Finite Group Databasebeta

← #28596673

Subgroups — #28596673

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×#16837759C211yesmaximal, characteristic
2#125×Dic59C211yesmaximal, characteristic, complemented
3#21426635C211yesmaximal, characteristic
4#21424907C211yesmaximal, characteristic
5#21424780C212yesmaximal
6#21424773C212yesmaximal, complemented
7#21424226C211yesmaximal, characteristic, complemented
8#21423915C212yesmaximal, complemented
9#21423910C212yesmaximal
10C4×C8×Dic59C211yesmaximal, characteristic, complemented
11C59×#1991C211yesmaximal, characteristic, complemented
12C2×#14273715C2211yescharacteristic
13C2×#14273692C2211yescharacteristic
14C2×#14273689C2211yescharacteristic
15#16837759C418yescomplemented
16C2×#14273652C2212yes—
17C2×#14273650C2212yes—
18C2×#14273649C2212yes—
19C2×#14273647C2212yes—
20C2×C4×C59⋊C8C2211yescharacteristic
21C2×C8×Dic59C2212yes—
22C2×C8×Dic59C2212yes—
23#16837307C2211yescharacteristic
24#16837303C2211yescharacteristic
25#16837301C2212yes—
26#16837291—22no—
27#16837288—22no—
28#16837268C2211yescharacteristic
29C4⋊C4×Dic59—22no—
30C4⋊C4×Dic59C2212yes—
31#16837249—22no—
32C42×Dic59C2211yescharacteristic
33#16837164C2212yes—
34#16837164C2212yes—
35C2×#125×C59C2211yescharacteristic
36C2×C4×C8×C59C2211yescharacteristic
37C59×#382C2211yescharacteristic
38C59×#340C2212yes—
39C59×#340C2212yes—
40C22×Dic118C2311yescharacteristic
41C22×D236C2311yescharacteristic
42C22×C4×D59—21no—
43C22×C59⋊C8C2312yes—
44#14273715—22no—
45#14273715C2×C414yes—
46#14273692—24no—
47#14273689—24no—
48C2×C4⋊Dic59—22no—
49C2×C4⋊Dic59C2312yes—
50C2×Dic59⋊C4—21no—
51C2×(C2×D59)⋊C4—22no—
52C2×(C2×D59)⋊C4—22no—
53C2×(C2×D59)⋊C4—21no—
54C2×C4×Dic59—21no—
55C2×C4×Dic59—22no—
56C2×C4×Dic59D412yes—
57C2×C4×Dic59D412yes—
58C2×C4×Dic59C2311yescharacteristic
59C2×C4×Dic59C2311yescharacteristic
60#14273652—24no—
61#14273650C2×C418yescomplemented
62#14273649—24no—
63#14273647C2×C418yes—
64C4×C59⋊C8C2×C414yescomplemented
65C8×Dic59—24no—
66C8×Dic59—24no—
67C59⋊#95—22no—
68C59⋊#95—22no—
69C59⋊#95—22no—
70C59⋊#95—22no—
71C22×C8×C59C2312yes—
72#125×C59—22no—
73#125×C59C2×C414yescomplemented
74C2×C4⋊C4×C59—22no—
75C2×C4⋊C4×C59C2311yescharacteristic
76C2×C4⋊C4×C59C2311yescharacteristic
77C2×C4⋊C4×C59C2312yes—
78C2×C42×C59C2311yescharacteristic
79C4×C8×C59—22no—
80#95×C59—22no—
81C23×D59—21no—
82C22×C4×C59—22no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59C2×D412yes—
86C22×C4×C59C2411yescharacteristic
87C22×Dic59—21no—
88C22×Dic59—22no—
89C22×Dic59C2×D411yescharacteristic
90C22×Dic59C2×D412yes—
91C22×Dic59C2×D411yescharacteristic
92C2×D236—22no—
93C2×D236—22no—
94C2×D236—21no—
95C2×D236C22×C412yes—
96C2×C4×D59—42no—
97C2×C4×D59—44no—
98C2×Dic118—22no—
99C2×Dic118—22no—
100C2×Dic118—21no—
101C2×Dic118C22×C412yes—
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C2×C8×C59#4114yes—
105C4⋊C4×C59—28no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—22no—
109C4⋊C4×C59C22×C414yes—
110C42×C59C22×C412yes—
111C42×C59#4112yes—
112(C2×D59)⋊C4—28no—
113(C2×D59)⋊C4—42no—
114(C2×D59)⋊C4—44no—
115C4⋊Dic59—22no—
116C4⋊Dic59—22no—
117C4⋊Dic59—44no—
118Dic59⋊C4—42no—
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×C59C22×D411yescharacteristic
129C22×D59—22no—
130C22×D59—41no—
131C22×D59—41no—
132C22×D59—42no—
133C22×D59—42no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—24no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59C4×D414yes—
144C2×C4×C59C23×C411yescharacteristic
145C2×C4×C59C2×#4112yes—
146C2×C4×C59C2×#4111yescharacteristic
147C2×C4×C59C2×#4112yes—
148C2×Dic59—22no—
149C2×Dic59—22no—
150C2×Dic59—22no—
151C2×Dic59—22no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—42no—
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59#12412yes—
158C2×Dic59#12412yes—
159C2×Dic59#13514yes—
160C2×Dic59#13514yes—
161D236—24no—
162D236—42no—
163C4×D59—84no—
164Dic118—24no—
165Dic118—42no—
166C8×C59—24no—
167C59⋊C8—24no—
168C4×#488—591nomaximal, Sylow
169C22×C59C2×C4×D411yescharacteristic
170C22×C59C2×#12411yescharacteristic
171C22×C59#53211yescharacteristic
172C22×C59C2×#13512yes—
173C22×C59C2×#13512yes—
174C2×D59—44no—
