Finite Group Databasebeta

← #28742097

Subgroups — #28742097

Order 15104.

652 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
1#21437229C211yesmaximal, characteristic
2#443×D59C212yesmaximal, complemented
3#21425982C211yesmaximal, characteristic, complemented
4#21425981C211yesmaximal, characteristic, complemented
5#21425979C212yesmaximal, complemented
6#21425978C211yesmaximal, characteristic, complemented
7#21425975C212yesmaximal, complemented
8#21425889C211yesmaximal, characteristic, complemented
9#21425822C212yesmaximal, complemented
10C4×#14273666C211yesmaximal, characteristic, complemented
11C59×#2178C211yesmaximal, characteristic, complemented
12#135×D59C2212yes—
13#131×D59—22no—
14#16837844C412yescomplemented
15#16837841C412yescomplemented
16#16837839C412yescomplemented
17#16837836C412yescomplemented
18#16837834C2211yescharacteristic
19#16837832C2212yes—
20C2×#14273666C2211yescharacteristic
21C2×#14273666C2211yescharacteristic
22#16837520C2212yescomplemented
23#16837519C2212yescomplemented
24#16837518C2212yescomplemented
25#16837517—22no—
26#16837516—22no—
27#108×D59C2212yes—
28#104×D59—22no—
29C4×C8×D59C2211yescharacteristic
30#16837232C2211yescharacteristic, complemented
31#16837231C2212yescomplemented
32#16837210C2212yescomplemented
33#16837209C2212yescomplemented
34#16837208C2212yescomplemented
35#16837177C2211yescharacteristic, complemented
36#16837177C2211yescharacteristic, complemented
37C4×C59⋊C16C2211yescharacteristic, complemented
38C59×#508C2211yescharacteristic
39C59×#443C2212yescomplemented
40C59×#368C2211yescharacteristic, complemented
41C59×#359C2212yescomplemented
42C4×C16×C59C2211yescharacteristic, complemented
43#41×D59—22no—
44#14273750C2×C412yes—
45#14273749C2312yes—
46#14273748—22no—
47#14273747—22no—
48#14273746C2×C412yes—
49Q16×D59C2×C412yes—
50SD16×D59—22no—
51D8×D59C2×C412yes—
52#14273735—22no—
53#14273735—42no—
54#14273734—22no—
55#14273733—22no—
56#14273733—42no—
57M16×D59—22no—
58M16×D59—42no—
59C2×C8×D59—42no—
60C2×C8×D59D411yescharacteristic
61C2×C8×D59C2311yescharacteristic
62C42×D59D411yescharacteristic
63Dic472C2×C411yescharacteristic, complemented
64#14273673C2×C412yescomplemented
65#14273672C2×C412yescomplemented
66#14273671C2×C412yescomplemented
67#14273670C2×C412yescomplemented
68D944C2×C411yescharacteristic, complemented
69#14273668C2×C412yescomplemented
70#14273666—22no—
71#14273666C2×C412yes—
72#14273666C2×C412yes—
73C2×C59⋊C16C2311yescharacteristic
74C2×C59⋊C16C2311yescharacteristic
75#14273663C2312yes—
76#14273662—22no—
77#14273655—22no—
78#14273654—22no—
79#14273653—22no—
80C4×C59⋊C8D411yescharacteristic
81C8×Dic59D411yescharacteristic
82C8×Dic59C2311yescharacteristic
83#135×C59C2312yes—
84#131×C59—22no—
85Q32×C59C2×C411yescharacteristic, complemented
86SD32×C59C2×C412yescomplemented
87D16×C59C2×C411yescharacteristic, complemented
88C2×C16×C59C2311yescharacteristic
89C2×C16×C59C2311yescharacteristic
90#108×C59C2312yes—
91#104×C59—22no—
92C4×C8×C59C2311yescharacteristic
93#41×C59—22no—
94#11745689—22no—
95Q8×D59—22no—
96#11745687—22no—
97#11745687—42no—
98D4×D59—22no—
99D4×D59—42no—
100#11745685—22no—
101#11745685—42no—
102C2×C4×D59—21no—
103C2×C4×D59—42no—
104C2×C4×D59C2×D411yescharacteristic
105Q16×C59C22×C412yes—
106SD16×C59—22no—
107D8×C59C22×C412yes—
108M16×C59—22no—
109M16×C59—42no—
110C2×C8×C59—42no—
111C2×C8×C59C2×D411yescharacteristic
112C2×C8×C59C2411yescharacteristic
113C42×C59C2×D411yescharacteristic
114#11745672—22no—
115#11745671—22no—
116#11745670—22no—
117#11745669—22no—
118C4×Dic59—21no—
119C4×Dic59C2×D411yescharacteristic
120#11745664—22no—
121#11745664—42no—
122C2×C59⋊C8—42no—
123C2×C59⋊C8C2×D411yescharacteristic
124C2×C59⋊C8C2×D411yescharacteristic
125Dic236C22×C412yes—
126D472C22×C412yes—
127#11745660—22no—
128#11745659—42no—
129#11745659—42no—
130#11745659—42no—
131C8×D59—42no—
132C8×D59—42no—
133C8×D59—42no—
134C8×D59D812yes—
135C8×D59D812yes—
136C8×D59C22×C412yes—
137C8×D59#4112yes—
138C16×C59—21no—
139C16×C59C22×C411yescharacteristic
140C16×C59C22×C411yescharacteristic
141C59⋊C16—21no—
142C59⋊C16C22×C412yes—
143C22×D59—42no—
144C22×D59#13511yescharacteristic
145Q8×C59—22no—
146D4×C59—22no—
147D4×C59—42no—
148C2×C4×C59—21no—
149C2×C4×C59—42no—
150C2×C4×C59C22×D411yescharacteristic
151C22⋊D59—42no—
152C22⋊D59—42no—
153C22⋊D59—42no—
154C2×Dic59—21no—
155C2×Dic59—42no—
156C2×Dic59#13511yescharacteristic
157D236—22no—
158D236—42no—
159C4×D59—42no—
160C4×D59—42no—
161C4×D59—42no—
162C4×D59—42no—
163C4×D59C4×D412yes—
164C4×D59C2×D812yes—
