Finite Group Databasebeta

← #28486269

Subgroups — #28486269

Order 15104.

436 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433092C211yesmaximal, characteristic, complemented
2#21422707C211yesmaximal, characteristic, complemented
3#21422656C211yesmaximal, characteristic, complemented
4#21422634C211yesmaximal, characteristic, complemented
5#21421995C211yesmaximal, characteristic, complemented
6#21421978C211yesmaximal, characteristic, complemented
7C59×#3180C211yesmaximal, characteristic
8#16837533—21no—
9#16837474—21no—
10#16837474—21no—
11D4×C59⋊C8—21no—
12#16837463—21no—
13#16837459C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837358—21no—
16C2×#14273638C2211yescharacteristic
17#16837268C2211yescharacteristic
18#16837142—21no—
19#16837142—21no—
20#16837139—21no—
21#16837136—21no—
22#16837136—21no—
23#16837136—21no—
24C59×#532C2211yescharacteristic
25C59×#529C2211yescharacteristic
26C59×#501C411yescharacteristic, complemented
27C59×#496C411yescharacteristic, complemented
28C59×#490C411yescharacteristic, complemented
29C59×#489C411yescharacteristic, complemented
30C59×#466—21no—
31C59×#465—21no—
32C59×#442—21no—
33C59×#440—21no—
34C2×#97×C59C2211yescharacteristic
35C2×#11745664—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38C2×C4⋊Dic59—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)D411yescharacteristic
47C22⋊(C59⋊C8)D411yescharacteristic
48#14273638—21no—
49#14273638—21no—
50#14273638—21no—
51#14273638D411yescharacteristic
52#14273638D411yescharacteristic
53C4×C59⋊C8—21no—
54C4×C59⋊C8—21no—
55C59⋊#95—21no—
56C59⋊#95—21no—
57C2×#41×C59—21no—
58C2×SD16×C59—21no—
59C2×D8×C59—21no—
60C22×C8×C59—21no—
61C4⋊Q8×C59C2×C411yescharacteristic
62#127×C59C2×C411yescharacteristic
63#126×C59—21no—
64#125×C59C2×C411yescharacteristic
65#124×C59—21no—
66#124×C59C2×C411yescharacteristic
67#123×C59—21no—
68C22⋊Q8×C59D411yescharacteristic
69#121×C59—21no—
70#121×C59—21no—
71#121×C59D411yescharacteristic
72C4×Q8×C59D411yescharacteristic
73C4×D4×C59—21no—
74C4×D4×C59—21no—
75C4×D4×C59—21no—
76C4×D4×C59D411yescharacteristic
77#117×C59—21no—
78C2×C42×C59C2311yescharacteristic
79#107×C59—21no—
80#106×C59—21no—
81#103×C59—21no—
82#102×C59—21no—
83#102×C59—21no—
84#102×C59—21no—
85#97×C59—21no—
86#97×C59C2×C411yescharacteristic
87#97×C59C2×C411yescharacteristic
88#41×C59—81no—
89#41×C59—81no—
90C2×Q8×C59—21no—
91C2×Q8×C59C22⋊C411yescharacteristic
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—41no—
97C2×D4×C59C22⋊C411yescharacteristic
98C22×C4×C59—21no—
99C22×C4×C59—41no—
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59#4111yescharacteristic
103C22×Dic59—41no—
104SD16×C59—42no—
105D8×C59—42no—
106C2×C8×C59—21no—
107C2×C8×C59—21no—
108C2×C8×C59—42no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59C22⋊C411yescharacteristic
116C4⋊C4×C59C22⋊C411yescharacteristic
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—21no—
120C22⋊C4×C59—21no—
121C22⋊C4×C59—21no—
122C22⋊C4×C59—41no—
123C22⋊C4×C59—41no—
124C42×C59—21no—
125C42×C59C22×C411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128C42×C59#4111yescharacteristic
129C4⋊Dic59—81no—
130#11745664—42no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—21no—
135C2×C59⋊C8—42no—
136C2×C59⋊C8—42no—
137C2×C59⋊C8—41no—
138C23×C59—21no—
139C23×C59—41no—
140C23×C59#12111yescharacteristic
141Q8×C59—41no—
142Q8×C59—81no—
143D4×C59—41no—
144D4×C59—41no—
145D4×C59—41no—
146D4×C59—41no—
147D4×C59—41no—
148D4×C59—81no—
149D4×C59—81no—
150D4×C59—81no—
151D4×C59—81no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—82no—
163C2×C4×C59C2×C22⋊C411yescharacteristic
164C2×C4×C59C4×D411yescharacteristic
165C2×C4×C59C4×D411yescharacteristic
166C2×C4×C59#12011yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#12311yescharacteristic
169C2×Dic59—81no—
170C2×Dic59—81no—
171C2×Dic59—81no—
172C8×C59—22no—
173C8×C59—41no—
174C59⋊C8—22no—
175C59⋊C8—41no—
176C59⋊C8—41no—
177C59⋊C8—41no—
178C59⋊C8—81no—
179#9432—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—41no—
183C22×C59—41no—
184C22×C59—41no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59—81no—
188C22×C59#38611yescharacteristic
189C22×C59#47111yescharacteristic
190C22×C59#47211yescharacteristic
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—41no—
200C4×C59—81no—
201Dic59—161no—
202#3180D5911yescharacteristic
203#2968—591no—
204#1530—591no—
205#1517—591no—
206#1512—591no—
207#1302—591no—
