Finite Group Databasebeta

← #18004073

Subgroups — #18004073

Order 4736.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11924563C212yesmaximal, complemented
2#127×D37C211yesmaximal, characteristic, complemented
3C4×#10596966C211yesmaximal, characteristic, complemented
4#14557733C211yesmaximal, characteristic, complemented
5#14557731C212yesmaximal, complemented
6#14557729C211yesmaximal, characteristic, complemented
7#14557727C212yesmaximal, complemented
8#14557725C211yesmaximal, characteristic, complemented
9#14557555C211yesmaximal, characteristic, complemented
10#14557536C212yesmaximal, complemented
11C37×#460C211yesmaximal, characteristic, complemented
12C2×#10596966C2211yescharacteristic
13C2×D4×D37—21no—
14C2×D4×D37—21no—
15C2×D4×D37C2212yes—
16#11924563—28no—
17C2×#10596941C2212yescomplemented
18C2×D296C2212yescomplemented
19C2×#10596939C2212yescomplemented
20C2×#10596929C2212yes—
21#11924533C2211yescharacteristic
22#11924532—21no—
23#11924532—21no—
24#11924532C2212yes—
25Q8×Dic37C2211yescharacteristic, complemented
26C4×D148—21no—
27C4×D148C2211yescharacteristic, complemented
28#11924498—21no—
29#11924498C2211yescharacteristic, complemented
30#11924494—21no—
31#11924494C2211yescharacteristic
32C42×D37—21no—
33C42×D37C2211yescharacteristic
34C4⋊#64436C2211yescharacteristic, complemented
35#11924465C2212yescomplemented
36#11924458C2212yescomplemented
37#11924457C2212yescomplemented
38#11924456C2212yescomplemented
39C2×SD16×C37C2212yescomplemented
40#127×C37C2211yescharacteristic
41C4×Q8×C37C2211yescharacteristic, complemented
42C4⋊C8×C37C2211yescharacteristic, complemented
43#102×C37C2212yescomplemented
44C23×D37—22no—
45C2×Q8×C37C2311yescharacteristic
46C2×D4×C37—21no—
47C2×D4×C37—21no—
48C2×D4×C37C2312yes—
49C2×C22⋊D37—22no—
50C2×C22⋊D37—22no—
51C2×C22⋊D37—22no—
52#10596966—22no—
53#10596966D414yescomplemented
54D4×D37—24no—
55D4×D37—24no—
56D4×D37—44no—
57D4×D37—44no—
58C2×D148—21no—
59C2×D148—21no—
60C2×D148—21no—
61C2×D148C2312yes—
62C2×D148C2311yescharacteristic
63C2×C4×D37—21no—
64C2×C4×D37—21no—
65C2×C4×D37—21no—
66C2×C4×D37D411yescharacteristic
67C2×C4×D37D411yescharacteristic
68C2×C4×D37C2311yescharacteristic
69SD16×C37—24no—
70C2×C8×C37C2312yes—
71C4⋊C4×C37—21no—
72C4⋊C4×C37C2311yescharacteristic
73C42×C37—21no—
74C42×C37C2311yescharacteristic
75#10596941—24no—
76#10596939—24no—
77#10596938—21no—
78#10596938—21no—
79#10596938—21no—
80C4⋊Dic37—21no—
81C4⋊Dic37C2311yescharacteristic
82C4×Dic37—21no—
83C4×Dic37—21no—
84C4×Dic37D411yescharacteristic
85C4×Dic37D411yescharacteristic
86C4×Dic37C2311yescharacteristic
87C2×#64436C2312yes—
88D296—24no—
89#10596929—24no—
90C23×C37—22no—
91C22×D37—21no—
92C22×D37—22no—
93C22×D37—44no—
94C22×D37—44no—
95C22×D37—44no—
96C22×D37C2×D411yescharacteristic
97Q8×C37—21no—
98Q8×C37C2×D412yescomplemented
