Finite Group Databasebeta

← #28596679

Subgroups — #28596679

Order 15104.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837756C211yesmaximal, characteristic
2#21426651C211yesmaximal, characteristic, complemented
3#124×Dic59C211yesmaximal, characteristic
4#21425061C211yesmaximal, characteristic, complemented
5#21424776C212yesmaximal
6#21424769C212yesmaximal, complemented
7#21424226C211yesmaximal, characteristic, complemented
8#103×Dic59C212yesmaximal, complemented
9#102×Dic59C212yesmaximal
10C4×C8×Dic59C211yesmaximal, characteristic, complemented
11C59×#1958C211yesmaximal, characteristic
12C2×#14273719C2211yescharacteristic
13C2×Q8×Dic59C2211yescharacteristic
14C2×D4×Dic59C2211yescharacteristic
15#16837756C418yescomplemented
16C2×#14273651C2212yes—
17C2×#14273648C2212yes—
18C2×C4×C59⋊C8C2211yescharacteristic
19C2×C8×Dic59C2212yes—
20C2×C8×Dic59C2212yes—
21#16837309C2211yescharacteristic
22#16837305C2211yescharacteristic
23#16837298C2212yes—
24#16837295—22no—
25#16837284—22no—
26#16837268C2211yescharacteristic
27C4⋊C4×Dic59C2212yes—
28#16837253—22no—
29C22⋊C4×Dic59—22no—
30C42×Dic59C2211yescharacteristic
31#16837165C2212yes—
32#16837165C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35C2×#124×C59C2211yescharacteristic
36C2×#103×C59C2212yes—
37C2×#102×C59C2212yes—
38C2×C4×C8×C59C2211yescharacteristic
39C59×#382C2211yescharacteristic
40C23×Dic59—21no—
41C22×C59⋊C8C2312yes—
42#14273719—22no—
43#14273719C2×C414yescomplemented
44Q8×Dic59—24no—
45D4×Dic59—24no—
46C2×C4⋊Dic59—21no—
47C2×C4⋊Dic59C2311yescharacteristic
48C2×C4⋊Dic59C2311yescharacteristic
49C2×C4⋊Dic59C2312yes—
50C2×Dic59⋊C4—22no—
51C2×Dic59⋊C4—22no—
52C2×C22⋊Dic59—22no—
53C2×C22⋊Dic59—21no—
54C2×C4×Dic59—21no—
55C2×C4×Dic59—22no—
56C2×C4×Dic59D412yes—
57C2×C4×Dic59D412yes—
58C2×C4×Dic59C2311yescharacteristic
59C2×C4×Dic59C2311yescharacteristic
60#14273651C2×C418yescomplemented
61#14273648C2×C418yes—
62C4×C59⋊C8C2×C414yescomplemented
63C8×Dic59—24no—
64C8×Dic59—24no—
65C59⋊#95—22no—
66C59⋊#95—22no—
67C59⋊#95—22no—
68C59⋊#95—22no—
69C22×Q8×C59C2311yescharacteristic
70C22×D4×C59C2311yescharacteristic
71C22×C8×C59C2312yes—
72#124×C59—22no—
73#124×C59C2×C414yes—
74C2×C4⋊C4×C59C2312yes—
75C2×C22⋊C4×C59—22no—
76C2×C42×C59C2311yescharacteristic
77#103×C59—24no—
78#102×C59—24no—
79C4×C8×C59—22no—
80#95×C59—22no—
81C24×C59—21no—
82C2×Q8×C59—21no—
83C2×Q8×C59—22no—
84C2×Q8×C59—22no—
85C2×Q8×C59C22×C412yes—
86C2×D4×C59—22no—
87C2×D4×C59—22no—
88C2×D4×C59—21no—
89C2×D4×C59C22×C412yes—
90C22×C4×C59—21no—
91C22×C4×C59—22no—
92C22×C4×C59C2×D412yes—
93C22×C4×C59C2411yescharacteristic
94C22×Dic59—22no—
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—42no—
98C22×Dic59—44no—
99C22×Dic59C2×D411yescharacteristic
100C22×Dic59C2×D412yes—
101C22×Dic59C2×D411yescharacteristic
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C2×C8×C59#4114yes—
105C4⋊C4×C59—22no—
106C4⋊C4×C59—22no—
107C22⋊C4×C59—28no—
108C42×C59C22×C412yes—
109C42×C59#4112yes—
110C22⋊Dic59—42no—
111C22⋊Dic59—44no—
112C4⋊Dic59—22no—
