Finite Group Databasebeta

← #18004538

Subgroups — #18004538

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×#11924511C211yesmaximal, characteristic
2#117×D37C211yesmaximal, characteristic, complemented
3#14557860C212yesmaximal, complemented
4#14557859C212yesmaximal, complemented
5#14557853C212yesmaximal, complemented
6#14557848C212yesmaximal, complemented
7#14557573C211yesmaximal, characteristic, complemented
8#14557539C211yesmaximal, characteristic, complemented
9#14557537C211yesmaximal, characteristic, complemented
10#14557496C211yesmaximal, characteristic, complemented
11C37×#428C211yesmaximal, characteristic, complemented
12C22×D148—21no—
13C22×D148C2211yescharacteristic
14C22×C4×D37C2211yescharacteristic
15C2×#10596934C2211yescharacteristic
16#11924511—21no—
17#11924511—21no—
18#11924511C2212yes—
19#11924511C2212yes—
20#11924495C2211yescharacteristic
21#11924494C2211yescharacteristic, complemented
22#11924494C2211yescharacteristic, complemented
23#11924493—21no—
24#11924492—21no—
25C4⋊C4×D37C2211yescharacteristic, complemented
26C4⋊C4×D37C2211yescharacteristic, complemented
27C2×#10596938—21no—
28C22⋊C4×D37—21no—
29C42×D37—21no—
30#11924469C2212yescomplemented
31#11924467C2212yescomplemented
32#11924458C2212yescomplemented
33#11924458C2212yescomplemented
34#11924456C2212yescomplemented
35#11924456C2212yescomplemented
36#11924448C2212yescomplemented
37#11924448C2212yescomplemented
38C2×M16×C37C2211yescharacteristic
39#117×C37C2211yescharacteristic
40C2×C4⋊C4×C37C2211yescharacteristic
41#107×C37C2212yescomplemented
42#106×C37C2212yescomplemented
43C23×D37—21no—
44C23×D37D411yescharacteristic
45C22×C4×C37—21no—
46C22×C4×C37C2311yescharacteristic
47C22×Dic37D411yescharacteristic
48C2×D148—22no—
49C2×D148—22no—
50C2×D148—22no—
51C2×D148—22no—
52C2×D148—21no—
53C2×D148—44no—
54C2×D148C2312yes—
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×C37—21no—
66C4⋊C4×C37C2311yescharacteristic
67C4⋊C4×C37C2311yescharacteristic
68C4⋊C4×C37C2312yes—
69C22⋊C4×C37—21no—
70C42×C37—21no—
71C22⋊Dic37—21no—
72#10596938—21no—
73#10596938—21no—
74#10596938—22no—
75#10596938—22no—
76C4⋊Dic37C2311yescharacteristic
77C4⋊Dic37C2311yescharacteristic
78#10596936—21no—
79#10596936—21no—
80C4×Dic37—21no—
81C4×Dic37—21no—
82C4×Dic37—21no—
83#10596934—22no—
84C2×#64436C2312yes—
85C23×C37C2×D411yescharacteristic
86C22×D37—22no—
87C22×D37—22no—
88C22×D37—22no—
89C22×D37—41no—
90C22×D37—42no—
91C22×D37—42no—
92C22×D37—42no—
93C22×D37C2×D411yescharacteristic
94C22×D37C2×D411yescharacteristic
95C2×C4×C37—21no—
96C2×C4×C37—21no—
97C2×C4×C37—22no—
98C2×C4×C37—42no—
99C2×C4×C37C2×D411yescharacteristic
100C2×C4×C37#4112yes—
101C2×C4×C37#4112yes—
102C2×C4×C37C2411yescharacteristic
103C2×Dic37—22no—
104C2×Dic37—21no—
105C2×Dic37—21no—
106C2×Dic37C2×D411yescharacteristic
107C2×Dic37C2×D411yescharacteristic
108D148—22no—
109D148—22no—
110D148—44no—
111D148—44no—
112D148—42no—
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#12312yes—
126C22×C37C22×D411yescharacteristic
127C2×D37—21no—
128C2×D37—21no—
129C2×D37—21no—
130C2×D37—21no—
131C2×D37—21no—
132C2×D37—44no—
133C2×D37—42no—
134C2×D37—42no—
135C2×D37—42no—
136C2×D37—84no—
137C2×D37AGL(1,ℤ/8)12yes—
138C4×C37—41no—
139C4×C37—42no—
140C4×C37—41no—
141C4×C37#12312yes—
142C4×C37C2×#4111yescharacteristic
143C4×C37C2×#4111yescharacteristic
144Dic37—21no—
145Dic37—21no—
146Dic37—41no—
147Dic37—41no—
148#3094—371nomaximal, Sylow
149C2×C37—22no—
150C2×C37C2×#12311yescharacteristic
151C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C37#57511yescharacteristic
153D37—22no—
154D37—41no—
155D37—82no—
