Finite Group Databasebeta

← #18005350

Subgroups — #18005350

Order 4736.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D37C211yesmaximal, characteristic, complemented
2C2×#11924507C211yesmaximal, characteristic
3#14557864C211yesmaximal, characteristic, complemented
4#14557861C211yesmaximal, characteristic, complemented
5#14557850C212yesmaximal, complemented
6#14557761C212yesmaximal, complemented
7#14557759C212yesmaximal, complemented
8#14557684C211yesmaximal, characteristic, complemented
9#14557670C212yesmaximal, complemented
10C22⋊C8×D37C211yesmaximal, characteristic, complemented
11C37×#450C211yesmaximal, characteristic, complemented
12C2×Q8×D37C2211yescharacteristic, complemented
13C22×C22⋊D37—21no—
14C22×C4×D37C2211yescharacteristic
15C2×#10596940C2212yescomplemented
16C2×C8×D37C2212yescomplemented
17#11924516C2211yescharacteristic, complemented
18#11924515C2211yescharacteristic
19#11924514—21no—
20#11924513—21no—
21#11924512C2211yescharacteristic, complemented
22#11924507—21no—
23#11924507—21no—
24#11924507C2212yes—
25#11924507C2212yes—
26C2×C4⋊Dic37C2211yescharacteristic
27C4⋊C4×D37—21no—
28C4⋊C4×D37C2211yescharacteristic, complemented
29C2×C22⋊Dic37—21no—
30C22⋊C4×D37—21no—
31#11924468C2212yescomplemented
32#11924467C2212yescomplemented
33#11924461C2212yescomplemented
34#11924460C2212yescomplemented
35#11924457C2212yescomplemented
36C22⋊#64436C2211yescharacteristic
37#11924448C2212yescomplemented
38C22×D4×C37C2211yescharacteristic
39C2×SD16×C37C2212yescomplemented
40C22⋊Q8×C37C2211yescharacteristic
41#102×C37C2212yescomplemented
42C22⋊C8×C37C2211yescharacteristic
43C24×C37—21no—
44C23×D37D411yescharacteristic
45C2×Q8×C37C2311yescharacteristic
46C2×D4×C37—21no—
47C2×D4×C37—42no—
48C2×D4×C37C2312yes—
49C22×C4×C37C2311yescharacteristic
50C2×C22⋊D37—22no—
51C2×C22⋊D37—22no—
52C2×C22⋊D37—44no—
53C22×Dic37—21no—
54C22×Dic37D411yescharacteristic
55Q8×D37—42no—
56C2×C4×D37—21no—
57C2×C4×D37—21no—
58C2×C4×D37—42no—
59C2×C4×D37D411yescharacteristic
60C2×C4×D37D411yescharacteristic
61C2×C4×D37C2311yescharacteristic
62C2×C4×D37C2311yescharacteristic
63C2×Dic74—21no—
64C2×Dic74C2311yescharacteristic
65SD16×C37—42no—
66C2×C8×C37C2312yes—
67C4⋊C4×C37—21no—
68C4⋊C4×C37C2311yescharacteristic
69C22⋊C4×C37—21no—
70C22⋊Dic37—21no—
71C22⋊Dic37—22no—
72C22⋊Dic37—22no—
73#10596940—42no—
74#10596938—21no—
75#10596938—21no—
76C4⋊Dic37—21no—
77C4⋊Dic37—21no—
78C4⋊Dic37C2311yescharacteristic
79C4⋊Dic37C2312yes—
80#10596936—21no—
81#10596936—21no—
82#10596936—21no—
83C2×#64436C2312yes—
84C8×D37—42no—
85C23×C37—22no—
86C23×C37—42no—
87C23×C37—42no—
88C23×C37—42no—
89C23×C37C2×D411yescharacteristic
90C22×D37—41no—
91C22×D37—41no—
92C22×D37SD1614yescomplemented
93C22×D37C2×D411yescharacteristic
94C22×D37C2×D411yescharacteristic
95Q8×C37—22no—
96D4×C37—22no—
97D4×C37—22no—
98D4×C37—42no—
99C2×C4×C37—22no—
100C2×C4×C37—21no—
101C2×C4×C37—21no—
102C2×C4×C37C2×D411yescharacteristic
103C2×C4×C37C2411yescharacteristic
104C22⋊D37—44no—
105C22⋊D37—44no—
106C2×Dic37—21no—
107C2×Dic37—21no—
108C2×Dic37—22no—
109C2×Dic37—41no—
110C2×Dic37—42no—
111C2×Dic37C2×D411yescharacteristic
112C2×Dic37C2×D411yescharacteristic
113C4×D37—22no—
114C4×D37—22no—
115C4×D37—41no—
116C4×D37—41no—
117C4×D37—42no—
118C4×D37—42no—
119Dic74—22no—
120Dic74—42no—
121C8×C37—22no—
122#64436—22no—
123C22×C37—21no—
124C22×C37—21no—
125C22×C37—42no—
126C22×C37—42no—
127C22×C37—42no—
128C22×C37—84no—
129C22×C37C2×SD1612yes—
130C22×C37C22×D411yescharacteristic
131C2×D37—21no—
132C2×D37—21no—
133C2×D37—21no—
134C2×D37—21no—
135C2×D37—44no—
136C2×D37C2×SD1612yes—
137C2×D37C2×SD1612yes—
138C4×C37—21no—
139C4×C37—41no—
140C4×C37—41no—
141C4×C37H(5,𝔽2)12yes—
142Dic37—21no—
143Dic37—21no—
144Dic37—41no—
145Dic37—42no—
146Dic37—41no—
147#3187—371nomaximal, Sylow
148C2×C37—22no—
149C2×C37—82no—
150C2×C37#53411yescharacteristic
151C2×C37C22×SD1611yescharacteristic
