Finite Group Databasebeta

← #28487840

Subgroups — #28487840

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#21425041C211yesmaximal, characteristic, complemented
2#21425016C211yesmaximal, characteristic
3C4×#14273659C211yesmaximal, characteristic, complemented
4#21422595C211yesmaximal, characteristic, complemented
5#21422591C211yesmaximal, characteristic
6C8×#11745664C211yesmaximal, characteristic, complemented
7C59×#3167C211yesmaximal, characteristic
8#16837455C2211yescharacteristic
9C2×#14273659—21no—
10C2×#14273654—22no—
11C4×#11745664C2211yescharacteristic
12C4×C4⋊Dic59C2211yescharacteristic
13C4×C4⋊Dic59C2211yescharacteristic
14#16837165—21no—
15#16837161—22no—
16#16837139—22no—
17#16837136—22no—
18#16837124—21no—
19C8×C59⋊C8—21no—
20C59×#533C2211yescharacteristic
21C59×#532C2211yescharacteristic
22C59×#498C411yescharacteristic, complemented
23C59×#492C411yescharacteristic, complemented
24C59×#487C411yescharacteristic, complemented
25C59×#486C411yescharacteristic, complemented
26C59×#465—21no—
27C59×#465—21no—
28C4×SD16×C59—21no—
29C4×SD16×C59—21no—
30C2×C4×C8×C59C2211yescharacteristic
31C2×#11745664D412yes—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36#14273659—24no—
37C4⋊(C59⋊C8)—21no—
38#14273654—44no—
39C22⋊(C59⋊C8)D412yes—
40#14273638—21no—
41C4×C59⋊C8—21no—
42C4×C59⋊C8—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C2×#41×C59—21no—
46C2×SD16×C59—21no—
47C2×SD16×C59—21no—
48C22×C8×C59D411yescharacteristic
49C22×C8×C59D411yescharacteristic
50C4⋊Q8×C59C2×C411yescharacteristic
51C4⋊Q8×C59C2×C411yescharacteristic
52C4⋊Q8×C59C2×C411yescharacteristic
53#127×C59C2×C411yescharacteristic
54#125×C59—21no—
55#124×C59—21no—
56C22⋊Q8×C59—21no—
57C22⋊Q8×C59—21no—
58#121×C59—21no—
59#121×C59—21no—
60C4×Q8×C59—21no—
61C4×Q8×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64#117×C59—21no—
65C2×C42×C59C2311yescharacteristic
66#106×C59—21no—
67#106×C59—21no—
68#103×C59—21no—
69#103×C59—21no—
70#102×C59—21no—
71#102×C59—21no—
72C4×C8×C59C2×C411yescharacteristic
73C4×C8×C59C2×C411yescharacteristic
74C4×C8×C59D411yescharacteristic
75C4×C8×C59D411yescharacteristic
76#41×C59—42no—
77#41×C59—42no—
78C2×Q8×C59—21no—
79C2×Q8×C59—21no—
80C2×Q8×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×Dic59—41no—
90C22×Dic59—41no—
91SD16×C59—42no—
92SD16×C59—42no—
93C2×C8×C59—22no—
94C2×C8×C59—41no—
95C2×C8×C59C22⋊C411yescharacteristic
96C2×C8×C59C22⋊C411yescharacteristic
97C2×C8×C59C22⋊C411yescharacteristic
98C2×C8×C59C22⋊C411yescharacteristic
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C22⋊C4×C59—41no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C42×C59—41no—
111C42×C59C22×C411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59#4111yescharacteristic
114C42×C59#4111yescharacteristic
115C4⋊Dic59—22no—
116C4⋊Dic59—22no—
117C4⋊Dic59—41no—
118C4⋊Dic59—41no—
119C4×Dic59—42no—
120C4×Dic59—42no—
121#11745664—22no—
122#11745664SD1614yescomplemented
123C2×C59⋊C8—22no—
124C2×C59⋊C8—22no—
125C23×C59—41no—
126C23×C59#12711yescharacteristic
127Q8×C59—41no—
128Q8×C59—41no—
129Q8×C59—42no—
130Q8×C59—81no—
131D4×C59—41no—
132D4×C59—41no—
133D4×C59—42no—
134D4×C59—81no—
135D4×C59—81no—
136D4×C59—81no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—82no—
144C2×C4×C59C2×C22⋊C411yescharacteristic
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59#12111yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59#12411yescharacteristic
150C2×C4×C59C2×SD1612yes—
151C2×C4×C59#13512yes—
152C2×Dic59—42no—
153C2×Dic59—42no—
154C2×Dic59—81no—
155C2×Dic59—81no—
156C2×Dic59—81no—
157C2×Dic59—81no—
158C8×C59—21no—
159C8×C59—21no—
160C8×C59—21no—
161C8×C59#10412yescomplemented
162C59⋊C8—24no—
163C59⋊C8—42no—
164#9717—591nomaximal, Sylow
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59#39011yescharacteristic
171C22×C59#49211yescharacteristic
172C22×C59#49511yescharacteristic
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—81no—
177C4×C59—81no—
178C4×C59—81no—
179C4×C59C2×#10411yescharacteristic
180C4×C59#42111yescharacteristic
181C4×C59C4×SD1612yes—
182C4×C59#46512yes—
183Dic59—82no—
184Dic59—82no—
185#3167D5911yescharacteristic
186#1986—591no—
187#1980—591no—
188C4×#106—591no—
189#1503—591no—
190#1502—591no—
191C8×M16—591no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—161no—
195C2×C59#196811yescharacteristic
196C2×C59#197611yescharacteristic
