Finite Group Databasebeta

← #29455440

Subgroups — #29455440

Order 15104.

5128 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 5128 conjugacy classes of subgroups — too many to draw. Only its 807 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437727C213yesmaximal, complemented
2#21433498C214yesmaximal, complemented
3#21433493C218yesmaximal, complemented
4#21433488C214yesmaximal, complemented
5C2×#16837589C218yesmaximal, complemented
6#21432557C213yesmaximal, complemented
7C59×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×D4×D59C2211yescharacteristic, complemented
9#16838040—212no—
10D4×C22⋊D59—212no—
11#16837966C2216yescomplemented
12#16837961C22112yescomplemented
13#16837909C2216yescomplemented
14C2×#14273695C22112yescomplemented
15#16837877C2213yescomplemented
16#16837592C22116yescomplemented
17#16837589C22116yescomplemented
18#16837589C22132yescomplemented
19#16837376C22112yescomplemented
20C2×#14273642C22124yescomplemented
21C59×#585—212no—
22C59×#585C2214yescomplemented
23C2×H(5,𝔽2)×C59—212no—
24C2×H(5,𝔽2)×C59C2213yes—
25C2×H(5,𝔽2)×C59C2218yescomplemented
26C24×D59C2312yescomplemented
27C22×D236C2311yescharacteristic, complemented
28C22×C22⋊D59—212no—
29C22×C22⋊D59C2312yescomplemented
30C2×D4×D59—212no—
31C22×C4×D59C2311yescharacteristic, complemented
32#14273723—26no—
33#14273716—26no—
34#14273710—212no—
35#14273708C23112yescomplemented
36#14273703—26no—
37#14273701—212no—
38#14273699—224no—
39#14273698—26no—
40#14273696—212no—
41#14273695—224no—
42#14273695C23112yescomplemented
43#14273695C23124yescomplemented
44D4×Dic59—212no—
45C4×C22⋊D59—212no—
46#14273686C2316yescomplemented
47C2×C22⋊Dic59C23112yescomplemented
48C2×(C2×D59)⋊C4—212no—
49#14273642C23148yescomplemented
502−1+4×C59—216no—
512−1+4×C59—212no—
52H(5,𝔽2)×C59—248no—
53H(5,𝔽2)×C59—224no—
54H(5,𝔽2)×C59—212no—
55H(5,𝔽2)×C59—248no—
56H(5,𝔽2)×C59C23116yes—
57C2×#41×C59—248no—
58C2×#41×C59—224no—
59C2×#41×C59—224no—
60C2×#41×C59D4112yescomplemented
61C2×#41×C59C2316yes—
62C22×D4×C59—216no—
63C22×D4×C59—224no—
64C22×D4×C59D4112yescomplemented
65C22×D4×C59C23112yes—
66C22×D4×C59C2311yescharacteristic
67C24×C59—28no—
68C24×C59C2×D4112yescomplemented
69C24×C59C2412yes—
70C23×D59—212no—
71C23×D59—84no—
72C23×D59C2411yescharacteristic
73C23×D59C2412yes—
74#41×C59—296no—
75#41×C59—248no—
76#41×C59—216no—
77#41×C59—248no—
78#41×C59—248no—
79#41×C59—224no—
80C2×Q8×C59—224no—
81C2×Q8×C59—28no—
82C2×Q8×C59C2×D416yescomplemented
83C2×D4×C59—296no—
84C2×D4×C59—224no—
85C2×D4×C59—224no—
86C2×D4×C59—248no—
87C2×D4×C59—248no—
88C2×D4×C59—248no—
89C2×D4×C59C2×D4112yes—
90C2×D4×C59C2×D416yescomplemented
91C2×D4×C59C2×D4124yescomplemented
92C2×D4×C59C24112yes—
93C22×C4×C59—224no—
94C22×C4×C59—216no—
95C22×C4×C59C2×D4112yescomplemented
96C22×C4×C59C2×D4112yescomplemented
97C22×C4×C59C2411yescharacteristic
98C2×C22⋊D59—212no—
99C2×C22⋊D59—212no—
100C2×C22⋊D59—224no—
101C2×C22⋊D59—824no—
102C22×Dic59—212no—
103C22×Dic59C2411yescharacteristic
104D4×D59—416no—
105C2×D236—26no—
106C2×C4×D59—26no—
107C22⋊Dic59—212no—
108C22⋊Dic59—212no—
109C22⋊Dic59C24112yes—
110(C2×D59)⋊C4—212no—
