Finite Group Databasebeta

← #28511395

Subgroups — #28511395

Order 15104.

2078 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273686C211yesmaximal, characteristic
2C22×C59⋊#95C211yesmaximal, characteristic
3#21422365C212yesmaximal, complemented
4#21422363C212yesmaximal, complemented
5#21422320C214yesmaximal, complemented
6#21422318C214yesmaximal
7C2×C59×#381C211yesmaximal, characteristic
8C24×Dic59C2212yes—
9C2×#14273686—24no—
10C2×#14273686—24no—
11C2×#14273686C418yescomplemented
12C22×C4⋊Dic59C2211yescharacteristic
13C22×C4⋊Dic59C2211yescharacteristic
14C22×C22⋊Dic59C2212yes—
15C22×C4×Dic59C2212yes—
16C2×C59⋊#95—24no—
17C2×C59⋊#95C2212yes—
18C2×C59⋊#95C2218yes—
19C2×C59⋊#95C2212yes—
20C2×C59⋊#95C2218yes—
21C24×C4×C59C2211yescharacteristic
22C22×C22⋊C4×C59C2212yes—
23C59×#381—24no—
24C2×#95×C59C2212yes—
25C2×#95×C59C2212yes—
26C23×Dic59—24no—
27C23×Dic59—28no—
28C23×Dic59—28no—
29C23×Dic59—44no—
30C23×Dic59C2312yes—
31C23×Dic59C2312yes—
32C23×Dic59C2314yes—
33#14273686—232no—
34C2×C4⋊Dic59—22no—
35C2×C4⋊Dic59—22no—
36C2×C4⋊Dic59—22no—
37C2×C4⋊Dic59—22no—
38C2×C4⋊Dic59C2×C414yescomplemented
39C2×C4⋊Dic59C2×C414yescomplemented
40C2×C22⋊Dic59—24no—
41C2×C22⋊Dic59—24no—
42C2×C22⋊Dic59C2×C418yes—
43C2×C4×Dic59—24no—
44C2×C4×Dic59—24no—
45C2×C4×Dic59C2×C418yescomplemented
46C59⋊#95—24no—
47C59⋊#95—24no—
48C59⋊#95—216no—
49C59⋊#95—48no—
50C25×C59C2311yescharacteristic
51C23×C4×C59—28no—
52C23×C4×C59C2×C414yes—
53C23×C4×C59D414yes—
54C23×C4×C59D414yes—
55C23×C4×C59C2314yes—
56C23×C4×C59C2311yescharacteristic
57C23×C4×C59C2311yescharacteristic
58C2×C22⋊C4×C59—24no—
59C2×C22⋊C4×C59—28no—
60#95×C59—24no—
61#95×C59—24no—
62C24×C59—24no—
63C24×C59—22no—
64C24×C59—22no—
65C24×C59C22×C412yes—
66C24×C59C2×D412yes—
67C24×C59C2×D412yes—
68C24×C59#4118yes—
69C24×C59C2411yescharacteristic
70C22×C4×C59—216no—
71C22×C4×C59—216no—
72C22×C4×C59—216no—
73C22×C4×C59—24no—
74C22×C4×C59—24no—
75C22×C4×C59—24no—
76C22×C4×C59—28no—
77C22×C4×C59—28no—
78C22×C4×C59—24no—
79C22×C4×C59—24no—
80C22×C4×C59C22⋊C4116yes—
81C22×C4×C59C22×C412yes—
82C22×C4×C59C22×C412yes—
83C22×C4×C59C2×D412yes—
84C22×C4×C59C2×D412yes—
85C22×C4×C59C2×D412yes—
86C22×C4×C59C2×D412yes—
87C22×Dic59—216no—
88C22×Dic59—24no—
89C22×Dic59—22no—
90C22×Dic59—22no—
91C22×Dic59—22no—
92C22×Dic59—22no—
93C22×Dic59—28no—
94C22×Dic59—24no—
95C22×Dic59—28no—
96C22×Dic59—24no—
97C22×Dic59—28no—
98C22×Dic59—24no—
99C22×Dic59—432no—
100C22×Dic59—48no—
101C22×Dic59—48no—
102C22×Dic59C22×C414yes—
103C22×Dic59C22×C414yes—
104C22×Dic59#4118yes—
105C22⋊C4×C59—48no—
106C22⋊Dic59—216no—
107C4⋊Dic59—28no—
108C4⋊Dic59—28no—
109C4×Dic59—216no—
110C23×C59—24no—
111C23×C59—24no—
112C23×C59—24no—
113C23×C59—24no—
114C23×C59—22no—
115C23×C59—22no—
116C23×C59—24no—
117C23×C59—216no—
118C23×C59—216no—
119C23×C59C2×C22⋊C414yes—
120C23×C59#11718yes—
121C23×C59#12318yes—
122C23×C59#12318yes—
123C23×C59C23×C411yescharacteristic
