Finite Group Databasebeta

← #28559229

Subgroups — #28559229

Order 15104.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434669C211yesmaximal, characteristic
2#21433070C212yesmaximal, complemented
3#21422986C212yesmaximal, complemented
4#21422982C211yesmaximal, characteristic, complemented
5Q16×C59⋊C8C211yesmaximal, characteristic, complemented
6#21422893C212yesmaximal, complemented
7#21422890C211yesmaximal, characteristic, complemented
8#21422858C211yesmaximal, characteristic, complemented
9#21422555C212yesmaximal, complemented
10C82×D59C211yesmaximal, characteristic, complemented
11C59×#1591C211yesmaximal, characteristic, complemented
12#16837939C2212yes—
13#16837803C411yescharacteristic
14#16837787C411yescharacteristic
15#16837758C412yes—
16#16837640C411yescharacteristic
17#16837639C411yescharacteristic
18#16837634C412yes—
19Q8×C59⋊C8C2212yes—
20C4×#11745671C2212yes—
21C4×D472C2211yescharacteristic
22#16837481C2212yes—
23#16837475—22no—
24#16837467—22no—
25#16837467C2212yes—
26#16837462—22no—
27#16837458C2212yes—
28#16837359—22no—
29C4×C8×D59C2211yescharacteristic
30C4×C8×D59C2211yescharacteristic
31C4×C8×D59C2211yescharacteristic
32#16837155C2211yescharacteristic
33#16837152C2212yes—
34Q8⋊(C59⋊C8)C2212yes—
35#16837131C2212yes—
36#16837129C2212yes—
37C8×C59⋊C8C2211yescharacteristic
38C8×C59⋊C8C2211yescharacteristic
39C8×C59⋊C8C2211yescharacteristic
40C59×#446C2212yes—
41C4×Q16×C59C2211yescharacteristic
42C59×#335C2211yescharacteristic
43C59×Q8⋊C8C2212yes—
44C82×C59C2211yescharacteristic
45C2×#11745671C2×C412yes—
46C2×D472C2×C411yescharacteristic
47C2×#11745659—22no—
48C2×C8×D59—21no—
49C2×C8×D59—22no—
50C2×C8×D59C2×C411yescharacteristic
51C2×C8×D59C2×C411yescharacteristic
52C2×C8×D59D411yescharacteristic
53C2×C8×D59D411yescharacteristic
54#14273722—22no—
55#14273718C2×C412yes—
56#14273716C2×C412yes—
57#14273712—22no—
58#14273703C2×C412yes—
59#14273702C2×C412yes—
60C4×D236—22no—
61C4×D236C2312yes—
62#14273684—22no—
63C42×D59C2311yescharacteristic
64#14273661C2×C411yescharacteristic
65#14273658C2×C412yes—
66C4⋊(C59⋊C8)—22no—
67C4⋊(C59⋊C8)C2312yes—
68#14273651C2×C412yes—
69#14273649C2×C412yes—
70#14273647C2×C412yes—
71(C2×D59)⋊C8—22no—
72(C2×D59)⋊C8—22no—
73(C2×D59)⋊C8—22no—
74#14273637—22no—
75C4×C59⋊C8—22no—
76C4×C59⋊C8C2311yescharacteristic
77C4×C59⋊C8C2311yescharacteristic
78C4×C59⋊C8C2311yescharacteristic
79C8×Dic59—21no—
80C8×Dic59C2×C411yescharacteristic
81C8×Dic59C2×C411yescharacteristic
82C8×Dic59D411yescharacteristic
83C8×Dic59D411yescharacteristic
84C2×Q16×C59C2×C411yescharacteristic
85C4×Q8×C59C2312yes—
86#107×C59C2×C411yescharacteristic
87C4⋊C8×C59—22no—
88C4⋊C8×C59C2312yes—
89#103×C59C2×C412yes—
90#97×C59—22no—
91C4×C8×C59C2311yescharacteristic
92C4×C8×C59C2311yescharacteristic
93C4×C8×C59C2311yescharacteristic
94C2×Q8×C59C22×C412yes—
95C2×D236—22no—
96C2×D236C22×C412yes—
97C2×C4×D59—22no—
98C2×C4×D59C22×C411yescharacteristic
99C2×C4×D59C2×D411yescharacteristic
100C2×C4×D59#4111yescharacteristic
101Q16×C59—22no—
102C2×C8×C59—21no—
103C2×C8×C59—22no—
104C2×C8×C59C22×C411yescharacteristic
105C2×C8×C59C22×C411yescharacteristic
106C2×C8×C59C2×D411yescharacteristic
107C2×C8×C59C2×D411yescharacteristic
108C4⋊C4×C59—22no—
109C4⋊C4×C59C22×C412yes—
110C42×C59—22no—
