Finite Group Databasebeta

← #18004534

Subgroups — #18004534

Order 4736.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#11924507C211yesmaximal, characteristic
2#117×D37C211yesmaximal, characteristic, complemented
3#14557856C212yesmaximal, complemented
4#14557855C212yesmaximal, complemented
5#14557850C212yesmaximal, complemented
6#14557845C212yesmaximal, complemented
7#14557576C211yesmaximal, characteristic, complemented
8#14557575C211yesmaximal, characteristic, complemented
9#14557538C211yesmaximal, characteristic, complemented
10#14557496C211yesmaximal, characteristic, complemented
11C37×#418C211yesmaximal, characteristic, complemented
12C22×C22⋊D37—21no—
13C22×C4×D37C2211yescharacteristic
14C2×#10596934C2211yescharacteristic
15#11924507—21no—
16#11924507—21no—
17#11924507C2212yes—
18#11924507C2212yes—
19#11924495C2211yescharacteristic
20#11924494C2211yescharacteristic, complemented
21#11924494C2211yescharacteristic, complemented
22#11924493—21no—
23#11924492—21no—
24C2×C4⋊Dic37C2211yescharacteristic
25C4⋊C4×D37C2211yescharacteristic, complemented
26C4⋊C4×D37C2211yescharacteristic, complemented
27C2×C22⋊Dic37—21no—
28C22⋊C4×D37—21no—
29C42×D37—21no—
30#11924470C2212yescomplemented
31#11924469C2212yescomplemented
32#11924468C2212yescomplemented
33#11924467C2212yescomplemented
34#11924457C2212yescomplemented
35#11924457C2212yescomplemented
36#11924448C2212yescomplemented
37#11924448C2212yescomplemented
38C22×D4×C37C2211yescharacteristic
39C2×M16×C37C2211yescharacteristic
40#117×C37C2211yescharacteristic
41#102×C37C2212yescomplemented
42#102×C37C2212yescomplemented
43C24×C37—21no—
44C23×D37D411yescharacteristic
45C2×D4×C37—21no—
46C2×D4×C37—22no—
47C2×D4×C37—22no—
48C2×D4×C37C2312yes—
49C22×C4×C37C2311yescharacteristic
50C2×C22⋊D37—22no—
51C2×C22⋊D37—22no—
52C2×C22⋊D37—44no—
53C22×Dic37—21no—
54C22×Dic37D411yescharacteristic
55C2×C4×D37—22no—
56C2×C4×D37—22no—
57C2×C4×D37—21no—
58C2×C4×D37—21no—
59C2×C4×D37D411yescharacteristic
60C2×C4×D37D411yescharacteristic
61C2×C4×D37C2311yescharacteristic
62C2×C4×D37C2311yescharacteristic
63M16×C37—22no—
64C2×C8×C37C2312yes—
65C4⋊C4×C37C2311yescharacteristic
66C4⋊C4×C37C2311yescharacteristic
67C22⋊C4×C37—21no—
68C42×C37—21no—
69C22⋊Dic37—21no—
70C22⋊Dic37—22no—
71C22⋊Dic37—22no—
72#10596938—21no—
73#10596938—21no—
74C4⋊Dic37—21no—
75C4⋊Dic37C2311yescharacteristic
76C4⋊Dic37C2311yescharacteristic
77C4⋊Dic37C2312yes—
78#10596936—21no—
79#10596936—21no—
80C4×Dic37—21no—
81C4×Dic37—21no—
82C4×Dic37—21no—
83#10596934—22no—
84C2×#64436C2312yes—
85C23×C37—22no—
86C23×C37—42no—
87C23×C37—42no—
88C23×C37—42no—
89C23×C37C2×D411yescharacteristic
90C22×D37—22no—
91C22×D37—22no—
92C22×D37—41no—
93C22×D37C2×D411yescharacteristic
94C22×D37C2×D411yescharacteristic
95D4×C37—22no—
96D4×C37—22no—
97D4×C37—42no—
98C2×C4×C37—21no—
99C2×C4×C37—21no—
100C2×C4×C37C2×D411yescharacteristic
101C2×C4×C37#4112yes—
102C2×C4×C37#4112yes—
103C2×C4×C37C2411yescharacteristic
104C22⋊D37—44no—
105C22⋊D37—44no—
106C2×Dic37—22no—
107C2×Dic37—21no—
108C2×Dic37—21no—
109C2×Dic37—22no—
110C2×Dic37—42no—
111C2×Dic37C2×D411yescharacteristic
112C2×Dic37C2×D411yescharacteristic
113C4×D37—22no—
114C4×D37—22no—
115C4×D37—21no—
116C4×D37—21no—
117C4×D37—21no—
118C4×D37—21no—
119C4×D37—42no—
120C4×D37—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×C37#12312yes—
130C22×C37C22×D411yescharacteristic
131C2×D37—21no—
132C2×D37—21no—
133C2×D37—21no—
134C2×D37—21no—
135C2×D37—21no—
136C2×D37—44no—
137C2×D37AGL(1,ℤ/8)12yes—
138C4×C37—41no—
139C4×C37—41no—
140C4×C37#12312yes—
141C4×C37C2×#4111yescharacteristic
142C4×C37C2×#4111yescharacteristic
143Dic37—21no—
144Dic37—21no—
145Dic37—41no—
146Dic37—42no—
147Dic37—41no—
148#3094—371nomaximal, Sylow
149C2×C37—22no—
150C2×C37—82no—
151C2×C37C2×#12311yescharacteristic
152C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C37#57511yescharacteristic