99D4×C37—22no—
100D4×C37—22no—
101D4×C37—42no—
102D4×C37—42no—
103C2×C4×C37—21no—
104C2×C4×C37—21no—
105C2×C4×C37C2×D411yescharacteristic
106C2×C4×C37C2×D411yescharacteristic
107C2×C4×C37C2411yescharacteristic
108C22⋊D37—44no—
109C22⋊D37—44no—
110C22⋊D37—44no—
111C2×Dic37—21no—
112C2×Dic37C2×D411yescharacteristic
113C2×Dic37C2×D411yescharacteristic
114C2×Dic37C2×D411yescharacteristic
115D148—24no—
116D148—22no—
117D148—21no—
118D148—22no—
119D148—42no—
120D148—42no—
121D148C2×D412yescomplemented
122C4×D37—22no—
123C4×D37—24no—
124C4×D37—22no—
125C4×D37—42no—
126C4×D37—42no—
127C4×D37C2×D412yes—
128C4×D37#4112yes—
129C8×C37—22no—
130#64436—22no—
131C22×C37—42no—
132C22×C37—42no—
133C22×C37—42no—
134C22×C37C22×D411yescharacteristic
135C2×D37—21no—
136C2×D37—21no—
137C2×D37—22no—
138C2×D37—41no—
139C2×D37—41no—
140C2×D37—42no—
141C2×D37—42no—
142C2×D37—42no—
143C2×D37—88no—
144C2×D37#12112yes—
145C4×C37—22no—
146C4×C37—21no—
147C4×C37—41no—
148C4×C37AGL(1,ℤ/8)12yescomplemented
149C4×C37C22×D411yescharacteristic
150C4×C37C2×#4111yescharacteristic
151Dic37—21no—
152Dic37—41no—
153Dic37#12112yes—
154Dic37AGL(1,ℤ/8)12yescomplemented
155#3038—371nomaximal, Sylow
156C2×C37—82no—
157C2×C37C2×#12111yescharacteristic
158C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
160D37—22no—
161D37—42no—
162D37—82no—
163C2×AGL(1,ℤ/8)—372no—
164C2×#127—371no—
165C4×#41—371no—
166#460—372no—
167#460—371no—
168#460D3711yescharacteristic, complemented
169#459—371no—
170#459—371no—
171#459—372no—
172#423—371no—
173#418—372no—
174C37#303811yescharacteristic, complemented, Sylow
175C2×#41—371no—
176C22×D4—372no—
177C22×D4—741no—
178C22×D4—741no—
179AGL(1,ℤ/8)—748no—
180C2×SD16—372no—
181C2×SD16C2×D3712yescomplemented
182C2×D8—372no—
183C2×D8—372no—
184C2×M16—372no—
185#127—372no—
186#127—371no—
187#127—741no—
188#127—741no—
189#127C2×D3711yescharacteristic
190C4×Q8—371no—
191C4×Q8C2×D3711yescharacteristic, complemented
192C4×D4—371no—
193C4×D4—371no—
194C4×D4—741no—
195C4×D4—741no—
196#117—371no—
197#117—741no—
198C2×C42—371no—
199C2×C42—741no—
200C4⋊C8—372no—
201C4⋊C8—371no—
202C4⋊C8C2×D3711yescharacteristic, complemented
203#102—372no—
204#102—372no—
205#102—372no—
206#102C2×D3712yescomplemented
207C24—742no—
208#41—374no—
209#41—742no—
210C2×Q8C22×D3711yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—371no—
214C2×D4—372no—
215C2×D4—741no—
216C2×D4—744no—
217C2×D4—744no—
218C2×D4—742no—
219C2×D4—742no—
220C2×D4—742no—
221C2×D4—741no—
222C2×D4—741no—
223C2×D4—1484no—
224C2×D4—1484no—
225C2×D4C22×D3712yes—
226C22×C4—371no—
227C22×C4—371no—
228C22×C4—371no—
229C22×C4—741no—
230C22×C4—741no—
231C22×C4—741no—
232SD16—24no—
233SD16—744no—
234D8—744no—
235D8—744no—
236M16—744no—
237C2×C8—372no—
238C2×C8C22×D3712yes—
239C4⋊C4—21no—
240C4⋊C4—371no—
241C4⋊C4—741no—
242C4⋊C4C22×D3711yescharacteristic