111C42×C59C2411yescharacteristic
112#11745671—24no—
113(C2×D59)⋊C4—22no—
114(C2×D59)⋊C4—22no—
115(C2×D59)⋊C4—22no—
116(C2×D59)⋊C4—22no—
117(C2×D59)⋊C4—22no—
118C4⋊Dic59—22no—
119C4⋊Dic59C22×C412yes—
120Dic59⋊C4—22no—
121Dic59⋊C4—22no—
122C4×Dic59C22×C411yescharacteristic
123C4×Dic59C2×D411yescharacteristic
124C4×Dic59#4111yescharacteristic
125C2×C59⋊C8—21no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8C22×C411yescharacteristic
128C2×C59⋊C8C22×C411yescharacteristic
129C2×C59⋊C8C2×D411yescharacteristic
130C2×C59⋊C8C2×D411yescharacteristic
131D472—22no—
132#11745659—44no—
133C8×D59—24no—
134C8×D59—22no—
135C8×D59—22no—
136C8×D59—22no—
137C8×D59—22no—
138C8×D59—44no—
139C22×D59—22no—
140C22×D59C4×D411yescharacteristic
141Q8×C59—22no—
142Q8×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—22no—
145C2×C4×C59C23×C411yescharacteristic
146C2×C4×C59C22×D411yescharacteristic
147C2×C4×C59C2×#4111yescharacteristic
148C2×Dic59—22no—
149C2×Dic59C4×D411yescharacteristic
150C2×Dic59C4×D411yescharacteristic
151C2×Dic59C4×D411yescharacteristic
152D236—22no—
153D236—22no—
154D236—44no—
155C4×D59—22no—
156C4×D59—21no—
157C4×D59—21no—
158C4×D59—21no—
159C4×D59—21no—
160C4×D59—42no—
161C4×D59—42no—
162C8×C59—22no—
163C8×C59—21no—
164C8×C59—21no—
165C8×C59—42no—
166C8×C59#13112yes—
167C8×C59#13512yes—
168C59⋊C8—22no—
169C59⋊C8—21no—
170C59⋊C8—21no—
171C59⋊C8—42no—
172C59⋊C8#13112yes—
173C59⋊C8#13512yes—
174#17959—591nomaximal, Sylow
175C22×C59C2×C4×D411yescharacteristic
176C2×D59—21no—
177C2×D59—21no—
178C2×D59—42no—
179C2×D59—42no—
180C2×D59—42no—
181C2×D59#44312yes—
182C4×C59—21no—
183C4×C59—42no—
184C4×C59—42no—
185C4×C59C2×#13111yescharacteristic
186C4×C59C2×#13111yescharacteristic
187C4×C59C2×#13511yescharacteristic
188C4×C59C2×#13511yescharacteristic
189Dic59—21no—
190Dic59—21no—
191Dic59—21no—
192Dic59—42no—
193#3168—591no—
194#2966—592no—
195#1591D5911yescharacteristic, complemented
196#1590—592no—
197#1589—591no—
198C8×Q16—591no—
199C8×SD16—592no—
200C8×D8—591no—
201#1568—591no—
202#1494—592no—
203C2×C82—591no—
204C2×C59#293111yescharacteristic
205C2×C59C4×#13511yescharacteristic
206C2×C59C2×#44311yescharacteristic
207D59—22no—
208D59—82no—
209#529—592no—
210#499—591no—
211#495—591no—
212#488—592no—
213#467—591no—
214#466—591no—
215#465—592no—
216#446C2×D5912yes—
217C8×Q8—592no—
218C4×Q16C2×D5911yescharacteristic
219C4×SD16—592no—
220C4×D8—591no—
221#436—592no—
222#435—1182no—
223C8×D4—592no—
224C8×D4—1182no—
225#433—1182no—
226#432—592no—
227#405—1182no—
228C2×C4×C8—591no—
229C2×C4×C8—591no—
230C2×C4×C8—591no—
231#335—591no—
232#335C2×D5911yescharacteristic
233#334—592no—
234Q8⋊C8—592no—
235Q8⋊C8C2×D5912yes—
236D4⋊C8—592no—
237D4⋊C8—592no—
238C82—591no—
239C82—591no—
240C82—591no—
241C82C2×D5911yescharacteristic
242C59#1795911yescharacteristic, complemented, Sylow
243C2×Q16C4×D5911yescharacteristic
244C2×SD16—592no—
245C2×D8—591no—
246C2×M16—1182no—
247C22×C8—591no—
248C22×C8—591no—
249C22×C8—591no—
250C22×C8—591no—
251C22×C8—1182no—
252C22×C8—1181no—
253#126—1182no—
254#125—592no—
255#124—592no—
256#123—1182no—
257C22⋊Q8—592no—
258#121—592no—
259C4×Q8C22×D5912yes—
260C4×D4—592no—
261C4×D4—1182no—
262#117—1182no—
263C2×C42—591no—
264#107—591no—
265#107C4×D5911yescharacteristic
