prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenter#2276D59
hypocenterC59#9324
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2276

Complements

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

NH ≅ G/Ntype
10. #16837687C4semidirect
11. #16837612C4semidirect
201. C59#9324semidirect

Maximal subgroups

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

subgroupindex
#214342262
#214250392
#214226072
#214225432
#214219892
#214219522
C59×#22762
#932459