Finite Group Databasebeta

← #18006110

Subgroups — #18006110

Order 4736.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924565C211yesmaximal, characteristic, complemented
2C22×#10596940C211yesmaximal, characteristic, complemented
3#14558196C211yesmaximal, characteristic, complemented
4#14558177C211yesmaximal, characteristic
5#14557924C211yesmaximal, characteristic, complemented
6#14557778C211yesmaximal, characteristic, complemented
7#14557772C211yesmaximal, characteristic, complemented
8#14557754C211yesmaximal, characteristic, complemented
9#14557736C211yesmaximal, characteristic, complemented
10#14557696C211yesmaximal, characteristic, complemented
11#14557691C211yesmaximal, characteristic, complemented
12#14557670C211yesmaximal, characteristic, complemented
13#14557630C211yesmaximal, characteristic, complemented
14#14557601C211yesmaximal, characteristic, complemented
15C37×#450C211yesmaximal, characteristic, complemented
16C2×#10596964C2211yescharacteristic
17C22×Dic74C2211yescharacteristic
18C2×Q8×D37C2211yescharacteristic, complemented
19#11924565—24no—
20C2×Dic148C2211yescharacteristic, complemented
21C2×Dic148C2211yescharacteristic, complemented
22C2×#10596942C2211yescharacteristic, complemented
23C2×#10596940—24no—
24C2×#10596940C2211yescharacteristic
25C2×#10596940C2211yescharacteristic
26C2×#10596940C2211yescharacteristic
27C2×#10596940C2211yescharacteristic
28C2×#10596929C2211yescharacteristic, complemented
29C22×#64436C2211yescharacteristic
30#11924535—21no—
31#11924534C2211yescharacteristic, complemented
32#11924523C2211yescharacteristic
33#11924517—21no—
34#11924516C2211yescharacteristic, complemented
35#11924515C2211yescharacteristic
36#11924514—21no—
37#11924513—21no—
38#11924508—21no—
39#11924507C2211yescharacteristic
40#11924505—21no—
41#11924501C2211yescharacteristic, complemented
42D4×Dic37C2211yescharacteristic
43C4×C22⋊D37C2211yescharacteristic
44#11924496—21no—
45C2×C22⋊Dic37—21no—
46#11924469C2211yescharacteristic, complemented
47#11924465C2211yescharacteristic, complemented
48#11924461C2211yescharacteristic, complemented
49#11924461C2211yescharacteristic, complemented
50#11924460C2211yescharacteristic, complemented
51#11924459C2211yescharacteristic, complemented
52#11924448C2211yescharacteristic, complemented
53#11924446C2211yescharacteristic, complemented
54C22×D4×C37C2211yescharacteristic
55C2×SD16×C37C2211yescharacteristic
56C2×SD16×C37C2211yescharacteristic
57C22⋊Q8×C37C2211yescharacteristic
58#102×C37C2211yescharacteristic
59#102×C37C2211yescharacteristic
60C22⋊C8×C37C2211yescharacteristic
61C24×C37—21no—
62C2×Q8×C37C2311yescharacteristic
63C2×D4×C37—22no—
64C2×D4×C37—21no—
65C2×D4×C37—41no—
66C2×D4×C37C2311yescharacteristic
67C2×D4×C37C2311yescharacteristic
68C22×C4×C37C2311yescharacteristic
69C2×C22⋊D37—21no—
70C2×C22⋊D37—21no—
71C2×C22⋊D37D411yescharacteristic
72C22×Dic37—21no—
73C22×Dic37—21no—
74C22×Dic37—21no—
75Q8×D37—22no—
76Q8×D37—41no—
77#10596964—22no—
78#10596964—41no—
79C2×C4×D37—21no—
80C2×C4×D37C2311yescharacteristic
81C2×Dic74—22no—
82C2×Dic74—21no—
83C2×Dic74—21no—
84C2×Dic74—41no—
85C2×Dic74C2311yescharacteristic
86C2×Dic74C2311yescharacteristic
87C2×Dic74C2311yescharacteristic
88SD16×C37—22no—
89SD16×C37—41no—
90C2×C8×C37C2311yescharacteristic
91C2×C8×C37C2311yescharacteristic
92C4⋊C4×C37—21no—
93C4⋊C4×C37C2311yescharacteristic
94C22⋊C4×C37—21no—
95C22⋊Dic37—21no—
96C22⋊Dic37—21no—
97C22⋊Dic37—21no—
98C22⋊Dic37—21no—
99C22⋊Dic37—21no—
100C22⋊Dic37D411yescharacteristic
101#10596942—22no—
