Finite Group Databasebeta

← #18004172

Subgroups — #18004172

Order 4736.

520 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924557C211yesmaximal, characteristic, complemented
2C22×C8×D37C211yesmaximal, characteristic, complemented
3#14558046C211yesmaximal, characteristic, complemented
4#14558043C211yesmaximal, characteristic, complemented
5#14557773C211yesmaximal, characteristic, complemented
6#14557772C211yesmaximal, characteristic, complemented
7#14557768C211yesmaximal, characteristic, complemented
8#14557767C211yesmaximal, characteristic, complemented
9#14557763C211yesmaximal, characteristic, complemented
10#14557762C211yesmaximal, characteristic, complemented
11#14557759C211yesmaximal, characteristic, complemented
12#14557568C211yesmaximal, characteristic, complemented
13#14557528C211yesmaximal, characteristic, complemented
14#14557526C211yesmaximal, characteristic, complemented
15C37×#465C211yesmaximal, characteristic, complemented
16C2×#10596966C2211yescharacteristic
17C2×#10596964C2211yescharacteristic
18C22×C4×D37C2211yescharacteristic
19#11924557—24no—
20C2×#10596942C2211yescharacteristic, complemented
21C2×#10596930C2211yescharacteristic, complemented
22C2×#10596939C2211yescharacteristic, complemented
23C22×#64436C2211yescharacteristic
24C2×C8×D37—24no—
25C2×C8×D37C2211yescharacteristic
26C2×C8×D37C2211yescharacteristic
27C2×C8×D37C2211yescharacteristic
28C2×C8×D37C2211yescharacteristic
29#11924521—21no—
30#11924518—21no—
31#11924516C2211yescharacteristic
32#11924515C2211yescharacteristic
33#11924512C2211yescharacteristic
34#11924511C2211yescharacteristic
35#11924510C2211yescharacteristic
36#11924507C2211yescharacteristic
37#11924506—21no—
38#11924504—21no—
39D4×Dic37—21no—
40#11924498—21no—
41#11924496—21no—
42#11924496—21no—
43#11924494C2211yescharacteristic
44#11924494C2211yescharacteristic
45#11924469C2211yescharacteristic, complemented
46#11924469C2211yescharacteristic, complemented
47#11924468C2211yescharacteristic, complemented
48#11924461C2211yescharacteristic, complemented
49#11924460C2211yescharacteristic, complemented
50#11924459C2211yescharacteristic, complemented
51#11924458C2211yescharacteristic, complemented
52#11924457C2211yescharacteristic, complemented
53#11924456C2211yescharacteristic, complemented
54C2×SD16×C37C2211yescharacteristic, complemented
55C22×C8×C37C2211yescharacteristic
56C22⋊Q8×C37C2211yescharacteristic
57#121×C37C2211yescharacteristic
58#106×C37C2211yescharacteristic, complemented
59#103×C37C2211yescharacteristic, complemented
60#102×C37C2211yescharacteristic, complemented
61C23×D37D411yescharacteristic
62C2×Q8×C37C2311yescharacteristic
63C2×D4×C37—21no—
64C2×D4×C37C2311yescharacteristic
65C22×C4×C37C2311yescharacteristic
66C2×C22⋊D37—21no—
67C2×C22⋊D37—21no—
68C2×C22⋊D37—21no—
69C22×Dic37—21no—
70C22×Dic37D411yescharacteristic
71#10596966—22no—
72#10596966—22no—
73#10596964—22no—
74#10596964—22no—
75C2×D148—21no—
76C2×D148—21no—
77C2×D148C2311yescharacteristic
78C2×C4×D37—22no—
79C2×C4×D37—22no—
80C2×C4×D37—21no—
81C2×C4×D37D411yescharacteristic
82C2×C4×D37D411yescharacteristic
83C2×C4×D37C2311yescharacteristic
84C2×C4×D37C2311yescharacteristic
85C2×Dic74C2311yescharacteristic
86SD16×C37—22no—
87C2×C8×C37—22no—
88C2×C8×C37C2311yescharacteristic
89C2×C8×C37C2311yescharacteristic
90C4⋊C4×C37—21no—
91C4⋊C4×C37C2311yescharacteristic
92C4⋊C4×C37C2311yescharacteristic
93C22⋊C4×C37—21no—
94C22⋊C4×C37—21no—
95C22⋊Dic37—21no—
96C22⋊Dic37—21no—
97#10596942—22no—
98#10596939—22no—
99#10596938—21no—
100#10596938—21no—
101#10596938—21no—
102#10596938—21no—
103#10596938—21no—