156C2×#121—371no—
157C2×#117—371no—
158#482—372no—
159#482—372no—
160#481—372no—
161#480—372no—
162#428—371no—
163#428D3711yescharacteristic, complemented
164#418—371no—
165#418—371no—
166#407—371no—
167C37#309411yescharacteristic, complemented, Sylow
168C22×D4—371no—
169C22×D4—741no—
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⋊C4C2×D3711yescharacteristic
186C2×C22⋊C4—741no—
187C2×C22⋊C4—741no—
188C2×C42—741no—
189#107—372no—
190#107C2×D3712yescomplemented
191#106—372no—
192#106C2×D3712yescomplemented
193#102—372no—
194#102—372no—
195#102—372no—
196#102—372no—
197C22⋊C8—372no—
198C22⋊C8—372no—
199C24—371no—
200C24—741no—
201C2×D4—372no—
202C2×D4—742no—
203C2×D4—742no—
204C2×D4—742no—
205C2×D4—741no—
206C2×D4—742no—
207C2×D4—1484no—
208C22×C4—21no—
209C22×C4—371no—
210C22×C4—371no—
211C22×C4—371no—
212C22×C4—371no—
213C22×C4—371no—
214C22×C4—741no—
215C22×C4—741no—
216C22×C4—742no—
217C22×C4—742no—
218C22×C4C22×D3711yescharacteristic
219M16—22no—
220M16—742no—
221C2×C8—372no—
222C2×C8C22×D3712yes—
223C4⋊C4—21no—
224C4⋊C4—371no—
225C4⋊C4—371no—
226C4⋊C4—741no—
227C4⋊C4—741no—
228C4⋊C4C22×D3711yescharacteristic
229C4⋊C4C22×D3711yescharacteristic
230C4⋊C4C22×D3712yes—
231C22⋊C4—21no—
232C22⋊C4—742no—
233C22⋊C4—742no—
234C22⋊C4—741no—
235C22⋊C4—741no—
236C22⋊C4—741no—
237C42—21no—
238C42—741no—
239C42—741no—
240C42—741no—
241C23—371no—
242C23—371no—
243C23—742no—
244C23—742no—
245C23—742no—
246C23—1482no—
247C23—1482no—
248C23—1481no—
249C23—1482no—
250C23D4×D3711yescharacteristic
251D4—742no—
252D4—742no—
253D4—1484no—
254D4—1484no—
255D4—1482no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—371no—
261C2×C4—371no—
262C2×C4—742no—
263C2×C4—742no—
264C2×C4—742no—
265C2×C4—741no—
266C2×C4—741no—
267C2×C4—741no—
268C2×C4—741no—
269C2×C4—741no—
270C2×C4—741no—
271C2×C4—1482no—
272C2×C4—1482no—
273C2×C4#1059696212yes—
274C2×C4D4×D3711yescharacteristic
275C2×C4#1059696612yes—
276C2×C4C23×D3711yescharacteristic
277C8—22no—
278C8—742no—
279C22—21no—
280C22—21no—
281C22—372no—
282C22—2964no—
283C22—741no—
284C22—741no—
285C22—741no—
286C22—741no—
287C22—741no—
288C22—1484no—
289C22—1482no—
290C22—1482no—
291C22—1482no—
292C22#1192452112yes—
293C22C2×D4×D3711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—741no—
298C4—741no—
299C4—1481no—
300C4—1481no—
301C4#1192452112yes—
302C4C2×#1059696211yescharacteristic
303C4C2×#1059696611yescharacteristic
304C2—22no—
305C2—2962no—
306C2—742no—
307C2—1481no—
308C2C2×#1192452111yescharacteristic
309C2AGL(1,ℤ/8)×D3711yescharacteristic
310C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#428C2
socleC22×C37C22×D4
prime radicalC2×#11924511C2
hypercenter#428D37
hypocenterC37#3094
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. #117×D37C2semidirect
3. #14557860C2semidirect
4. #14557859C2semidirect
5. #14557853C2semidirect
6. #14557848C2semidirect
7. #14557573C2semidirect
8. #14557539C2semidirect
9. #14557537C2semidirect
10. #14557496C2semidirect
11. C37×#428C2semidirect
21. #11924494C22semidirect
22. #11924494C22semidirect
25. C4⋊C4×D37C22semidirect
26. C4⋊C4×D37C22semidirect
30. #11924469C22semidirect
31. #11924467C22semidirect
32. #11924458C22semidirect
33. #11924458C22semidirect
34. #11924456C22semidirect
35. #11924456C22semidirect
36. #11924448C22semidirect
37. #11924448C22semidirect
41. #107×C37C22semidirect
42. #106×C37C22semidirect
163. #428D37semidirect
167. C37#3094semidirect
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
C2×#119245112
#117×D372
#145578602
#145578592
#145578532
#145578482
#145575732
#145575392
#145575372
#145574962
C37×#4282
#309437