Finite Group Databasebeta

← #28590779

Subgroups — #28590779

Order 15104.

914 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×Dic59⋊C8C211yesmaximal, characteristic
2C2×#16837358C211yesmaximal, characteristic
3#21426578C211yesmaximal, characteristic
4#21424658C214yesmaximal, complemented
5#21424651C214yesmaximal, complemented
6#21423794C211yesmaximal, characteristic
7#21423792C211yesmaximal, characteristic
8#21423789C211yesmaximal, characteristic
9C59×#1729C211yesmaximal, characteristic
10C22×#11745664C2211yescharacteristic
11C23×C59⋊C8C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13C22×C4×Dic59C2211yescharacteristic
14C2×Dic59⋊C8—24no—
15C2×Dic59⋊C8C2212yes—
16C2×Dic59⋊C8C2212yes—
17#16837358—24no—
18C2×#14273638C2212yes—
19C2×C4×C59⋊C8C2212yes—
20#16837298C412yes—
21#16837298C412yes—
22#16837274C412yes—
23#16837274C412yes—
24C4×Dic59⋊C4C2214yes—
25#16837156C2212yes—
26#16837156C2214yes—
27#16837156C2214yes—
28#16837156C2212yes—
29C22×M16×C59C2211yescharacteristic
30C23×C8×C59C2211yescharacteristic
31C2×#117×C59C2211yescharacteristic
32C59×#336C2214yes—
33C23×Dic59C2×C411yescharacteristic
34C23×Dic59C2×C411yescharacteristic
35C2×#11745664—22no—
36C2×#11745664—22no—
37C2×#11745664Q814yes—
38C22×C59⋊C8—22no—
39C22×C59⋊C8—22no—
40C22×C59⋊C8D414yes—
41C22×C59⋊C8C2311yescharacteristic
42C22×C59⋊C8C2311yescharacteristic
43C22×C59⋊C8C2312yes—
44#14273686—22no—
45#14273686—22no—
46C2×C4⋊Dic59C2×C412yes—
47C2×Dic59⋊C4C2×C414yes—
48C2×Dic59⋊C4C2×C414yes—
49C2×C22⋊Dic59C2×C412yes—
50C2×C4×Dic59—22no—
51C2×C4×Dic59—22no—
52C2×C4×Dic59C2×C412yes—
53C2×C4×Dic59C2312yes—
54C2×C4×Dic59C2312yes—
55Dic59⋊C8—24no—
56Dic59⋊C8—24no—
57#14273638—24no—
58C4×C59⋊C8—24no—
59C59⋊#95C2×C414yes—
60C59⋊#95C2×C414yes—
61C23×C4×C59C2311yescharacteristic
62C2×M16×C59—22no—
63C2×M16×C59—22no—
64C2×M16×C59—22no—
65C22×C8×C59—22no—
66C22×C8×C59—22no—
67C22×C8×C59—22no—
68C22×C8×C59C2312yes—
69C22×C8×C59C2312yes—
70#117×C59—22no—
71#117×C59—22no—
72C2×C4⋊C4×C59C2×C411yescharacteristic
73C2×C4⋊C4×C59C2×C411yescharacteristic
74C2×C22⋊C4×C59C2×C412yes—
75C2×C42×C59C2312yes—
76C24×C59C22×C411yescharacteristic
77C22×C4×C59C22×C412yes—
78C22×C4×C59C22×C412yes—
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2×Q812yes—
82C22×C4×C59#4112yes—
83C22×C4×C59#4112yes—
84C22×C4×C59#4112yes—
85C22×C4×C59#4112yes—
86C22×C4×C59C2411yescharacteristic
87C22×Dic59—22no—
88C22×Dic59—24no—
89C22×Dic59—22no—
90C22×Dic59—42no—
91C22×Dic59C22×C412yes—
92C22×Dic59C22×C414yes—
93C22×Dic59C22×C412yes—
94M16×C59—44no—
95C2×C8×C59—22no—
96C2×C8×C59—22no—
97C2×C8×C59—22no—
98C2×C8×C59—22no—
99C2×C8×C59—22no—
100C2×C8×C59—22no—
101C2×C8×C59—44no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C22⋊C4×C59—24no—
105C42×C59—22no—
106C42×C59—22no—
107C22⋊Dic59—24no—
108C4⋊Dic59—24no—
109Dic59⋊C4—28no—
110Dic59⋊C4—44no—
111C4×Dic59—24no—
112C4×Dic59—22no—
113C4×Dic59—22no—
114C4×Dic59—22no—
115C4×Dic59—22no—
116#11745664—28no—
117C2×C59⋊C8—28no—
118C2×C59⋊C8—22no—
119C2×C59⋊C8—21no—