154D37—22no—
155D37—41no—
156C2×#121—371no—
157C2×#117—371no—
158#482—372no—
159#482—372no—
160#481—372no—
161#480—372no—
162#428—371no—
163#428—371no—
164#418—371no—
165#418D3711yescharacteristic, complemented
166#407—371no—
167C37#309411yescharacteristic, complemented, Sylow
168C22×D4—741no—
169C22×D4C2×D3711yescharacteristic
170C23×C4—371no—
171C2×M16—371no—
172C2×M16C2×D3711yescharacteristic
173#121—372no—
174#121—372no—
175#121—741no—
176#121—741no—
177#117—371no—
178#117—371no—
179#117—371no—
180#117—741no—
181#117—741no—
182#117C2×D3711yescharacteristic
183C2×C4⋊C4—371no—
184C2×C4⋊C4—371no—
185C2×C4⋊C4—371no—
186C2×C22⋊C4—741no—
187C2×C22⋊C4—741no—
188C2×C42—741no—
189#107—372no—
190#107—372no—
191#106—372no—
192#106—372no—
193#102—372no—
194#102—372no—
195#102C2×D3712yescomplemented
196#102C2×D3712yescomplemented
197C22⋊C8—372no—
198C22⋊C8—372no—
199C24—21no—
200C24—371no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—742no—
205C2×D4—742no—
206C2×D4—1484no—
207C2×D4C22×D3712yes—
208C22×C4—371no—
209C22×C4—371no—
210C22×C4—371no—
211C22×C4—371no—
212C22×C4—371no—
213C22×C4—741no—
214C22×C4—741no—
215C22×C4—742no—
216C22×C4—742no—
217C22×C4—741no—
218C22×C4C22×D3711yescharacteristic
219M16—22no—
220M16—742no—
221C2×C8—372no—
222C2×C8C22×D3712yes—
223C4⋊C4—371no—
224C4⋊C4—371no—
225C4⋊C4—372no—
226C4⋊C4—741no—
227C4⋊C4—741no—
228C4⋊C4—741no—
229C4⋊C4C22×D3711yescharacteristic
230C4⋊C4C22×D3711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—741no—
233C22⋊C4—741no—
234C22⋊C4—742no—
235C22⋊C4—742no—
236C22⋊C4—741no—
237C42—21no—
238C42—741no—
239C42—741no—
240C42—741no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—371no—
246C23—371no—
247C23—742no—
248C23—742no—
249C23—1481no—
250C23D4×D3711yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—1484no—
255D4—1484no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—371no—
259C2×C4—371no—
260C2×C4—742no—
261C2×C4—742no—
262C2×C4—742no—
263C2×C4—741no—
264C2×C4—741no—
265C2×C4—741no—
266C2×C4—741no—
267C2×C4—741no—
268C2×C4—741no—
269C2×C4—742no—
270C2×C4—1482no—
271C2×C4—1482no—
272C2×C4—1482no—
273C2×C4#1059696212yes—
274C2×C4D4×D3711yescharacteristic
275C2×C4#1059696412yes—
276C2×C4C23×D3711yescharacteristic
277C8—22no—
278C8—742no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—372no—
286C22—741no—
287C22—741no—
288C22—741no—
289C22—741no—
290C22—741no—
291C22—1484no—
292C22#1192451812yes—
293C22C2×D4×D3711yescharacteristic
294C4—41no—
295C4—41no—
296C4—741no—
297C4—741no—
298C4—1482no—
299C4—1481no—
300C4—1481no—
301C4#1192451812yes—
302C4C2×#1059696411yescharacteristic
303C4C2×#1059696211yescharacteristic
304C2—22no—
305C2—82no—
306C2—742no—
307C2—1481no—
308C2C2×#1192451811yescharacteristic
309C2AGL(1,ℤ/8)×D3711yescharacteristic
310C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#418C2
socleC22×C37C22×D4
prime radicalC2×#11924507C2
hypercenter#418D37
hypocenterC37#3094
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D37C2semidirect
3. #14557856C2semidirect
4. #14557855C2semidirect
5. #14557850C2semidirect
6. #14557845C2semidirect
7. #14557576C2semidirect
8. #14557575C2semidirect
9. #14557538C2semidirect
10. #14557496C2semidirect
11. C37×#418C2semidirect
20. #11924494C22semidirect
21. #11924494C22semidirect
25. C4⋊C4×D37C22semidirect
26. C4⋊C4×D37C22semidirect
30. #11924470C22semidirect
31. #11924469C22semidirect
32. #11924468C22semidirect
33. #11924467C22semidirect
34. #11924457C22semidirect
35. #11924457C22semidirect
36. #11924448C22semidirect
37. #11924448C22semidirect
41. #102×C37C22semidirect
42. #102×C37C22semidirect
165. #418D37semidirect
167. C37#3094semidirect
195. #102C2×D37semidirect
196. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245072
#117×D372
#145578562
#145578552
#145578502
#145578452
#145575762
#145575752
#145575382
#145574962
C37×#4182
#309437