Finite Group Databasebeta

← #18003905

Subgroups — #18003905

Order 4736.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924581C211yesmaximal, characteristic, complemented
2C2×#11924562C211yesmaximal, characteristic, complemented
3C2×#11924558C211yesmaximal, characteristic, complemented
4#14558114C211yesmaximal, characteristic, complemented
5#14557696C212yesmaximal, complemented
6#14557685C212yesmaximal, complemented
7#14557679C212yesmaximal, complemented
8#14557668C212yesmaximal, complemented
9#14557532C211yesmaximal, characteristic, complemented
10#14557499C211yesmaximal, characteristic, complemented
11C2×#102×C37C211yesmaximal, characteristic, complemented
12#11924581—24no—
13#11924581—24no—
14C2×#10596962C2211yescharacteristic
15C2×#10596962C2211yescharacteristic
16C2×#10596962C2211yescharacteristic, complemented
17C2×#10596964—21no—
18C2×#10596964C2212yes—
19C22×C22⋊D37—22no—
20C2×D4×D37—21no—
21C2×D4×D37C2212yes—
22#11924562—24no—
23#11924558—24no—
24C2×Dic148C2211yescharacteristic, complemented
25C2×#10596930C2211yescharacteristic, complemented
26C2×#10596930C2211yescharacteristic, complemented
27C2×#10596940C2212yescomplemented
28C2×D296C2211yescharacteristic, complemented
29C2×#10596939C2212yescomplemented
30C2×#10596934C2211yescharacteristic, complemented
31#11924516—21no—
32#11924516C2212yescomplemented
33#11924511—21no—
34#11924511C2212yescomplemented
35#11924510—22no—
36#11924459C2212yescomplemented
37#11924456C2212yescomplemented
38#11924448C2212yescomplemented
39#11924448C2212yescomplemented
40C22×D4×C37C2211yescharacteristic
41C22×C8×C37C2211yescharacteristic, complemented
42C2×C4⋊C4×C37C2211yescharacteristic, complemented
43#102×C37C2212yescomplemented
44#102×C37C2212yescomplemented
45C24×C37—21no—
46C23×D37—22no—
47C2×D4×C37—21no—
48C2×D4×C37—22no—
49C2×D4×C37D414yescomplemented
50C2×D4×C37C2312yes—
51C22×C4×C37—21no—
52C22×C4×C37C2311yescharacteristic
53C2×C22⋊D37—216no—
54C2×C22⋊D37—21no—
55C2×C22⋊D37—22no—
56C2×C22⋊D37—21no—
57C2×C22⋊D37—22no—
58C2×C22⋊D37D412yes—
59C22×Dic37—22no—
60#10596964—28no—
61#10596964—24no—
62#10596964—24no—
63D4×D37—28no—
64D4×D37—24no—
65D4×D37—24no—
66#10596962—22no—
67#10596962—22no—
68#10596962—22no—
69#10596962—22no—
70#10596962—22no—
71#10596962D414yescomplemented
72C2×D148—22no—
73C2×D148D411yescharacteristic
74C2×D148C2311yescharacteristic
75C2×D148C2311yescharacteristic
76C2×C4×D37—21no—
77C2×C4×D37—21no—
78C2×C4×D37C2312yes—
79C2×Dic74D411yescharacteristic
80C2×Dic74C2311yescharacteristic
81C2×Dic74C2311yescharacteristic
82C2×C8×C37—22no—
83C2×C8×C37C2311yescharacteristic
84C2×C8×C37C2311yescharacteristic
85C4⋊C4×C37—21no—
86C4⋊C4×C37C2312yes—
87#10596940—24no—
88#10596939—24no—
89#10596938—22no—
90#10596938—22no—
91C4⋊Dic37—22no—
92#10596934—22no—
93C2×#64436C2312yes—
94Dic148—22no—
95D296—22no—
96#10596930—22no—
97#10596930—22no—
98C23×C37—22no—
99C23×C37—24no—
100C23×C37—81no—
101C23×C37—81no—
102C23×C37C2×D411yescharacteristic
103C22×D37—21no—
104C22×D37—28no—