124C23×C59C22×D411yescharacteristic
125C23×C59C22×D411yescharacteristic
126C23×C59C2×#4114yes—
127C23×C59C2×#4114yes—
128C23×C59H(5,𝔽2)12yes—
129C23×C592−1+412yes—
130C2×C4×C59—24no—
131C2×C4×C59—24no—
132C2×C4×C59—24no—
133C2×C4×C59—24no—
134C2×C4×C59—24no—
135C2×C4×C59—24no—
136C2×C4×C59—232no—
137C2×C4×C59—44no—
138C2×C4×C59—44no—
139C2×C4×C59—44no—
140C2×C4×C59—48no—
141C2×C4×C59—48no—
142C2×C4×C59C2×C22⋊C414yes—
143C2×C4×C59C2×C22⋊C414yes—
144C2×Dic59—24no—
145C2×Dic59—24no—
146C2×Dic59—28no—
147C2×Dic59—28no—
148C2×Dic59—28no—
149C2×Dic59—216no—
150C2×Dic59—432no—
151C2×Dic59—44no—
152C2×Dic59—44no—
153C2×Dic59—44no—
154C2×Dic59—42no—
155C2×Dic59—44no—
156C2×Dic59—42no—
157C2×Dic59—44no—
158C2×Dic59—48no—
159#13958—591nomaximal, Sylow
160C22×C59—24no—
161C22×C59—216no—
162C22×C59—28no—
163C22×C59—28no—
164C22×C59—216no—
165C22×C59—24no—
166C22×C59#38118yes—
167C22×C59C22×C22⋊C411yescharacteristic
168C22×C59C2×#11712yes—
169C22×C59C4×#4112yes—
170C22×C59#52012yes—
171C22×C59C2×#12312yes—
172C22×C59C2×#12312yes—
173C22×C59#52512yes—
174C22×C59#53212yes—
175C22×C59#53511yescharacteristic
176C22×C59#53611yescharacteristic
177C22×C59#53712yes—
178C22×C59#55118yes—
179C22×C59#55218yes—
180C4×C59—24no—
181C4×C59—24no—
182C4×C59—84no—
183Dic59—44no—
184Dic59—48no—
185Dic59—44no—
186C22×#117—591no—
187C2×#381D5911yescharacteristic
188C22×#95—591no—
189#1445—592no—
190#1445—592no—
191#1434—594no—
192C22⋊#95—594no—
193C2×C59—28no—
194C2×C59C2×#38111yescharacteristic
195C2×C59#229711yescharacteristic
196C2×C59#231111yescharacteristic
197C2×C59#234314yes—
198C2×C59#248212yes—
199C2×C59#248412yes—
200C2×C59#249814yes—
201C24×C4—592no—
202C24×C4C2×D5911yescharacteristic
203C2×#117—598no—
204C2×#117—1184no—
205C2×#117—1184no—
206C22×C4⋊C4—591no—
207C22×C4⋊C4—591no—
208C22×C22⋊C4—592no—
209C22×C22⋊C4C2×D5912yes—
210C22×C42—592no—
211#381—24no—
212C2×#95—598no—
213C2×#95—592no—
214C2×#95—592no—
215C2×#95—598no—
216C2×#95—1184no—
217C2×#95C2×D5912yes—
218C2×#95C2×D5912yes—
219C59#1395811yescharacteristic, complemented, Sylow
220C25C22×D5911yescharacteristic
221C23×C4—28no—
222C23×C4—592no—
223C23×C4—592no—
224C23×C4—594no—
225C23×C4—1188no—
226C23×C4—1184no—
227C23×C4—1188no—
228C23×C4—2364no—
229C23×C4C4×D5914yes—
230C23×C4D23614yes—
231C23×C4C22⋊D5914yes—
232C23×C4C22×D5914yes—
233C23×C4C22×D5911yescharacteristic
234C23×C4C22×D5911yescharacteristic
235#117—11832no—
236C2×C4⋊C4—594no—
237C2×C4⋊C4—594no—
238C2×C4⋊C4—1182no—
239C2×C4⋊C4—1182no—
240C2×C4⋊C4—1182no—
241C2×C4⋊C4—1182no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—598no—
245C2×C22⋊C4—1184no—
246C2×C22⋊C4—1184no—
247C2×C42—598no—
248C2×C42—1184no—
249C2×C42—1184no—
250#95—24no—
251#95—24no—
252#95—11816no—
253#95—1184no—
254#95—1184no—
255#95—2368no—
256C24—24no—
257C24—22no—
258C24—22no—
259C24C2×C4×D5912yes—
260C24C2×D23612yes—
261C24#1174568718yes—
262C24C2×C22⋊D5912yes—
