Finite Group Databasebeta

← #29458227

Subgroups — #29458227

Order 15104.

3250 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
1C2×#16838210C214yesmaximal, complemented
2#21437874C211yesmaximal, characteristic, complemented
3#21434574C2112yesmaximal, complemented
4#21434553C213yesmaximal, complemented
5#21434549C213yesmaximal, complemented
6#21434522C213yesmaximal, complemented
7C2×#16837776C213yesmaximal, complemented
8C4×#14273734C211yesmaximal, characteristic, complemented
9C59×#3499C211yesmaximal, characteristic, complemented
10C2×#14273773C2212yes—
11#16838210—232no—
12C2×#14273758C22112yescomplemented
13C2×#14273752C22112yescomplemented
14C2×#14273747C22112yescomplemented
15C22×#11745671C22112yescomplemented
16C2×#14273734C2212yes—
17#16838074—22no—
18D4×D236—26no—
19#16837971C2216yes—
20#16837952C2213yes—
21C2×#14273724C2213yes—
22#16837794C22112yescomplemented
23#16837792C22112yescomplemented
24#16837784C22112yescomplemented
25#16837781C2213yescomplemented
26#16837778C2213yescomplemented
27#16837776C2213yescomplemented
28#16837776C2216yescomplemented
29#16837776C22112yescomplemented
30#16837774C2211yescharacteristic, complemented
31#16837358C2216yescomplemented
32C4×#11745664C2213yescomplemented
33C2×C4×C59⋊C8C2213yescomplemented
34C2×2−1+4×C59C2212yes—
35D4×Q8×C59—22no—
36C59×#547—26no—
37C59×#536C2216yescomplemented
38C59×#532C2213yescomplemented
39C2×C4⋊Q8×C59C2213yescomplemented
40C4×#41×C59C2211yescharacteristic
41#14273773—216no—
42C2×#11745689—22no—
43C2×#11745685C2316yes—
44C22×D236—26no—
45C22×D236C2316yes—
46C2×D4×D59—26no—
47#14273758—248no—
48#14273752—248no—
49#14273747—248no—
50C2×#11745672C23112yes—
51C2×#11745670C2314yes—
52C2×#11745671—248no—
53C2×#11745671C23112yes—
54C2×#11745671C23124yes—
55C2×#11745669C23112yes—
56#14273734D4116yescomplemented
57C2×#11745664C2316yes—
58C22×C59⋊C8C2316yes—
59#14273727C2311yescharacteristic
60#14273724—26no—
61#14273724C2313yes—
62#14273724C2316yes—
63#14273715C2316yes—
64#14273702—212no—
65#14273701—26no—
66#14273697—212no—
67C4×D236—26no—
68#14273638C2316yes—
69C4×C59⋊C8C2311yescharacteristic
70C4×C59⋊C8C2313yes—
71C4×C59⋊C8C2316yes—
722−1+4×C59—216no—
73C2×#41×C59—26no—
74C2×#41×C59C2316yes—
75C2×#41×C59C2311yescharacteristic
76C22×Q8×C59—22no—
77C22×Q8×C59C2316yes—
78C4⋊Q8×C59—26no—
79C4⋊Q8×C59C2313yes—
80C4⋊Q8×C59C2316yes—
81#127×C59C2311yescharacteristic
82#124×C59C2316yes—
83#123×C59—212no—
84C22⋊Q8×C59—212no—
85#121×C59—26no—
86C4×Q8×C59D412yes—
87C4×D4×C59—26no—
88C4×D4×C59D416yes—
89#117×C59C2313yes—
90C2×C4⋊C4×C59—26no—
91C2×C42×C59C2313yes—
92C23×D59—26no—
93#41×C59—224no—
94#41×C59—424no—
95#41×C59C2×D418yes—
96C2×Q8×C59—26no—
97C2×Q8×C59—224no—
98C2×Q8×C59—48no—
99C2×Q8×C59C2×D411yescharacteristic
100C2×Q8×C59C2416yes—
101C2×D4×C59—26no—
102C2×D4×C59—22no—
103C2×D4×C59C2×D413yes—
104C2×D4×C59C2412yes—
105C22×C4×C59—26no—
106C22×C4×C59—26no—
107C22×C4×C59C2×D416yes—
108C22×C4×C59C2413yes—
109C2×C22⋊D59—26no—
110#11745689—48no—
111D4×D59—424no—
112#11745685—224no—
113C2×D236—224no—
114C2×D236—26no—
115C2×D236—26no—
116C2×D236—212no—
117C2×D236—812no—
118C2×D236C2416yes—
119C2×C4×D59—26no—
120C2×Dic118C2412yes—
121C4⋊C4×C59—26no—
122C4⋊C4×C59—212no—
123C4⋊C4×C59C2×D416yes—
