Finite Group Databasebeta

← #28519722

Subgroups — #28519722

Order 15104.

1702 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
1C22×#14273706C211yesmaximal, characteristic
2C2×#16837265C211yesmaximal, characteristic
3#21422366C212yesmaximal, complemented
4#21422365C212yesmaximal, complemented
5#21422346C214yesmaximal, complemented
6#21422343C214yesmaximal
7C2×C59×#380C211yesmaximal, characteristic
8C23×Dic118C2211yescharacteristic
9C2×#14273706—24no—
10C2×#14273706—24no—
11C2×#14273706C418yescomplemented
12C22×C4⋊Dic59C2211yescharacteristic
13C22×Dic59⋊C4C2212yes—
14C22×C22⋊Dic59C2212yes—
15C22×C22⋊Dic59C2212yes—
16#16837265—24no—
17C2×C59⋊#95C2212yes—
18C2×C59⋊#95C2212yes—
19C2×C59⋊#95C2218yes—
20C2×C59⋊#95C2218yes—
21C24×C4×C59C2211yescharacteristic
22C22×C4⋊C4×C59C2212yes—
23C22×C22⋊C4×C59C2212yes—
24C59×#380—24no—
25C2×#95×C59C2212yes—
26C22×Dic118—22no—
27C22×Dic118—22no—
28C22×Dic118—44no—
29C22×Dic118C2×C414yes—
30C23×Dic59C2314yes—
31C23×Dic59C2312yes—
32C23×Dic59C2312yes—
33#14273706—232no—
34C2×C4⋊Dic59—22no—
35C2×C4⋊Dic59—22no—
36C2×C4⋊Dic59C2×C414yes—
37C2×Dic59⋊C4—24no—
38C2×Dic59⋊C4—24no—
39C2×Dic59⋊C4C2×C418yes—
40C2×C22⋊Dic59—24no—
41C2×C22⋊Dic59—24no—
42C2×C22⋊Dic59—24no—
43C2×C22⋊Dic59—44no—
44C2×C22⋊Dic59C2×C418yes—
45C59⋊#95—24no—
46C59⋊#95—24no—
47C59⋊#95—48no—
48C59⋊#95—48no—
49C25×C59C2311yescharacteristic
50C23×C4×C59—44no—
51C23×C4×C59C2×C414yes—
52C23×C4×C59D414yes—
53C23×C4×C59D414yes—
54C23×C4×C59C2314yes—
55C23×C4×C59C2311yescharacteristic
56C23×C4×C59C2311yescharacteristic
57C2×C4⋊C4×C59—24no—
58C2×C4⋊C4×C59—44no—
59C2×C22⋊C4×C59—24no—
60C2×C22⋊C4×C59—44no—
61#95×C59—24no—
62C24×C59—22no—
63C24×C59—24no—
64C24×C59—24no—
65C24×C59—22no—
66C24×C59C22×C412yes—
67C24×C59C2×D412yes—
68C24×C59C2×D412yes—
69C24×C59C2411yescharacteristic
70C22×C4×C59—24no—
71C22×C4×C59—24no—
72C22×C4×C59—24no—
73C22×C4×C59—28no—
74C22×C4×C59—28no—
75C22×C4×C59—28no—
76C22×C4×C59—48no—
77C22×C4×C59—48no—
78C22×C4×C59—48no—
79C22×C4×C59C22⋊C4116yes—
80C22×C4×C59C22×C412yes—
81C22×C4×C59C22×C412yes—
82C22×C4×C59C2×D412yes—
83C22×C4×C59C2×D412yes—
84C22×C4×C59C2×D412yes—
85C22×C4×C59C2×D412yes—
86C22×Dic59—22no—
87C22×Dic59—22no—
88C22×Dic59—22no—
89C22×Dic59—22no—
90C22×Dic59—24no—
91C22×Dic59—28no—
92C22×Dic59—24no—
93C22×Dic59—24no—
94C22×Dic59—24no—
95C22×Dic59—28no—
96C22×Dic59—28no—
97C22×Dic59—24no—
98C22×Dic59C22×C414yes—
99C22×Dic59C22×C414yes—
100C2×Dic118—28no—
101C2×Dic118—48no—
102C2×Dic118—48no—
103C2×Dic118—48no—
104C4⋊C4×C59—48no—
105C22⋊C4×C59—48no—
106C22⋊Dic59—216no—
107C22⋊Dic59—48no—
108C4⋊Dic59—28no—
109Dic59⋊C4—216no—
110C23×C59—24no—
111C23×C59—24no—
112C23×C59—24no—
113C23×C59—24no—
114C23×C59—24no—
115C23×C59—22no—
116C23×C59—24no—
117C23×C59—22no—
118C23×C59—24no—
