Finite Group Databasebeta

← #28743584

Subgroups — #28743584

Order 15104.

556 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433701C211yesmaximal, characteristic
2C2×#16837449C211yesmaximal, characteristic, complemented
3#21426314C214yesmaximal, complemented
4#21426309C214yesmaximal, complemented
5#21425870C211yesmaximal, characteristic, complemented
6#21425869C211yesmaximal, characteristic, complemented
7#21425845C211yesmaximal, characteristic, complemented
8C2×#16837183C211yesmaximal, characteristic, complemented
9C59×#2148C211yesmaximal, characteristic, complemented
10C22×#11745659C2211yescharacteristic
11C2×M16×D59—21no—
12#16837909—21no—
13C2×#14273667C2211yescharacteristic
14#16837657C2212yes—
15#16837652C2212yes—
16#16837645—22no—
17#16837449—22no—
18#16837449C2214yes—
19C2×#14273663C2211yescharacteristic
20C2×#14273662—21no—
21#16837440C2211yescharacteristic
22#16837398—21no—
23#16837226C2214yescomplemented
24#16837225C2214yescomplemented
25#16837218C2212yescomplemented
26#16837215C2212yescomplemented
27#16837183C2214yes—
28C2×#135×C59C2211yescharacteristic
29C2×M32×C59C2211yescharacteristic
30C2×#108×C59C2211yescharacteristic
31C59×#362C2212yescomplemented
32C59×#361C2212yescomplemented
33C22×C4×D59D411yescharacteristic
34C2×#11745664—21no—
35C2×#11745659—21no—
36C2×#11745659—21no—
37C2×#11745659—21no—
38C2×#11745659—21no—
39C2×#11745659—41no—
40C2×#11745659D412yes—
41C2×#11745659C2312yes—
42M16×D59—24no—
43M16×D59—42no—
44C22×C59⋊C8D411yescharacteristic
45C2×C8×D59—41no—
46#14273708—41no—
47#14273703—22no—
48#14273698—22no—
49#14273695—41no—
50C4×C22⋊D59—42no—
51#14273686—21no—
52#14273667—21no—
53#14273667C2312yes—
54C2×C59⋊C16—21no—
55#14273663—21no—
56#14273663C2312yes—
57#14273662—24no—
58#14273661—21no—
59#14273659C2312yes—
60#14273651—22no—
61#14273648—22no—
62C2×#41×C59—21no—
63#135×C59—21no—
64#135×C59—21no—
65#135×C59—21no—
66#135×C59—21no—
67C2×Q16×C59C2311yescharacteristic
68C2×SD16×C59—21no—
69C2×D8×C59C2311yescharacteristic
70C2×M16×C59—21no—
71C22×C8×C59C2311yescharacteristic
72M32×C59—21no—
73M32×C59C2312yes—
74C2×C16×C59—21no—
75#108×C59—21no—
76#108×C59C2312yes—
77C23×D59SD1611yescharacteristic
78#41×C59—41no—
79#41×C59—41no—
80#41×C59—41no—
81#41×C59—41no—
82C2×Q8×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59C2×D411yescharacteristic
87C2×C22⋊D59—42no—
88C22×Dic59SD1611yescharacteristic
89C2×C4×D59—21no—
90C2×C4×D59—21no—
91C2×C4×D59—21no—
92C2×C4×D59—21no—
93C2×C4×D59SD1612yes—
94C2×C4×D59C2×D412yes—
95Q16×C59—21no—
96Q16×C59—21no—
97SD16×C59—41no—
98SD16×C59—41no—
99D8×C59—21no—
100D8×C59—21no—
101M16×C59—22no—
102M16×C59—41no—
103C2×C8×C59—41no—
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59#4111yescharacteristic
106C2×C8×C59#4111yescharacteristic
107C2×C8×C59#4111yescharacteristic
108C2×C8×C59#4111yescharacteristic
109C2×C8×C59C2411yescharacteristic
110C22⋊Dic59—41no—
111C22⋊Dic59—42no—
112(C2×D59)⋊C4—42no—
113C4⋊Dic59—22no—
114Dic59⋊C4—42no—
115C4×Dic59—41no—
116#11745664—22no—
117#11745664—41no—
118C2×C59⋊C8—21no—