152C2×C37#57511yescharacteristic
153D37—22no—
154D37—22no—
155C2×C22⋊Q8—371no—
156C2×#121—371no—
157#483—371no—
158#483—371no—
159#481—372no—
160#465—372no—
161#465—372no—
162#450—371no—
163#450D3711yescharacteristic, complemented
164#448—372no—
165C2×C22⋊C8—371no—
166C37#318711yescharacteristic, complemented, Sylow
167C22×Q8—371no—
168C22×D4—741no—
169C22×D4C2×D3711yescharacteristic
170C23×C4—371no—
171C2×SD16—372no—
172C2×SD16C2×D3712yescomplemented
173C22×C8—372no—
174C22⋊Q8—371no—
175C22⋊Q8—371no—
176C22⋊Q8—371no—
177C22⋊Q8—741no—
178C22⋊Q8—741no—
179C22⋊Q8C2×D3711yescharacteristic
180#121—372no—
181#121—372no—
182#121—741no—
183#121—741no—
184C2×C4⋊C4—371no—
185C2×C4⋊C4—371no—
186C2×C4⋊C4—741no—
187C2×C22⋊C4—741no—
188C2×C22⋊C4—741no—
189#106—372no—
190#106—372no—
191#103—372no—
192#103—372no—
193#102—372no—
194#102C2×D3712yescomplemented
195C22⋊C8—371no—
196C22⋊C8—372no—
197C22⋊C8C2×D3711yescharacteristic
198C24—21no—
199C24—371no—
200C2×Q8—371no—
201C2×Q8—741no—
202C2×Q8—1482no—
203C2×Q8C22×D3711yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—742no—
207C2×D4—742no—
208C2×D4—1484no—
209C2×D4C22×D3712yes—
210C22×C4—371no—
211C22×C4—371no—
212C22×C4—371no—
213C22×C4—371no—
214C22×C4—371no—
215C22×C4—741no—
216C22×C4—741no—
217C22×C4—741no—
218C22×C4—1482no—
219C22×C4C22×D3711yescharacteristic
220SD16—42no—
221SD16—1482no—
222C2×C8—372no—
223C2×C8—1482no—
224C2×C8C22×D3712yes—
225C4⋊C4—21no—
226C4⋊C4—371no—
227C4⋊C4—372no—
228C4⋊C4—741no—
229C4⋊C4—741no—
230C4⋊C4—741no—
231C4⋊C4—741no—
232C4⋊C4—741no—
233C4⋊C4C22×D3711yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—741no—
236C22⋊C4—741no—
237C22⋊C4—742no—
238C22⋊C4—742no—
239C22⋊C4—741no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—371no—
245C23—371no—
246C23—374no—
247C23—1481no—
248C23—1481no—
249C23D4×D3711yescharacteristic
250Q8—22no—
251Q8—742no—
252Q8—1482no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—1484no—
257D4—1484no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—371no—
262C2×C4—371no—
263C2×C4—742no—
264C2×C4—742no—
265C2×C4—741no—
266C2×C4—742no—
267C2×C4—741no—
268C2×C4—1481no—
269C2×C4—1481no—
270C2×C4—1482no—
271C2×C4—1482no—
272C2×C4—1481no—
273C2×C4—1482no—
274C2×C4D4×D3711yescharacteristic
275C2×C4C23×D3711yescharacteristic
276C8—22no—
277C8—742no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—372no—
285C22—372no—
286C22—741no—
287C22—741no—
288C22—741no—
289C22—741no—
290C22—1484no—
291C22SD16×D3712yes—
292C22C2×D4×D3711yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—741no—
297C4—741no—
298C4—1482no—
299C4—1481no—
300C4—1481no—
301C4#1192458112yes—
302C2—22no—
303C2—82no—
304C2—742no—
305C2—742no—
306C2#1455809311yescharacteristic
307C2C2×SD16×D3711yescharacteristic
308C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#450C2
socleC22×C37C22×D4
prime radicalC2×#11924507C2
hypercenter#450D37
hypocenterC37#3187
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. C22⋊Q8×D37C2semidirect
3. #14557864C2semidirect
4. #14557861C2semidirect
5. #14557850C2semidirect
6. #14557761C2semidirect
7. #14557759C2semidirect
8. #14557684C2semidirect
9. #14557670C2semidirect
10. C22⋊C8×D37C2semidirect
11. C37×#450C2semidirect
12. C2×Q8×D37C22semidirect
15. C2×#10596940C22semidirect
16. C2×C8×D37C22semidirect
17. #11924516C22semidirect
21. #11924512C22semidirect
28. C4⋊C4×D37C22semidirect
31. #11924468C22semidirect
32. #11924467C22semidirect
33. #11924461C22semidirect
34. #11924460C22semidirect
35. #11924457C22semidirect
37. #11924448C22semidirect
39. C2×SD16×C37C22semidirect
41. #102×C37C22semidirect
92. C22×D37SD16semidirect
163. #450D37semidirect
166. C37#3187semidirect
172. C2×SD16C2×D37semidirect
194. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D372
C2×#119245072
#145578642
#145578612
#145578502
#145577612
#145577592
#145576842
#145576702
C22⋊C8×D372
C37×#4502
#318737