Finite Group Databasebeta

← #18006160

Subgroups — #18006160

Order 4736.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×D37C211yesmaximal, characteristic, complemented
2Q8×C22⋊D37C211yesmaximal, characteristic, complemented
3#14558190C211yesmaximal, characteristic
4#14557924C211yesmaximal, characteristic, complemented
5#14557791C211yesmaximal, characteristic, complemented
6#14557759C211yesmaximal, characteristic, complemented
7#14557753C211yesmaximal, characteristic, complemented
8#14557741C211yesmaximal, characteristic, complemented
9#14557695C211yesmaximal, characteristic, complemented
10#14557689C211yesmaximal, characteristic, complemented
11#14557687C211yesmaximal, characteristic, complemented
12#14557670C211yesmaximal, characteristic, complemented
13#14557640C211yesmaximal, characteristic, complemented
14#14557602C211yesmaximal, characteristic, complemented
15C2×#137×C37C211yesmaximal, characteristic, complemented
16C2×Q8×D37C2211yescharacteristic
17C2×Q8×D37C2211yescharacteristic
18C2×Dic148C2211yescharacteristic
19C2×#10596942C2211yescharacteristic
20C2×#10596942C2211yescharacteristic
21C2×#10596942C2211yescharacteristic
22Q16×D37—24no—
23C2×#10596940C2211yescharacteristic
24C2×C8×D37C2211yescharacteristic
25#11924534—21no—
26#11924534C2211yescharacteristic
27#11924523—21no—
28#11924523C2211yescharacteristic
29#11924515—21no—
30#11924515—21no—
31#11924515C2211yescharacteristic
32#11924515C2211yescharacteristic
33#11924512—21no—
34#11924512C2211yescharacteristic
35#11924507—21no—
36#11924507C2211yescharacteristic
37Q8×Dic37C2211yescharacteristic
38Q8×Dic37C2211yescharacteristic
39C4×C22⋊D37—21no—
40C4×C22⋊D37—21no—
41C4×C22⋊D37C2211yescharacteristic
42#11924468C2211yescharacteristic
43#11924465C2211yescharacteristic
44#11924461C2211yescharacteristic
45#11924461C2211yescharacteristic
46#11924460C2211yescharacteristic
47#11924460C2211yescharacteristic
48#11924460C2211yescharacteristic
49#11924457C2211yescharacteristic
50C22⋊#64436C2211yescharacteristic
51#11924448C2211yescharacteristic
52#11924446C2211yescharacteristic
53C2×#41×C37C2211yescharacteristic
54C22×Q8×C37C2211yescharacteristic
55#137×C37—24no—
56C2×Q16×C37C2211yescharacteristic
57C2×Q16×C37C2211yescharacteristic
58C2×SD16×C37C2211yescharacteristic
59C2×SD16×C37C2211yescharacteristic
60C2×M16×C37C2211yescharacteristic
61#41×C37—22no—
62#41×C37—41no—
63C2×Q8×C37—21no—
64C2×Q8×C37—22no—
65C2×Q8×C37—41no—
66C2×Q8×C37C2311yescharacteristic
67C2×Q8×C37C2311yescharacteristic
68C2×Q8×C37C2311yescharacteristic
69C2×D4×C37—21no—
70C2×D4×C37C2311yescharacteristic
71C22×C4×C37—21no—
72C22×C4×C37—21no—
73C22×C4×C37C2311yescharacteristic
74C2×C22⋊D37—21no—
75C2×C22⋊D37D411yescharacteristic
76Q8×D37—22no—
77Q8×D37—22no—
78Q8×D37—41no—
79Q8×D37—41no—
80C2×C4×D37—21no—
81C2×C4×D37—21no—
82C2×C4×D37C2311yescharacteristic
83C2×Dic74—21no—
84C2×Dic74—21no—
85C2×Dic74C2311yescharacteristic
86C2×Dic74C2311yescharacteristic
87Q16×C37—22no—
88Q16×C37D414yescomplemented
89SD16×C37—22no—
90SD16×C37—22no—
91M16×C37—22no—
92C2×C8×C37C2311yescharacteristic
93C2×C8×C37C2311yescharacteristic
94C22⋊Dic37—21no—
95C22⋊Dic37—21no—
96C22⋊Dic37—21no—
97C22⋊Dic37—21no—
