Finite Group Databasebeta

← #28485983

Subgroups — #28485983

Order 15104.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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#21435302C211yesmaximal, characteristic
2#21425039C211yesmaximal, characteristic
3#21422698C211yesmaximal, characteristic
4#21422646C211yesmaximal, characteristic
5#21421960C211yesmaximal, characteristic
6#21421952C211yesmaximal, characteristic
7C59×#2336C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837812C412yescomplemented
10#16837772C412yescomplemented
11#16837737—21no—
12#16837732—21no—
13#16837706—22no—
14#16837459C2211yescharacteristic
15#16837459C2211yescharacteristic
16C4×#11745664C2211yescharacteristic
17#16837162—21no—
18#16837162—21no—
19#16837144—22no—
20#16837138—21no—
21#16837138—21no—
22C2×#125×C59C2211yescharacteristic
23C59×#389—21no—
24C59×#387—21no—
25C59×#387—21no—
26C59×#387—21no—
27C59×#387—21no—
28C59×#383C2211yescharacteristic
29C4×C4⋊C4×C59—21no—
30C4×C4⋊C4×C59C2211yescharacteristic
31C2×#11745664—21no—
32#14273727C2×C411yescharacteristic
33#14273727C2×C411yescharacteristic
34#14273719—21no—
35#14273706—21no—
36#14273706—21no—
37C4×Dic118—21no—
38C4×Dic118—21no—
39#14273686—21no—
40#14273660—22no—
41#14273658—22no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44#14273650—22no—
45#14273650—22no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48#14273638—21no—
49#14273638—21no—
50#14273638C2×C412yes—
51C4×C59⋊C8—21no—
52#125×C59—21no—
53#125×C59—21no—
54#125×C59C2×C412yes—
55#125×C59D412yes—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59D411yescharacteristic
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C42×C59—21no—
64C2×C42×C59—21no—
65C2×C42×C59D411yescharacteristic
66C2×C42×C59D411yescharacteristic
67C2×C42×C59C2311yescharacteristic
68#95×C59—21no—
69#95×C59—21no—
70#95×C59—21no—
71#95×C59—21no—
72#95×C59—21no—
73#95×C59—21no—
74#95×C59—21no—
75#95×C59D411yescharacteristic
76#95×C59D411yescharacteristic
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C2×Dic118—21no—
87C2×Dic118—21no—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—22no—
91C4⋊C4×C59—22no—
92C4⋊C4×C59—21no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—42no—
95C4⋊C4×C59—42no—
96C4⋊C4×C59—81no—
97C4⋊C4×C59C22⋊C412yes—
98C4⋊C4×C59C22⋊C412yes—
99C42×C59—42no—
100C42×C59—41no—
101C42×C59—41no—
102C42×C59—81no—
103C42×C59C22×C411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59#4111yescharacteristic
106C42×C59#4111yescharacteristic
107C22⋊Dic59—41no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110C4⋊Dic59—21no—
111C4⋊Dic59—21no—
112Dic59⋊C4—41no—
113Dic59⋊C4—41no—
114C4×Dic59—41no—
115#11745664—22no—
116#11745664—41no—
117C2×C59⋊C8—22no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C23×C59#12011yescharacteristic
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—22no—
124C2×C4×C59—22no—
125C2×C4×C59—22no—
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×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59C2×C22⋊C411yescharacteristic
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#12111yescharacteristic
148C2×C4×C59#12311yescharacteristic
149C2×Dic59—41no—
150C2×Dic59—41no—
151Dic118—42no—
152Dic118—81no—
153C59⋊C8—42no—
154C59⋊C8—42no—
155C59⋊C8—81no—
156#9380—591nomaximal, Sylow
157C22×C59—21no—
158C22×C59—21no—
159C22×C59#38611yescharacteristic
160C22×C59#45511yescharacteristic
161C22×C59#45611yescharacteristic
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—21no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—42no—
169C4×C59—42no—
