Finite Group Databasebeta

← #28487841

Subgroups — #28487841

Order 15104.

468 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434656C211yesmaximal, characteristic
2#21425015C211yesmaximal, characteristic
3C4×#14273658C211yesmaximal, characteristic, complemented
4#21422595C211yesmaximal, characteristic, complemented
5#21422590C211yesmaximal, characteristic
6M16×C59⋊C8C211yesmaximal, characteristic, complemented
7C59×#1986C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837798C411yescharacteristic, complemented
10#16837776C411yescharacteristic, complemented
11#16837752C411yescharacteristic, complemented
12#16837750C411yescharacteristic, complemented
13#16837633—21no—
14#16837633—21no—
15C4×#11745671—21no—
16C4×#11745671—21no—
17#16837455C2211yescharacteristic
18C2×#14273658—21no—
19C2×#14273653—22no—
20C2×C4×C59⋊C8C2211yescharacteristic
21C4×C4⋊Dic59C2211yescharacteristic
22#16837164—21no—
23#16837139—22no—
24#16837135—22no—
25#16837123—21no—
26C8×C59⋊C8—21no—
27C59×#533C2211yescharacteristic
28C4×M16×C59C2211yescharacteristic
29C4×C4⋊C4×C59C2211yescharacteristic
30C59×#339—22no—
31C2×#11745685—21no—
32C2×#11745671—21no—
33C2×#11745671—21no—
34C22×C59⋊C8D411yescharacteristic
35C22×C59⋊C8D411yescharacteristic
36#14273727C2×C411yescharacteristic
37#14273724C2×C411yescharacteristic
38#14273715—21no—
39#14273701—21no—
40#14273701—21no—
41C4×D236—21no—
42C4×D236—21no—
43C2×C4⋊Dic59—21no—
44C2×C4×Dic59—21no—
45#14273658—24no—
46#14273658—21no—
47#14273658—21no—
48C4⋊(C59⋊C8)—21no—
49#14273653—44no—
50#14273651—21no—
51#14273651—21no—
52#14273647—21no—
53#14273647—21no—
54C22⋊(C59⋊C8)D412yes—
55C4×C59⋊C8—21no—
56C4×C59⋊C8C2×C411yescharacteristic
57C4×C59⋊C8C2×C411yescharacteristic
58C4×C59⋊C8D411yescharacteristic
59C4×C59⋊C8D411yescharacteristic
60C59⋊#95—21no—
61C2×M16×C59D412yes—
62C4⋊Q8×C59C2×C411yescharacteristic
63C4⋊Q8×C59C2×C411yescharacteristic
64#125×C59—21no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—21no—
67C4×Q8×C59—21no—
68C4×Q8×C59—21no—
69#117×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C42×C59—21no—
72C2×C42×C59C2311yescharacteristic
73#97×C59—21no—
74C4×C8×C59—21no—
75#95×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C2×C22⋊D59—41no—
83C22×Dic59—41no—
84#11745685—42no—
85#11745685—42no—
86C2×D236—21no—
87C2×D236—21no—
88C2×D236—21no—
89C2×C4×D59—41no—
90C2×Dic118—21no—
91M16×C59—22no—
92M16×C59SD1614yescomplemented
93C2×C8×C59—22no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—41no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C42×C59—41no—
104C42×C59—42no—
105C42×C59C22×C411yescharacteristic
106C42×C59C2×D411yescharacteristic
107C42×C59#4111yescharacteristic
108C42×C59#4111yescharacteristic
109#11745671—42no—
110#11745671—42no—
111(C2×D59)⋊C4—41no—
112(C2×D59)⋊C4—41no—
113C4⋊Dic59—22no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116C4⋊Dic59—41no—
117C4×Dic59—42no—
118C2×C59⋊C8—22no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8C22⋊C411yescharacteristic
122C2×C59⋊C8C22⋊C411yescharacteristic
123C2×C59⋊C8C22⋊C411yescharacteristic
124C2×C59⋊C8C22⋊C411yescharacteristic
125C23×C59#12711yescharacteristic
126C22×D59—41no—
127Q8×C59—41no—
128Q8×C59—41no—
129Q8×C59—81no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—41no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59C2×C22⋊C411yescharacteristic
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12411yescharacteristic
144C2×C4×C59C2×SD1612yes—
145C2×C4×C59#13512yes—
146C22⋊D59—81no—
147C22⋊D59—81no—
148C2×Dic59—42no—
149C2×Dic59—41no—
150C2×Dic59—81no—
151C2×Dic59—81no—
152D236—41no—
153D236—41no—
154D236—42no—
155D236—81no—
156C4×D59—82no—
157Dic118—42no—
158C8×C59—24no—
159C59⋊C8—21no—
160C59⋊C8—21no—
161C59⋊C8—21no—
162C59⋊C8—42no—
163C59⋊C8#10412yescomplemented
164#9717—591nomaximal, Sylow
165C22×C59—21no—
166C22×C59—21no—
167C22×C59#39011yescharacteristic
168C22×C59#49211yescharacteristic
169C22×C59#49511yescharacteristic
170C2×D59—81no—
171C2×D59—81no—
172C2×D59—81no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—82no—
177C4×C59—81no—
178C4×C59—81no—
179C4×C59C2×#10411yescharacteristic
180C4×C59#42111yescharacteristic
181C4×C59C4×SD1612yes—
182C4×C59#46512yes—
183Dic59—82no—
184Dic59—81no—
185#3167—591no—
186#1986D5911yescharacteristic, complemented
187#1980—591no—
188C4×#106—591no—
