Finite Group Databasebeta

← #28597142

Subgroups — #28597142

Order 15104.

1398 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837778C211yesmaximal, characteristic
2C2×#16837512C214yesmaximal, complemented
3C4×#14273724C211yesmaximal, characteristic, complemented
4#21424971C214yesmaximal, complemented
5#21424903C211yesmaximal, characteristic, complemented
6#21424901C211yesmaximal, characteristic, complemented
7#21424252C211yesmaximal, characteristic, complemented
8C4×#14273638C211yesmaximal, characteristic, complemented
9C4×#127×C59C211yesmaximal, characteristic, complemented
10C22×#11745669C2214yes—
11C2×#14273724C2211yescharacteristic
12C2×C4×D236—22no—
13C2×C4×D236C2212yes—
14#16837778C418yescomplemented
15#16837512—216no—
16C2×#14273658C2214yes—
17C2×#14273648C2214yes—
18C2×#14273647C2214yes—
19C2×#14273638C2212yes—
20C2×#14273638C2212yes—
21C2×C4×C59⋊C8C2211yescharacteristic
22C2×C4×C59⋊C8C2211yescharacteristic
23#16837304—22no—
24#16837304C2212yes—
25#16837272C2211yescharacteristic
26C2×#127×C59C2211yescharacteristic
27C2×C4×D4×C59—22no—
28C2×C4×D4×C59C2212yes—
29C59×#390—22no—
30C59×#390C2212yes—
31C59×#384C2211yescharacteristic
32C43×C59C2211yescharacteristic
33C22×D236—22no—
34C22×D236C2312yes—
35C22×C4×D59—22no—
36C2×#11745669—28no—
37C2×#11745669—28no—
38C2×#11745669C2×C4116yescomplemented
39C22×C59⋊C8C2312yes—
40C22×C59⋊C8C2312yes—
41#14273724—22no—
42#14273724C2×C414yescomplemented
43C4×D236—28no—
44C4×D236—48no—
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59C2312yes—
47C2×(C2×D59)⋊C4—22no—
48C2×(C2×D59)⋊C4—24no—
49#14273658—28no—
50#14273648—28no—
51#14273647—28no—
52#14273638—24no—
53#14273638D418yescomplemented
54C4×C59⋊C8—22no—
55C4×C59⋊C8C2×C414yescomplemented
56C22×D4×C59—22no—
57C22×D4×C59C2312yes—
58C23×C4×C59—22no—
59#127×C59—22no—
60#127×C59C2×C414yescomplemented
61C4×D4×C59—28no—
62C4×D4×C59—48no—
63C2×C4⋊C4×C59—22no—
64C2×C4⋊C4×C59C2312yes—
65C2×C22⋊C4×C59—22no—
66C2×C22⋊C4×C59—24no—
67C2×C42×C59D412yes—
68C2×C42×C59D412yes—
69C2×C42×C59C2311yescharacteristic
70C2×C42×C59C2311yescharacteristic
71C2×C42×C59C2311yescharacteristic
72C24×C59—22no—
73C23×D59—22no—
74C2×D4×C59—28no—
75C2×D4×C59—24no—
76C2×D4×C59—24no—
77C2×D4×C59—22no—
78C2×D4×C59—44no—
79C2×D4×C59—44no—
80C2×D4×C59C22×C414yes—
81C22×C4×C59—22no—
82C22×C4×C59—48no—
83C22×C4×C59—44no—
84C22×C4×C59C2×D412yes—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D412yes—
88C22×C4×C59C2411yescharacteristic
89C22×Dic59—22no—
90C2×D236—24no—
91C2×D236—24no—
92C2×D236—28no—
93C2×D236—22no—
94C2×D236—44no—
95C2×D236—44no—
96C2×D236C22×C414yes—
97C2×C4×D59—48no—
98C2×C4×D59—44no—
99C4⋊C4×C59—24no—
100C4⋊C4×C59—44no—
101C22⋊C4×C59—48no—
102C22⋊C4×C59—44no—
103C42×C59—24no—
104C42×C59—24no—
105C42×C59—22no—
106C42×C59C22×C412yes—
107C42×C59C2×D414yes—
108C42×C59#4112yes—
109#11745669—232no—
110(C2×D59)⋊C4—48no—
111(C2×D59)⋊C4—44no—
112C4⋊Dic59—24no—
113C4⋊Dic59—44no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—22no—
116C2×C59⋊C8C22×C414yes—
117C2×C59⋊C8C2×D414yes—