124C22⋊C4×C59—212no—
125C22⋊C4×C59C2×D416yes—
126C42×C59C2×D416yes—
127C42×C59C2411yescharacteristic
128C42×C59C2413yes—
129#11745672—224no—
130#11745671—224no—
131#11745671—248no—
132#11745670—28no—
133#11745669—224no—
134(C2×D59)⋊C4—212no—
135C4⋊Dic59—22no—
136#11745664C2×D4124yescomplemented
137C2×C59⋊C8C2×D4124yescomplemented
138C2×C59⋊C8C2412yes—
139C2×C59⋊C8C2416yes—
140C23×C59—22no—
141C23×C59C22×D413yes—
142C22×D59—26no—
143C22×D59—412no—
144C22×D59—812no—
145Q8×C59—212no—
146Q8×C59—412no—
147Q8×C59#12714yes—
148D4×C59—24no—
149D4×C59—412no—
150D4×C59—44no—
151D4×C59#127112yes—
152C2×C4×C59—212no—
153C2×C4×C59—26no—
154C2×C4×C59—412no—
155C2×C4×C59—412no—
156C2×C4×C59C22×D4112yes—
157C2×C4×C59C22×D416yes—
158C2×C4×C59C22×D413yes—
159C2×C4×C59C22×D412yes—
160C2×C4×C59C2511yescharacteristic
161C22⋊D59—412no—
162C2×Dic59—22no—
163D236—212no—
164D236—412no—
165D236—412no—
166D236—812no—
167C4×D59—412no—
168Dic118—24no—
169C59⋊C8C22×D414yes—
170C59⋊C8C22×D4112yes—
171#60646—591nomaximal, Sylow
172C22×C59—26no—
173C22×C59—42no—
174C22×C59—44no—
175C22×C59C2×#12716yes—
176C22×C59C23×D411yescharacteristic
177C2×D59—46no—
178C2×D59—412no—
179C2×D59—1612no—
180C4×C59—26no—
181C4×C59—46no—
182C4×C59C2×#12716yes—
183C4×C59#57714yescomplemented
184C4×C59C23×D412yes—
185Dic59—42no—
186C2×#577—594no—
187#3499D5911yescharacteristic, complemented
188#3498—591no—
189#3154—5912no—
190#3152—593no—
191#3151—593no—
192#3147—593no—
193C2×#492—593no—
194C4×#131—591no—
195C2×C59—26no—
196C2×C59—82no—
197C2×C59C22×#12711yescharacteristic
198C2×C59C2×#57712yes—
199D59—86no—
200C2×2−1+4C2×D5912yes—
201C2×H(5,𝔽2)—592no—
202#577—11832no—
203C2×#137—5912no—
204C2×AGL(1,ℤ/8)—5912no—
205C2×#135—5912no—
206C22×SD16—5912no—
207C2×#131—592no—
208#550—1182no—
209D4×Q8—22no—
210#547—26no—
211D42—1186no—
212#536C2×D5916yescomplemented
213#535—596no—
214#532—593no—
215#532C2×D5913yescomplemented
216C2×C4⋊Q8C2×D5913yescomplemented
217C2×#127—593no—
218C4×#41C2×D5911yescharacteristic
219#497—5912no—
220#496—5912no—
221#495—5912no—
222#494—593no—
223#493—593no—
224#492—593no—
225#492—596no—
226#492—5912no—
227#492—591no—
228#405—596no—
229C4×M16—593no—
230C2×C4×C8—593no—
231C59#6064611yescharacteristic, complemented, Sylow
2322−1+4—216no—
233H(5,𝔽2)—11816no—
234C2×#41—26no—
235C2×#41—596no—
236C2×#41—1182no—
237C2×#41C22×D5916yes—
238C2×#41C22×D5911yescharacteristic
239C22×Q8—22no—
240C22×Q8C22×D5916yes—
241C22×D4—596no—
242C22×D4—1186no—
243C22×D4—1186no—
244#137—11848no—
245AGL(1,ℤ/8)—11848no—
246#135—11848no—
247C2×Q16—5912no—
248C2×SD16—5912no—
249C2×SD16—5924no—
250C2×SD16—594no—
251C2×SD16—11848no—
252C2×D8—5912no—
253#131—5916no—
254C2×M16—596no—
255C22×C8—596no—
256C4⋊Q8—26no—
257C4⋊Q8—591no—
258C4⋊Q8C22×D5913yes—
259C4⋊Q8C22×D5916yes—
260#127—593no—
261#127—596no—
262#127—1186no—
263#127C22×D5911yescharacteristic
264#124—596no—
265#124C22×D5916yes—
266#123—212no—
267C22⋊Q8—212no—
268#121—26no—
269#121—1186no—
270#121—11812no—
271#120—11812no—
272C4×Q8C22⋊D5912yes—
273C4×D4—26no—
274C4×D4—1186no—
275C4×D4C22⋊D5916yes—
276#117C22×D5913yes—
277C2×C4⋊C4—26no—
278C2×C42C22×D5913yes—
279#97—596no—
280C4×C8—591no—
281C4×C8—593no—
282C4×C8—596no—
283C24—1186no—
284#41—224no—