105C22×D37—21no—
106C22×D37—82no—
107C22×D37—82no—
108C22×D37C2×D412yes—
109D4×C37—24no—
110D4×C37—22no—
111D4×C37C2×D414yes—
112C2×C4×C37—22no—
113C2×C4×C37—81no—
114C2×C4×C37C2×D412yes—
115C2×C4×C37C2×D411yescharacteristic
116C2×C4×C37C2×D412yes—
117C2×C4×C37C2411yescharacteristic
118C22⋊D37—28no—
119C22⋊D37—28no—
120C22⋊D37—24no—
121C22⋊D37—24no—
122C22⋊D37—42no—
123C2×Dic37—28no—
124C2×Dic37—21no—
125C2×Dic37—21no—
126C2×Dic37C2×D412yes—
127D148—22no—
128D148—21no—
129D148—21no—
130D148—21no—
131D148—82no—
132D148C2×D412yes—
133C4×D37—24no—
134C4×D37—22no—
135C4×D37—22no—
136C4×D37—42no—
137Dic74—22no—
138Dic74—21no—
139Dic74—21no—
140Dic74—21no—
141Dic74C2×D412yes—
142C8×C37—21no—
143C8×C37—21no—
144#64436—22no—
145C22×C37—21no—
146C22×C37—21no—
147C22×C37—24no—
148C22×C37—42no—
149C22×C37—42no—
150C22×C37—84no—
151C22×C37#12012yes—
152C22×C37C22×D411yescharacteristic
153C2×D37—22no—
154C2×D37—22no—
155C2×D37—22no—
156C2×D37—22no—
157C2×D37—41no—
158C2×D37—41no—
159C2×D37—41no—
160C2×D37—41no—
161C2×D37—41no—
162C2×D37—88no—
163C4×C37—42no—
164C4×C37#12012yes—
165C4×C37C22×D411yescharacteristic
166C4×C37C22×D411yescharacteristic
167Dic37—22no—
168Dic37—22no—
169Dic37—41no—
170#3012—371nomaximal, Sylow
171C2×C37—22no—
172C2×C37—44no—
173C2×C37C2×#12011yescharacteristic
174C2×C37#57611yescharacteristic
175C2×C37#57711yescharacteristic
176D37—42no—
177D37—42no—
178D37—81no—
179C2×H(5,𝔽2)—371no—
180C2×AGL(1,ℤ/8)—371no—
181C2×#135—371no—
182#537—371no—
183#452—372no—
184#450—372no—
185#449—372no—
186#447—372no—
187#417—371no—
188C2×#102D3711yescharacteristic, complemented
189#408—371no—
190C37#301211yescharacteristic, complemented, Sylow
191H(5,𝔽2)—744no—
192H(5,𝔽2)—744no—
193C2×#41—371no—
194C2×#41—372no—
195C2×#41—371no—
196C2×#41—371no—
197C2×#41—741no—
198C22×D4—372no—
199C22×D4—741no—
200C22×D4—742no—
201C22×D4C2×D3711yescharacteristic
202AGL(1,ℤ/8)—744no—
203#135—744no—
204C2×Q16—371no—
205C2×SD16—372no—
206C2×SD16—371no—
207C2×SD16—371no—
208C2×D8—372no—
209C2×D8—371no—
210C2×M16—371no—
211C22×C8C2×D3711yescharacteristic, complemented
212C22⋊Q8—372no—
213C22⋊Q8—741no—
214#121—372no—
215#121—741no—
216#121—742no—
217C2×C4⋊C4C2×D3711yescharacteristic, complemented
218#103—372no—
219#102—372no—
220#102C2×D3712yescomplemented
221#102C2×D3712yescomplemented
222C22⋊C8—372no—
223C22⋊C8—372no—
224C24—21no—
225C24—742no—
226#41—374no—
227#41—744no—
228#41—748no—
229#41—744no—
230#41—742no—
231#41—742no—
232#41—742no—
233#41—742no—
234#41—742no—
235C2×Q8—371no—
236C2×Q8—371no—
237C2×Q8—371no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—372no—
241C2×D4—371no—
242C2×D4—371no—
243C2×D4—371no—
244C2×D4—742no—
245C2×D4—742no—
246C2×D4—744no—
247C2×D4—748no—
248C2×D4—742no—
249C2×D4—7416no—
250C2×D4—741no—
251C2×D4—744no—
252C2×D4—741no—
253C2×D4D14814yescomplemented
254C2×D4C22×D3712yes—
255C22×C4—21no—
256C22×C4—372no—
257C22×C4—742no—
258C22×C4—741no—
259C22×C4—741no—
260C22×C4C22×D3711yescharacteristic
261Q16—742no—
262SD16—742no—
