Finite Group Databasebeta

← #18004209

Subgroups — #18004209

Order 4736.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D37C211yesmaximal, characteristic, complemented
2#14558040C211yesmaximal, characteristic, complemented
3C2×#11924507C211yesmaximal, characteristic, complemented
4#14557783C212yesmaximal, complemented
5#14557776C212yesmaximal, complemented
6#14557766C212yesmaximal, complemented
7#14557759C212yesmaximal, complemented
8#14557575C211yesmaximal, characteristic, complemented
9#14557538C211yesmaximal, characteristic, complemented
10#14557534C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12C22×C22⋊D37—21no—
13C22×C4×D37C2211yescharacteristic
14C2×#10596934C2211yescharacteristic
15C2×#10596929C2212yescomplemented
16M16×D37—24no—
17C2×C8×D37C2212yescomplemented
18#11924517—21no—
19#11924512C2211yescharacteristic, complemented
20#11924512C2211yescharacteristic, complemented
21#11924507—21no—
22#11924507—21no—
23#11924507C2212yes—
24#11924507C2212yes—
25#11924507C2211yescharacteristic, complemented
26#11924507C2211yescharacteristic, complemented
27#11924504—21no—
28C4×C22⋊D37—21no—
29C4×C22⋊D37—21no—
30#11924495C2211yescharacteristic
31C2×C4⋊Dic37C2211yescharacteristic
32C2×C22⋊Dic37—21no—
33#11924470C2212yescomplemented
34#11924468C2212yescomplemented
35#11924460C2212yescomplemented
36#11924457C2212yescomplemented
37#11924457C2212yescomplemented
38#11924457C2212yescomplemented
39C2×#41×C37C2211yescharacteristic
40C22×D4×C37C2211yescharacteristic
41AGL(1,ℤ/8)×C37—24no—
42C2×SD16×C37C2212yescomplemented
43C2×D8×C37C2212yescomplemented
44C2×M16×C37C2211yescharacteristic
45C24×C37—21no—
46C23×D37D411yescharacteristic
47#41×C37—22no—
48#41×C37—22no—
49C2×Q8×C37C2311yescharacteristic
50C2×D4×C37—21no—
51C2×D4×C37—22no—
52C2×D4×C37—21no—
53C2×D4×C37—22no—
54C2×D4×C37C2312yes—
55C2×D4×C37C2311yescharacteristic
56C22×C4×C37—21no—
57C22×C4×C37C2311yescharacteristic
58C2×C22⋊D37—21no—
59C2×C22⋊D37—21no—
60C2×C22⋊D37—22no—
61C2×C22⋊D37—22no—
62C2×C22⋊D37—44no—
63C22×Dic37—21no—
64C22×Dic37D411yescharacteristic
65C2×C4×D37—22no—
66C2×C4×D37—22no—
67C2×C4×D37D411yescharacteristic
68C2×C4×D37D411yescharacteristic
69C2×C4×D37C2311yescharacteristic
70C2×C4×D37C2311yescharacteristic
71SD16×C37—24no—
72D8×C37—24no—
73M16×C37D414yescomplemented
74C2×C8×C37C2312yes—
75C22⋊Dic37—21no—
76C22⋊Dic37—22no—
77C22⋊Dic37—22no—
78C22⋊Dic37—21no—
79C22⋊Dic37—21no—
80#10596938—21no—
81#10596938—21no—
82C4⋊Dic37—21no—
83C4⋊Dic37C2312yes—
84C4⋊Dic37C2311yescharacteristic
85C4⋊Dic37C2311yescharacteristic
86#10596936—21no—
87#10596936—21no—
88C4×Dic37—21no—
89#10596934—22no—
90C2×#64436C2312yes—
91#10596929—24no—
92C8×D37—24no—
93C23×C37—21no—
94C23×C37—22no—
95C23×C37—42no—
96C23×C37—42no—
97C23×C37—42no—
98C23×C37C2×D411yescharacteristic
99C22×D37—22no—
100C22×D37—22no—
