Finite Group Databasebeta

← #28558175

Subgroups — #28558175

Order 15104.

714 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433206C212yesmaximal, complemented
2C4×#14273750C211yesmaximal, characteristic
3#21422921C211yesmaximal, characteristic, complemented
4#21422919C211yesmaximal, characteristic, complemented
5#21422915C211yesmaximal, characteristic, complemented
6C8×#11745671C212yesmaximal, complemented
7#21422893C212yesmaximal, complemented
8#334×D59C211yesmaximal, characteristic, complemented
9#21422482C212yesmaximal, complemented
10#21422381C211yesmaximal, characteristic, complemented
11C59×#1579C211yesmaximal, characteristic, complemented
12C2×#14273750C411yescharacteristic
13C4×#11745689C2212yes—
14Q8×C59⋊C8C2212yes—
15Q16×Dic59C411yescharacteristic
16C4×#11745671C2212yes—
17#16837487C412yes—
18C4×D472C2211yescharacteristic
19#16837484C411yescharacteristic
20C8×D236—22no—
21C8×D236C2212yes—
22#16837467—22no—
23#16837467C2212yes—
24#16837453—22no—
25#16837453C2211yescharacteristic
26#16837453C2211yescharacteristic
27#16837439C411yescharacteristic
28C4⋊C8×D59—22no—
29C4⋊C8×D59C2212yes—
30#16837397C412yes—
31C4×C8×D59C2211yescharacteristic
32#16837153C2211yescharacteristic
33#16837152C2212yes—
34Q8⋊(C59⋊C8)C2212yes—
35#16837131C2212yes—
36#16837129C2212yes—
37#16837123C2211yescharacteristic
38C8×C59⋊C8C2211yescharacteristic
39C8×C59⋊C8C2211yescharacteristic
40C8×Q8×C59C2212yes—
41C4×Q16×C59C2211yescharacteristic
42C59×#334C2211yescharacteristic
43C59×Q8⋊C8C2212yes—
44C59×#322C2211yescharacteristic
45C2×#11745689C2×C412yes—
46#14273750C818yescomplemented
47C2×#11745671C2×C412yes—
48C2×D472C2×C411yescharacteristic
49C2×C8×D59—22no—
50C2×C8×D59—22no—
51C2×C8×D59—22no—
52C2×C8×D59C2×C411yescharacteristic
53C2×C8×D59C2×C411yescharacteristic
54Q8×Dic59C2×C412yes—
55C4×D236—22no—
56C4×D236C2312yes—
57#14273689—22no—
58#14273689C2×C412yes—
59#14273685—22no—
60#14273685C2×C412yes—
61C42×D59—22no—
62C42×D59C2311yescharacteristic
63#14273661C2×C411yescharacteristic
64#14273658C2×C412yes—
65C4⋊(C59⋊C8)—22no—
66C4⋊(C59⋊C8)C2312yes—
67Dic59⋊C8—22no—
68Dic59⋊C8—21no—
69Dic59⋊C8D412yes—
70#14273651C2×C412yes—
71#14273649C2×C412yes—
72#14273647C2×C412yes—
73(C2×D59)⋊C8—21no—
74(C2×D59)⋊C8—22no—
75(C2×D59)⋊C8—22no—
76(C2×D59)⋊C8D412yes—
77C4×C59⋊C8—22no—
78C4×C59⋊C8C2311yescharacteristic
79C4×C59⋊C8C2311yescharacteristic
80C4×C59⋊C8C2311yescharacteristic
81C8×Dic59C2×C411yescharacteristic
82C8×Dic59C2×C411yescharacteristic
83C2×Q16×C59C2×C411yescharacteristic
84C4×Q8×C59C2312yes—
85#107×C59C2×C411yescharacteristic
86C4⋊C8×C59—22no—
87C4⋊C8×C59C2312yes—
88#103×C59C2×C412yes—
89C4×C8×C59—22no—
90C4×C8×C59C2311yescharacteristic
91C4×C8×C59C2311yescharacteristic
92C4×C8×C59C2311yescharacteristic
93C2×Q8×C59C22×C412yes—
94#11745689—24no—
95#11745689C2×C818yescomplemented
96C2×D236—22no—
97C2×D236C22×C412yes—
98C2×C4×D59—22no—
99C2×C4×D59—22no—
100C2×C4×D59—22no—
101C2×C4×D59C22×C411yescharacteristic
102C2×C4×D59C2×D411yescharacteristic
103C2×C4×D59#4111yescharacteristic
104Q16×C59C2×C814yescomplemented
105C2×C8×C59—22no—
106C2×C8×C59—21no—
107C2×C8×C59C22×C411yescharacteristic
108C2×C8×C59C22×C411yescharacteristic
109C2×C8×C59C2×D412yes—
110C4⋊C4×C59—22no—
