Finite Group Databasebeta

← #28620644

Subgroups — #28620644

Order 15104.

1318 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C2×#16837552C212yesmaximal
2#21426676C211yesmaximal, characteristic
3#21425239C212yesmaximal, complemented
4#21425179C212yesmaximal, complemented
5#21424451C214yesmaximal, complemented
6#21424409C212yesmaximal, complemented
7#21424167C211yesmaximal, characteristic, complemented
8C2×C59×#541C211yesmaximal, characteristic, complemented
9C23×D236C2211yescharacteristic
10C22×#11745671C2214yes—
11C2×#14273686C2211yescharacteristic
12#16837552—28no—
13C2×#14273651C2214yescomplemented
14C2×#14273647C2214yes—
15C2×C22⋊(C59⋊C8)C2212yes—
16#16837304C2212yes—
17#16837304C2212yes—
18#16837303—22no—
19#16837257—22no—
20#16837164C2214yescomplemented
21#16837157C2212yescomplemented
22#16837157C2212yescomplemented
23C59×#541—28no—
24C2×C4⋊Q8×C59C2211yescharacteristic, complemented
25C2×#126×C59—22no—
26C2×#125×C59—22no—
27C2×#125×C59C2211yescharacteristic, complemented
28C2×C22⋊Q8×C59C2212yes—
29C2×C4×Q8×C59C2212yescomplemented
30C2×#117×C59C2211yescharacteristic
31C24×D59—21no—
32C22×D236—21no—
33C22×D236—24no—
34C22×D236—24no—
35C22×D236—42no—
36C22×D236C2312yes—
37C2×#11745671—28no—
38C2×#11745671—28no—
39C2×#11745671—44no—
40C22×C59⋊C8C2314yes—
41#14273686—42no—
42C2×C4⋊Dic59C2312yes—
43C2×C22⋊Dic59—21no—
44C2×(C2×D59)⋊C4—24no—
45C2×(C2×D59)⋊C4—24no—
46C2×C4×Dic59—21no—
47#14273651—44no—
48#14273647—28no—
49C22⋊(C59⋊C8)—24no—
50C22×Q8×C59C2312yes—
51C23×C4×C59C2311yescharacteristic
52C4⋊Q8×C59—24no—
53#126×C59—216no—
54#125×C59—216no—
55#125×C59—24no—
56C22⋊Q8×C59—24no—
57C22⋊Q8×C59D418yes—
58C4×Q8×C59—24no—
59C4×Q8×C59—24no—
60#117×C59—24no—
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59—22no—
63C2×C4⋊C4×C59—22no—
64C2×C4⋊C4×C59—22no—
65C2×C4⋊C4×C59D412yes—
66C2×C4⋊C4×C59C2312yes—
67C2×C22⋊C4×C59—22no—
68C2×C22⋊C4×C59D412yes—
69C2×C42×C59—22no—
70C2×C42×C59C2311yescharacteristic
71C2×C42×C59C2311yescharacteristic
72C24×C59C2×D411yescharacteristic
73C23×D59—22no—
74C23×D59—24no—
75C23×D59—42no—
76C23×D59—42no—
77C23×D59—84no—
78C2×Q8×C59—22no—
79C2×Q8×C59—24no—
80C2×Q8×C59—24no—
81C2×Q8×C59C2×D414yes—
82C22×C4×C59—22no—
83C22×C4×C59—22no—
84C22×C4×C59—22no—
85C22×C4×C59—22no—
86C22×C4×C59C2×D412yes—
87C22×C4×C59C2×D412yes—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D414yes—
90C22×C4×C59C2411yescharacteristic
91C22×Dic59—22no—
92C2×D236—24no—
93C2×D236—24no—
94C2×D236—24no—
95C2×D236—22no—
96C2×D236—416no—
97C2×D236—42no—
98C2×D236—42no—
99C2×D236#4114yes—
100C4⋊C4×C59—28no—
101C4⋊C4×C59—28no—
102C4⋊C4×C59—28no—
103C4⋊C4×C59—28no—
104C4⋊C4×C59—24no—
105C4⋊C4×C59—22no—
106C4⋊C4×C59C2×D414yes—
107C22⋊C4×C59—28no—
108C22⋊C4×C59—24no—
109C42×C59—28no—
110C42×C59—22no—
111C42×C59—22no—
112C22⋊Dic59—42no—