101C22×D37—41no—
102C22×D37C2×D411yescharacteristic
103C22×D37C2×D411yescharacteristic
104Q8×C37—21no—
105Q8×C37—21no—
106D4×C37—21no—
107D4×C37—22no—
108D4×C37—22no—
109D4×C37—21no—
110D4×C37—42no—
111D4×C37—42no—
112C2×C4×C37—21no—
113C2×C4×C37—42no—
114C2×C4×C37C2×D411yescharacteristic
115C2×C4×C37C2×D412yes—
116C2×C4×C37#4112yes—
117C2×C4×C37C2411yescharacteristic
118C22⋊D37—42no—
119C22⋊D37—44no—
120C22⋊D37—42no—
121C22⋊D37—44no—
122C2×Dic37—22no—
123C2×Dic37—21no—
124C2×Dic37—21no—
125C2×Dic37—22no—
126C2×Dic37—42no—
127C2×Dic37C2×D411yescharacteristic
128C2×Dic37C2×D411yescharacteristic
129C4×D37—22no—
130C4×D37—22no—
131C4×D37—21no—
132C4×D37—21no—
133C4×D37—21no—
134C4×D37—21no—
135C8×C37C2×D414yescomplemented
136#64436—22no—
137C22×C37—21no—
138C22×C37—21no—
139C22×C37—41no—
140C22×C37—42no—
141C22×C37—42no—
142C22×C37—41no—
143C22×C37—41no—
144C22×C37—42no—
145C22×C37—84no—
146C22×C37#12112yes—
147C22×C37C22×D411yescharacteristic
148C2×D37—21no—
149C2×D37—21no—
150C2×D37—21no—
151C2×D37—21no—
152C2×D37—21no—
153C2×D37—44no—
154C2×D37AGL(1,ℤ/8)12yescomplemented
155C4×C37—41no—
156C4×C37#12112yes—
157C4×C37C22×D411yescharacteristic
158C4×C37C2×#4111yescharacteristic
159Dic37—21no—
160Dic37—21no—
161Dic37—42no—
162Dic37—41no—
163Dic37—41no—
164#3055—371nomaximal, Sylow
165C2×C37—22no—
166C2×C37—81no—
167C2×C37—82no—
168C2×C37C2×#12111yescharacteristic
169C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C37#57511yescharacteristic
171D37—22no—
172D37—41no—
173C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
174C22×M16—371no—
175#525—371no—
176C2×#121—371no—
177#469—372no—
178#468—372no—
179#466—372no—
180#465—372no—
181#428—371no—
182#418—371no—
183#417—371no—
184C37#305511yescharacteristic, complemented, Sylow
185C2×#41C2×D3711yescharacteristic
186C22×D4—741no—
187C22×D4C2×D3711yescharacteristic
188C23×C4—371no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D3712yescomplemented
191C2×D8C2×D3712yescomplemented
192C2×M16—371no—
193C2×M16—372no—
194C2×M16—744no—
195C2×M16C2×D3711yescharacteristic
196C22×C8—372no—
197#123—741no—
198C22⋊Q8—371no—
199C22⋊Q8—371no—
200#121—372no—
201#121—372no—
202#121—371no—
203#121—371no—
204#121—741no—
205#121—741no—
206#120—741no—
207C4×D4—741no—
208C4×D4—741no—
209#117—371no—
210C2×C4⋊C4—371no—
211C2×C22⋊C4—741no—
212#107—372no—
213#106—372no—
214#103—372no—
215#102—372no—
216#102—372no—
217#102—372no—
218C24—21no—
219C24—371no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D3711yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—741no—
228C2×D4—741no—
229C2×D4—742no—
230C2×D4—742no—
231C2×D4—1484no—
232C2×D4C22×D3711yescharacteristic
233C2×D4C22×D3712yes—
234C22×C4—21no—
235C22×C4—371no—
236C22×C4—371no—
237C22×C4—371no—
238C22×C4—371no—
239C22×C4—371no—
240C22×C4—742no—
241C22×C4—742no—
242C22×C4—741no—