119C23×C59—24no—
120C23×C59—48no—
121C23×C59—48no—
122C23×C59C2×C22⋊C414yes—
123C23×C59C23×C411yescharacteristic
124C23×C59C22×D411yescharacteristic
125C23×C59C22×D411yescharacteristic
126C23×C59H(5,𝔽2)14yes—
127C23×C59H(5,𝔽2)12yes—
128C23×C592−1+414yes—
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—416no—
137C2×C4×C59—44no—
138C2×C4×C59—44no—
139C2×C4×C59—44no—
140C2×C4×C59—416no—
141C2×C4×C59C2×C22⋊C414yes—
142C2×C4×C59C2×C22⋊C414yes—
143C2×Dic59—24no—
144C2×Dic59—24no—
145C2×Dic59—28no—
146C2×Dic59—28no—
147C2×Dic59—44no—
148C2×Dic59—44no—
149C2×Dic59—42no—
150C2×Dic59—42no—
151C2×Dic59—44no—
152C2×Dic59—44no—
153C2×Dic59—44no—
154C2×Dic59—44no—
155C2×Dic59—48no—
156C2×Dic59—48no—
157Dic118—416no—
158#14660—591nomaximal, Sylow
159C22×C59—24no—
160C22×C59—24no—
161C22×C59—24no—
162C22×C59—44no—
163C22×C59—44no—
164C22×C59—48no—
165C22×C59—48no—
166C22×C59C22×C22⋊C411yescharacteristic
167C22×C59#51812yes—
168C22×C59#51912yes—
169C22×C59#52012yes—
170C22×C59#53412yes—
171C22×C59#53412yes—
172C22×C59#53511yescharacteristic
173C22×C59#53612yes—
174C22×C59#53611yescharacteristic
175C22×C59#53612yes—
176C22×C59#53712yes—
177C22×C59#56218yes—
178C22×C59#56318yes—
179C4×C59—24no—
180C4×C59—24no—
181C4×C59—84no—
182Dic59—44no—
183Dic59—44no—
184Dic59—84no—
185C22×C22⋊Q8—591no—
186C2×#381—591no—
187C2×#380D5911yescharacteristic
188#1446—592no—
189#1445—592no—
190#1441—594no—
191#1440—594no—
192C2×C59—44no—
193C2×C59#230811yescharacteristic
194C2×C59#231011yescharacteristic
195C2×C59#231111yescharacteristic
196C2×C59#237814yes—
197C2×C59#281414yes—
198C2×C59#281612yes—
199C2×C59#281712yes—
200C23×Q8—591no—
201C24×C4C2×D5911yescharacteristic
202C2×C22⋊Q8—598no—
203C2×C22⋊Q8—1184no—
204C2×C22⋊Q8—1184no—
205C22×C4⋊C4—592no—
206C22×C4⋊C4—591no—
207C22×C4⋊C4C2×D5912yes—
208C22×C22⋊C4—592no—
209C22×C22⋊C4—592no—
210C22×C22⋊C4C2×D5912yes—
211#381—1184no—
212#380—24no—
213C2×#95—592no—
214C2×#95—592no—
215C2×#95—598no—
216C2×#95—598no—
217C2×#95C2×D5912yes—
218C59#1466011yescharacteristic, complemented, Sylow
219C25C22×D5911yescharacteristic
220C22×Q8—594no—
221C22×Q8—1182no—
222C22×Q8—1182no—
223C22×Q8—2364no—
224C23×C4—44no—
225C23×C4—592no—
226C23×C4—592no—
227C23×C4—594no—
228C23×C4C4×D5914yes—
229C23×C4D23614yes—
230C23×C4C22⋊D5914yes—
231C23×C4C22×D5914yes—
232C23×C4C22×D5911yescharacteristic
233C23×C4C22×D5911yescharacteristic
234C22⋊Q8—11832no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—598no—
238C2×C4⋊C4—594no—
239C2×C4⋊C4—1184no—
240C2×C4⋊C4—1182no—
241C2×C4⋊C4—1184no—
242C2×C4⋊C4—1182no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—598no—
246C2×C22⋊C4—1184no—
247C2×C22⋊C4—1184no—
248C2×C22⋊C4—1184no—
249C2×C22⋊C4—2364no—
250#95—24no—
251#95—1184no—
252#95—1184no—
253#95—2368no—
254#95—2368no—
