Finite Group Databasebeta

← #18004213

Subgroups — #18004213

Order 4736.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#11924562C211yesmaximal, characteristic, complemented
2C2×M16×D37C211yesmaximal, characteristic, complemented
3#14558045C211yesmaximal, characteristic, complemented
4C2×#11924507C211yesmaximal, characteristic, complemented
5#14557785C212yesmaximal, complemented
6#14557778C212yesmaximal, complemented
7#14557768C212yesmaximal, complemented
8#14557761C212yesmaximal, complemented
9#14557576C211yesmaximal, characteristic, complemented
10#14557535C211yesmaximal, characteristic, complemented
11C37×#418C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×C22⋊D37—21no—
14C22×C4×D37C2211yescharacteristic
15#11924562—24no—
16C2×#10596940C2212yescomplemented
17C2×#10596939C2212yescomplemented
18C2×#10596934C2211yescharacteristic
19C2×#10596929C2212yescomplemented
20M16×D37—24no—
21C2×C8×D37C2212yescomplemented
22#11924520—21no—
23#11924516C2211yescharacteristic, complemented
24#11924516C2211yescharacteristic, complemented
25#11924511C2211yescharacteristic, complemented
26#11924511C2211yescharacteristic, complemented
27#11924507—21no—
28#11924507—21no—
29#11924507C2212yes—
30#11924507C2212yes—
31#11924506—21no—
32C4×D148—21no—
33C4×D148—21no—
34C2×C4⋊Dic37C2211yescharacteristic
35C2×C22⋊Dic37—21no—
36#11924469C2212yescomplemented
37#11924467C2212yescomplemented
38#11924461C2212yescomplemented
39#11924458C2212yescomplemented
40C22×D4×C37C2211yescharacteristic
41C2×M16×C37C2211yescharacteristic
42#117×C37C2211yescharacteristic
43#102×C37C2212yescomplemented
44#102×C37C2212yescomplemented
45C24×C37—21no—
46C23×D37D411yescharacteristic
47C2×D4×C37—22no—
48C2×D4×C37—21no—
49C2×D4×C37—22no—
50C2×D4×C37C2312yes—
51C22×C4×C37C2311yescharacteristic
52C2×C22⋊D37—21no—
53C2×C22⋊D37—22no—
54C2×C22⋊D37—22no—
55C2×C22⋊D37—44no—
56C22×Dic37—21no—
57C22×Dic37D411yescharacteristic
58#10596962—22no—
59#10596962—22no—
60C2×D148—21no—
61C2×D148—21no—
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×C37—22no—
72C2×C8×C37C2312yes—
73C4⋊C4×C37C2311yescharacteristic
74C4⋊C4×C37C2311yescharacteristic
75C22⋊C4×C37—21no—
76C42×C37—21no—
77C22⋊Dic37—22no—
78C22⋊Dic37—22no—
79#10596940—24no—
80#10596939—24no—
81#10596938—21no—
82#10596938—21no—
83#10596938—21no—
84#10596938—21no—
85C4⋊Dic37—21no—
86C4⋊Dic37—21no—
87C4⋊Dic37—21no—
88C4⋊Dic37C2312yes—
89#10596934D414yescomplemented
90C2×#64436C2312yes—
91#10596929—24no—
92C8×D37—24no—
93C23×C37—22no—
94C23×C37—42no—
95C23×C37—42no—
96C23×C37—42no—
97C23×C37C2×D411yescharacteristic
98C22×D37—22no—
99C22×D37—22no—
100C22×D37—21no—
101C22×D37—41no—
102C22×D37C2×D411yescharacteristic
103C22×D37C2×D411yescharacteristic
104D4×C37—22no—
105D4×C37—22no—
106D4×C37—42no—
107C2×C4×C37—21no—
108C2×C4×C37—21no—
109C2×C4×C37C2×D411yescharacteristic
110C2×C4×C37C2×D412yes—
111C2×C4×C37#4112yes—
112C2×C4×C37C2411yescharacteristic
113C22⋊D37—44no—
114C22⋊D37—42no—
115C22⋊D37—44no—
116C2×Dic37—22no—
117C2×Dic37—21no—
118C2×Dic37—22no—
119C2×Dic37—42no—
120C2×Dic37C2×D411yescharacteristic
121C2×Dic37C2×D411yescharacteristic
122D148—21no—
123D148—21no—
124D148—42no—
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×C37—22no—
136#64436C2×D414yescomplemented
137C22×C37—21no—
138C22×C37—21no—
139C22×C37—42no—
140C22×C37—42no—
141C22×C37—42no—
142C22×C37—84no—
143C22×C37#12112yes—
144C22×C37C22×D411yescharacteristic
145C2×D37—21no—
146C2×D37—21no—
147C2×D37—21no—
148C2×D37—21no—
149C2×D37—21no—
150C2×D37—44no—
151C2×D37—41no—
152C2×D37—41no—
153C2×D37—41no—
154C2×D37AGL(1,ℤ/8)12yescomplemented
155C4×C37—41no—
156C4×C37—41no—
157C4×C37#12112yes—
158C4×C37C22×D411yescharacteristic
159C4×C37C2×#4111yescharacteristic
160Dic37—21no—
161Dic37—21no—
162Dic37—41no—
163Dic37—42no—
164#3055—371nomaximal, Sylow
165C2×C37—22no—