243C22⋊C4—741no—
244C22⋊C4—741no—
245C22⋊C4—741no—
246C42—21no—
247C42—371no—
248C42—371no—
249C42—371no—
250C42—741no—
251C42—741no—
252C42C22×D3711yescharacteristic
253C23—22no—
254C23—371no—
255C23—741no—
256C23—742no—
257C23—1484no—
258C23—1484no—
259C23—1484no—
260Q8—21no—
261Q8D4×D3712yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—372no—
267D4—744no—
268D4—742no—
269D4—741no—
270D4—742no—
271D4—1482no—
272D4—1484no—
273D4—1484no—
274D4—1484no—
275D4—1482no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—371no—
279C2×C4—371no—
280C2×C4—372no—
281C2×C4—372no—
282C2×C4—371no—
283C2×C4—742no—
284C2×C4—744no—
285C2×C4—742no—
286C2×C4—741no—
287C2×C4—1482no—
288C2×C4—1482no—
289C2×C4D4×D3711yescharacteristic
290C2×C4D4×D3711yescharacteristic
291C2×C4C23×D3711yescharacteristic
292C8—22no—
293C8—742no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—372no—
298C22—2968no—
299C22—741no—
300C22—741no—
301C22—742no—
302C22—1482no—
303C22—1482no—
304C22—1482no—
305C22—1481no—
306C22—1481no—
307C22C2×D4×D3711yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—372no—
312C4—372no—
313C4—741no—
314C4—1481no—
315C4#1192456312yescomplemented
316C4C2×D4×D3711yescharacteristic
317C4#41×D3711yescharacteristic
318C2—82no—
319C2—2962no—
320C2—742no—
321C2—1482no—
322C2#121×D3711yescharacteristic
323C2AGL(1,ℤ/8)×D3711yescharacteristic
324C2C2×#1192456311yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#460C2
socleC22×C37C22×D4
hypercenter#460D37
hypocenterC37#3038
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#11924563C2semidirect
2. #127×D37C2semidirect
3. C4×#10596966C2semidirect
4. #14557733C2semidirect
5. #14557731C2semidirect
6. #14557729C2semidirect
7. #14557727C2semidirect
8. #14557725C2semidirect
9. #14557555C2semidirect
10. #14557536C2semidirect
11. C37×#460C2semidirect
17. C2×#10596941C22semidirect
18. C2×D296C22semidirect
19. C2×#10596939C22semidirect
25. Q8×Dic37C22semidirect
27. C4×D148C22semidirect
29. #11924498C22semidirect
34. C4⋊#64436C22semidirect
35. #11924465C22semidirect
36. #11924458C22semidirect
37. #11924457C22semidirect
38. #11924456C22semidirect
39. C2×SD16×C37C22semidirect
41. C4×Q8×C37C22semidirect
42. C4⋊C8×C37C22semidirect
43. #102×C37C22semidirect
53. #10596966D4semidirect
98. Q8×C37C2×D4semidirect
121. D148C2×D4semidirect
148. C4×C37AGL(1,ℤ/8)semidirect
154. Dic37AGL(1,ℤ/8)semidirect
168. #460D37semidirect
174. C37#3038semidirect
181. C2×SD16C2×D37semidirect
191. C4×Q8C2×D37semidirect
202. C4⋊C8C2×D37semidirect
206. #102C2×D37semidirect
261. Q8D4×D37semidirect
315. C4#11924563semidirect

Maximal subgroups

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

subgroupindex
C2×#119245632
#127×D372
C4×#105969662
#145577332
#145577312
#145577292
#145577272
#145577252
#145575552
#145575362
C37×#4602
#303837