111C4⋊C4×C59C22×C412yes—
112C42×C59—22no—
113C42×C59C2411yescharacteristic
114#11745671C2×C818yescomplemented
115(C2×D59)⋊C4—22no—
116(C2×D59)⋊C4—22no—
117(C2×D59)⋊C4—22no—
118C4⋊Dic59—22no—
119C4⋊Dic59C22×C412yes—
120C4×Dic59—22no—
121C4×Dic59—22no—
122C4×Dic59C22×C411yescharacteristic
123C4×Dic59C2×D411yescharacteristic
124C4×Dic59#4111yescharacteristic
125C2×C59⋊C8—22no—
126C2×C59⋊C8—21no—
127C2×C59⋊C8C22×C411yescharacteristic
128C2×C59⋊C8C22×C411yescharacteristic
129C2×C59⋊C8C2×D412yes—
130D472C2×C814yescomplemented
131C8×D59—22no—
132C8×D59—82no—
133C8×D59—82no—
134C8×D59—82no—
135C8×D59C2×C814yescomplemented
136C22×D59—22no—
137C22×D59C4×D411yescharacteristic
138Q8×C59—22no—
139Q8×C59C22×C814yes—
140C2×C4×C59—22no—
141C2×C4×C59—22no—
142C2×C4×C59C23×C411yescharacteristic
143C2×C4×C59C22×D411yescharacteristic
144C2×C4×C59C2×#4111yescharacteristic
145C2×Dic59—22no—
146C2×Dic59C4×D411yescharacteristic
147C2×Dic59C4×D411yescharacteristic
148C2×Dic59C4×D411yescharacteristic
149D236—28no—
150D236—22no—
151D236C22×C814yes—
152C4×D59—22no—
153C4×D59—28no—
154C4×D59—41no—
155C4×D59—42no—
156C4×D59—42no—
157C4×D59—82no—
158C4×D59C22×C812yes—
159C4×D59#13112yes—
160C8×C59—21no—
161C8×C59—22no—
162C8×C59—41no—
163C8×C59—42no—
164C8×C59C22×C812yes—
165C59⋊C8—21no—
166C59⋊C8—22no—
167C59⋊C8—41no—
168C59⋊C8—42no—
169C59⋊C8C22×C812yes—
170#17858—591nomaximal, Sylow
171C22×C59C2×C4×D411yescharacteristic
172C2×D59—21no—
173C2×D59—21no—
174C2×D59—24no—
175C2×D59—42no—
176C2×D59—42no—
177C2×D59C8×D412yes—
178C4×C59—24no—
179C4×C59—21no—
180C4×C59—42no—
181C4×C59C23×C811yescharacteristic
182C4×C59C2×#13111yescharacteristic
183C4×C59#57712yes—
184Dic59—21no—
185Dic59—21no—
186Dic59—42no—
187Dic59C8×D412yes—
188#2988—592no—
189C4×#135—591no—
190#1580—591no—
191#1580—591no—
192#1579—591no—
193#1579D5911yescharacteristic, complemented
194C8×SD16—592no—
195C8×SD16—592no—
196C2×#334—591no—
197#1477—592no—
198#1450—591no—
199C2×C59C2×C8×D411yescharacteristic
200C2×C59#295211yescharacteristic
201C2×C59#295711yescharacteristic
202D59—22no—
203D59—44no—
204C2×#135—591no—
205C4×#41—592no—
206C8×Q8—592no—
207C8×Q8C2×D5912yes—
208C4×Q16—591no—
209C4×Q16C2×D5911yescharacteristic
210C4×SD16—592no—
211C4×SD16—592no—
212C4×D8—591no—
213C4×D8—591no—
214C8×D4—592no—
215C8×D4—592no—
216C8×D4—1182no—
217C8×D4—1182no—
218#431—591no—
219#431—591no—
220#431—1182no—
221#427—591no—
222C2×C4⋊C8—592no—
223C2×C4⋊C8—1182no—
224#416—592no—
225C2×C4×C8—591no—
226#334—592no—
227#334—591no—
228#334C2×D5911yescharacteristic
229Q8⋊C8—592no—
230Q8⋊C8C2×D5912yes—
231D4⋊C8—592no—
232D4⋊C8—592no—
233#322—591no—
234#322C2×D5911yescharacteristic
235C82—591no—
236C82—591no—
237C59#1785811yescharacteristic, complemented, Sylow
238C2×#41—592no—
239#135—598no—
240C2×Q16C4×D5911yescharacteristic
241C2×SD16—592no—
242C2×D8—591no—
243C22×C8—591no—
244C22×C8—591no—
245C22×C8—1182no—
246C22×C8—1182no—
247C22×C8—1182no—
248C4×Q8—592no—
249C4×Q8C22×D5912yes—
250C4×D4—592no—
251C4×D4—592no—
252C4×D4—1182no—
253C4×D4—1182no—
254#117—592no—
255#117—1182no—
256C2×C42—591no—
257C2×C42—1182no—
258#107—591no—
259#107C4×D5911yescharacteristic
260#106—592no—
261C4⋊C8—22no—
262C4⋊C8—592no—
263C4⋊C8—592no—
264C4⋊C8—1182no—