266#106—592no—
267C4⋊C8—22no—
268C4⋊C8—592no—
269C4⋊C8—1182no—
270C4⋊C8C22×D5912yes—
271#103—592no—
272#103C4×D5912yes—
273#102—592no—
274#102—592no—
275C22⋊C8—1182no—
276C22⋊C8—1182no—
277C22⋊C8—1182no—
278#97—22no—
279#97—1182no—
280C4×C8—591no—
281C4×C8—591no—
282C4×C8—591no—
283C4×C8—591no—
284C4×C8—591no—
285C4×C8—591no—
286C4×C8—591no—
287C4×C8—1182no—
288C4×C8—1181no—
289C4×C8C22×D5911yescharacteristic
290C4×C8C22×D5911yescharacteristic
291C4×C8C22×D5911yescharacteristic
292C2×Q8C2×C4×D5912yes—
293C2×D4—592no—
294C2×D4—1182no—
295C22×C4—591no—
296C22×C4—591no—
297C22×C4—591no—
298C22×C4—1182no—
299Q16—22no—
300SD16—1184no—
301D8—1182no—
302M16—2364no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—591no—
306C2×C8—591no—
307C2×C8—591no—
308C2×C8—591no—
309C2×C8—1182no—
310C2×C8—1182no—
311C2×C8—1184no—
312C2×C8—1181no—
313C2×C8—1182no—
314C2×C8—1182no—
315C2×C8—1182no—
316C2×C8—2364no—
317C2×C8C2×C4×D5911yescharacteristic
318C2×C8C2×C4×D5911yescharacteristic
319C2×C8D4×D5911yescharacteristic
320C2×C8D4×D5911yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—592no—
323C4⋊C4—1182no—
324C4⋊C4—1182no—
325C4⋊C4—1182no—
326C4⋊C4C2×C4×D5912yes—
327C22⋊C4—1182no—
328C22⋊C4—1182no—
329C22⋊C4—1182no—
330C22⋊C4—1182no—
331C22⋊C4—1182no—
332C42—22no—
333C42—591no—
334C42—591no—
335C42—591no—
336C42C23×D5911yescharacteristic
337C23—591no—
338C23—1182no—
339Q8—22no—
340Q8—22no—
341D4—1182no—
342D4—1182no—
343D4—2364no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—591no—
347C2×C4—591no—
348C2×C4—591no—
349C2×C4—1182no—
350C2×C4—1181no—
351C2×C4—1181no—
352C2×C4—1181no—
353C2×C4—1181no—
354C2×C4—1182no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4C22×C4×D5911yescharacteristic
358C2×C4C2×D4×D5911yescharacteristic
359C2×C4#41×D5911yescharacteristic
360C8—22no—
361C8—21no—
362C8—21no—
363C8—42no—
364C8—592no—
365C8—592no—
366C8—1182no—
367C8—1181no—
368C8—1181no—
369C8—2362no—
370C8#1427373512yes—
371C8#1427375012yes—
372C22—592no—
373C22—1181no—
374C22—1181no—
375C22—2362no—
376C22—2362no—
377C22—2362no—
378C22C4×D4×D5911yescharacteristic
379C4—21no—
380C4—42no—
381C4—42no—
382C4—1181no—
383C4—1181no—
384C4—1181no—
385C4—2362no—
386C4#131×D5911yescharacteristic
387C4C2×#1427373511yescharacteristic
388C4#135×D5911yescharacteristic
389C4C2×#1427375011yescharacteristic
390C2—1182no—
391C2—4722no—
392C2#2143280511yescharacteristic
393C2C4×#1427375011yescharacteristic
394C2#443×D5911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1591C2
socleC22×C59C2×C4×D4
prime radical#16837639C4
cosocleC42×C59C24
hypercenter#1591D59
hypocenterC59#17959
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

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

NH ≅ G/Ntype
2. #21433070C2semidirect
3. #21422986C2semidirect
4. #21422982C2semidirect
5. Q16×C59⋊C8C2semidirect
6. #21422893C2semidirect
7. #21422890C2semidirect
8. #21422858C2semidirect
9. #21422555C2semidirect
10. C82×D59C2semidirect
11. C59×#1591C2semidirect
195. #1591D59semidirect
242. C59#17959semidirect

Maximal subgroups

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

subgroupindex
#214346692
#214330702
#214229862
#214229822
Q16×C59⋊C82
#214228932
#214228902
#214228582
#214225552
C82×D592
C59×#15912
#1795959