285#41—424no—
286#41—11824no—
287#41—2368no—
288#41D4×D5918yes—
289C2×Q8—26no—
290C2×Q8—224no—
291C2×Q8—48no—
292C2×Q8—592no—
293C2×Q8C2×C22⋊D5911yescharacteristic
294C2×Q8C23×D5916yes—
295C2×D4—26no—
296C2×D4—22no—
297C2×D4—596no—
298C2×D4—1186no—
299C2×D4—11824no—
300C2×D4—11812no—
301C2×D4—1186no—
302C2×D4—1186no—
303C2×D4—47212no—
304C2×D4—23624no—
305C2×D4C2×C22⋊D5913yes—
306C2×D4C23×D5912yes—
307C22×C4—26no—
308C22×C4—26no—
309C22×C4—1186no—
310C22×C4C2×C22⋊D5916yes—
311C22×C4C23×D5913yes—
312Q16—11824no—
313SD16—1188no—
314SD16—11824no—
315SD16—11848no—
316D8—11824no—
317M16—5924no—
318C2×C8—5924no—
319C2×C8—592no—
320C2×C8—596no—
321C4⋊C4—26no—
322C4⋊C4—212no—
323C4⋊C4—1182no—
324C4⋊C4C2×C22⋊D5916yes—
325C22⋊C4—212no—
326C22⋊C4—11812no—
327C22⋊C4C2×C22⋊D5916yes—
328C42C2×C22⋊D5916yes—
329C42C23×D5911yescharacteristic
330C42C23×D5913yes—
331C23—22no—
332C23—1186no—
333C23—47212no—
334C23—23612no—
335C23C22×C22⋊D5913yes—
336Q8—212no—
337Q8—412no—
338Q8—1184no—
339Q8#1427372314yes—
340D4—24no—
341D4—412no—
342D4—44no—
343D4—11812no—
344D4—47212no—
345D4—23612no—
346D4—23612no—
347D4—23612no—
348D4#14273723112yes—
349C2×C4—212no—
350C2×C4—26no—
351C2×C4—412no—
352C2×C4—412no—
353C2×C4—1182no—
354C2×C4—23612no—
355C2×C4C2×D4×D59112yes—
356C2×C4C22×C22⋊D5916yes—
357C2×C4C22×C22⋊D5913yes—
358C2×C4C22×C22⋊D5912yes—
359C2×C4C24×D5911yescharacteristic
360C8—594no—
361C8—5912no—
362C22—26no—
363C22—44no—
364C22—42no—
365C22—94412no—
366C22—23612no—
367C22—2366no—
368C22C2×#1427372316yes—
369C22C23×C22⋊D5911yescharacteristic
370C4—26no—
371C4—46no—
372C4—2362no—
373C4C2×#1427372316yes—
374C4#1683821014yescomplemented
375C4C22×D4×D5912yes—
376C2—26no—
377C2—82no—
378C2—4726no—
379C2C22×#1427372311yescharacteristic
380C2C2×#1683821012yes—
subgroupG/H
centerC22C23×C22⋊D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC59×#3499C2
socleC22×C59C23×D4
hypercenter#3499D59
hypocenterC59#60646
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. C2×#16838210C2semidirect
2. #21437874C2semidirect
3. #21434574C2semidirect
4. #21434553C2semidirect
5. #21434549C2semidirect
6. #21434522C2semidirect
7. C2×#16837776C2semidirect
8. C4×#14273734C2semidirect
9. C59×#3499C2semidirect
12. C2×#14273758C22semidirect
13. C2×#14273752C22semidirect
14. C2×#14273747C22semidirect
15. C22×#11745671C22semidirect
22. #16837794C22semidirect
23. #16837792C22semidirect
24. #16837784C22semidirect
25. #16837781C22semidirect
26. #16837778C22semidirect
27. #16837776C22semidirect
28. #16837776C22semidirect
29. #16837776C22semidirect
30. #16837774C22semidirect
31. #16837358C22semidirect
32. C4×#11745664C22semidirect
33. C2×C4×C59⋊C8C22semidirect
37. C59×#536C22semidirect
38. C59×#532C22semidirect
39. C2×C4⋊Q8×C59C22semidirect
56. #14273734D4semidirect
136. #11745664C2×D4semidirect
137. C2×C59⋊C8C2×D4semidirect
183. C4×C59#577semidirect
187. #3499D59semidirect
212. #536C2×D59semidirect
215. #532C2×D59semidirect
216. C2×C4⋊Q8C2×D59semidirect
231. C59#60646semidirect
374. C4#16838210semidirect

Maximal subgroups

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

subgroupindex
C2×#168382102
#214378742
#214345742
#214345532
#214345492
#214345222
C2×#168377762
C4×#142737342
C59×#34992
#6064659