102#10596940—22no—
103#10596940—22no—
104#10596940—22no—
105#10596940D414yescomplemented
106#10596938—21no—
107#10596938D411yescharacteristic
108C4⋊Dic37—21no—
109C4⋊Dic37C2311yescharacteristic
110#10596936—21no—
111#10596936—21no—
112#10596936—21no—
113#10596936—21no—
114#10596936D411yescharacteristic
115C4×Dic37—21no—
116C4×Dic37C2311yescharacteristic
117C2×#64436—22no—
118C2×#64436C2311yescharacteristic
119C2×#64436C2311yescharacteristic
120Dic148—22no—
121Dic148—41no—
122#10596929—22no—
123C23×C37—21no—
124C23×C37—21no—
125C23×C37—42no—
126C23×C37—42no—
127C23×C37—81no—
128C23×C37C2×D411yescharacteristic
129C22×D37C2×D411yescharacteristic
130Q8×C37—21no—
131Q8×C37—21no—
132D4×C37—21no—
133D4×C37—21no—
134D4×C37—21no—
135D4×C37—42no—
136D4×C37C2×D412yes—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37—21no—
140C2×C4×C37—21no—
141C2×C4×C37C2×D411yescharacteristic
142C2×C4×C37C2411yescharacteristic
143C22⋊D37—22no—
144C22⋊D37—42no—
145C22⋊D37—81no—
146C2×Dic37—21no—
147C2×Dic37—21no—
148C2×Dic37—21no—
149C2×Dic37—21no—
150C2×Dic37—42no—
151C2×Dic37—42no—
152C2×Dic37—81no—
153C2×Dic37C2×D411yescharacteristic
154C2×Dic37C2×D411yescharacteristic
155C2×Dic37C2×D411yescharacteristic
156C4×D37—21no—
157C4×D37—21no—
158C4×D37—42no—
159Dic74—21no—
160Dic74—21no—
161Dic74—21no—
162Dic74—21no—
163Dic74—21no—
164Dic74—42no—
165Dic74—42no—
166Dic74C2×D412yes—
167C8×C37—21no—
168C8×C37—21no—
169#64436—21no—
170#64436C2×D412yes—
171C22×C37—21no—
172C22×C37—21no—
173C22×C37—22no—
174C22×C37—41no—
175C22×C37—41no—
176C22×C37—41no—
177C22×C37—41no—
178C22×C37—41no—
179C22×C37—84no—
180C22×C37#13712yescomplemented
181C22×C37C22×D411yescharacteristic
182C2×D37—21no—
183C2×D37—21no—
184C2×D37—21no—
185C4×C37—21no—
186C4×C37—41no—
187C4×C37—41no—
188C4×C37C22×D411yescharacteristic
189C4×C37H(5,𝔽2)11yescharacteristic
190Dic37—22no—
191Dic37—21no—
192Dic37—21no—
193Dic37—21no—
194Dic37—41no—
195Dic37—41no—
196Dic37—41no—
197#3266—371nomaximal, Sylow
198C2×C37—22no—
199C2×C37—42no—
200C2×C37—81no—
201C2×C37D4211yescharacteristic
202C2×C37C2×#13711yescharacteristic
203C2×C37#57711yescharacteristic
204D37—41no—
205C2×#137—371no—
206C22×SD16—371no—
207D4×Q8—371no—
208#546—371no—
209#497—371no—
210#468—371no—
211#467—371no—
212#464—371no—
213#461—371no—
214#452—371no—
215#451—371no—
216#450D3711yescharacteristic, complemented
217#448—371no—
218#440—371no—
219#435—371no—
220C37#326611yescharacteristic, complemented, Sylow
221C2×#41—371no—
222C22×Q8—371no—
223C22×Q8—371no—
224C22×D4C2×D3711yescharacteristic
225#137—744no—
226C2×Q16—371no—
227C2×Q16—371no—
228C2×Q16—371no—
229C2×SD16—371no—
230C2×SD16—371no—
231C2×SD16—371no—
232C2×SD16—371no—
233C2×SD16—744no—
234C2×SD16C2×D3711yescharacteristic
235C2×SD16C2×D3711yescharacteristic
236C2×M16—371no—
237C22×C8—371no—
238C4⋊Q8—371no—
239C4⋊Q8—741no—
240#124—371no—
241#123—741no—
242C22⋊Q8—371no—
243C22⋊Q8—371no—
244C22⋊Q8—741no—
245C22⋊Q8—741no—
246C22⋊Q8C2×D3711yescharacteristic
247#121—371no—
248#121—741no—
249#120—741no—
250C4×Q8—371no—
251C4×D4—371no—
252C4×D4—371no—
253C4×D4—741no—
254C2×C22⋊C4—741no—
255#107—371no—
256C4⋊C8—371no—
257#103—371no—
258#103—371no—
259#103—371no—
260#103—371no—
261#102C2×D3711yescharacteristic
262#102C2×D3711yescharacteristic
263C22⋊C8—371no—
264C22⋊C8C2×D3711yescharacteristic
265#97—371no—
266C24—21no—
267#41—742no—
268#41—1481no—
269C2×Q8—371no—
270C2×Q8—371no—
271C2×Q8—371no—
272C2×Q8—741no—
273C2×Q8—742no—
274C2×Q8—741no—
275C2×Q8—742no—
276C2×Q8—1481no—
277C2×Q8—1481no—