120C2×C59⋊C8—21no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8C2×D412yes—
125C2×C59⋊C8C2×D412yes—
126C2×C59⋊C8C2×Q814yes—
127C23×C59—22no—
128C23×C59—22no—
129C23×C59#11712yes—
130C23×C59C4×D412yes—
131C23×C59C4×Q812yes—
132C23×C59C23×C411yescharacteristic
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—24no—
136C2×C4×C59—24no—
137C2×C4×C59#11712yes—
138C2×C4×C59C4×D412yes—
139C2×C4×C59C4×Q812yes—
140C2×C4×C59C22⋊Q814yes—
141C2×C4×C59#12314yes—
142C2×C4×C59#12514yes—
143C2×C4×C59#12614yes—
144C2×C4×C59C22×D411yescharacteristic
145C2×C4×C59C22×Q811yescharacteristic
146C2×C4×C59C2×#4111yescharacteristic
147C2×C4×C59C2×#4111yescharacteristic
148C2×C4×C59C2×#4111yescharacteristic
149C2×C4×C59C2×#4111yescharacteristic
150C2×Dic59—28no—
151C2×Dic59—22no—
152C2×Dic59—22no—
153C2×Dic59—24no—
154C2×Dic59—22no—
155C2×Dic59—24no—
156C2×Dic59—22no—
157C2×Dic59—24no—
158C2×Dic59—22no—
159C2×Dic59—44no—
160C2×Dic59#13114yes—
161C8×C59—42no—
162C8×C59—42no—
163C59⋊C8—24no—
164C59⋊C8—22no—
165C59⋊C8—22no—
166#21024—591nomaximal, Sylow
167C22×C59—24no—
168C22×C59—22no—
169C22×C59—22no—
170C22×C59—24no—
171C22×C59#38714yes—
172C22×C59C2×#11711yescharacteristic
173C22×C59C2×C4×D411yescharacteristic
174C22×C59C2×C4×Q811yescharacteristic
175C22×C59C2×#13112yes—
176C22×C59#56812yes—
177C4×C59—42no—
178C4×C59—42no—
179C4×C59#38714yes—
180C4×C59C2×C22⋊Q811yescharacteristic
181C4×C59C2×#12311yescharacteristic
182C4×C59C2×#12511yescharacteristic
183C4×C59C2×#12611yescharacteristic
184Dic59—24no—
185Dic59—44no—
186Dic59—42no—
187C22×C4⋊C8—591no—
188C2×#405—591no—
189#2296—591no—
190#1917—594no—
191#1916—594no—
192#1729—591no—
193#1729—591no—
194#1729D5911yescharacteristic
195#1728—591no—
196C2×C59—24no—
197C2×C59C2×#38711yescharacteristic
198C2×C59#292512yes—
199C2×C59#292911yescharacteristic
200C2×C59#293011yescharacteristic
201C2×C59#296212yes—
202C22×M16—591no—
203C22×M16C2×D5911yescharacteristic
204C23×C8—591no—
205C23×C8C2×D5911yescharacteristic
206C2×#117—591no—
207C2×#117C2×D5911yescharacteristic
208C22×C42—591no—
209C2×C4⋊C8—592no—
210C2×C4⋊C8—592no—
211C2×C4⋊C8—1184no—
212#405—1184no—
213C2×#97—592no—
214C2×C4×C8—592no—
215#389—592no—
216#389—592no—
217#385—592no—
218#385—592no—
219C4×C4⋊C4—594no—
220#336—592no—
221#336—594no—
222#336—594no—
223#336—592no—
224#336C2×D5914yes—
225C59#2102411yescharacteristic, complemented, Sylow
226C23×C4—591no—
227C23×C4—591no—
228C23×C4C22×D5911yescharacteristic
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—594no—
233C2×M16—1182no—
234C2×M16—1182no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—594no—
239C22×C8—591no—
240C22×C8—591no—
241C22×C8—592no—
242C22×C8—1182no—
243C22×C8—1182no—
244C22×C8C22×D5912yes—
245C22×C8C22×D5912yes—
246#117—22no—
247#117—22no—
248#117—1182no—
249#117—1182no—
250C2×C4⋊C4—592no—
251C2×C4⋊C4—594no—
252C2×C4⋊C4—594no—
253C2×C4⋊C4C4×D5911yescharacteristic
254C2×C4⋊C4C4×D5911yescharacteristic
255C2×C22⋊C4—592no—
256C2×C22⋊C4C4×D5912yes—
257C2×C42—592no—
258C2×C42—592no—
259C2×C42—592no—
260C2×C42—1182no—
261C2×C42—1182no—
