Finite Group Databasebeta

← #18004216

Subgroups — #18004216

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×#11924564C211yesmaximal, characteristic, complemented
2C2×M16×D37C211yesmaximal, characteristic, complemented
3#14558045C211yesmaximal, characteristic, complemented
4C2×#11924511C211yesmaximal, characteristic, complemented
5#14557787C212yesmaximal, complemented
6#14557780C212yesmaximal, complemented
7#14557770C212yesmaximal, complemented
8#14557763C212yesmaximal, complemented
9#14557537C211yesmaximal, characteristic, complemented
10#14557532C211yesmaximal, characteristic, complemented
11C37×#428C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×D148—21no—
14C22×D148C2211yescharacteristic
15C22×C4×D37C2211yescharacteristic
16#11924564—24no—
17C2×#10596930C2212yescomplemented
18C2×D296C2212yescomplemented
19C2×#10596934C2211yescharacteristic
20C2×#10596929C2212yescomplemented
21M16×D37—24no—
22C2×C8×D37C2212yescomplemented
23#11924520—21no—
24#11924516C2211yescharacteristic, complemented
25#11924516C2211yescharacteristic, complemented
26#11924511—21no—
27#11924511—21no—
28#11924511C2212yes—
29#11924511C2212yes—
30#11924511C2211yescharacteristic, complemented
31#11924511C2211yescharacteristic, complemented
32#11924506—21no—
33C4×D148—21no—
34C4×D148—21no—
35C2×#10596938—21no—
36#11924459C2212yescomplemented
37#11924456C2212yescomplemented
38#11924456C2212yescomplemented
39#11924456C2212yescomplemented
40C2×M16×C37C2211yescharacteristic
41#117×C37C2211yescharacteristic
42C2×C4⋊C4×C37C2211yescharacteristic
43#107×C37C2212yescomplemented
44#106×C37C2212yescomplemented
45C23×D37—21no—
46C23×D37D411yescharacteristic
47C22×C4×C37—21no—
48C22×C4×C37C2311yescharacteristic
49C2×C22⋊D37—21no—
50C22×Dic37D411yescharacteristic
51#10596962—22no—
52#10596962—22no—
53C2×D148—22no—
54C2×D148—22no—
55C2×D148—21no—
56C2×D148—21no—
57C2×D148—22no—
58C2×D148—22no—
59C2×D148—21no—
60C2×D148—44no—
61C2×D148C2312yes—
62C2×D148C2311yescharacteristic
63C2×C4×D37—22no—
64C2×C4×D37—22no—
65C2×C4×D37—21no—
66C2×C4×D37D411yescharacteristic
67C2×C4×D37D411yescharacteristic
68C2×C4×D37C2311yescharacteristic
69C2×C4×D37C2311yescharacteristic
70C2×Dic74C2311yescharacteristic
71M16×C37D414yescomplemented
72C2×C8×C37C2312yes—
73C4⋊C4×C37—21no—
74C4⋊C4×C37C2312yes—
75C4⋊C4×C37C2311yescharacteristic
76C4⋊C4×C37C2311yescharacteristic
77C22⋊C4×C37—21no—
78C42×C37—21no—
79#10596938—21no—
80#10596938—21no—
81#10596938—22no—
82#10596938—22no—
83#10596938—21no—
84#10596938—21no—
85C4⋊Dic37—21no—
86C4⋊Dic37—21no—
87#10596934—22no—
88C2×#64436C2312yes—
89D296—24no—
90#10596930—24no—
91#10596929—24no—
92C8×D37—24no—
93C23×C37C2×D411yescharacteristic
94C22×D37—22no—
95C22×D37—22no—
96C22×D37—21no—
97C22×D37—22no—
98C22×D37—41no—
99C22×D37—42no—
100C22×D37—42no—
101C22×D37—42no—
102C22×D37C2×D411yescharacteristic
103C22×D37C2×D411yescharacteristic
104C2×C4×C37—21no—
105C2×C4×C37—21no—
106C2×C4×C37—22no—
107C2×C4×C37—42no—
108C2×C4×C37C2×D412yes—
109C2×C4×C37C2×D411yescharacteristic
110C2×C4×C37#4112yes—
111C2×C4×C37C2411yescharacteristic
112C22⋊D37—42no—
113C2×Dic37—22no—
114C2×Dic37—21no—
115C2×Dic37C2×D411yescharacteristic
116C2×Dic37C2×D411yescharacteristic
117D148—22no—
118D148—21no—
119D148—22no—
120D148—21no—
121D148—42no—
122D148—44no—
123D148—42no—
124D148—44no—
125D148—42no—
126C4×D37—22no—
127C4×D37—22no—
128C4×D37—21no—
129C4×D37—21no—
130C4×D37—21no—
131C4×D37—21no—
132C4×D37—42no—
133Dic74—21no—
134Dic74—21no—
135C8×C37C2×D414yescomplemented
136#64436—22no—
137C22×C37—21no—
138C22×C37—21no—
139C22×C37#12112yes—
140C22×C37C22×D411yescharacteristic
141C2×D37—21no—
142C2×D37—21no—
143C2×D37—21no—
144C2×D37—21no—
145C2×D37—21no—
146C2×D37—44no—
147C2×D37—41no—
148C2×D37—42no—
149C2×D37—42no—
150C2×D37—41no—
151C2×D37—41no—
152C2×D37—42no—
153C2×D37—84no—
154C2×D37AGL(1,ℤ/8)12yescomplemented
155C4×C37—42no—
156C4×C37—41no—
157C4×C37—41no—
158C4×C37#12112yes—
159C4×C37C22×D411yescharacteristic
160C4×C37C2×#4111yescharacteristic
161Dic37—21no—
162Dic37—21no—
163Dic37—41no—
164#3055—371nomaximal, Sylow
165C2×C37—22no—
166C2×C37C2×#12111yescharacteristic