263SD16—742no—
264SD16—744no—
265D8—742no—
266D8—744no—
267M16—742no—
268C2×C8—22no—
269C2×C8—372no—
270C2×C8C22×D3711yescharacteristic
271C2×C8C22×D3711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—742no—
274C4⋊C4C22×D3712yes—
275C22⋊C4—742no—
276C22⋊C4—742no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—372no—
282C23—2962no—
283C23—2962no—
284C23—748no—
285C23—741no—
286C23—741no—
287C23D4×D3711yescharacteristic
288Q8—372no—
289Q8—742no—
290Q8—741no—
291Q8—741no—
292Q8—741no—
293D4—24no—
294D4—22no—
295D4—372no—
296D4—2962no—
297D4—748no—
298D4—748no—
299D4—744no—
300D4—744no—
301D4—742no—
302D4—741no—
303D4—741no—
304D4—741no—
305D4—1482no—
306D4C2×D14814yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—372no—
310C2×C4—748no—
311C2×C4—741no—
312C2×C4—744no—
313C2×C4—742no—
314C2×C4—742no—
315C2×C4—741no—
316C2×C4—1482no—
317C2×C4C2×D14812yes—
318C2×C4D4×D3711yescharacteristic
319C2×C4D4×D3712yes—
320C2×C4C23×D3711yescharacteristic
321C8—21no—
322C8—21no—
323C8—742no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—2968no—
331C22—742no—
332C22—742no—
333C22—742no—
334C22—742no—
335C22—1481no—
336C22—1481no—
337C22—1481no—
338C22—1481no—
339C22—1481no—
340C22#1192450612yes—
341C22C2×D4×D3711yescharacteristic
342C4—42no—
343C4—742no—
344C4—742no—
345C4—1481no—
346C4#1192450612yes—
347C4C2×D4×D3711yescharacteristic
348C4C22×D14811yescharacteristic
349C2—22no—
350C2—44no—
351C2—2961no—
352C2—1482no—
353C2—1482no—
354C2C2×#1192450611yescharacteristic
355C2#1455833411yescharacteristic
356C2#1455833911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×#102×C37C2
socleC22×C37C22×D4
hypercenterC2×#102D37
hypocenterC37#3012
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#11924581C2semidirect
2. C2×#11924562C2semidirect
3. C2×#11924558C2semidirect
4. #14558114C2semidirect
5. #14557696C2semidirect
6. #14557685C2semidirect
7. #14557679C2semidirect
8. #14557668C2semidirect
9. #14557532C2semidirect
10. #14557499C2semidirect
11. C2×#102×C37C2semidirect
16. C2×#10596962C22semidirect
24. C2×Dic148C22semidirect
25. C2×#10596930C22semidirect
26. C2×#10596930C22semidirect
27. C2×#10596940C22semidirect
28. C2×D296C22semidirect
29. C2×#10596939C22semidirect
30. C2×#10596934C22semidirect
32. #11924516C22semidirect
34. #11924511C22semidirect
36. #11924459C22semidirect
37. #11924456C22semidirect
38. #11924448C22semidirect
39. #11924448C22semidirect
41. C22×C8×C37C22semidirect
42. C2×C4⋊C4×C37C22semidirect
43. #102×C37C22semidirect
44. #102×C37C22semidirect
49. C2×D4×C37D4semidirect
71. #10596962D4semidirect
188. C2×#102D37semidirect
190. C37#3012semidirect
211. C22×C8C2×D37semidirect
217. C2×C4⋊C4C2×D37semidirect
220. #102C2×D37semidirect
221. #102C2×D37semidirect
253. C2×D4D148semidirect

Maximal subgroups

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

subgroupindex
C2×#119245812
C2×#119245622
C2×#119245582
#145581142
#145576962
#145576852
#145576792
#145576682
#145575322
#145574992
C2×#102×C372
#301237