113#11745671—416no—
114(C2×D59)⋊C4—84no—
115(C2×D59)⋊C4—84no—
116C4⋊Dic59—24no—
117C4×Dic59—42no—
118C2×C59⋊C8—24no—
119C2×C59⋊C8—24no—
120C2×C59⋊C8—24no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—24no—
124C23×C59#12012yes—
125C23×C59C22×D411yescharacteristic
126C22×D59—24no—
127C22×D59—48no—
128C22×D59—42no—
129C22×D59—42no—
130C22×D59—44no—
131C22×D59—44no—
132C22×D59—88no—
133C22×D59—88no—
134C22×D59—88no—
135Q8×C59—28no—
136Q8×C59—44no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—28no—
140C2×C4×C59—22no—
141C2×C4×C59—22no—
142C2×C4×C59—24no—
143C2×C4×C59—24no—
144C2×C4×C59—24no—
145C2×C4×C59—24no—
146C2×C4×C59—42no—
147C2×C4×C59—42no—
148C2×C4×C59—44no—
149C2×C4×C59#12012yes—
150C2×C4×C59C22×D412yes—
151C2×C4×C59C2×#4112yes—
152C2×C4×C59H(5,𝔽2)12yes—
153C2×Dic59—42no—
154C2×Dic59—44no—
155D236—28no—
156D236—48no—
157D236—44no—
158C59⋊C8—44no—
159#23942—591nomaximal, Sylow
160C22×C59—22no—
161C22×C59—24no—
162C22×C59—42no—
163C22×C59—42no—
164C22×C59C2×#12011yescharacteristic
165C22×C59#52511yescharacteristic
166C22×C59#53511yescharacteristic
167C22×C59#57714yes—
168C2×D59—48no—
169C2×D59—44no—
170C2×D59—816no—
171C2×D59—82no—
172C2×D59—82no—
173C2×D59—84no—
174C4×C59—22no—
175C4×C59—44no—
176C4×C59—42no—
177C4×C59—42no—
178C4×C59—44no—
179C4×C59#54614yes—
180Dic59—82no—
181C2×#541D5911yescharacteristic, complemented
182C2×#450—592no—
183#2309—591no—
184#2025—592no—
185#2013—592no—
186#1880—594no—
187#1873—592no—
188#1818—591no—
189C2×C59—42no—
190C2×C59#240411yescharacteristic
191C2×C59#301012yes—
192C2×C59#312312yes—
193C2×C59#322212yes—
194D59—84no—
195C23×D4—591no—
196C22×SD16—594no—
197#541—28no—
198C2×C4⋊Q8C2×D5911yescharacteristic, complemented
199C2×#126—22no—
200C2×#125—22no—
201C2×#125C2×D5911yescharacteristic, complemented
202C2×C22⋊Q8C2×D5912yes—
203C2×C4×Q8C2×D5912yescomplemented
204C2×#117—591no—
205C2×#117C2×D5911yescharacteristic
206#450—1188no—
207C2×#103—594no—
208C2×#102—594no—
209C2×C22⋊C8—592no—
210#390—592no—
211#390—592no—
212#390—1182no—
213#379—1182no—
214#340—594no—
215#336—592no—
216#336—592no—
217C59#2394211yescharacteristic, complemented, Sylow
218C25—1181no—
219C22×Q8C22×D5912yes—
220C22×D4—592no—
221C22×D4—1184no—
222C22×D4—1181no—
223C22×D4—1184no—
224C22×D4—2362no—
225C23×C4C22×D5911yescharacteristic
226C2×SD16—1188no—
227C2×SD16—1188no—
228C2×SD16—2364no—
229C22×C8—594no—
230C4⋊Q8—24no—
231#126—216no—
232#125—216no—
233#125—24no—
234C22⋊Q8—24no—
235C22⋊Q8C22⋊D5918yes—
236C4×Q8—24no—
237C4×Q8—24no—
238#117—24no—
239#117—2362no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—592no—
245C2×C4⋊C4C22⋊D5912yes—
246C2×C4⋊C4C22×D5912yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—1184no—
249C2×C22⋊C4—1184no—
250C2×C22⋊C4—1181no—
251C2×C22⋊C4C22⋊D5912yes—
252C2×C42—22no—
