Finite Group Databasebeta

← #18004517

Subgroups — #18004517

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#14558040C211yesmaximal, characteristic
2C2×C4⋊C4×D37C211yesmaximal, characteristic, complemented
3#14557867C212yesmaximal, complemented
4#14557862C212yesmaximal, complemented
5#14557850C212yesmaximal, complemented
6#14557845C212yesmaximal, complemented
7#14557575C211yesmaximal, characteristic, complemented
8#14557573C211yesmaximal, characteristic, complemented
9#14557534C211yesmaximal, characteristic, complemented
10#14557496C211yesmaximal, characteristic, complemented
11C37×#417C211yesmaximal, characteristic, complemented
12C22×C4×D37—21no—
13C22×C4×D37C2211yescharacteristic
14C2×#10596934C2211yescharacteristic
15#11924517—21no—
16#11924512C2211yescharacteristic
17#11924512C2211yescharacteristic
18#11924507C2211yescharacteristic
19#11924507C2211yescharacteristic
20#11924504—21no—
21C4×C22⋊D37—21no—
22C4×C22⋊D37—21no—
23#11924495C2211yescharacteristic
24C2×C4⋊Dic37C2211yescharacteristic
25C2×#10596936—21no—
26C4⋊C4×D37—21no—
27C4⋊C4×D37—21no—
28C4⋊C4×D37C2212yes—
29C4⋊C4×D37C2212yes—
30#11924470C2212yescomplemented
31#11924469C2212yescomplemented
32#11924468C2212yescomplemented
33#11924467C2212yescomplemented
34#11924460C2212yescomplemented
35#11924457C2212yescomplemented
36#11924448C2212yes—
37#11924448C2212yes—
38C2×#41×C37C2211yescharacteristic
39C2×M16×C37C2211yescharacteristic
40C2×C4⋊C4×C37C2211yescharacteristic
41#103×C37C2212yescomplemented
42#102×C37C2212yescomplemented
43C23×D37D411yescharacteristic
44#41×C37—22no—
45#41×C37—22no—
46C2×Q8×C37C2311yescharacteristic
47C2×D4×C37—21no—
48C2×D4×C37C2311yescharacteristic
49C22×C4×C37—21no—
50C22×C4×C37—21no—
51C22×C4×C37C2311yescharacteristic
52C2×C22⋊D37—21no—
53C2×C22⋊D37—21no—
54C22×Dic37—21no—
55C22×Dic37D411yescharacteristic
56C2×C4×D37—22no—
57C2×C4×D37—22no—
58C2×C4×D37—22no—
59C2×C4×D37—22no—
60C2×C4×D37—44no—
61C2×C4×D37D411yescharacteristic
62C2×C4×D37D411yescharacteristic
63C2×C4×D37C2311yescharacteristic
64C2×C4×D37C2311yescharacteristic
65M16×C37—22no—
66C2×C8×C37C2312yes—
67C4⋊C4×C37—21no—
68C4⋊C4×C37C2312yes—
69C22⋊Dic37—21no—
70C22⋊Dic37—21no—
71C22⋊Dic37—21no—
72#10596938—21no—
73#10596938—21no—
74C4⋊Dic37—21no—
75C4⋊Dic37C2312yes—
76C4⋊Dic37C2311yescharacteristic
77C4⋊Dic37C2311yescharacteristic
78#10596936—22no—
79#10596936—22no—
80#10596936—21no—
81#10596936—21no—
82C4×Dic37—21no—
83#10596934—22no—
84C2×#64436C2312yes—
85C23×C37—21no—
86C23×C37C2×D411yescharacteristic
87C22×D37—22no—
88C22×D37—22no—
89C22×D37—22no—
90C22×D37C2×D411yescharacteristic
91C22×D37C2×D411yescharacteristic
92Q8×C37—21no—
93Q8×C37—21no—
94D4×C37—21no—
95D4×C37—21no—
96D4×C37—42no—
97C2×C4×C37—21no—
98C2×C4×C37—22no—
99C2×C4×C37—42no—
100C2×C4×C37—42no—
101C2×C4×C37C2×D411yescharacteristic
102C2×C4×C37#4112yes—
103C2×C4×C37#4112yes—
104C2×C4×C37C2411yescharacteristic
105C22⋊D37—42no—
106C22⋊D37—42no—
107C2×Dic37—22no—
108C2×Dic37—21no—
109C2×Dic37—21no—
110C2×Dic37—22no—
111C2×Dic37—42no—
112C2×Dic37C2×D411yescharacteristic
113C2×Dic37C2×D411yescharacteristic
114C4×D37—22no—
115C4×D37—22no—
116C4×D37—21no—
117C4×D37—21no—
118C4×D37—21no—
119C4×D37—21no—
120C4×D37—44no—
121C4×D37—44no—
122C8×C37—22no—
123#64436—22no—
124C22×C37—21no—
125C22×C37—21no—
126C22×C37—41no—
127C22×C37—41no—
128C22×C37—41no—
129C22×C37#12312yes—
130C22×C37C22×D411yescharacteristic
131C2×D37—28no—
132C2×D37—21no—
133C2×D37—21no—
134C2×D37—21no—
135C2×D37—21no—
136C2×D37AGL(1,ℤ/8)12yes—
137C2×D37#13712yes—
138C4×C37—41no—