113C4⋊Dic59—22no—
114C4⋊Dic59—22no—
115C4⋊Dic59—42no—
116C4⋊Dic59C22×C414yes—
117Dic59⋊C4—28no—
118Dic59⋊C4—44no—
119C4×Dic59—22no—
120C4×Dic59—24no—
121C4×Dic59—24no—
122C4×Dic59—42no—
123C4×Dic59—44no—
124C4×Dic59#4114yes—
125C2×C59⋊C8—22no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8C22×C414yes—
128C23×C59—22no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—42no—
132C23×C59—42no—
133C23×C59C22×D411yescharacteristic
134Q8×C59—24no—
135Q8×C59—42no—
136D4×C59—24no—
137D4×C59—42no—
138C2×C4×C59—22no—
139C2×C4×C59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59C4×D414yes—
147C2×C4×C59C23×C411yescharacteristic
148C2×C4×C59C2×#4112yes—
149C2×C4×C59C2×#4112yes—
150C2×C4×C59C2×#4111yescharacteristic
151C2×Dic59—22no—
152C2×Dic59—22no—
153C2×Dic59—22no—
154C2×Dic59—24no—
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—42no—
160C2×Dic59—42no—
161C2×Dic59—84no—
162C2×Dic59#12412yes—
163C2×Dic59#12412yes—
164C2×Dic59#13514yes—
165C2×Dic59#13514yes—
166C8×C59—24no—
167C59⋊C8—24no—
168C4×#488—591nomaximal, Sylow
169C22×C59—44no—
170C22×C59—42no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59—82no—
174C22×C59C2×C4×D411yescharacteristic
175C22×C59C2×#12411yescharacteristic
176C22×C59#53211yescharacteristic
177C22×C59C2×#13512yes—
178C22×C59C2×#13512yes—
179C4×C59—24no—
180C4×C59—42no—
181C4×C59—82no—
182C4×C59C4×#4112yes—
183C4×C59#52912yes—
184Dic59—22no—
185Dic59—24no—
186Dic59—44no—
187Dic59—81no—
188Dic59—81no—
189Dic59#48814yescomplemented
190C2×#488—591no—
191C4×#125—591no—
192C4×#124—591no—
193#1991—591no—
194#1958D5911yescharacteristic
195#1934—592no—
196#1933—592no—
197#1835—591no—
198C4×#103—592no—
199C4×#102—592no—
200C42×C8—591no—
201C2×C59—82no—
202C2×C59C4×#12411yescharacteristic
203C2×C59C4×#13512yes—
204C2×C59C2×#48811yescharacteristic
205C2×C59#313111yescharacteristic
206C2×C59#316812yes—
207C2×#125—591no—
208C2×#124C2×D5911yescharacteristic
209C2×C4×Q8—591no—
210C2×C4×D4—591no—
211#488—598no—
212C2×#103—592no—
213C2×#103C2×D5912yes—
214C2×#102—592no—
215C2×#102C2×D5912yes—
216C2×C4×C8—591no—
217C2×C4×C8—592no—
218C2×C4×C8—592no—
219C2×C4×C8C2×D5911yescharacteristic
220#391—591no—
221#390—591no—
222#389—592no—
223#388—1182no—
224#387—1182no—
225#382—591no—
226#382C2×D5911yescharacteristic
227C4×C4⋊C4—592no—
228C4×C4⋊C4—1182no—
229C4×C22⋊C4—1182no—
230C43—591no—
231#340—592no—
232#340—592no—
233#340—592no—
234#340—592no—
235C59C4×#48811yescharacteristic, complemented, Sylow
236C22×Q8C22×D5911yescharacteristic
237C22×D4C22×D5911yescharacteristic
238C23×C4—1181no—
239C22×C8—592no—
240C22×C8C22×D5912yes—
241#125—594no—
242#125—1182no—
243#124—22no—
244#124C4×D5914yes—
245C4×Q8—1184no—
246C4×D4—1184no—
247C2×C4⋊C4—592no—
248C2×C4⋊C4—591no—
249C2×C4⋊C4—591no—
250C2×C4⋊C4—1182no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1182no—
253C2×C4⋊C4C22×D5912yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—1181no—
256C2×C22⋊C4—1182no—
257C2×C42—591no—
258C2×C42—592no—
259C2×C42—592no—
260C2×C42—591no—