189#1503—591no—
190#1502—591no—
191C8×M16—591no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59#196811yescharacteristic
195C2×C59#197611yescharacteristic
196C2×C59#198011yescharacteristic
197D59—161no—
198#533C2×D5911yescharacteristic
199#532—591no—
200#498—591no—
201#492—591no—
202#487—591no—
203#486—591no—
204#465—1181no—
205#465—1181no—
206C4×SD16—1181no—
207C4×SD16—1181no—
208#431—591no—
209C2×#106—1181no—
210C2×#104—1182no—
211C4×M16C2×D5911yescharacteristic
212C2×C4×C8—591no—
213C4×C4⋊C4—591no—
214C4×C4⋊C4C2×D5911yescharacteristic
215#340—1181no—
216#339—22no—
217#328—1182no—
218#327—1182no—
219#322—1181no—
220C82—1181no—
221C59#971711yescharacteristic, complemented, Sylow
222C2×#41—1181no—
223C2×SD16—1181no—
224C2×SD16—1181no—
225C2×M16C22⋊D5912yes—
226C22×C8—591no—
227C22×C8—591no—
228C4⋊Q8—591no—
229C4⋊Q8C4×D5911yescharacteristic
230C4⋊Q8C4×D5911yescharacteristic
231#127—591no—
232#125—21no—
233#124—1181no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236#121—1181no—
237#121—1181no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—1181no—
241C4×D4—1181no—
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1181no—
245C2×C42—21no—
246C2×C42—1181no—
247C2×C42C22×D5911yescharacteristic
248#106—1181no—
249#106—1181no—
250#106—1184no—
251C4⋊C8—1181no—
252#104—2364no—
253#103—1181no—
254#103—1181no—
255#102—1181no—
256#102—1181no—
257C22⋊C8—592no—
258#97—21no—
259C4×C8—21no—
260C4×C8—591no—
261C4×C8—591no—
262C4×C8—591no—
263C4×C8—591no—
264C4×C8—1181no—
265#95—21no—
266#95—1181no—
267#41—2362no—
268#41—2362no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—1181no—
272C2×D4—1181no—
273C2×D4—1181no—
274C2×D4—1181no—
275C2×D4—2361no—
276C22×C4—41no—
277C22×C4—2361no—
278C22×C4—2361no—
279C22×C4D4×D5911yescharacteristic
280C22×C4D4×D5911yescharacteristic
281C22×C4C2×C22⋊D5911yescharacteristic
282SD16—2362no—
283SD16—2362no—
284M16—22no—
285M16#1174567114yescomplemented
286C2×C8—22no—
287C2×C8—591no—
288C2×C8—591no—
289C2×C8—591no—
290C2×C8—591no—
291C2×C8—1182no—
292C2×C8—1182no—
293C2×C8—2361no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—1182no—
303C4⋊C4—1181no—
304C4⋊C4—1181no—
305C4⋊C4—2361no—
306C22⋊C4—41no—
307C22⋊C4—2361no—
308C22⋊C4—2361no—
309C42—41no—
310C42—42no—
311C42—2362no—
312C42C2×C4×D5911yescharacteristic
313C42#1174568511yescharacteristic
314C42D4×D5911yescharacteristic
315C42#1174568711yescharacteristic
316C23—2361no—
317C23#1427372311yescharacteristic
318Q8—41no—
319Q8—41no—
320Q8—81no—
321Q8—2362no—
322D4—4721no—
323D4—4721no—
324D4—4721no—
325D4—2361no—
326D4—2361no—
327D4—2362no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—4721no—
337C2×C4—4721no—
338C2×C4—4722no—
339C2×C4—2362no—
340C2×C4—2361no—
341C2×C4C22⋊C4×D5911yescharacteristic
342C2×C4(C22⋊D59)⋊C411yescharacteristic
343C2×C4C4×C22⋊D5911yescharacteristic
344C2×C4#1427369911yescharacteristic
345C2×C4#1427370011yescharacteristic
346C2×C4#1427371311yescharacteristic
347C2×C4C2×#1174567112yes—
348C2×C4#1427374712yes—
349C8—24no—
350C8—592no—
351C8—1181no—
352C8—1181no—
353C8—1181no—
354C8—2362no—
355C22—21no—
356C22—21no—
357C22—4721no—
358C22—4721no—
359C22—4721no—
360C22#1683730211yescharacteristic
361C22#1683777611yescharacteristic
362C22#1683778411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—4722no—
370C4—4721no—
371C4#104×D5911yescharacteristic
372C4#1683741711yescharacteristic
373C4C4×#1174567112yes—
374C4#1683763312yes—
375C2—21no—
376C2—21no—
377C2—9441no—
378C2#2142495311yescharacteristic
379C2#2142499311yescharacteristic
380C2#2142501311yescharacteristic
subgroupG/H
centerC2×C4#14273699
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1986C2
socleC22×C59#390
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1986D59
hypocenterC59#9717
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
3. C4×#14273658C2semidirect
4. #21422595C2semidirect
6. M16×C59⋊C8C2semidirect
7. C59×#1986C2semidirect
9. #16837798C4semidirect
10. #16837776C4semidirect
11. #16837752C4semidirect
12. #16837750C4semidirect
92. M16×C59SD16semidirect
163. C59⋊C8#104semidirect
186. #1986D59semidirect
221. C59#9717semidirect
285. M16#11745671semidirect

Maximal subgroups

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

subgroupindex
#214346562
#214250152
C4×#142736582
#214225952
#214225902
M16×C59⋊C82
C59×#19862
#971759