118C2×C59⋊C8C2×D414yes—
119C2×C59⋊C8#4114yes—
120C23×C59—24no—
121C23×C59—42no—
122C23×C59—42no—
123C23×C59—44no—
124C23×C59—44no—
125C23×C59C22×D411yescharacteristic
126C22×D59—24no—
127C22×D59—42no—
128C22×D59—42no—
129C22×D59—44no—
130C22×D59—44no—
131D4×C59—28no—
132D4×C59—44no—
133D4×C59—416no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—42no—
140C2×C4×C59—44no—
141C2×C4×C59—88no—
142C2×C4×C59C4×D414yes—
143C2×C4×C59#12712yes—
144C2×C4×C59#12712yes—
145C2×C4×C59AGL(1,ℤ/8)14yes—
146C2×C4×C59AGL(1,ℤ/8)14yes—
147C2×C4×C59C23×C411yescharacteristic
148C2×C4×C59C22×D412yes—
149C2×C4×C59C2×#4112yes—
150C2×C4×C59C2×#4111yescharacteristic
151C2×Dic59—42no—
152C2×Dic59—44no—
153D236—28no—
154D236—44no—
155D236—416no—
156C4×D59—88no—
157C59⋊C8—24no—
158C59⋊C8C4×D418yescomplemented
159#21690—591nomaximal, Sylow
160C22×C59—44no—
161C22×C59—48no—
162C22×C59—82no—
163C22×C59—82no—
164C22×C59—84no—
165C22×C59C2×C4×D411yescharacteristic
166C22×C59C2×#12711yescharacteristic
167C22×C59#53211yescharacteristic
168C22×C59C2×AGL(1,ℤ/8)12yes—
169C22×C59C2×AGL(1,ℤ/8)12yes—
170C2×D59—44no—
171C2×D59—48no—
172C2×D59—82no—
173C2×D59—82no—
174C2×D59—84no—
175C4×C59—22no—
176C4×C59—22no—
177C4×C59—42no—
178C4×C59—82no—
179C4×C59#44218yescomplemented
180C4×C59C2×C4×D412yes—
181C4×C59#53212yes—
182Dic59—82no—
183C2×#493—591no—
184C2×#442—594no—
185C4×#127—591no—
186C4×#127D5911yescharacteristic, complemented
187#1972—594no—
188#1957—591no—
189#1957—591no—
190#1844—591no—
191C4×#97—591no—
192C2×C59—84no—
193C2×C59C4×#12711yescharacteristic
194C2×C59C2×#44212yes—
195C2×C59#315111yescharacteristic
196C2×C59#315111yescharacteristic
197C2×C59#318412yes—
198D59—84no—
199C22×D8—594no—
200C2×#127—591no—
201C2×#127C2×D5911yescharacteristic
202C2×C4×D4—22no—
203C2×C4×D4—592no—
204C2×C4×D4—1182no—
205C2×C4×D4C2×D5912yes—
206#493—598no—
207#442—11816no—
208C2×#106—594no—
209C2×#102—594no—
210C2×#102—594no—
211C2×#97—592no—
212C2×#97—592no—
213C2×C4×C8—591no—
214C2×C4×C8—591no—
215#390—22no—
216#390—592no—
217#390—1182no—
218#390C2×D5912yes—
219#384—591no—
220#384C2×D5911yescharacteristic
221C43C2×D5911yescharacteristic
222C59#2169011yescharacteristic, complemented, Sylow
223C22×D4—22no—
224C22×D4—592no—
225C22×D4—1182no—
226C22×D4C22×D5912yes—
227C23×C4—22no—
228C23×C4—1182no—
229C2×D8—5916no—
230C2×D8—1188no—
231C2×D8—1188no—
232C22×C8—592no—
233C22×C8—592no—
234#127—22no—
235#127—594no—
236#127—1182no—
237#127C4×D5914yescomplemented
238C4×D4—28no—
239C4×D4—48no—
240C4×D4—1188no—
241C4×D4—2368no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—592no—
244C2×C4⋊C4—1182no—
245C2×C4⋊C4C22×D5912yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—1182no—
249C2×C22⋊C4—1184no—
250C2×C42C22⋊D5912yes—
251C2×C42C22⋊D5912yes—
252C2×C42C22×D5911yescharacteristic
253C2×C42C22×D5911yescharacteristic
254C2×C42C22×D5911yescharacteristic
255#106—1188no—
256#102—1188no—
257#102—1188no—
258#97—598no—
259#97—1184no—
260C4×C8—594no—
261C4×C8—1182no—
262C24—22no—
263C24—1182no—
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—594no—