255C24—22no—
256C24—24no—
257C24—24no—
258C24—22no—
259C24C2×C4×D5912yes—
260C24C2×D23612yes—
261C24C2×C22⋊D5912yes—
262C24C23×D5911yescharacteristic
263C2×Q8—1188no—
264C2×Q8—2368no—
265C2×Q8—2368no—
266C2×Q8—2368no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—594no—
277C22×C4—594no—
278C22×C4—1182no—
279C22×C4—1182no—
280C22×C4—1182no—
281C22×C4—1182no—
282C22×C4—1184no—
283C22×C4—1188no—
284C22×C4—1184no—
285C22×C4—1184no—
286C22×C4—1184no—
287C22×C4—1188no—
288C22×C4—1188no—
289C22×C4—1184no—
290C22×C4(C2×D59)⋊C4116yes—
291C22×C4C2×C4×D5912yes—
292C22×C4C2×C4×D5912yes—
293C22×C4C2×D23612yes—
294C22×C4C2×D23612yes—
295C22×C4C2×C22⋊D5912yes—
296C22×C4C2×C22⋊D5912yes—
297C4⋊C4—48no—
298C4⋊C4—11816no—
299C4⋊C4—1188no—
300C22⋊C4—48no—
301C22⋊C4—11816no—
302C22⋊C4—2368no—
303C23—24no—
304C23—24no—
305C23—22no—
306C23—24no—
307C23—22no—
308C23—24no—
309C23—24no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23—48no—
314C23—48no—
315C23C2×(C2×D59)⋊C414yes—
316C23C22×C4×D5911yescharacteristic
317C23C22×C22⋊D5911yescharacteristic
318C23C22×D23611yescharacteristic
319C23#1427377212yes—
320C23#1427377214yes—
321C23#1427377414yes—
322C23#1427377512yes—
323Q8—23616no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—416no—
331C2×C4—416no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—1184no—
336C2×C4—1184no—
337C2×C4—1188no—
338C2×C4—1188no—
339C2×C4—2368no—
340C2×C4—2368no—
341C2×C4—2364no—
342C2×C4—2364no—
343C2×C4—2364no—
344C2×C4—2364no—
345C2×C4—2364no—
346C2×C4—2362no—
347C2×C4—2362no—
348C2×C4—2364no—
349C2×C4C2×(C2×D59)⋊C414yes—
350C2×C4C2×(C2×D59)⋊C414yes—
351C22—24no—
352C22—24no—
353C22—24no—
354C22—48no—
355C22—48no—
356C22—44no—
357C22—44no—
358C22C22×(C2×D59)⋊C411yescharacteristic
359C22#1683787512yes—
360C22#1683787912yes—
361C22#1683788312yes—
362C22#1683796112yes—
363C22#1683796412yes—
364C22#1683796911yescharacteristic
365C22#1683797312yes—
366C22#1683797611yescharacteristic
367C22#1683797812yes—
368C22#1683798312yes—
369C22#1683814718yes—
370C22#1683815318yes—
371C4—24no—
372C4—24no—
373C4—84no—
374C4—4724no—
375C4—2364no—
376C4—2364no—
377C2—44no—
378C2#2142667111yescharacteristic
379C2#2142668011yescharacteristic
380C2#2142668911yescharacteristic
381C2#2142725914yes—
382C2#2143211114yes—
383C2#2143212412yes—
384C2#2143213112yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC2×C59×#380C2
socle, cosocleC24×C59C24
prime radicalC25×C59C23
hypercenterC2×#380D59
hypocenterC59#14660
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. #21422366C2semidirect
4. #21422365C2semidirect
5. #21422346C2semidirect
11. C2×#14273706C4semidirect
218. C59#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#142737062
C2×#168372652
#214223662
#214223652
#214223462
#214223432
C2×C59×#3802
#1466059