170C4×C59—42no—
171C4×C59—81no—
172C4×C59—81no—
173C4×C59—81no—
174Dic59—81no—
175Dic59—81no—
176#3236—591no—
177#2336D5911yescharacteristic
178#1986—591no—
179#1528—591no—
180#1515—591no—
181#1288—591no—
182#1285—591no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59#188211yescharacteristic
186C2×C59#190611yescharacteristic
187C2×C59#191211yescharacteristic
188#533—591no—
189C2×#125C2×D5911yescharacteristic
190#501—592no—
191#491—592no—
192#484—1181no—
193#483—1181no—
194#479—1182no—
195#432—591no—
196#432—591no—
197C4×M16—591no—
198#389—21no—
199#387—21no—
200#387—21no—
201#387—21no—
202#387—21no—
203#383C2×D5911yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2×D5911yescharacteristic
206#339—1181no—
207#339—1181no—
208#330—1182no—
209#328—1181no—
210#328—1181no—
211C59#938011yescharacteristic, complemented, Sylow
212C2×M16—1181no—
213C4⋊Q8—591no—
214C4⋊Q8—591no—
215#125—21no—
216#125—21no—
217#125—1181no—
218#125C4×D5912yes—
219#125D23612yes—
220C22⋊Q8—1181no—
221C22⋊Q8—1181no—
222C4×Q8—1181no—
223C4×Q8—1181no—
224#117—1181no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22⋊D5911yescharacteristic
231C2×C4⋊C4C22⋊D5911yescharacteristic
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22⋊D5911yescharacteristic
235C2×C42C22⋊D5911yescharacteristic
236C2×C42C22×D5911yescharacteristic
237#107—1182no—
238#106—1182no—
239C4⋊C8—1181no—
240C4⋊C8—1181no—
241#103—1182no—
242#103—1182no—
243C22⋊C8—1181no—
244C22⋊C8—1181no—
245#97—592no—
246#97—1181no—
247#97—1181no—
248C4×C8—1181no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95C22⋊D5911yescharacteristic
257#95C22⋊D5911yescharacteristic
258C2×Q8—1181no—
259C2×Q8—1181no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4C2×C22⋊D5911yescharacteristic
267C22×C4C2×C22⋊D5911yescharacteristic
268C22×C4C2×C22⋊D5911yescharacteristic
269M16—1182no—
270M16—2361no—
271C2×C8—1182no—
272C2×C8—2361no—
273C2×C8—2361no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—81no—
283C4⋊C4—1181no—
284C4⋊C4—1181no—
285C4⋊C4—1181no—
286C4⋊C4—1181no—
287C4⋊C4—2361no—
288C4⋊C4—2361no—
289C4⋊C4(C2×D59)⋊C412yes—
290C4⋊C4(C2×D59)⋊C412yes—
291C22⋊C4—2361no—
292C42—41no—
293C42—41no—
294C42—42no—
295C42—81no—
296C42—2361no—
297C42C2×C4×D5911yescharacteristic
298C42C2×D23611yescharacteristic
299C42#1174568511yescharacteristic
300C42#1174568511yescharacteristic
301C23#1427369611yescharacteristic
302Q8—4721no—
303Q8—2362no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4C2×(C2×D59)⋊C411yescharacteristic
329C2×C4C4×C22⋊D5911yescharacteristic
330C2×C4C4×C22⋊D5911yescharacteristic
331C2×C4#1427370111yescharacteristic
332C2×C4#1427370111yescharacteristic
333C2×C4#1427371011yescharacteristic
334C8—4721no—
335C8—2362no—
336C8—2362no—
337C22—21no—
338C22—21no—
339C22#1683728111yescharacteristic
340C22#1683758011yescharacteristic
341C22#1683758511yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—4721no—
355C4—4721no—
356C2—21no—
357C2—21no—
358C2#2142446511yescharacteristic
359C2#2142459711yescharacteristic
360C2#2142463311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2336C2
socleC22×C59#386
prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenter#2336D59
hypocenterC59#9380
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2336

Complements

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

NH ≅ G/Ntype
9. #16837812C4semidirect
10. #16837772C4semidirect
211. C59#9380semidirect

Maximal subgroups

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

subgroupindex
#214353022
#214250392
#214226982
#214226462
#214219602
#214219522
C59×#23362
#938059