167C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C37#57511yescharacteristic
169D37—22no—
170D37—41no—
171D37—81no—
172D37—82no—
173C2×AGL(1,ℤ/8)—371no—
174C22×M16—371no—
175#525—371no—
176C2×#121—371no—
177#469—372no—
178#468—372no—
179#466—372no—
180#465—372no—
181#428D3711yescharacteristic, complemented
182#418—371no—
183#417—371no—
184C37#305511yescharacteristic, complemented, Sylow
185C2×#41—371no—
186C22×D4—371no—
187C22×D4—741no—
188C23×C4—371no—
189AGL(1,ℤ/8)—744no—
190C2×SD16—372no—
191C2×D8—372no—
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#117C2×D3711yescharacteristic
210C2×C4⋊C4C2×D3711yescharacteristic
211C2×C22⋊C4—741no—
212#107C2×D3712yescomplemented
213#106C2×D3712yescomplemented
214#103—372no—
215#102—372no—
216#102—372no—
217#102—372no—
218C24—371no—
219C24—741no—
220#41—742no—
221#41—742no—
222C2×Q8—371no—
223C2×D4—371no—
224C2×D4—372no—
225C2×D4—741no—
226C2×D4—741no—
227C2×D4—742no—
228C2×D4—742no—
229C2×D4—741no—
230C2×D4—742no—
231C2×D4—741no—
232C2×D4—742no—
233C2×D4—1484no—
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—744no—
245D8—744no—
246M16—742no—
247M16—744no—
248M16D14814yescomplemented
249C2×C8—372no—
250C2×C8—744no—
251C2×C8C22×D3712yes—
252C4⋊C4—21no—
253C4⋊C4—741no—
254C4⋊C4—741no—
255C4⋊C4C22×D3711yescharacteristic
256C4⋊C4C22×D3711yescharacteristic
257C4⋊C4C22×D3712yes—
258C22⋊C4—21no—
259C22⋊C4—742no—
260C22⋊C4—742no—
261C22⋊C4—741no—
262C22⋊C4—741no—
263C22⋊C4—741no—
264C22⋊C4—741no—
265C42—21no—
266C23—371no—
267C23—371no—
268C23—742no—
269C23—742no—
270C23—741no—
271C23—742no—
272C23—1482no—
273C23—1482no—
274C23—1481no—
275C23—1482no—
276C23D4×D3711yescharacteristic
277Q8—741no—
278Q8—741no—
279D4—741no—
280D4—742no—
281D4—742no—
282D4—741no—
283D4—1484no—
284D4—1482no—
285D4—1484no—
286D4—1482no—
287D4—1482no—
288D4—1482no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—371no—
294C2×C4—371no—
295C2×C4—742no—
296C2×C4—742no—
297C2×C4—742no—
298C2×C4—741no—
299C2×C4—741no—
300C2×C4—741no—
301C2×C4—741no—
302C2×C4—741no—
303C2×C4—1482no—
304C2×C4C2×D14812yes—
305C2×C4D4×D3711yescharacteristic
306C2×C4#1059696612yes—
307C2×C4C23×D3711yescharacteristic
308C8—742no—
309C8C2×D14814yescomplemented
310C22—21no—
311C22—21no—
312C22—372no—
313C22—2964no—
314C22—741no—
315C22—741no—
316C22—741no—
317C22—741no—
318C22—741no—
319C22—1484no—
320C22—1481no—
321C22—1482no—
322C22—1482no—
323C22—1481no—
324C22—1481no—
325C22—1482no—
326C22#1192451112yes—
327C22C2×D4×D3711yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—741no—
332C4—741no—
333C4—1481no—
334C4#1192451112yes—
335C4C22×D14811yescharacteristic
336C4C2×#1059696611yescharacteristic
337C2—22no—
338C2—2962no—
339C2—2961no—
340C2—742no—
341C2—1481no—
342C2C2×#1192451111yescharacteristic
343C2AGL(1,ℤ/8)×D3711yescharacteristic
344C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#428C2
socleC22×C37C22×D4
hypercenter#428D37
hypocenterC37#3055
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
1. C2×#11924564C2semidirect
2. C2×M16×D37C2semidirect
3. #14558045C2semidirect
4. C2×#11924511C2semidirect
5. #14557787C2semidirect
6. #14557780C2semidirect
7. #14557770C2semidirect
8. #14557763C2semidirect
9. #14557537C2semidirect
10. #14557532C2semidirect
11. C37×#428C2semidirect
17. C2×#10596930C22semidirect
18. C2×D296C22semidirect
20. C2×#10596929C22semidirect
22. C2×C8×D37C22semidirect
24. #11924516C22semidirect
25. #11924516C22semidirect
30. #11924511C22semidirect
31. #11924511C22semidirect
36. #11924459C22semidirect
37. #11924456C22semidirect
38. #11924456C22semidirect
39. #11924456C22semidirect
43. #107×C37C22semidirect
44. #106×C37C22semidirect
71. M16×C37D4semidirect
135. C8×C37C2×D4semidirect
154. C2×D37AGL(1,ℤ/8)semidirect
181. #428D37semidirect
184. C37#3055semidirect
212. #107C2×D37semidirect
213. #106C2×D37semidirect
248. M16D148semidirect
309. C8C2×D148semidirect

Maximal subgroups

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

subgroupindex
C2×#119245642
C2×M16×D372
#145580452
C2×#119245112
#145577872
#145577802
#145577702
#145577632
#145575372
#145575322
C37×#4282
#305537