243C22×C4C22×D3711yescharacteristic
244SD16—24no—
245D8—24no—
246M16—742no—
247M16—744no—
248M16C22⋊D3714yescomplemented
249C2×C8—372no—
250C2×C8—744no—
251C2×C8C22×D3712yes—
252C4⋊C4—371no—
253C4⋊C4—371no—
254C4⋊C4—372no—
255C4⋊C4—741no—
256C4⋊C4—741no—
257C4⋊C4—741no—
258C22⋊C4—741no—
259C22⋊C4—741no—
260C22⋊C4—742no—
261C22⋊C4—742no—
262C22⋊C4—741no—
263C22⋊C4—741no—
264C22⋊C4—741no—
265C42—741no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—371no—
272C23—371no—
273C23—742no—
274C23—742no—
275C23—1481no—
276C23D4×D3711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—1484no—
286D4—1482no—
287D4—1484no—
288D4—1482no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—371no—
292C2×C4—371no—
293C2×C4—742no—
294C2×C4—742no—
295C2×C4—742no—
296C2×C4—741no—
297C2×C4—741no—
298C2×C4—741no—
299C2×C4—741no—
300C2×C4—741no—
301C2×C4—741no—
302C2×C4—742no—
303C2×C4—1482no—
304C2×C4D4×D3711yescharacteristic
305C2×C4#1059696412yes—
306C2×C4C2×C22⋊D3712yes—
307C2×C4C23×D3711yescharacteristic
308C8—742no—
309C8C2×C22⋊D3714yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—372no—
320C22—741no—
321C22—741no—
322C22—741no—
323C22—741no—
324C22—741no—
325C22—1484no—
326C22#1192450712yes—
327C22C2×D4×D3711yescharacteristic
328C4—41no—
329C4—741no—
330C4—741no—
331C4—1482no—
332C4—1481no—
333C4—1481no—
334C4#1192450712yes—
335C4C22×C22⋊D3711yescharacteristic
336C4C2×#1059696411yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—742no—
341C2—1481no—
342C2C2×#1192450711yescharacteristic
343C2AGL(1,ℤ/8)×D3711yescharacteristic
344C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC22×C37C22×D4
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#3055
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D37C2semidirect
2. #14558040C2semidirect
3. C2×#11924507C2semidirect
4. #14557783C2semidirect
5. #14557776C2semidirect
6. #14557766C2semidirect
7. #14557759C2semidirect
8. #14557575C2semidirect
9. #14557538C2semidirect
10. #14557534C2semidirect
11. C2×AGL(1,ℤ/8)×C37C2semidirect
15. C2×#10596929C22semidirect
17. C2×C8×D37C22semidirect
19. #11924512C22semidirect
20. #11924512C22semidirect
25. #11924507C22semidirect
26. #11924507C22semidirect
33. #11924470C22semidirect
34. #11924468C22semidirect
35. #11924460C22semidirect
36. #11924457C22semidirect
37. #11924457C22semidirect
38. #11924457C22semidirect
42. C2×SD16×C37C22semidirect
43. C2×D8×C37C22semidirect
73. M16×C37D4semidirect
135. C8×C37C2×D4semidirect
154. C2×D37AGL(1,ℤ/8)semidirect
173. C2×AGL(1,ℤ/8)D37semidirect
184. C37#3055semidirect
190. C2×SD16C2×D37semidirect
191. C2×D8C2×D37semidirect
248. M16C22⋊D37semidirect
309. C8C2×C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D372
#145580402
C2×#119245072
#145577832
#145577762
#145577662
#145577592
#145575752
#145575382
#145575342
C2×AGL(1,ℤ/8)×C372
#305537