104#10596938—21no—
105C4⋊Dic37—21no—
106C4⋊Dic37C2311yescharacteristic
107C4⋊Dic37C2311yescharacteristic
108#10596936—21no—
109#10596936—21no—
110C4×Dic37—21no—
111C4×Dic37—21no—
112C2×#64436—22no—
113C2×#64436C2311yescharacteristic
114C2×#64436C2311yescharacteristic
115#10596930—22no—
116C8×D37—22no—
117C8×D37—22no—
118C8×D37—22no—
119C8×D37D414yescomplemented
120C23×C37—21no—
121C23×C37C2×D411yescharacteristic
122C22×D37—22no—
123C22×D37—22no—
124C22×D37—22no—
125C22×D37—21no—
126C22×D37C2×D411yescharacteristic
127C22×D37C2×D411yescharacteristic
128Q8×C37—21no—
129Q8×C37—21no—
130D4×C37—21no—
131D4×C37—21no—
132D4×C37—42no—
133C2×C4×C37—21no—
134C2×C4×C37—21no—
135C2×C4×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37C2×D411yescharacteristic
139C2×C4×C37C2411yescharacteristic
140C22⋊D37—42no—
141C22⋊D37—42no—
142C22⋊D37—42no—
143C2×Dic37—22no—
144C2×Dic37—21no—
145C2×Dic37—21no—
146C2×Dic37—21no—
147C2×Dic37—42no—
148C2×Dic37C2×D411yescharacteristic
149C2×Dic37C2×D411yescharacteristic
150D148—21no—
151D148—21no—
152D148—42no—
153D148—42no—
154C4×D37—22no—
155C4×D37—21no—
156C4×D37—21no—
157C4×D37—22no—
158C4×D37—42no—
159C4×D37C2×D412yes—
160C4×D37#4112yes—
161Dic74—21no—
162Dic74—21no—
163C8×C37—21no—
164C8×C37C2×D412yes—
165#64436—21no—
166#64436C2×D412yes—
167C22×C37—21no—
168C22×C37—21no—
169C22×C37—41no—
170C22×C37—41no—
171C22×C37—41no—
172C22×C37#13512yescomplemented
173C22×C37C22×D411yescharacteristic
174C2×D37—28no—
175C2×D37—21no—
176C2×D37—21no—
177C2×D37—21no—
178C2×D37—21no—
179C2×D37—41no—
180C2×D37—41no—
181C2×D37—41no—
182C2×D37#12112yes—
183C2×D37#13512yescomplemented
184C4×C37—21no—
185C4×C37—41no—
186C4×C37—41no—
187C4×C37—41no—
188C4×C37C22×D411yescharacteristic
189C4×C37C2×#4111yescharacteristic
190Dic37—21no—
191Dic37—41no—
192Dic37—41no—
193Dic37—41no—
194Dic37#12112yes—
195#3050—371nomaximal, Sylow
196C2×C37—22no—
197C2×C37—81no—
198C2×C37C2×#12111yescharacteristic
199C2×C37C2×#13511yescharacteristic
200C2×C37C2×#13511yescharacteristic
201D37—22no—
202D37—22no—
203D37—81no—
204C2×#135—371no—
205C23×C8—371no—
206#525—371no—
207#525—371no—
208#467—371no—
209#467—371no—
210#466—371no—
211#466—371no—
212#465—371no—
213#465—371no—
214#465—371no—
215#465D3711yescharacteristic, complemented
216#427—371no—
217#416—371no—
218#416—371no—
219C37#305011yescharacteristic, complemented, Sylow
220C2×#41—371no—
221C2×#41—371no—
222C23×C4—371no—
223#135—744no—
224C2×Q16—371no—
225C2×SD16—371no—
226C2×SD16C2×D3711yescharacteristic, complemented
227C2×D8—371no—
228C22×C8—371no—
229C22×C8—371no—
230C22×C8—371no—
231C22×C8—371no—
232C22×C8—371no—
233C22×C8—744no—
234C22×C8C2×D3711yescharacteristic
235#123—741no—
236#123—741no—
237C22⋊Q8—371no—
238C22⋊Q8—371no—
239C22⋊Q8—371no—
240C22⋊Q8C2×D3711yescharacteristic
241#121—371no—
242#121—371no—
243#121—371no—
244#121C2×D3711yescharacteristic
245#120—741no—
246#120—741no—
247C4×D4—741no—
248C4×D4—741no—
249C4×D4—741no—
250C4×D4—741no—
251#117—371no—
252#117—371no—
253#107—371no—
254#107—371no—
255#106—371no—
256#106C2×D3711yescharacteristic, complemented
257#103—371no—
258#103—371no—
259#103—371no—
260#103C2×D3711yescharacteristic, complemented
261#102—371no—
262#102—371no—
263#102—371no—
264#102C2×D3711yescharacteristic, complemented
265C24—371no—
266#41—742no—
267#41—742no—
268#41—742no—
269#41—742no—
270C2×Q8—371no—
271C2×Q8C22×D3711yescharacteristic
272C2×D4—21no—
273C2×D4—371no—
274C2×D4—741no—
275C2×D4—741no—
276C2×D4—741no—
277C2×D4—741no—
278C2×D4—741no—
279C2×D4C22×D3711yescharacteristic
280C22×C4—371no—
281C22×C4—371no—
282C22×C4—371no—