98C22⋊Dic37—21no—
99C22⋊Dic37D411yescharacteristic
100#10596942—22no—
101#10596942—22no—
102#10596942—41no—
103#10596940—41no—
104#10596938—21no—
105#10596938D411yescharacteristic
106C4⋊Dic37—21no—
107C4⋊Dic37—21no—
108C4⋊Dic37C2311yescharacteristic
109C4⋊Dic37C2311yescharacteristic
110#10596936—21no—
111#10596936—21no—
112#10596936—21no—
113#10596936—21no—
114#10596936—21no—
115#10596936D411yescharacteristic
116C4×Dic37—21no—
117C4×Dic37—21no—
118C4×Dic37C2311yescharacteristic
119C2×#64436C2311yescharacteristic
120C2×#64436C2311yescharacteristic
121Dic148—22no—
122C8×D37—22no—
123C23×C37—21no—
124C23×C37C2×D411yescharacteristic
125C22×D37C2×D411yescharacteristic
126Q8×C37—21no—
127Q8×C37—21no—
128Q8×C37—21no—
129Q8×C37—21no—
130Q8×C37—42no—
131Q8×C37C2×D412yes—
132Q8×C37C2×D412yes—
133D4×C37—21no—
134D4×C37—21no—
135D4×C37—42no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37—21no—
140C2×C4×C37—21no—
141C2×C4×C37—42no—
142C2×C4×C37—42no—
143C2×C4×C37C2×D411yescharacteristic
144C2×C4×C37C2411yescharacteristic
145C22⋊D37—22no—
146C22⋊D37—42no—
147C2×Dic37—21no—
148C2×Dic37—21no—
149C2×Dic37—21no—
150C2×Dic37—21no—
151C2×Dic37C2×D411yescharacteristic
152C2×Dic37C2×D411yescharacteristic
153C2×Dic37C2×D411yescharacteristic
154C4×D37—21no—
155C4×D37—21no—
156C4×D37—42no—
157C4×D37—42no—
158Dic74—21no—
159Dic74—21no—
160Dic74—21no—
161Dic74—21no—
162Dic74—42no—
163Dic74—42no—
164C8×C37—21no—
165C8×C37C2×D412yes—
166#64436—21no—
167#64436—21no—
168C22×C37—21no—
169C22×C37—21no—
170C22×C37—21no—
171C22×C37—41no—
172C22×C37—41no—
173C22×C37—41no—
174C22×C37C22×D411yescharacteristic
175C2×D37—21no—
176C2×D37—21no—
177C2×D37#13712yescomplemented
178C4×C37—22no—
179C4×C37—22no—
180C4×C37—21no—
181C4×C37—41no—
182C4×C37C22×D411yescharacteristic
183C4×C37H(5,𝔽2)11yescharacteristic
184Dic37—21no—
185Dic37—21no—
186Dic37—21no—
187Dic37—41no—
188Dic37—41no—
189Dic37—41no—
190Dic37—41no—
191#3270—371nomaximal, Sylow
192C2×C37—41no—
193C2×C37—81no—
194C2×C37D4211yescharacteristic
195C2×C37C2×#13711yescharacteristic
196C2×C37#57811yescharacteristic
197D37—22no—
198C2×#137D3711yescharacteristic, complemented
199C22×Q16—371no—
200D4×Q8—371no—
201#548—371no—
202#497—371no—
203#470—371no—
204#465—371no—
205#464—371no—
206#462—371no—
207#452—371no—
208#451—371no—
209#451—371no—
210#448—371no—
211#441—371no—
212#435—371no—
213C37#327011yescharacteristic, complemented, Sylow
214C2×#41C2×D3711yescharacteristic
215C22×Q8—371no—
216C22×Q8—371no—
217C22×Q8C2×D3711yescharacteristic
218#137—24no—
219C2×Q16—371no—
220C2×Q16—371no—
221C2×Q16—371no—
222C2×Q16—371no—
223C2×Q16—744no—
224C2×Q16C2×D3711yescharacteristic
225C2×Q16C2×D3711yescharacteristic
226C2×SD16—371no—
227C2×SD16C2×D3711yescharacteristic
228C2×SD16C2×D3711yescharacteristic
229C2×M16C2×D3711yescharacteristic
230C22×C8—371no—
231C4⋊Q8—371no—
232C4⋊Q8—741no—
233#124—371no—
234#124—741no—
235C22⋊Q8—371no—
236C22⋊Q8—371no—
237C22⋊Q8—371no—
238C22⋊Q8—741no—
239C22⋊Q8—741no—
240C22⋊Q8—741no—
241#121—371no—
242#121—741no—
243C4×Q8—371no—
244C4×Q8—371no—
245C4×D4—371no—
246C4×D4—741no—
247C4×D4—741no—
248#106—371no—
249C4⋊C8—371no—
250#103—371no—
251#103—371no—