271C2×D4—1188no—
272C2×D4—1184no—
273C2×D4—1184no—
274C2×D4—1182no—
275C2×D4—2364no—
276C2×D4—2364no—
277C2×D4C2×C4×D5914yes—
278C22×C4—22no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—1182no—
282C22×C4—2368no—
283C22×C4—2364no—
284C22×C4C2×C22⋊D5912yes—
285C22×C4C2×C22⋊D5911yescharacteristic
286C22×C4C2×C22⋊D5911yescharacteristic
287C22×C4C2×C22⋊D5912yes—
288C22×C4C23×D5911yescharacteristic
289D8—11832no—
290C2×C8—594no—
291C2×C8—594no—
292C2×C8—594no—
293C2×C8—594no—
294C2×C8—1182no—
295C2×C8—1182no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—1184no—
299C4⋊C4—2364no—
300C22⋊C4—48no—
301C22⋊C4—44no—
302C22⋊C4—2368no—
303C22⋊C4—2364no—
304C42—24no—
305C42—24no—
306C42—22no—
307C42C2×C4×D5912yes—
308C42#1174568512yes—
309C42D4×D5914yes—
310C23—24no—
311C23—44no—
312C23—44no—
313C23—42no—
314C23—42no—
315C23—1184no—
316C23—2364no—
317C23—2364no—
318C23—2362no—
319C23—2362no—
320C23C22×C22⋊D5911yescharacteristic
321D4—28no—
322D4—44no—
323D4—416no—
324D4—1188no—
325D4—2364no—
326D4—23616no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—88no—
335C2×C4—4728no—
336C2×C4—2364no—
337C2×C4—2362no—
338C2×C4C4×C22⋊D5914yes—
339C2×C4#1427372312yes—
340C2×C4#1427372312yes—
341C2×C4#1427375214yes—
342C2×C4#1427375314yes—
343C2×C4C22×C4×D5911yescharacteristic
344C2×C4C2×D4×D5912yes—
345C2×C4#41×D5912yes—
346C2×C4C2×#1174568511yescharacteristic
347C8—598no—
348C8—1184no—
349C22—48no—
350C22—44no—
351C22—82no—
352C22—82no—
353C22—84no—
354C22—4722no—
355C22—4722no—
356C22—4724no—
357C22—2368no—
358C22—2364no—
359C22C2×C4×C22⋊D5911yescharacteristic
360C22C2×#1427372311yescharacteristic
361C22#1683794811yescharacteristic
362C22C2×#1427375212yes—
363C22C2×#1427375312yes—
364C4—22no—
365C4—22no—
366C4—42no—
367C4—82no—
368C4—4722no—
369C4#1683751218yescomplemented
370C4C4×D4×D5912yes—
371C4#1683794912yes—
372C2—84no—
373C2—4724no—
374C2C4×#1427372311yescharacteristic
375C2C2×#1683751212yes—
376C2#2143454611yescharacteristic
377C2#2143454911yescharacteristic
378C2#2143483012yes—
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#127×C59C2
socleC23×C59C22×D4
prime radicalC2×#16837778C2
cosocleC22×C4×C59C24
hypercenterC4×#127D59
hypocenterC59#21690
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C2×#16837512C2semidirect
3. C4×#14273724C2semidirect
4. #21424971C2semidirect
5. #21424903C2semidirect
6. #21424901C2semidirect
7. #21424252C2semidirect
8. C4×#14273638C2semidirect
9. C4×#127×C59C2semidirect
14. #16837778C4semidirect
38. C2×#11745669C2×C4semidirect
42. #14273724C2×C4semidirect
53. #14273638D4semidirect
55. C4×C59⋊C8C2×C4semidirect
60. #127×C59C2×C4semidirect
158. C59⋊C8C4×D4semidirect
179. C4×C59#442semidirect
186. C4×#127D59semidirect
222. C59#21690semidirect
237. #127C4×D59semidirect
369. C4#16837512semidirect

Maximal subgroups

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

subgroupindex
C2×#168377782
C2×#168375122
C4×#142737242
#214249712
#214249032
#214249012
#214242522
C4×#142736382
C4×#127×C592
#2169059