253C2×C42—1181no—
254C2×C42C22×D5911yescharacteristic
255C2×C42C22×D5911yescharacteristic
256#103—2364no—
257#102—1188no—
258C22⋊C8—1184no—
259C24—1184no—
260C24—1182no—
261C24—4724no—
262C24—2362no—
263C24—2362no—
264C24C2×C22⋊D5911yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8C2×C22⋊D5914yes—
269C2×D4—594no—
270C2×D4—1184no—
271C2×D4—1184no—
272C2×D4—1182no—
273C2×D4—1184no—
274C2×D4—23616no—
275C2×D4—2362no—
276C2×D4—2362no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—1182no—
282C22×C4C2×C22⋊D5912yes—
283C22×C4C2×C22⋊D5912yes—
284C22×C4C2×C22⋊D5911yescharacteristic
285C22×C4C2×C22⋊D5914yes—
286C22×C4C23×D5911yescharacteristic
287SD16—23616no—
288C2×C8—1184no—
289C2×C8—1184no—
290C2×C8—1184no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—1184no—
298C4⋊C4C2×C22⋊D5914yes—
299C22⋊C4—28no—
300C22⋊C4—24no—
301C22⋊C4—4724no—
302C22⋊C4—4724no—
303C22⋊C4—2362no—
304C42—28no—
305C42—22no—
306C42—22no—
307C42—2362no—
308C23—21no—
309C23—24no—
310C23—21no—
311C23—1184no—
312C23—4728no—
313C23—4728no—
314C23—4728no—
315C23—2368no—
316C23—2362no—
317C23—2364no—
318C23—2364no—
319C23—2362no—
320C23#1427369612yes—
321C23C22×C22⋊D5911yescharacteristic
322Q8—28no—
323Q8—44no—
324D4—1188no—
325D4—2368no—
326D4—2364no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—2364no—
340C2×C4—2362no—
341C2×C4#1427369612yes—
342C2×C4C22×C22⋊D5912yes—
343C2×C4#41×D5912yes—
344C2×C4#1427377312yes—
345C8—2364no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—47216no—
351C22—4722no—
352C22—4722no—
353C22—4724no—
354C22—2364no—
355C22—2368no—
356C22C2×#1427369611yescharacteristic
357C22#1683790911yescharacteristic
358C22#1683796611yescharacteristic
359C22#1683821014yes—
360C4—22no—
361C4—42no—
362C4—42no—
363C4—44no—
364C4—44no—
365C4—4722no—
366C4#1683804014yes—
367C2—42no—
368C2—4724no—
369C2#2142741111yescharacteristic
370C2#2143338212yes—
371C2#2143434312yes—
372C2#2143516212yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#541C2
socleC23×C59C22×D4
prime radicalC23×D236C22
hypercenterC2×#541D59
hypocenterC59#23942
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
3. #21425239C2semidirect
4. #21425179C2semidirect
5. #21424451C2semidirect
6. #21424409C2semidirect
7. #21424167C2semidirect
8. C2×C59×#541C2semidirect
13. C2×#14273651C22semidirect
20. #16837164C22semidirect
21. #16837157C22semidirect
22. #16837157C22semidirect
24. C2×C4⋊Q8×C59C22semidirect
27. C2×#125×C59C22semidirect
29. C2×C4×Q8×C59C22semidirect
181. C2×#541D59semidirect
198. C2×C4⋊Q8C2×D59semidirect
201. C2×#125C2×D59semidirect
203. C2×C4×Q8C2×D59semidirect
217. C59#23942semidirect

Maximal subgroups

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

subgroupindex
C2×#168375522
#214266762
#214252392
#214251792
#214244512
#214244092
#214241672
C2×C59×#5412
#2394259