166C2×C37—82no—
167C2×C37C2×#12111yescharacteristic
168C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C37#57511yescharacteristic
170D37—22no—
171D37—41no—
172D37—81no—
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#428—371no—
182#418D3711yescharacteristic, complemented
183#417—371no—
184C37#305511yescharacteristic, complemented, Sylow
185C2×#41—371no—
186C22×D4—741no—
187C22×D4C2×D3711yescharacteristic
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⋊C4—371no—
211C2×C22⋊C4—741no—
212#107—372no—
213#106—372no—
214#103—372no—
215#102—372no—
216#102C2×D3712yescomplemented
217#102C2×D3712yescomplemented
218C24—21no—
219C24—371no—
220#41—742no—
221#41—742no—
222C2×Q8—371no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—371no—
227C2×D4—741no—
228C2×D4—741no—
229C2×D4—742no—
230C2×D4—742no—
231C2×D4—741no—
232C2×D4—1484no—
233C2×D4C22×D3712yes—
234C22×C4—371no—
235C22×C4—371no—
236C22×C4—371no—
237C22×C4—371no—
238C22×C4—371no—
239C22×C4—742no—
240C22×C4—742no—
241C22×C4—741no—
242C22×C4—741no—
243C22×C4C22×D3711yescharacteristic
244SD16—744no—
245D8—744no—
246M16—22no—
247M16—374no—
248M16—744no—
249C2×C8—372no—
250C2×C8—744no—
251C2×C8C22×D3712yes—
252C4⋊C4—372no—
253C4⋊C4—741no—
254C4⋊C4—741no—
255C4⋊C4—741no—
256C4⋊C4C22×D3711yescharacteristic
257C4⋊C4C22×D3711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—741no—
260C22⋊C4—741no—
261C22⋊C4—741no—
262C22⋊C4—741no—
263C22⋊C4—742no—
264C22⋊C4—742no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—371no—
271C23—371no—
272C23—742no—
273C23—742no—
274C23—741no—
275C23—1481no—
276C23D4×D3711yescharacteristic
277Q8—741no—
278Q8—741no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—741no—
283D4—741no—
284D4—1482no—
285D4—1482no—
286D4—1484no—
287D4—1484no—
288D4—1482no—
289C2×C4—21no—
290C2×C4—21no—
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—742no—
301C2×C4—741no—
302C2×C4—1482no—
303C2×C4—1482no—
304C2×C4#1059696212yes—
305C2×C4D4×D3711yescharacteristic
306C2×C4D4×D3712yes—
307C2×C4C23×D3711yescharacteristic
308C8—22no—
309C8—374no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—372no—
317C22—741no—
318C22—741no—
319C22—741no—
320C22—741no—
321C22—741no—
322C22—1484no—
323C22—1481no—
324C22—1481no—
325C22—1481no—
326C22#1192450912yes—
327C22C2×D4×D3711yescharacteristic
328C4—41no—
329C4—41no—
330C4—741no—
331C4—741no—
332C4—1481no—
333C4—1482no—
334C4#1192450912yes—
335C4C2×D4×D3711yescharacteristic
336C4C2×#1059696211yescharacteristic
337C2—22no—
338C2—82no—
339C2—2961no—
340C2—742no—
341C2—1481no—
342C2C2×#1192450911yescharacteristic
343C2AGL(1,ℤ/8)×D3711yescharacteristic
344C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#418C2
socleC22×C37C22×D4
hypercenter#418D37
hypocenterC37#3055
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
1. C2×#11924562C2semidirect
2. C2×M16×D37C2semidirect
3. #14558045C2semidirect
4. C2×#11924507C2semidirect
5. #14557785C2semidirect
6. #14557778C2semidirect
7. #14557768C2semidirect
8. #14557761C2semidirect
9. #14557576C2semidirect
10. #14557535C2semidirect
11. C37×#418C2semidirect
16. C2×#10596940C22semidirect
17. C2×#10596939C22semidirect
19. C2×#10596929C22semidirect
21. C2×C8×D37C22semidirect
23. #11924516C22semidirect
24. #11924516C22semidirect
25. #11924511C22semidirect
26. #11924511C22semidirect
36. #11924469C22semidirect
37. #11924467C22semidirect
38. #11924461C22semidirect
39. #11924458C22semidirect
43. #102×C37C22semidirect
44. #102×C37C22semidirect
89. #10596934D4semidirect
136. #64436C2×D4semidirect
154. C2×D37AGL(1,ℤ/8)semidirect
182. #418D37semidirect
184. C37#3055semidirect
216. #102C2×D37semidirect
217. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245622
C2×M16×D372
#145580452
C2×#119245072
#145577852
#145577782
#145577682
#145577612
#145575762
#145575352
C37×#4182
#305537