Finite Group Databasebeta

← #18004522

Subgroups — #18004522

Order 4736.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#14558045C211yesmaximal, characteristic
2C2×C4⋊C4×D37C211yesmaximal, characteristic, complemented
3#14557870C212yesmaximal, complemented
4#14557865C212yesmaximal, complemented
5#14557853C212yesmaximal, complemented
6#14557848C212yesmaximal, complemented
7#14557576C211yesmaximal, characteristic, complemented
8#14557535C211yesmaximal, characteristic, complemented
9#14557532C211yesmaximal, characteristic, complemented
10#14557496C211yesmaximal, characteristic, complemented
11C37×#428C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×C4×D37—21no—
14C22×C4×D37C2211yescharacteristic
15C2×#10596934C2211yescharacteristic
16#11924520—21no—
17#11924516C2211yescharacteristic
18#11924516C2211yescharacteristic
19#11924511C2211yescharacteristic
20#11924511C2211yescharacteristic
21#11924506—21no—
22C4×D148—21no—
23C4×D148—21no—
24C2×C4⋊Dic37C2211yescharacteristic
25C2×#10596936—21no—
26C4⋊C4×D37—21no—
27C4⋊C4×D37—21no—
28C4⋊C4×D37C2212yes—
29C4⋊C4×D37C2212yes—
30#11924469C2212yescomplemented
31#11924467C2212yescomplemented
32#11924461C2212yescomplemented
33#11924459C2212yescomplemented
34#11924458C2212yescomplemented
35#11924456C2212yescomplemented
36#11924448C2212yes—
37#11924448C2212yes—
38C2×M16×C37C2211yescharacteristic
39#117×C37C2211yescharacteristic
40C2×C4⋊C4×C37C2211yescharacteristic
41#107×C37C2212yescomplemented
42#106×C37C2212yescomplemented
43C23×D37D411yescharacteristic
44C22×C4×C37—21no—
45C22×C4×C37C2311yescharacteristic
46C2×C22⋊D37—21no—
47C22×Dic37—21no—
48C22×Dic37D411yescharacteristic
49#10596962—22no—
50#10596962—22no—
51C2×D148—21no—
52C2×D148—21no—
53C2×D148C2311yescharacteristic
54C2×C4×D37—22no—
55C2×C4×D37—22no—
56C2×C4×D37—22no—
57C2×C4×D37—22no—
58C2×C4×D37—21no—
59C2×C4×D37—44no—
60C2×C4×D37D411yescharacteristic
61C2×C4×D37D411yescharacteristic
62C2×C4×D37C2311yescharacteristic
63C2×C4×D37C2311yescharacteristic
64C2×Dic74C2311yescharacteristic
65M16×C37—22no—
66C2×C8×C37C2312yes—
67C4⋊C4×C37—21no—
68C4⋊C4×C37C2312yes—
69C4⋊C4×C37C2311yescharacteristic
70C4⋊C4×C37C2311yescharacteristic
71C22⋊C4×C37—21no—
72C42×C37—21no—
73#10596938—21no—
74#10596938—21no—
75#10596938—21no—
76#10596938—21no—
77C4⋊Dic37—21no—
78C4⋊Dic37—21no—
79C4⋊Dic37—21no—
80C4⋊Dic37C2312yes—
81#10596936—22no—
82#10596936—22no—
83#10596934—22no—
84C2×#64436C2312yes—
85C23×C37C2×D411yescharacteristic
86C22×D37—22no—
87C22×D37—22no—
88C22×D37—22no—
89C22×D37—21no—
90C22×D37C2×D411yescharacteristic
91C22×D37C2×D411yescharacteristic
92C2×C4×C37—22no—
93C2×C4×C37—21no—
94C2×C4×C37—21no—
95C2×C4×C37—42no—
96C2×C4×C37C2×D411yescharacteristic
97C2×C4×C37#4112yes—
98C2×C4×C37#4112yes—
99C2×C4×C37C2411yescharacteristic
100C22⋊D37—42no—
101C2×Dic37—22no—
102C2×Dic37—22no—
103C2×Dic37—21no—
104C2×Dic37—42no—
105C2×Dic37C2×D411yescharacteristic
106C2×Dic37C2×D411yescharacteristic
107D148—21no—
108D148—21no—
109D148—42no—
110D148—42no—
111C4×D37—22no—
112C4×D37—22no—
113C4×D37—21no—
114C4×D37—21no—
115C4×D37—21no—
116C4×D37—21no—
117C4×D37—44no—
118C4×D37—44no—
119C4×D37—42no—
120Dic74—21no—
121Dic74—21no—
122C8×C37—22no—
123#64436—22no—
124C22×C37—21no—
125C22×C37—21no—
126C22×C37#12312yes—
127C22×C37C22×D411yescharacteristic
128C2×D37—28no—
129C2×D37—21no—
130C2×D37—21no—
131C2×D37—21no—
132C2×D37—21no—
133C2×D37—41no—
134C2×D37—41no—
135C2×D37—41no—
136C2×D37AGL(1,ℤ/8)12yes—
137C2×D37#13712yes—
138C4×C37—42no—
139C4×C37—41no—
140C4×C37—41no—
141C4×C37#12312yes—
142C4×C37C2×#4111yescharacteristic
143C4×C37C2×#4111yescharacteristic
144Dic37—21no—
145Dic37—21no—
146Dic37—41no—
147Dic37—42no—
148#3092—371nomaximal, Sylow
149C2×C37—22no—
150C2×C37C2×#12311yescharacteristic
151C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C37C2×#13711yescharacteristic