175C2×D59—42no—
176C2×D59—81no—
177C2×D59—81no—
178C2×D59—82no—
179C4×C59—24no—
180C4×C59—44no—
181C4×C59—81no—
182C4×C59—81no—
183C4×C59C4×#4112yes—
184C4×C59#52912yes—
185Dic59—22no—
186Dic59—24no—
187Dic59—42no—
188Dic59—82no—
189Dic59#48814yescomplemented
190C2×#488—591no—
191C4×#125—591no—
192C4×#124—591no—
193#1991D5911yescharacteristic, complemented
194#1958—591no—
195#1934—592no—
196#1933—592no—
197#1835—591no—
198C4×#103—592no—
199C4×#102—592no—
200C42×C8—591no—
201C2×C59C4×#12411yescharacteristic
202C2×C59C4×#13512yes—
203C2×C59C2×#48811yescharacteristic
204C2×C59#313111yescharacteristic
205C2×C59#316812yes—
206D59—82no—
207C2×#125C2×D5911yescharacteristic
208C2×#124—591no—
209C2×C4×Q8—591no—
210C2×C4×D4—591no—
211#488—598no—
212C2×#103—592no—
213C2×#103—592no—
214C2×#102—592no—
215C2×#102—592no—
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#340C2×D5912yes—
234#340C2×D5912yes—
235C59C4×#48811yescharacteristic, complemented, Sylow
236C22×Q8—591no—
237C22×D4—591no—
238C23×C4—1181no—
239C22×C8—592no—
240C22×C8C22×D5912yes—
241#125—22no—
242#125C4×D5914yescomplemented
243#124—594no—
244#124—1182no—
245C4×Q8—1184no—
246C4×D4—1184no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—592no—
249C2×C4⋊C4—1181no—
250C2×C4⋊C4—1182no—
251C2×C4⋊C4C22×D5911yescharacteristic
252C2×C4⋊C4C22×D5911yescharacteristic
253C2×C4⋊C4C22×D5912yes—
254C2×C22⋊C4—1182no—
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—598no—
265#103—1184no—
266#102—598no—
267#102—1184no—
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—1181no—
278C2×Q8—592no—
279C2×Q8—1181no—
280C2×Q8—1182no—
281C2×Q8—1182no—
282C2×D4—592no—
283C2×D4—1182no—
284C2×D4—1181no—
285C2×D4—1182no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—591no—
290C22×C4—591no—
291C22×C4—592no—
292C22×C4—1181no—
293C22×C4—1182no—
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#1174568914yes—
304C4⋊C4—28no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—1182no—
309C4⋊C4—1182no—
310C4⋊C4—2362no—
311C4⋊C4—2364no—
312C4⋊C4C2×C4×D5914yes—
313C22⋊C4—1188no—
314C22⋊C4—2362no—
315C22⋊C4—2364no—
316C42—594no—
317C42—1184no—
318C42—1184no—
319C42—1182no—
320C42—2362no—
321C42—2364no—
322C42C2×C4×D5912yes—
323C42#1174568712yes—
324C23—1182no—
325C23—2361no—
326C23—2362no—
327C23—2362no—
328C23—2361no—
329C23C2×D4×D5911yescharacteristic
330Q8—1184no—
331Q8—2362no—
332D4—1184no—
333D4—2362no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—592no—
344C2×C4—594no—
345C2×C4—592no—
346C2×C4—594no—
347C2×C4—1182no—
348C2×C4—1182no—
349C2×C4—1182no—
350C2×C4—1182no—
351C2×C4—4724no—
352C2×C4—2362no—
353C2×C4—2362no—
354C2×C4—2362no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4(C22⋊D59)⋊C414yes—
358C2×C4C22×C4×D5911yescharacteristic
359C2×C4#41×D5912yes—
360C2×C4C2×#1174568711yescharacteristic
361C2×C4C2×#1174568912yes—
362C8—24no—
363C8—1184no—
364C22—4721no—
365C22—4722no—
366C22—4721no—
367C22—2364no—
368C22—2362no—
369C22C2×(C22⋊D59)⋊C411yescharacteristic
370C22#124×D5911yescharacteristic
371C22#1683795311yescharacteristic
372C22#135×D5912yes—
373C22#135×D5912yes—
374C4—24no—
375C4—44no—
376C4—81no—
377C4—81no—
378C4—594no—
379C4—1184no—
380C4—1182no—
381C4—4722no—
382C4—2362no—
383C4C4×#1174568912yes—
384C4#1683793812yes—
385C2—4722no—
386C2#2142663911yescharacteristic
387C2#2143286412yes—
388C2#488×D5911yescharacteristic
389C2#2143441711yescharacteristic
390C2#2143466812yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1991C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1991D59
hypocenterC59C4×#488
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. #125×Dic59C2semidirect
6. #21424773C2semidirect
7. #21424226C2semidirect
8. #21423915C2semidirect
10. C4×C8×Dic59C2semidirect
11. C59×#1991C2semidirect
15. #16837759C4semidirect
61. #14273650C2×C4semidirect
64. C4×C59⋊C8C2×C4semidirect
73. #125×C59C2×C4semidirect
189. Dic59#488semidirect
193. #1991D59semidirect
235. C59C4×#488semidirect
242. #125C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377592
#125×Dic592
#214266352
#214249072
#214247802
#214247732
#214242262
#214239152
#214239102
C4×C8×Dic592
C59×#19912
C4×#48859