252#103—371no—
253#103—371no—
254#103—371no—
255#102—371no—
256C22⋊C8—371no—
257C22⋊C8—371no—
258#97—371no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—371no—
265C2×Q8—371no—
266C2×Q8—741no—
267C2×Q8—741no—
268C2×Q8—742no—
269C2×Q8—742no—
270C2×Q8—1481no—
271C2×Q8—1481no—
272C2×Q8C22×D3711yescharacteristic
273C2×Q8C22×D3711yescharacteristic
274C2×Q8C22×D3711yescharacteristic
275C2×D4—21no—
276C2×D4—371no—
277C2×D4—741no—
278C2×D4C22×D3711yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—371no—
282C22×C4—741no—
283C22×C4—741no—
284C22×C4C22×D3711yescharacteristic
285Q16—22no—
286Q16—742no—
287Q16—742no—
288Q16—742no—
289Q16—1481no—
290Q16C22⋊D3714yescomplemented
291SD16—22no—
292SD16—22no—
293SD16—1481no—
294M16—22no—
295C2×C8—371no—
296C2×C8—371no—
297C2×C8—742no—
298C2×C8C22×D3711yescharacteristic
299C2×C8C22×D3711yescharacteristic
300C4⋊C4—371no—
301C4⋊C4—371no—
302C4⋊C4—371no—
303C4⋊C4—741no—
304C4⋊C4—741no—
305C4⋊C4—741no—
306C4⋊C4—741no—
307C4⋊C4—741no—
308C4⋊C4—741no—
309C4⋊C4—741no—
310C22⋊C4—371no—
311C22⋊C4—371no—
312C22⋊C4—741no—
313C22⋊C4—741no—
314C22⋊C4—741no—
315C22⋊C4—741no—
316C22⋊C4—741no—
317C22⋊C4—741no—
318C42—371no—
319C42—741no—
320C42—741no—
321C23—21no—
322C23—371no—
323C23D4×D3711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—42no—
329Q8—741no—
330Q8—741no—
331Q8—741no—
332Q8—741no—
333Q8—1482no—
334Q8—1482no—
335Q8C2×C22⋊D3712yes—
336Q8C2×C22⋊D3712yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—742no—
341D4—1482no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—371no—
350C2×C4—371no—
351C2×C4—371no—
352C2×C4—741no—
353C2×C4—741no—
354C2×C4—741no—
355C2×C4—741no—
356C2×C4—741no—
357C2×C4—741no—
358C2×C4—1482no—
359C2×C4—1482no—
360C2×C4D4×D3711yescharacteristic
361C2×C4C23×D3711yescharacteristic
362C8—21no—
363C8—741no—
364C8—741no—
365C8C2×C22⋊D3712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—372no—
373C22—741no—
374C22—741no—
375C22C2×D4×D3711yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—741no—
381C4—741no—
382C4—741no—
383C4—1481no—
384C4—1481no—
385C4—1481no—
386C4—1481no—
387C4C22×C22⋊D3711yescharacteristic
388C4#1192458111yescharacteristic
389C2—41no—
390C2—81no—
391C2—742no—
392C2D4×C22⋊D3711yescharacteristic
393C2#137×D3711yescharacteristic
394C2#1455834911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×#137×C37C2
socleC22×C37C22×D4
prime radical#14558190C2
hypercenterC2×#137D37
hypocenterC37#3270
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D37C2semidirect
2. Q8×C22⋊D37C2semidirect
4. #14557924C2semidirect
5. #14557791C2semidirect
6. #14557759C2semidirect
7. #14557753C2semidirect
8. #14557741C2semidirect
9. #14557695C2semidirect
10. #14557689C2semidirect
11. #14557687C2semidirect
12. #14557670C2semidirect
13. #14557640C2semidirect
14. #14557602C2semidirect
15. C2×#137×C37C2semidirect
88. Q16×C37D4semidirect
177. C2×D37#137semidirect
198. C2×#137D37semidirect
213. C37#3270semidirect
290. Q16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D372
Q8×C22⋊D372
#145581902
#145579242
#145577912
#145577592
#145577532
#145577412
#145576952
#145576892
#145576872
#145576702
#145576402
#145576022
C2×#137×C372
#327037