263C24C23×D5911yescharacteristic
264C22×C4—216no—
265C22×C4—216no—
266C22×C4—216no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—594no—
275C22×C4—594no—
276C22×C4—598no—
277C22×C4—11816no—
278C22×C4—1182no—
279C22×C4—1182no—
280C22×C4—1184no—
281C22×C4—1182no—
282C22×C4—1182no—
283C22×C4—1188no—
284C22×C4—1184no—
285C22×C4—1188no—
286C22×C4—1184no—
287C22×C4—1188no—
288C22×C4—1184no—
289C22×C4—23632no—
290C22×C4—2368no—
291C22×C4—2368no—
292C22×C4(C2×D59)⋊C4116yes—
293C22×C4C2×C4×D5912yes—
294C22×C4C2×C4×D5912yes—
295C22×C4C2×D23612yes—
296C22×C4C2×D23612yes—
297C22×C4C2×C22⋊D5912yes—
298C22×C4C2×C22⋊D5912yes—
299C4⋊C4—1188no—
300C4⋊C4—1188no—
301C22⋊C4—48no—
302C22⋊C4—11816no—
303C42—11816no—
304C23—24no—
305C23—24no—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—216no—
310C23—216no—
311C23—24no—
312C23—24no—
313C23C2×(C2×D59)⋊C414yes—
314C23#1427368318yes—
315C23#1427370818yes—
316C23#1427371118yes—
317C23C22×C4×D5911yescharacteristic
318C23C22×C22⋊D5911yescharacteristic
319C23C22×D23611yescharacteristic
320C23#41×D5914yes—
321C23C2×#1174568714yes—
322C23#1427377312yes—
323C23#1427377412yes—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—232no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—1188no—
337C2×C4—11816no—
338C2×C4—1184no—
339C2×C4—1184no—
340C2×C4—1188no—
341C2×C4—1188no—
342C2×C4—23632no—
343C2×C4—2364no—
344C2×C4—2364no—
345C2×C4—2364no—
346C2×C4—2362no—
347C2×C4—2364no—
348C2×C4—2362no—
349C2×C4—2364no—
350C2×C4—2368no—
351C2×C4C2×(C2×D59)⋊C414yes—
352C2×C4C2×(C2×D59)⋊C414yes—
353C22—216no—
354C22—216no—
355C22—28no—
356C22—24no—
357C22—28no—
358C22—24no—
359C22#1683726318yes—
360C22C22×(C2×D59)⋊C411yescharacteristic
361C22C2×#1427368312yes—
362C22#1683787012yes—
363C22#1683788212yes—
364C22C2×#1427370812yes—
365C22C2×#1427371112yes—
366C22#1683791412yes—
367C22#1683794812yes—
368C22#1683797111yescharacteristic
369C22#1683797811yescharacteristic
370C22#1683798012yes—
371C22#1683807518yes—
372C22#1683808918yes—
373C4—24no—
374C4—24no—
375C4—84no—
376C4—2368no—
377C4—2364no—
378C4—2364no—
379C2—28no—
380C2C2×#1683726311yescharacteristic
381C2#2142658411yescharacteristic
382C2#2142668411yescharacteristic
383C2#2142695814yes—
384C2#2142837212yes—
385C2#2142838512yes—
386C2#2142854214yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC2×C59×#381C2
socle, cosocleC24×C59C24
prime radicalC25×C59C23
hypercenterC2×#381D59
hypocenterC59#13958
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
3. #21422365C2semidirect
4. #21422363C2semidirect
5. #21422320C2semidirect
11. C2×#14273686C4semidirect
38. C2×C4⋊Dic59C2×C4semidirect
39. C2×C4⋊Dic59C2×C4semidirect
45. C2×C4×Dic59C2×C4semidirect
219. C59#13958semidirect

Maximal subgroups

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

subgroupindex
C22×#142736862
C22×C59⋊#952
#214223652
#214223632
#214223202
#214223182
C2×C59×#3812
#1395859