165Dic118—22no—
166Dic118—42no—
167C8×C59—42no—
168C8×C59C2×D811yescharacteristic
169C8×C59C2×D811yescharacteristic
170C8×C59C23×C411yescharacteristic
171C8×C59C2×#4111yescharacteristic
172C59⋊C8—42no—
173C59⋊C8C4×D412yes—
174C59⋊C8C2×D812yes—
175#33463—591nomaximal, Sylow
176C22×C59—42no—
177C22×C59C2×#13511yescharacteristic
178C2×D59—41no—
179C2×D59—42no—
180C2×D59—84no—
181C2×D59C4×D812yes—
182C4×C59—22no—
183C4×C59—42no—
184C4×C59C2×C4×D411yescharacteristic
185C4×C59C22×D811yescharacteristic
186Dic59—41no—
187Dic59—42no—
188Dic59C4×D812yes—
189#3414—591no—
190C2×#443—592no—
191#2179—591no—
192#2179—591no—
193#2179—592no—
194#2178—592no—
195#2178—591no—
196#2178D5911yescharacteristic, complemented
197#2161—591no—
198#2143—592no—
199C4×M32—591no—
200C2×C59—21no—
201C2×C59—82no—
202C2×C59C2×C4×D811yescharacteristic
203D59—22no—
204D59—82no—
205C2×#135—592no—
206C2×#131—1182no—
207#510—592no—
208#510—592no—
209#509—592no—
210#509—592no—
211#508—592no—
212#508—591no—
213#508C2×D5911yescharacteristic
214C2×M32—591no—
215C2×M32—591no—
216#443—592no—
217#443—592no—
218#443—592no—
219#443—1182no—
220#443—1182no—
221#443C2×D5912yescomplemented
222C2×#108—592no—
223C2×#104—1182no—
224C2×C4×C8—591no—
225#368—591no—
226#368—592no—
227#368C2×D5911yescharacteristic, complemented
228#359—592no—
229#359—592no—
230#359—592no—
231#359C2×D5912yescomplemented
232#346—591no—
233#346—591no—
234C4×C16—591no—
235C4×C16C2×D5911yescharacteristic, complemented
236C59#3346311yescharacteristic, complemented, Sylow
237C2×#41—1182no—
238#135—592no—
239#135—592no—
240#135—592no—
241#135—1182no—
242#135—1182no—
243#135C22×D5912yes—
244C2×Q16—592no—
245C2×SD16—1182no—
246C2×D8—592no—
247#131—22no—
248#131—1182no—
249#131—1182no—
250#131—1182no—
251#131—2362no—
252#131—2362no—
253C2×M16—1182no—
254C2×M16—2362no—
255C22×C8—591no—
256C22×C8—591no—
257C22×C8—2362no—
258C2×C42—591no—
259Q32—592no—
260Q32—591no—
261Q32C4×D5911yescharacteristic, complemented
262SD32—592no—
263SD32—592no—
264SD32—592no—
265SD32C4×D5912yescomplemented
266D16—592no—
267D16—591no—
268D16C4×D5911yescharacteristic, complemented
269M32—592no—
270M32—592no—
271M32—1182no—
272C2×C16—591no—
273C2×C16—591no—
274C2×C16C22×D5911yescharacteristic
275C2×C16C22×D5911yescharacteristic
276#108—592no—
277#108—1182no—
278#108C22×D5912yes—
279#104—22no—
280#104—1182no—
281#104—1182no—
282#104—1182no—
283C4×C8—591no—
284C4×C8—591no—
285C4×C8—591no—
286C4×C8C22×D5911yescharacteristic
287#41—22no—
288#41—1182no—
289#41—1182no—
290#41—1182no—
291#41—2362no—
292#41—2362no—
293C2×Q8—1182no—
294C2×D4—1182no—
295C2×D4—2362no—
296C22×C4—591no—
297C22×C4—1181no—
298C22×C4—2362no—
299Q16—592no—
300Q16—1182no—
301Q16C2×C4×D5912yes—
302SD16—22no—
303SD16—1182no—
304SD16—1182no—
305SD16—1182no—
306D8—592no—
307D8—1182no—
308D8C2×C4×D5912yes—
309M16—22no—
310M16—42no—
311M16—1182no—
312M16—2362no—
313M16—2362no—
314M16—2362no—
315M16—2362no—
316C2×C8—42no—
317C2×C8—591no—
318C2×C8—591no—
319C2×C8—592no—
320C2×C8—592no—
321C2×C8—592no—
322C2×C8—592no—
323C2×C8—2362no—
324C2×C8—2362no—
325C2×C8—2362no—
326C2×C8—2362no—
327C2×C8D4×D5911yescharacteristic
328C2×C8C23×D5911yescharacteristic
329C42—591no—
330C42—1181no—
331C42D4×D5911yescharacteristic
332C16—21no—
333C16—592no—
334C16—1181no—
335C16C2×C4×D5911yescharacteristic
336C16C2×C4×D5911yescharacteristic
337C23—591no—
338C23—2362no—
339Q8—22no—
340Q8—1182no—
341Q8—2362no—
342D4—22no—
343D4—42no—
344D4—1182no—
345D4—2362no—
346D4—2362no—
347D4—2362no—
348D4—2362no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—591no—
352C2×C4—592no—
353C2×C4—592no—
354C2×C4—1181no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4—2362no—
358C2×C4—2362no—
359C2×C4—2362no—
360C2×C4C2×D4×D5911yescharacteristic
361C8—42no—
362C8—592no—
363C8—592no—
364C8—2362no—
365C8D8×D5911yescharacteristic
366C8D8×D5911yescharacteristic
367C8C22×C4×D5911yescharacteristic
368C8#41×D5911yescharacteristic
369C22—42no—
370C22—592no—
371C22—4724no—
372C22—2361no—
373C22—2362no—
374C22#135×D5911yescharacteristic
375C4—22no—
376C4—42no—
377C4—592no—
378C4—2361no—
379C4—2362no—
380C4C4×D4×D5911yescharacteristic
381C4C2×D8×D5911yescharacteristic
382C2—21no—
383C2—82no—
384C2—1182no—
385C2—4722no—
386C2C4×D8×D5911yescharacteristic
subgroupG/H
centerC8D8×D59
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC59×#2178C2
socleC2×C59C2×C4×D8
prime radical#21437229C2
cosocleC2×C8×C59C24
hypercenter#2178D59
hypocenterC59#33463
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ #2178