111(C2×D59)⋊C4—224no—
112C4⋊Dic59—26no—
113Dic59⋊C4—212no—
114C4×Dic59—26no—
115C23×C59—216no—
116C23×C59—224no—
117C23×C59—82no—
118C23×C59—812no—
119C23×C59—168no—
120C23×C59C22×D4112yes—
121C23×C59C22×D4112yes—
122C23×C59C2511yescharacteristic
123C22×D59—23no—
124C22×D59—26no—
125C22×D59—42no—
126C22×D59—424no—
127C22×D59—48no—
128C22×D59—82no—
129C22×D59—812no—
130Q8×C59—216no—
131Q8×C59—212no—
132D4×C59—248no—
133D4×C59—248no—
134D4×C59—224no—
135D4×C59—212no—
136D4×C59#120116yes—
137C2×C4×C59—248no—
138C2×C4×C59—224no—
139C2×C4×C59—224no—
140C2×C4×C59C22×D4112yes—
141C2×C4×C59C22×D416yes—
142C22⋊D59—424no—
143C22⋊D59—48no—
144C22⋊D59—824no—
145C2×Dic59—212no—
146C2×Dic59—23no—
147C2×Dic59—812no—
148D236—44no—
149C4×D59—44no—
150#60399—591nomaximal, Sylow
151C22×C59—212no—
152C22×C59—412no—
153C22×C59—41no—
154C22×C59—412no—
155C22×C59—824no—
156C22×C59—816no—
157C22×C59C2×#12018yes—
158C22×C59C23×D413yes—
159C2×D59—26no—
160C2×D59—43no—
161C2×D59—41no—
162C2×D59—412no—
163C2×D59—42no—
164C2×D59—816no—
165C4×C59—212no—
166C4×C59C2×#12014yes—
167Dic59—412no—
168Dic59—41no—
169H(7,𝔽2)D5911yescharacteristic, complemented
170#3484—593no—
171#3027—594no—
172#3026—598no—
173#3025—594no—
174C2×B(4,𝔽2)—598no—
175#2883—593no—
176C2×C59—23no—
177C2×C59—28no—
178C2×C59—412no—
179C2×C59C22×#12011yescharacteristic
180D59—42no—
181D59—82no—
182#585—212no—
183#585C2×D5914yescomplemented
184C2×H(5,𝔽2)—212no—
185C2×H(5,𝔽2)C2×D5913yes—
186C2×H(5,𝔽2)C2×D5918yescomplemented
187C23×D4—591no—
188#546—11812no—
189D42—11812no—
190#535—596no—
191#534—5912no—
192#525—596no—
193C2×#120—5912no—
194#518—593no—
195#458—5916no—
196B(4,𝔽2)—5916no—
197B(4,𝔽2)—5932no—
198#410—5912no—
199C2×#99—5924no—
200C59#6039911yescharacteristic, complemented, Sylow
201C25—592no—
2022−1+4—216no—
2032−1+4—212no—
204H(5,𝔽2)—248no—
205H(5,𝔽2)—224no—
206H(5,𝔽2)—212no—
207H(5,𝔽2)—248no—
208H(5,𝔽2)C22×D59116yes—
209C2×#41—248no—
210C2×#41—224no—
211C2×#41—224no—
212C2×#41C22⋊D59112yescomplemented
213C2×#41C22×D5916yes—
214C22×D4—216no—
215C22×D4—224no—
216C22×D4—592no—
217C22×D4—591no—
218C22×D4—11812no—
219C22×D4—11812no—
220C22×D4C22⋊D59112yescomplemented
221C22×D4C22×D59112yes—
222C22×D4C22×D5911yescharacteristic
223C23×C4—591no—
224#127—1186no—
225#124—1186no—
226#123—5912no—
227#123—11812no—
228C22⋊Q8—1186no—
229#121—1186no—
230#121—11824no—
231#121—11812no—
232#120—5912no—
233#120—5924no—
234#120—11812no—
235#120—11824no—
236C4×D4—11812no—
237C4×D4—11812no—
238#117—596no—
239C2×C22⋊C4—5912no—
240C2×C22⋊C4—11812no—
241#99—5948no—
242C24—28no—
243C24—592no—
244C24—591no—
245C24—11812no—
246C24—4724no—
247C24C2×C22⋊D59112yescomplemented
248C24C23×D5912yes—
249#41—296no—
250#41—248no—
251#41—216no—
252#41—248no—
253#41—248no—
254#41—224no—
255C2×Q8—224no—
256C2×Q8—28no—
257C2×Q8C2×C22⋊D5916yescomplemented
258C2×D4—296no—
259C2×D4—224no—
260C2×D4—224no—
261C2×D4—248no—
262C2×D4—248no—
263C2×D4—248no—
264C2×D4—11824no—
265C2×D4—11812no—
266C2×D4—11812no—
267C2×D4—1186no—
268C2×D4—47224no—
269C2×D4—23616no—
270C2×D4C2×C22⋊D59112yes—
271C2×D4C2×C22⋊D59124yescomplemented
272C2×D4C2×C22⋊D5916yescomplemented