261C2×C42—1181no—
262C2×C42—1182no—
263C2×C42C22×D5911yescharacteristic
264#103—24no—
265#103—598no—
266#102—24no—
267#102—598no—
268C4×C8—22no—
269C4×C8—594no—
270C4×C8—1184no—
271C4×C8—1184no—
272#95—22no—
273#95—1182no—
274#95—1182no—
275#95—1182no—
276#95—1182no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8C2×C4×D5912yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4C2×C4×D5912yes—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—591no—
289C22×C4—591no—
290C22×C4—592no—
291C22×C4—1182no—
292C22×C4—1181no—
293C22×C4—1181no—
294C22×C4—2362no—
295C22×C4—2364no—
296C22×C4D4×D5912yes—
297C22×C4C23×D5911yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—594no—
301C2×C8—1182no—
302C2×C8—1182no—
303C2×C8#1174568714yes—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—594no—
307C4⋊C4—1188no—
308C4⋊C4—1182no—
309C4⋊C4—1182no—
310C4⋊C4—1182no—
311C4⋊C4—2362no—
312C4⋊C4—2364no—
313C22⋊C4—28no—
314C22⋊C4—2362no—
315C22⋊C4—2364no—
316C42—594no—
317C42—1184no—
318C42—1184no—
319C42—1182no—
320C42—2362no—
321C42—2364no—
322C42C2×C4×D5912yes—
323C42#1174568712yes—
324C23—22no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23C2×D4×D5911yescharacteristic
330Q8—24no—
331Q8—42no—
332D4—24no—
333D4—42no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—592no—
343C2×C4—594no—
344C2×C4—592no—
345C2×C4—594no—
346C2×C4—1182no—
347C2×C4—1182no—
348C2×C4—1182no—
349C2×C4—1184no—
350C2×C4—4724no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4—2362no—
354C2×C4—2362no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4(C22⋊D59)⋊C414yes—
358C2×C4C22×C4×D5911yescharacteristic
359C2×C4#41×D5912yes—
360C2×C4C2×#1174568711yescharacteristic
361C2×C4C2×#1174568712yes—
362C8—24no—
363C8—1184no—
364C22—44no—
365C22—42no—
366C22—82no—
367C22—81no—
368C22—81no—
369C22C2×(C22⋊D59)⋊C411yescharacteristic
370C22#124×D5911yescharacteristic
371C22#1683795011yescharacteristic
372C22#135×D5912yes—
373C22#135×D5912yes—
374C4—24no—
375C4—42no—
376C4—82no—
377C4—594no—
378C4—1184no—
379C4—1182no—
380C4—4721no—
381C4—4721no—
382C4—2364no—
383C4C4×#1174568712yes—
384C4#1683793312yes—
385C2—82no—
386C2#2142664311yescharacteristic
387C2#2143286412yes—
388C2#488×D5911yescharacteristic
389C2#2143441711yescharacteristic
390C2#2143466312yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1958C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1958D59
hypocenterC59C4×#488
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. #21426651C2semidirect
4. #21425061C2semidirect
6. #21424769C2semidirect
7. #21424226C2semidirect
8. #103×Dic59C2semidirect
10. C4×C8×Dic59C2semidirect
15. #16837756C4semidirect
43. #14273719C2×C4semidirect
60. #14273651C2×C4semidirect
62. C4×C59⋊C8C2×C4semidirect
189. Dic59#488semidirect
235. C59C4×#488semidirect

Maximal subgroups

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

subgroupindex
C2×#168377562
#214266512
#124×Dic592
#214250612
#214247762
#214247692
#214242262
#103×Dic592
#102×Dic592
C4×C8×Dic592
C59×#19582
C4×#48859