265C4⋊C8—1182no—
266C4⋊C8—1181no—
267C4⋊C8C22×D5912yes—
268#103—592no—
269#103C4×D5912yes—
270#102—592no—
271#102—592no—
272C22⋊C8—592no—
273C22⋊C8—1182no—
274C22⋊C8—1181no—
275C22⋊C8—1182no—
276C4×C8—22no—
277C4×C8—591no—
278C4×C8—591no—
279C4×C8—591no—
280C4×C8—591no—
281C4×C8—591no—
282C4×C8—1182no—
283C4×C8C22×D5911yescharacteristic
284C4×C8C22×D5911yescharacteristic
285C4×C8C22×D5911yescharacteristic
286#41—598no—
287#41—1184no—
288C2×Q8C2×C4×D5912yes—
289C2×D4—592no—
290C2×D4—1182no—
291C22×C4—591no—
292C22×C4—591no—
293C22×C4—591no—
294C22×C4—1182no—
295C22×C4—1182no—
296C22×C4—1182no—
297Q16C8×D5914yescomplemented
298SD16—598no—
299D8—594no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—591no—
303C2×C8—594no—
304C2×C8—591no—
305C2×C8—592no—
306C2×C8—1182no—
307C2×C8—1182no—
308C2×C8—1181no—
309C2×C8—4722no—
310C2×C8—4722no—
311C2×C8—4722no—
312C2×C8C2×C4×D5911yescharacteristic
313C2×C8C2×C4×D5911yescharacteristic
314C2×C8D4×D5912yes—
315C4⋊C4—22no—
316C4⋊C4—592no—
317C4⋊C4—1182no—
318C4⋊C4C2×C4×D5912yes—
319C22⋊C4—1182no—
320C22⋊C4—1182no—
321C22⋊C4—1182no—
322C42—22no—
323C42—591no—
324C42—591no—
325C42—591no—
326C42—1182no—
327C42—1182no—
328C42C23×D5911yescharacteristic
329C23—591no—
330C23—1182no—
331Q8—22no—
332Q8C2×C8×D5914yes—
333D4—594no—
334D4—1188no—
335D4—1182no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—591no—
339C2×C4—592no—
340C2×C4—592no—
341C2×C4—591no—
342C2×C4—591no—
343C2×C4—1188no—
344C2×C4—1182no—
345C2×C4—1182no—
346C2×C4—4722no—
347C2×C4—2361no—
348C2×C4—2362no—
349C2×C4—2362no—
350C2×C4C22×C4×D5911yescharacteristic
351C2×C4C2×D4×D5911yescharacteristic
352C2×C4#41×D5911yescharacteristic
353C8—21no—
354C8—22no—
355C8—41no—
356C8—42no—
357C8—592no—
358C8—1181no—
359C8—1182no—
360C8—2361no—
361C8—2362no—
362C8C2×C8×D5912yes—
363C22—592no—
364C22—1181no—
365C22—1181no—
366C22—1184no—
367C22—2362no—
368C22—2362no—
369C22C4×D4×D5911yescharacteristic
370C4—21no—
371C4—24no—
372C4—42no—
373C4—592no—
374C4—1181no—
375C4—1181no—
376C4—2362no—
377C4C22×C8×D5911yescharacteristic
378C4#131×D5911yescharacteristic
379C4#1683821112yes—
380C2—1182no—
381C2—2364no—
382C2C8×D4×D5911yescharacteristic
383C2#2143297611yescharacteristic
384C2#2143301011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC4×C59C23×C8
FrattiniC42C23×D59
FittingC59×#1579C2
socleC22×C59C2×C4×D4
prime radicalC2×#14273750C4
cosocleC42×C59C24
hypercenter#1579D59
hypocenterC59#17858
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
1. #21433206C2semidirect
3. #21422921C2semidirect
4. #21422919C2semidirect
5. #21422915C2semidirect
6. C8×#11745671C2semidirect
7. #21422893C2semidirect
8. #334×D59C2semidirect
9. #21422482C2semidirect
10. #21422381C2semidirect
11. C59×#1579C2semidirect
46. #14273750C8semidirect
95. #11745689C2×C8semidirect
104. Q16×C59C2×C8semidirect
114. #11745671C2×C8semidirect
130. D472C2×C8semidirect
135. C8×D59C2×C8semidirect
193. #1579D59semidirect
237. C59#17858semidirect
297. Q16C8×D59semidirect

Maximal subgroups

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

subgroupindex
#214332062
C4×#142737502
#214229212
#214229192
#214229152
C8×#117456712
#214228932
#334×D592
#214224822
#214223812
C59×#15792
#1785859