197C2×C59#198011yescharacteristic
198#533C2×D5911yescharacteristic
199#532C2×D5911yescharacteristic
200#498Dic5911yescharacteristic, complemented
201#492Dic5911yescharacteristic, complemented
202#487Dic5911yescharacteristic, complemented
203#486Dic5911yescharacteristic, complemented
204#465—21no—
205#465—21no—
206C4×SD16—21no—
207C4×SD16—21no—
208#431—591no—
209C2×#106—1181no—
210C2×#104—1182no—
211C4×M16—591no—
212C2×C4×C8C2×D5911yescharacteristic
213C4×C4⋊C4—591no—
214C4×C4⋊C4—591no—
215#340—1181no—
216#339—1182no—
217#328—1182no—
218#327—1182no—
219#322—1181no—
220C82—1181no—
221C59#971711yescharacteristic, complemented, Sylow
222C2×#41—21no—
223C2×SD16—21no—
224C2×SD16—21no—
225C2×M16—592no—
226C22×C8C22⋊D5911yescharacteristic
227C22×C8C22⋊D5911yescharacteristic
228C4⋊Q8C2×Dic5911yescharacteristic
229C4⋊Q8C2×Dic5911yescharacteristic
230C4⋊Q8C2×Dic5911yescharacteristic
231#127C2×Dic5911yescharacteristic
232#125—21no—
233#124—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236#121—21no—
237#121—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—21no—
241C4×D4—21no—
242#117—21no—
243C2×C4⋊C4—1181no—
244C2×C4⋊C4—1181no—
245C2×C42—1181no—
246C2×C42—1181no—
247C2×C42C22×D5911yescharacteristic
248#106—21no—
249#106—21no—
250#106—1184no—
251C4⋊C8—1181no—
252#104—2364no—
253#103—21no—
254#103—21no—
255#102—21no—
256#102—21no—
257C22⋊C8—592no—
258#97—1181no—
259C4×C8—1181no—
260C4×C8—1181no—
261C4×C8C2×Dic5911yescharacteristic
262C4×C8C2×Dic5911yescharacteristic
263C4×C8C22⋊D5911yescharacteristic
264C4×C8C22⋊D5911yescharacteristic
265#95—1181no—
266#95—1181no—
267#41—42no—
268#41—42no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4—2361no—
278C22×C4—2361no—
279C22×C4D4×D5911yescharacteristic
280C22×C4D4×D5911yescharacteristic
281C22×C4C2×C22⋊D5911yescharacteristic
282SD16—42no—
283SD16—42no—
284M16—594no—
285M16—1182no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—1182no—
289C2×C8—1182no—
290C2×C8C22⋊Dic5911yescharacteristic
291C2×C8C22⋊Dic5911yescharacteristic
292C2×C8C22⋊Dic5911yescharacteristic
293C2×C8C22⋊Dic5911yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—1182no—
303C4⋊C4—1182no—
304C4⋊C4—2361no—
305C4⋊C4—2361no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—41no—
310C42—2362no—
311C42—2362no—
312C42#1174568711yescharacteristic
313C42#1174568711yescharacteristic
314C42C22×Dic5911yescharacteristic
315C42C2×C22⋊D5911yescharacteristic
316C23—41no—
317C23#1427372311yescharacteristic
318Q8—41no—
319Q8—41no—
320Q8—42no—
321Q8—81no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—82no—
335C2×C4—4721no—
336C2×C4—4721no—
337C2×C4—4721no—
338C2×C4—4721no—
339C2×C4—2362no—
340C2×C4—2362no—
341C2×C4C2×C22⋊Dic5911yescharacteristic
342C2×C4D4×Dic5911yescharacteristic
343C2×C4D4×Dic5911yescharacteristic
344C2×C4#1427369811yescharacteristic
345C2×C4#1427369811yescharacteristic
346C2×C4#1427371411yescharacteristic
347C2×C4SD16×D5912yes—
348C2×C4#1427374812yes—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—1184no—
353C8—2362no—
354C8#1427365412yescomplemented
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1683730511yescharacteristic
361C22#1683777311yescharacteristic
362C22#1683778311yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—4722no—
370C4—4722no—
371C4C2×#1427365411yescharacteristic
372C4#1683741911yescharacteristic
373C4SD16×Dic5912yes—
374C4#1683762812yes—
375C2—21no—
376C2—21no—
377C2—161no—
378C2#2142495211yescharacteristic
379C2#2142499511yescharacteristic
380C2#2142501611yescharacteristic
subgroupG/H
centerC2×C4#14273698
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3167C2
socleC22×C59#390
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#3167D59
hypocenterC59#9717
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21425041C2semidirect
3. C4×#14273659C2semidirect
4. #21422595C2semidirect
6. C8×#11745664C2semidirect
22. C59×#498C4semidirect
23. C59×#492C4semidirect
24. C59×#487C4semidirect
25. C59×#486C4semidirect
122. #11745664SD16semidirect
161. C8×C59#104semidirect
200. #498Dic59semidirect
201. #492Dic59semidirect
202. #487Dic59semidirect
203. #486Dic59semidirect
221. C59#9717semidirect
354. C8#14273654semidirect

Maximal subgroups

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

subgroupindex
#214250412
#214250162
C4×#142736592
#214225952
#214225912
C8×#117456642
C59×#31672
#971759