273C2×D4C23×D59112yes—
274C22×C4—224no—
275C22×C4—216no—
276C22×C4—591no—
277C22×C4—11812no—
278C22×C4—1186no—
279C22×C4C2×C22⋊D59112yescomplemented
280C22×C4C2×C22⋊D59112yescomplemented
281C22×C4C23×D5911yescharacteristic
282C4⋊C4—1186no—
283C4⋊C4—11812no—
284C22⋊C4—5912no—
285C22⋊C4—11812no—
286C22⋊C4—11812no—
287C22⋊C4—11812no—
288C22⋊C4—11824no—
289C42—1186no—
290C23—216no—
291C23—224no—
292C23—82no—
293C23—812no—
294C23—168no—
295C23—1183no—
296C23—1186no—
297C23—47212no—
298C23—4722no—
299C23—2362no—
300C23—23624no—
301C23—2368no—
302C23C22×C22⋊D59112yes—
303C23C22×C22⋊D59112yes—
304C23C24×D5911yescharacteristic
305Q8—216no—
306Q8—212no—
307D4—248no—
308D4—248no—
309D4—224no—
310D4—212no—
311D4—47224no—
312D4—23624no—
313D4—2368no—
314D4—2364no—
315D4#14273696116yes—
316C2×C4—248no—
317C2×C4—224no—
318C2×C4—224no—
319C2×C4—11812no—
320C2×C4—1183no—
321C2×C4—47212no—
322C2×C4—2364no—
323C2×C4C22×C22⋊D59112yes—
324C2×C4C22×C22⋊D5916yes—
325C22—212no—
326C22—412no—
327C22—41no—
328C22—412no—
329C22—824no—
330C22—816no—
331C22—1186no—
332C22—47216no—
333C22—23612no—
334C22—2363no—
335C22—2362no—
336C22—2361no—
337C22C2×#1427369618yes—
338C22C23×C22⋊D5913yes—
339C4—212no—
340C4—23612no—
341C4—2361no—
342C4C2×#1427369614yes—
343C2—23no—
344C2—28no—
345C2—412no—
346C2—4722no—
347C2—2362no—
348C2C22×#1427369611yescharacteristic
subgroupG/H
centerC2C22×#14273696
commutator, cosocleC23×C59C25
FrattiniC23C24×D59
FittingC59×H(7,𝔽2)C2
socleC2×C59C22×#120
hypercenterH(7,𝔽2)D59
hypocenterC59#60399
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #21437727C2semidirect
2. #21433498C2semidirect
3. #21433493C2semidirect
4. #21433488C2semidirect
5. C2×#16837589C2semidirect
6. #21432557C2semidirect
7. C59×H(7,𝔽2)C2semidirect
8. C22×D4×D59C22semidirect
11. #16837966C22semidirect
12. #16837961C22semidirect
13. #16837909C22semidirect
14. C2×#14273695C22semidirect
15. #16837877C22semidirect
16. #16837592C22semidirect
17. #16837589C22semidirect
18. #16837589C22semidirect
19. #16837376C22semidirect
20. C2×#14273642C22semidirect
22. C59×#585C22semidirect
25. C2×H(5,𝔽2)×C59C22semidirect
26. C24×D59C23semidirect
27. C22×D236C23semidirect
29. C22×C22⋊D59C23semidirect
31. C22×C4×D59C23semidirect
35. #14273708C23semidirect
42. #14273695C23semidirect
43. #14273695C23semidirect
46. #14273686C23semidirect
47. C2×C22⋊Dic59C23semidirect
49. #14273642C23semidirect
60. C2×#41×C59D4semidirect
64. C22×D4×C59D4semidirect
68. C24×C59C2×D4semidirect
82. C2×Q8×C59C2×D4semidirect
90. C2×D4×C59C2×D4semidirect
91. C2×D4×C59C2×D4semidirect
95. C22×C4×C59C2×D4semidirect
96. C22×C4×C59C2×D4semidirect
169. H(7,𝔽2)D59semidirect
183. #585C2×D59semidirect
186. C2×H(5,𝔽2)C2×D59semidirect
200. C59#60399semidirect
212. C2×#41C22⋊D59semidirect
220. C22×D4C22⋊D59semidirect
247. C24C2×C22⋊D59semidirect
257. C2×Q8C2×C22⋊D59semidirect
271. C2×D4C2×C22⋊D59semidirect
272. C2×D4C2×C22⋊D59semidirect
279. C22×C4C2×C22⋊D59semidirect
280. C22×C4C2×C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214377272
#214334982
#214334932
#214334882
C2×#168375892
#214325572
C59×H(7,𝔽2)2
#6039959