208#1295—591no—
209C2×C59—21no—
210C2×C59—21no—
211C2×C59—81no—
212C2×C59—161no—
213C2×C59#188411yescharacteristic
214C2×C59#190611yescharacteristic
215C2×C59#190711yescharacteristic
216#532C2×D5911yescharacteristic
217#529C2×D5911yescharacteristic
218#501Dic5911yescharacteristic, complemented
219#496Dic5911yescharacteristic, complemented
220#490Dic5911yescharacteristic, complemented
221#489Dic5911yescharacteristic, complemented
222#466—21no—
223#465—21no—
224#446—1181no—
225#442—21no—
226#440—21no—
227#435—1181no—
228#435—1181no—
229C8×D4—1181no—
230#433—1181no—
231#432—591no—
232#431—591no—
233#405—1181no—
234C2×#97—591no—
235C2×#97C2×D5911yescharacteristic
236#382—591no—
237#329—1181no—
238#329—1181no—
239#328—1181no—
240#327—1181no—
241#327—1181no—
242#327—1181no—
243C59#943211yescharacteristic, complemented, Sylow
244C2×#41—21no—
245C2×SD16—21no—
246C2×D8—21no—
247C2×M16—1181no—
248C22×C8—21no—
249C22×C8—1181no—
250C22×C8—1181no—
251C4⋊Q8C2×Dic5911yescharacteristic
252#127C2×Dic5911yescharacteristic
253#126—21no—
254#125C2×Dic5911yescharacteristic
255#124—21no—
256#124C2×Dic5911yescharacteristic
257#123—21no—
258C22⋊Q8C22⋊D5911yescharacteristic
259#121—21no—
260#121—21no—
261#121C22⋊D5911yescharacteristic
262C4×Q8C22⋊D5911yescharacteristic
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4C22⋊D5911yescharacteristic
267#117—21no—
268C2×C4⋊C4—1181no—
269C2×C42C22×D5911yescharacteristic
270#107—21no—
271#106—21no—
272C4⋊C8—1181no—
273C4⋊C8—1181no—
274C4⋊C8—1181no—
275C4⋊C8—1181no—
276#103—21no—
277#102—21no—
278#102—21no—
279#102—21no—
280C22⋊C8—591no—
281C22⋊C8—591no—
282C22⋊C8—1181no—
283C22⋊C8—1181no—
284C22⋊C8—1181no—
285#97—21no—
286#97—591no—
287#97—591no—
288#97—1181no—
289#97—1181no—
290#97—1181no—
291#97C2×Dic5911yescharacteristic
292#97C2×Dic5911yescharacteristic
293C4×C8—1181no—
294C4×C8—1181no—
295#95—1181no—
296#95—1181no—
297#41—81no—
298#41—81no—
299C2×Q8—21no—
300C2×Q8C22⋊Dic5911yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4C22⋊Dic5911yescharacteristic
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—2361no—
310C22×C4D4×D5911yescharacteristic
311C22×C4#1174568711yescharacteristic
312C22×C4C2×C22⋊D5911yescharacteristic
313SD16—42no—
314D8—42no—
315M16—2362no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—42no—
319C2×C8—1181no—
320C2×C8—1181no—
321C2×C8—1181no—
322C2×C8—1181no—
323C2×C8—2362no—
324C2×C8—2362no—
325C2×C8—2361no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—4721no—
333C4⋊C4C22⋊Dic5911yescharacteristic
334C4⋊C4C22⋊Dic5911yescharacteristic
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—21no—
343C42D4×D5911yescharacteristic
344C42#1174568711yescharacteristic
345C42C22×Dic5911yescharacteristic
346C42C2×C22⋊D5911yescharacteristic
347C23—21no—
348C23—41no—
349C23#1427369911yescharacteristic
350Q8—41no—
351Q8—81no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—4721no—
373C2×C4—4721no—
374C2×C4—4721no—
375C2×C4C2×C22⋊Dic5911yescharacteristic
376C2×C4D4×Dic5911yescharacteristic
377C2×C4D4×Dic5911yescharacteristic
378C2×C4#1427369511yescharacteristic
379C2×C4#1427369911yescharacteristic
380C2×C4#1427370811yescharacteristic
381C8—22no—
382C8—41no—
383C8—1182no—
384C8—4721no—
385C8—2361no—
386C8—2361no—
387C8—2361no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1683728211yescharacteristic
397C22#1683766311yescharacteristic
398C22#1683767011yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4—9441no—
410C2—21no—
411C2—21no—
412C2—81no—
413C2—161no—
414C2#2142448011yescharacteristic
415C2#2142459911yescharacteristic
416C2#2142460611yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3180C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3180D59
hypocenterC59#9432
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3180

Complements

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

NH ≅ G/Ntype
1. #21433092C2semidirect
2. #21422707C2semidirect
3. #21422656C2semidirect
4. #21422634C2semidirect
5. #21421995C2semidirect
6. #21421978C2semidirect
26. C59×#501C4semidirect
27. C59×#496C4semidirect
28. C59×#490C4semidirect
29. C59×#489C4semidirect
218. #501Dic59semidirect
219. #496Dic59semidirect
220. #490Dic59semidirect
221. #489Dic59semidirect
243. C59#9432semidirect

Maximal subgroups

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

subgroupindex
#214330922
#214227072
#214226562
#214226342
#214219952
#214219782
C59×#31802
#943259