283C22×C4—371no—
284C22×C4—371no—
285C22×C4—742no—
286C22×C4—742no—
287C22×C4—741no—
288C22×C4—741no—
289C22×C4C22×D3711yescharacteristic
290Q16—742no—
291SD16—22no—
292SD16—742no—
293D8—742no—
294C2×C8—22no—
295C2×C8—371no—
296C2×C8—374no—
297C2×C8—371no—
298C2×C8—742no—
299C2×C8—742no—
300C2×C8—742no—
301C2×C8—742no—
302C2×C8C22×D3711yescharacteristic
303C2×C8C22×D3711yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—371no—
306C4⋊C4—371no—
307C4⋊C4—741no—
308C4⋊C4—741no—
309C4⋊C4—741no—
310C4⋊C4C22×D3711yescharacteristic
311C4⋊C4C22×D3711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—741no—
315C22⋊C4—741no—
316C22⋊C4—741no—
317C22⋊C4—741no—
318C22⋊C4—741no—
319C22⋊C4—741no—
320C22⋊C4—741no—
321C22⋊C4—741no—
322C42—741no—
323C42—741no—
324C23—21no—
325C23—371no—
326C23—371no—
327C23—742no—
328C23—742no—
329C23—742no—
330C23—741no—
331C23D4×D3711yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—741no—
335Q8—741no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—741no—
340D4—741no—
341D4—1482no—
342D4—1482no—
343D4—1482no—
344D4—1482no—
345D4—1482no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—371no—
352C2×C4—372no—
353C2×C4—372no—
354C2×C4—371no—
355C2×C4—742no—
356C2×C4—742no—
357C2×C4—742no—
358C2×C4—741no—
359C2×C4—741no—
360C2×C4—741no—
361C2×C4—741no—
362C2×C4—741no—
363C2×C4—1482no—
364C2×C4—1482no—
365C2×C4D4×D3711yescharacteristic
366C2×C4C23×D3711yescharacteristic
367C8—21no—
368C8—372no—
369C8—741no—
370C8D4×D3712yes—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—372no—
377C22—372no—
378C22—748no—
379C22—741no—
380C22—741no—
381C22—741no—
382C22—741no—
383C22—1481no—
384C22—1481no—
385C22—1481no—
386C22#1192455712yescomplemented
387C22C2×D4×D3711yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—372no—
393C4—741no—
394C4—1481no—
395C4—1481no—
396C4—1481no—
397C4C2×D4×D3711yescharacteristic
398C4#41×D3711yescharacteristic
399C2—22no—
400C2—81no—
401C2—2961no—
402C2—742no—
403C2—742no—
404C2#121×D3711yescharacteristic
405C2#135×D3711yescharacteristic
406C2C2×#1192455711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#465C2
socleC22×C37C22×D4
hypercenter#465D37
hypocenterC37#3050
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#11924557C2semidirect
2. C22×C8×D37C2semidirect
3. #14558046C2semidirect
4. #14558043C2semidirect
5. #14557773C2semidirect
6. #14557772C2semidirect
7. #14557768C2semidirect
8. #14557767C2semidirect
9. #14557763C2semidirect
10. #14557762C2semidirect
11. #14557759C2semidirect
12. #14557568C2semidirect
13. #14557528C2semidirect
14. #14557526C2semidirect
15. C37×#465C2semidirect
20. C2×#10596942C22semidirect
21. C2×#10596930C22semidirect
22. C2×#10596939C22semidirect
45. #11924469C22semidirect
46. #11924469C22semidirect
47. #11924468C22semidirect
48. #11924461C22semidirect
49. #11924460C22semidirect
50. #11924459C22semidirect
51. #11924458C22semidirect
52. #11924457C22semidirect
53. #11924456C22semidirect
54. C2×SD16×C37C22semidirect
58. #106×C37C22semidirect
59. #103×C37C22semidirect
60. #102×C37C22semidirect
119. C8×D37D4semidirect
172. C22×C37#135semidirect
183. C2×D37#135semidirect
215. #465D37semidirect
219. C37#3050semidirect
226. C2×SD16C2×D37semidirect
256. #106C2×D37semidirect
260. #103C2×D37semidirect
264. #102C2×D37semidirect
386. C22#11924557semidirect

Maximal subgroups

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

subgroupindex
C2×#119245572
C22×C8×D372
#145580462
#145580432
#145577732
#145577722
#145577682
#145577672
#145577632
#145577622
#145577592
#145575682
#145575282
#145575262
C37×#4652
#305037