278C2×Q8C22×D3711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—371no—
283C2×D4—741no—
284C2×D4—741no—
285C2×D4C22×D3711yescharacteristic
286C2×D4C22×D3711yescharacteristic
287C22×C4—371no—
288C22×C4—741no—
289C22×C4—741no—
290C22×C4—741no—
291C22×C4—741no—
292C22×C4C22×D3711yescharacteristic
293Q16—742no—
294Q16—742no—
295Q16—1481no—
296SD16—22no—
297SD16—41no—
298SD16—374no—
299SD16—742no—
300SD16—742no—
301SD16—742no—
302M16—742no—
303C2×C8—371no—
304C2×C8—371no—
305C2×C8—742no—
306C2×C8C22×D3711yescharacteristic
307C2×C8C22×D3711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—371no—
310C4⋊C4—371no—
311C4⋊C4—741no—
312C4⋊C4—741no—
313C4⋊C4—741no—
314C4⋊C4—741no—
315C4⋊C4—741no—
316C4⋊C4C22×D3711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—371no—
319C22⋊C4—371no—
320C22⋊C4—741no—
321C22⋊C4—741no—
322C22⋊C4—741no—
323C22⋊C4—741no—
324C22⋊C4—741no—
325C22⋊C4—741no—
326C42—371no—
327C42—741no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—371no—
334C23D4×D3711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—372no—
338Q8—741no—
339Q8—741no—
340Q8—741no—
341Q8—741no—
342Q8—741no—
343Q8—1482no—
344Q8—1482no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—2961no—
350D4—742no—
351D4—1482no—
352D4D4×D3712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—371no—
358C2×C4—371no—
359C2×C4—371no—
360C2×C4—2961no—
361C2×C4—741no—
362C2×C4—741no—
363C2×C4—741no—
364C2×C4—741no—
365C2×C4—741no—
366C2×C4—741no—
367C2×C4—1482no—
368C2×C4—1482no—
369C2×C4—1482no—
370C2×C4D4×D3711yescharacteristic
371C2×C4C23×D3711yescharacteristic
372C8—21no—
373C8—21no—
374C8—372no—
375C8—741no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—741no—
386C22—741no—
387C22—741no—
388C22#1192456512yescomplemented
389C22C2×D4×D3711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—741no—
394C4—741no—
395C4—741no—
396C4—742no—
397C4—1481no—
398C4—1481no—
399C4—1481no—
400C4C2×D4×D3711yescharacteristic
401C4#1192458111yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—1481no—
406C2#1455816511yescharacteristic
407C2C2×#1192456511yescharacteristic
408C2#1455834311yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#450C2
socleC22×C37C22×D4
prime radical#14558177C2
hypercenter#450D37
hypocenterC37#3266
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#11924565C2semidirect
2. C22×#10596940C2semidirect
3. #14558196C2semidirect
5. #14557924C2semidirect
6. #14557778C2semidirect
7. #14557772C2semidirect
8. #14557754C2semidirect
9. #14557736C2semidirect
10. #14557696C2semidirect
11. #14557691C2semidirect
12. #14557670C2semidirect
13. #14557630C2semidirect
14. #14557601C2semidirect
15. C37×#450C2semidirect
18. C2×Q8×D37C22semidirect
20. C2×Dic148C22semidirect
21. C2×Dic148C22semidirect
22. C2×#10596942C22semidirect
28. C2×#10596929C22semidirect
31. #11924534C22semidirect
34. #11924516C22semidirect
41. #11924501C22semidirect
46. #11924469C22semidirect
47. #11924465C22semidirect
48. #11924461C22semidirect
49. #11924461C22semidirect
50. #11924460C22semidirect
51. #11924459C22semidirect
52. #11924448C22semidirect
53. #11924446C22semidirect
105. #10596940D4semidirect
180. C22×C37#137semidirect
216. #450D37semidirect
220. C37#3266semidirect
388. C22#11924565semidirect

Maximal subgroups

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

subgroupindex
C2×#119245652
C22×#105969402
#145581962
#145581772
#145579242
#145577782
#145577722
#145577542
#145577362
#145576962
#145576912
#145576702
#145576302
#145576012
C37×#4502
#326637