119C2×C59⋊C8—21no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8C2×D412yes—
122#11745659—22no—
123#11745659—22no—
124#11745659—22no—
125#11745659—22no—
126#11745659—44no—
127C8×D59—44no—
128C16×C59—22no—
129C59⋊C16—22no—
130C23×C59—41no—
131C23×C59C2×SD1611yescharacteristic
132C22×D59—21no—
133C22×D59—21no—
134C22×D59—81no—
135C22×D59C2×SD1612yes—
136Q8×C59—41no—
137Q8×C59—41no—
138D4×C59—41no—
139D4×C59—41no—
140D4×C59—81no—
141D4×C59—81no—
142C2×C4×C59—41no—
143C2×C4×C59—81no—
144C2×C4×C59—81no—
145C2×C4×C59#12311yescharacteristic
146C2×C4×C59#12311yescharacteristic
147C2×C4×C59C2×SD1611yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C59C22×D411yescharacteristic
151C22⋊D59—84no—
152C2×Dic59—21no—
153C2×Dic59—21no—
154C2×Dic59—42no—
155C2×Dic59C2×SD1612yes—
156C4×D59—22no—
157C4×D59—22no—
158C4×D59—22no—
159C4×D59—22no—
160C8×C59—42no—
161C8×C59#12311yescharacteristic
162C8×C59#12311yescharacteristic
163C8×C59C2×#4111yescharacteristic
164C8×C59C2×#4111yescharacteristic
165C59⋊C8—22no—
166C59⋊C8—42no—
167#33656—591nomaximal, Sylow
168C22×C59—41no—
169C22×C59—82no—
170C22×C59—81no—
171C22×C59#48111yescharacteristic
172C22×C59#48111yescharacteristic
173C22×C59C22×SD1611yescharacteristic
174C2×D59—22no—
175C2×D59—42no—
176C2×D59—42no—
177C2×D59—81no—
178C4×C59—81no—
179C4×C59#48111yescharacteristic
180C4×C59#48111yescharacteristic
181C4×C59C2×#12311yescharacteristic
182C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
183Dic59—22no—
184Dic59—82no—
185#3054—591no—
186C2×#430—591no—
187#2239—594no—
188#2238—594no—
189#2155—591no—
190#2155—591no—
191#2148—591no—
192#2148D5911yescharacteristic, complemented
193C2×#349—591no—
194C2×C59—21no—
195C2×C59—21no—
196C2×C59—21no—
197C2×C59—161no—
198C2×C59C2×#48111yescharacteristic
199D59—42no—
200C2×#135C2×D5911yescharacteristic
201C22×M16—591no—
202C22×M16—1181no—
203#525—1181no—
204C2×M32—591no—
205C2×M32C2×D5911yescharacteristic
206#470—592no—
207#469—592no—
208#468—1182no—
209#430—594no—
210#430—1182no—
211C2×#108—591no—
212C2×#108—1181no—
213C2×#108C2×D5911yescharacteristic
214#427—591no—
215#416—1181no—
216#365—594no—
217#365—594no—
218#362—592no—
219#362C2×D5912yescomplemented
220#361—592no—
221#361C2×D5912yescomplemented
222#349—594no—
223C59#3365611yescharacteristic, complemented, Sylow
224C2×#41—21no—
225C23×C4—591no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135—21no—
230C2×Q16C22×D5911yescharacteristic
231C2×SD16—21no—
232C2×D8C22×D5911yescharacteristic
233C2×M16—21no—
234C2×M16—592no—
235C2×M16—592no—
236C2×M16—1181no—
237C2×M16—1181no—
238C2×M16—1181no—
239C2×M16—1181no—
240C2×M16—1184no—
241C2×M16—1181no—
242C2×M16—2361no—
243C2×M16—2362no—
244C22×C8—591no—
245C22×C8—2361no—
246C22×C8C22×D5911yescharacteristic
247#123—2361no—
248C22⋊Q8—1182no—
249#121—1182no—
250#120—2361no—
251C4×D4—2362no—
252#117—1181no—
253M32—21no—
254M32—592no—
255M32—1181no—
256M32C22×D5912yes—
257C2×C16—21no—
258C2×C16—1181no—
259#108—21no—
260#108—592no—
261#108—1184no—
262#108—1181no—
263#108C22×D5912yes—
264#107—1181no—
265#106—592no—
266#103—1182no—
267#102—1182no—