Complements

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

NH ≅ G/Ntype
2. #443×D59C2semidirect
3. #21425982C2semidirect
4. #21425981C2semidirect
5. #21425979C2semidirect
6. #21425978C2semidirect
7. #21425975C2semidirect
8. #21425889C2semidirect
9. #21425822C2semidirect
10. C4×#14273666C2semidirect
11. C59×#2178C2semidirect
14. #16837844C4semidirect
15. #16837841C4semidirect
16. #16837839C4semidirect
17. #16837836C4semidirect
22. #16837520C22semidirect
23. #16837519C22semidirect
24. #16837518C22semidirect
30. #16837232C22semidirect
31. #16837231C22semidirect
32. #16837210C22semidirect
33. #16837209C22semidirect
34. #16837208C22semidirect
35. #16837177C22semidirect
36. #16837177C22semidirect
37. C4×C59⋊C16C22semidirect
39. C59×#443C22semidirect
40. C59×#368C22semidirect
41. C59×#359C22semidirect
42. C4×C16×C59C22semidirect
63. Dic472C2×C4semidirect
64. #14273673C2×C4semidirect
65. #14273672C2×C4semidirect
66. #14273671C2×C4semidirect
67. #14273670C2×C4semidirect
68. D944C2×C4semidirect
69. #14273668C2×C4semidirect
85. Q32×C59C2×C4semidirect
86. SD32×C59C2×C4semidirect
87. D16×C59C2×C4semidirect
196. #2178D59semidirect
221. #443C2×D59semidirect
227. #368C2×D59semidirect
231. #359C2×D59semidirect
235. C4×C16C2×D59semidirect
236. C59#33463semidirect
261. Q32C4×D59semidirect
265. SD32C4×D59semidirect
268. D16C4×D59semidirect

Maximal subgroups

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

subgroupindex
#214372292
#443×D592
#214259822
#214259812
#214259792
#214259782
#214259752
#214258892
#214258222
C4×#142736662
C59×#21782
#3346359