153D37—22no—
154D37—22no—
155D37—81no—
156#525—371no—
157C22×C4⋊C4—371no—
158#484—372no—
159#483—372no—
160#481—372no—
161#480—372no—
162#428—371no—
163#428D3711yescharacteristic, complemented
164#417—371no—
165#417—371no—
166#407—371no—
167C37#309211yescharacteristic, complemented, Sylow
168C2×#41—371no—
169C23×C4—371no—
170C23×C4—741no—
171C2×M16—371no—
172C2×M16C2×D3711yescharacteristic
173#123—741no—
174C22⋊Q8—371no—
175C22⋊Q8—371no—
176#121—371no—
177#121—371no—
178#120—741no—
179C4×D4—741no—
180C4×D4—741no—
181#117C2×D3711yescharacteristic
182C2×C4⋊C4—371no—
183C2×C4⋊C4—372no—
184C2×C4⋊C4—372no—
185C2×C4⋊C4—741no—
186C2×C4⋊C4—741no—
187C2×C4⋊C4—741no—
188C2×C4⋊C4C2×D3711yescharacteristic
189#107—372no—
190#107C2×D3712yescomplemented
191#106—372no—
192#106C2×D3712yescomplemented
193#103—372no—
194#103—372no—
195#102—372no—
196#102—372no—
197C22⋊C8—372no—
198C22⋊C8—372no—
199C24—371no—
200#41—742no—
201#41—742no—
202C2×Q8—371no—
203C2×D4—371no—
204C2×D4—741no—
205C2×D4—741no—
206C2×D4—741no—
207C22×C4—21no—
208C22×C4—371no—
209C22×C4—371no—
210C22×C4—371no—
211C22×C4—371no—
212C22×C4—371no—
213C22×C4—742no—
214C22×C4—742no—
215C22×C4—742no—
216C22×C4—742no—
217C22×C4—741no—
218C22×C4—741no—
219C22×C4—1484no—
220C22×C4C22×D3711yescharacteristic
221M16—22no—
222M16—742no—
223C2×C8—372no—
224C2×C8C22×D3712yes—
225C4⋊C4—21no—
226C4⋊C4—372no—
227C4⋊C4—742no—
228C4⋊C4—742no—
229C4⋊C4—741no—
230C4⋊C4—741no—
231C4⋊C4—741no—
232C4⋊C4C22×D3712yes—
233C4⋊C4C22×D3711yescharacteristic
234C4⋊C4C22×D3711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—741no—
237C22⋊C4—741no—
238C22⋊C4—741no—
239C22⋊C4—741no—
240C42—21no—
241C23—371no—
242C23—371no—
243C23—742no—
244C23—742no—
245C23—742no—
246C23—741no—
247C23D4×D3711yescharacteristic
248Q8—741no—
249Q8—741no—
250D4—741no—
251D4—741no—
252D4—1482no—
253D4—1482no—
254D4—1482no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—371no—
260C2×C4—371no—
261C2×C4—742no—
262C2×C4—742no—
263C2×C4—742no—
264C2×C4—741no—
265C2×C4—741no—
266C2×C4—741no—
267C2×C4—741no—
268C2×C4—741no—
269C2×C4—742no—
270C2×C4—1484no—
271C2×C4—1484no—
272C2×C4—1482no—
273C2×C4—1482no—
274C2×C4#1059696212yes—
275C2×C4D4×D3711yescharacteristic
276C2×C4#1059696612yes—
277C2×C4C23×D3711yescharacteristic
278C8—22no—
279C8—742no—
280C22—21no—
281C22—21no—
282C22—372no—
283C22—372no—
284C22—748no—
285C22—741no—
286C22—741no—
287C22—741no—
288C22—741no—
289C22—1481no—
290C22—1481no—
291C22—1481no—
292C22#1192452112yes—
293C22C2×D4×D3711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—741no—
298C4—741no—
299C4—1481no—
300C4—1482no—
301C4#1192452112yes—
302C4C2×#1059696211yescharacteristic
303C4C2×#1059696611yescharacteristic
304C2—22no—
305C2—2961no—
306C2—742no—
307C2—742no—
308C2C2×#1192452111yescharacteristic
309C2AGL(1,ℤ/8)×D3711yescharacteristic
310C2#137×D3711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#428C2
socleC22×C37C22×D4
prime radical#14558045C2
hypercenter#428D37
hypocenterC37#3092
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D37C2semidirect
3. #14557870C2semidirect
4. #14557865C2semidirect
5. #14557853C2semidirect
6. #14557848C2semidirect
7. #14557576C2semidirect
8. #14557535C2semidirect
9. #14557532C2semidirect
10. #14557496C2semidirect
11. C37×#428C2semidirect
30. #11924469C22semidirect
31. #11924467C22semidirect
32. #11924461C22semidirect
33. #11924459C22semidirect
34. #11924458C22semidirect
35. #11924456C22semidirect
41. #107×C37C22semidirect
42. #106×C37C22semidirect
163. #428D37semidirect
167. C37#3092semidirect
190. #107C2×D37semidirect
192. #106C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580452
C2×C4⋊C4×D372
#145578702
#145578652
#145578532
#145578482
#145575762
#145575352
#145575322
#145574962
C37×#4282
#309237