139C4×C37—42no—
140C4×C37#12312yes—
141C4×C37C2×#4111yescharacteristic
142C4×C37C2×#4111yescharacteristic
143Dic37—21no—
144Dic37—21no—
145Dic37—42no—
146Dic37—41no—
147Dic37—41no—
148#3092—371nomaximal, Sylow
149C2×C37—22no—
150C2×C37—81no—
151C2×C37C2×#12311yescharacteristic
152C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C37C2×#13711yescharacteristic
154D37—22no—
155D37—22no—
156#525—371no—
157C22×C4⋊C4—371no—
158#484—372no—
159#483—372no—
160#481—372no—
161#480—372no—
162#428—371no—
163#428—371no—
164#417—371no—
165#417D3711yescharacteristic, complemented
166#407—371no—
167C37#309211yescharacteristic, complemented, Sylow
168C2×#41C2×D3711yescharacteristic
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#117—371no—
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#107—372no—
191#106—372no—
192#106—372no—
193#103—372no—
194#103C2×D3712yescomplemented
195#102—372no—
196#102C2×D3712yescomplemented
197C22⋊C8—372no—
198C22⋊C8—372no—
199C24—371no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D3711yescharacteristic
203C2×D4—21no—
204C2×D4—741no—
205C2×D4—741no—
206C2×D4C22×D3711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—371no—
210C22×C4—371no—
211C22×C4—371no—
212C22×C4—371no—
213C22×C4—371no—
214C22×C4—742no—
215C22×C4—742no—
216C22×C4—742no—
217C22×C4—742no—
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—371no—
228C4⋊C4—371no—
229C4⋊C4—742no—
230C4⋊C4—742no—
231C4⋊C4—741no—
232C4⋊C4—741no—
233C4⋊C4—741no—
234C4⋊C4C22×D3712yes—
235C22⋊C4—741no—
236C22⋊C4—741no—
237C22⋊C4—741no—
238C22⋊C4—741no—
239C22⋊C4—741no—
240C42—741no—
241C23—21no—
242C23—371no—
243C23—371no—
244C23—742no—
245C23—742no—
246C23—742no—
247C23D4×D3711yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—1482no—
254D4—1482no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
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—741no—
270C2×C4—742no—
271C2×C4—1484no—
272C2×C4—1484no—
273C2×C4—1482no—
274C2×C4#1059696212yes—
275C2×C4D4×D3711yescharacteristic
276C2×C4#1059696412yes—
277C2×C4C23×D3711yescharacteristic
278C8—22no—
279C8—742no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—372no—
286C22—372no—
287C22—748no—
288C22—741no—
289C22—741no—
290C22—741no—
291C22—741no—
292C22#1192451812yes—
293C22C2×D4×D3711yescharacteristic
294C4—41no—
295C4—42no—
296C4—741no—
297C4—741no—
298C4—1482no—
299C4—1481no—
300C4—1481no—
301C4#1192451812yes—
302C4C2×#1059696411yescharacteristic
303C4C2×#1059696211yescharacteristic
304C2—22no—
305C2—81no—
306C2—742no—
307C2—742no—
308C2C2×#1192451811yescharacteristic
309C2AGL(1,ℤ/8)×D3711yescharacteristic
310C2#137×D3711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#417C2
socleC22×C37C22×D4
prime radical#14558040C2
hypercenter#417D37
hypocenterC37#3092
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D37C2semidirect
3. #14557867C2semidirect
4. #14557862C2semidirect
5. #14557850C2semidirect
6. #14557845C2semidirect
7. #14557575C2semidirect
8. #14557573C2semidirect
9. #14557534C2semidirect
10. #14557496C2semidirect
11. C37×#417C2semidirect
30. #11924470C22semidirect
31. #11924469C22semidirect
32. #11924468C22semidirect
33. #11924467C22semidirect
34. #11924460C22semidirect
35. #11924457C22semidirect
41. #103×C37C22semidirect
42. #102×C37C22semidirect
165. #417D37semidirect
167. C37#3092semidirect
194. #103C2×D37semidirect
196. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580402
C2×C4⋊C4×D372
#145578672
#145578622
#145578502
#145578452
#145575752
#145575732
#145575342
#145574962
C37×#4172
#309237