268C24—591no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—2362no—
277C22×C4—41no—
278C22×C4—591no—
279C22×C4—592no—
280C22×C4—592no—
281C22×C4—1181no—
282C22×C4—1181no—
283C22×C4—1181no—
284C22×C4—1181no—
285C22×C4D4×D5911yescharacteristic
286Q16—21no—
287Q16—21no—
288SD16—41no—
289SD16—41no—
290D8—21no—
291D8—21no—
292M16—22no—
293M16—41no—
294M16—1182no—
295M16—1182no—
296M16—1182no—
297M16—1182no—
298M16—1182no—
299M16—2364no—
300M16—2361no—
301C2×C8—41no—
302C2×C8—592no—
303C2×C8—1181no—
304C2×C8—1181no—
305C2×C8—2364no—
306C2×C8—2361no—
307C2×C8#1174568511yescharacteristic
308C2×C8#1174568511yescharacteristic
309C2×C8D4×D5911yescharacteristic
310C2×C8#1174568711yescharacteristic
311C2×C8#1174568711yescharacteristic
312C2×C8C23×D5911yescharacteristic
313C4⋊C4—1182no—
314C4⋊C4—2362no—
315C22⋊C4—2362no—
316C22⋊C4—2362no—
317C22⋊C4—2361no—
318C42—2361no—
319C16—22no—
320C16—1182no—
321C23—41no—
322C23—592no—
323C23—1181no—
324C23—1181no—
325C23—4721no—
326C23SD16×D5911yescharacteristic
327Q8—41no—
328Q8—41no—
329D4—41no—
330D4—41no—
331D4—81no—
332D4—81no—
333D4—4724no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—592no—
338C2×C4—1181no—
339C2×C4—1181no—
340C2×C4—1182no—
341C2×C4—1182no—
342C2×C4—1182no—
343C2×C4—1182no—
344C2×C4—2362no—
345C2×C4#1427370911yescharacteristic
346C2×C4#1427370911yescharacteristic
347C2×C4SD16×D5911yescharacteristic
348C2×C4#1427375111yescharacteristic
349C2×C4#1427375111yescharacteristic
350C2×C4C2×D4×D5911yescharacteristic
351C8—42no—
352C8—1182no—
353C8—2362no—
354C8#1427370911yescharacteristic
355C8#1427370911yescharacteristic
356C8C2×#1174568711yescharacteristic
357C8C2×#1174568511yescharacteristic
358C22—41no—
359C22—82no—
360C22—81no—
361C22—1182no—
362C22—4721no—
363C22—2362no—
364C22—2362no—
365C22#1683771911yescharacteristic
366C22#1683771911yescharacteristic
367C22C2×SD16×D5911yescharacteristic
368C4—81no—
369C4—1182no—
370C4—4722no—
371C4#1683771911yescharacteristic
372C4#1683771911yescharacteristic
373C4C2×#1427370911yescharacteristic
374C4C2×#1427375111yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—161no—
379C2—2362no—
380C2C2×#1683771911yescharacteristic
subgroupG/H
centerC4#16837719
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2148C2
socleC2×C59C2×#481
prime radical#21433701C2
hypercenter#2148D59
hypocenterC59#33656
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. C2×#16837449C2semidirect
3. #21426314C2semidirect
4. #21426309C2semidirect
5. #21425870C2semidirect
6. #21425869C2semidirect
7. #21425845C2semidirect
8. C2×#16837183C2semidirect
9. C59×#2148C2semidirect
23. #16837226C22semidirect
24. #16837225C22semidirect
25. #16837218C22semidirect
26. #16837215C22semidirect
31. C59×#362C22semidirect
32. C59×#361C22semidirect
192. #2148D59semidirect
219. #362C2×D59semidirect
221. #361C2×D59semidirect
223. C59#33656semidirect

Maximal subgroups

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

subgroupindex
#214337012
C2×#168374492
#214263142
#214263092
#214258702
#214258692
#214258452
C2×#168371832
C59×#21482
#3365659