262C2×C42C22×D5912yes—
263C4⋊C8—1184no—
264C4⋊C8—1184no—
265#97—1184no—
266C4×C8—1184no—
267#95—594no—
268#95—594no—
269C24C2×C4×D5911yescharacteristic
270C22×C4—592no—
271C22×C4—592no—
272C22×C4—594no—
273C22×C4—1184no—
274C22×C4—1182no—
275C22×C4—1182no—
276C22×C4—2362no—
277C22×C4C2×C4×D5912yes—
278C22×C4C2×C4×D5912yes—
279C22×C4C2×C4×D5911yescharacteristic
280C22×C4#1174568512yes—
281C22×C4#1174568512yes—
282C22×C4D4×D5912yes—
283C22×C4#1174568712yes—
284C22×C4#1174568712yes—
285C22×C4Q8×D5912yes—
286C22×C4C23×D5911yescharacteristic
287M16—44no—
288M16—1188no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—44no—
296C2×C8—592no—
297C2×C8—592no—
298C2×C8—594no—
299C2×C8—1188no—
300C2×C8—1182no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—1181no—
304C2×C8—1181no—
305C2×C8—1182no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—1188no—
309C4⋊C4—1184no—
310C4⋊C4—2364no—
311C22⋊C4—24no—
312C22⋊C4—1184no—
313C42—22no—
314C42—22no—
315C42—1184no—
316C42—1182no—
317C42—1182no—
318C42—1182no—
319C42—1182no—
320C23—22no—
321C23—22no—
322C23#1427368412yes—
323C23(C22⋊D59)⋊C412yes—
324C23#1427369212yes—
325C23C22×C4×D5911yescharacteristic
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—594no—
331C2×C4—1188no—
332C2×C4—1182no—
333C2×C4—1182no—
334C2×C4—1184no—
335C2×C4—1182no—
336C2×C4—1184no—
337C2×C4—1182no—
338C2×C4—1184no—
339C2×C4—1182no—
340C2×C4—2364no—
341C2×C4#1427368412yes—
342C2×C4(C22⋊D59)⋊C412yes—
343C2×C4#1427369212yes—
344C2×C4#1427370514yes—
345C2×C4#1427370914yes—
346C2×C4#1427371714yes—
347C2×C4#1427372014yes—
348C2×C4C2×D4×D5911yescharacteristic
349C2×C4C2×Q8×D5911yescharacteristic
350C2×C4C2×#1174568711yescharacteristic
351C2×C4C2×#1174568711yescharacteristic
352C2×C4C2×#1174568511yescharacteristic
353C2×C4C2×#1174568511yescharacteristic
354C8—42no—
355C8—42no—
356C8—1184no—
357C8—1182no—
358C8—1182no—
359C22—24no—
360C22—24no—
361C22—22no—
362C22—22no—
363C22#1683728414yes—
364C22C2×#1427368411yescharacteristic
365C22C2×(C22⋊D59)⋊C411yescharacteristic
366C22C2×#1427369211yescharacteristic
367C22#131×D5912yes—
368C22#1683816712yes—
369C4—42no—
370C4—42no—
371C4—1184no—
372C4—2364no—
373C4—2362no—
374C4#1683728414yes—
375C4C2×#1427370511yescharacteristic
376C4C2×#1427370911yescharacteristic
377C4C2×#1427371711yescharacteristic
378C4C2×#1427372011yescharacteristic
379C2—24no—
380C2C2×#1683728411yescharacteristic
381C2#2143276012yes—
382C2#2143279311yescharacteristic
383C2#2143280211yescharacteristic
384C2#2143304212yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutator, socleC23×C59C23×C4
FittingC59×#1729C2
prime radicalC24×C59C22×C4
cosocleC22×C4×C59C24
hypercenter#1729D59
hypocenterC59#21024
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
4. #21424658C2semidirect
5. #21424651C2semidirect
225. C59#21024semidirect

Maximal subgroups

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

subgroupindex
C22×Dic59⋊C82
C2×#168373582
#214265782
#214246582
#214246512
#214237942
#214237922
#214237892
C59×#17292
#2102459