Finite Group Databasebeta

← #18004055

Subgroups — #18004055

Order 4736.

548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 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×#11924560C212yesmaximal, complemented
2#14558081C211yesmaximal, characteristic, complemented
3C4×D4×D37C211yesmaximal, characteristic, complemented
4#14557747C212yesmaximal, complemented
5#14557746C212yesmaximal, complemented
6#14557739C211yesmaximal, characteristic, complemented
7#14557738C211yesmaximal, characteristic, complemented
8#14557725C211yesmaximal, characteristic, complemented
9#14557555C211yesmaximal, characteristic, complemented
10#14557531C212yesmaximal, complemented
11C37×#459C211yesmaximal, characteristic, complemented
12C2×#10596964C2212yes—
13C2×D4×D37C2211yescharacteristic
14C22×C4×D37—21no—
15#11924560—28no—
16C2×#10596930C2212yes—
17C2×#10596940C2212yes—
18C2×#10596939C2212yes—
19C2×#10596929C2212yes—
20#11924536C2211yescharacteristic, complemented
21#11924523C2212yescomplemented
22#11924507—21no—
23#11924507—21no—
24C4×D148C2211yescharacteristic
25D4×Dic37—21no—
26D4×Dic37—21no—
27D4×Dic37C2211yescharacteristic
28#11924498C2211yescharacteristic
29C4×C22⋊D37—21no—
30#11924496—21no—
31C4⋊C4×D37C2211yescharacteristic
32C22⋊C4×D37—21no—
33C42×D37C2211yescharacteristic
34C4⋊#64436C2211yescharacteristic, complemented
35#11924465C2212yescomplemented
36#11924461C2212yescomplemented
37#11924459C2212yescomplemented
38#11924457C2212yescomplemented
39C2×D8×C37C2212yes—
40#127×C37C2211yescharacteristic, complemented
41C4×D4×C37C2211yescharacteristic
42C4⋊C8×C37C2211yescharacteristic, complemented
43#102×C37C2212yescomplemented
44C23×D37—21no—
45C2×D4×C37—21no—
46C2×D4×C37—21no—
47C2×D4×C37C2312yes—
48C2×D4×C37C2311yescharacteristic
49C22×C4×C37—21no—
50C2×C22⋊D37—22no—
51C2×C22⋊D37—21no—
52C22×Dic37—21no—
53C22×Dic37—22no—
54#10596964—24no—
55#10596964—24no—
56D4×D37—22no—
57D4×D37D414yescomplemented
58C2×D148C2311yescharacteristic
59C2×C4×D37—21no—
60C2×C4×D37—21no—
61C2×C4×D37—21no—
62C2×C4×D37—44no—
63C2×C4×D37D411yescharacteristic
64C2×C4×D37D411yescharacteristic
65C2×C4×D37C2311yescharacteristic
66C2×Dic74C2312yes—
67D8×C37—24no—
68C2×C8×C37C2312yes—
69C4⋊C4×C37C2311yescharacteristic
70C22⋊C4×C37—21no—
71C42×C37C2311yescharacteristic
72C22⋊Dic37—21no—
73C22⋊Dic37—22no—
74C22⋊Dic37—22no—
75#10596940—24no—
76#10596939—24no—
77#10596938—21no—
78#10596938—21no—
79C4⋊Dic37—21no—
80C4⋊Dic37—21no—
81C4⋊Dic37C2311yescharacteristic
82#10596936—21no—
83C4×Dic37D411yescharacteristic
84C4×Dic37D411yescharacteristic
85C4×Dic37C2311yescharacteristic
86C2×#64436C2312yes—
87#10596930—24no—
88#10596929—24no—
89C23×C37—21no—
90C23×C37—22no—
91C22×D37—21no—
92C22×D37—24no—
93C22×D37—42no—
94C22×D37—42no—
95C22×D37C2×D411yescharacteristic
96D4×C37—22no—
97D4×C37—21no—
98D4×C37—22no—
99D4×C37—42no—
100D4×C37—42no—
101D4×C37C2×D412yes—
102C2×C4×C37—21no—
103C2×C4×C37—42no—
104C2×C4×C37C2×D411yescharacteristic
105C2×C4×C37C2×D411yescharacteristic
106C2×C4×C37C2411yescharacteristic
107C22⋊D37—24no—
108C22⋊D37—44no—
109C2×Dic37—21no—
110C2×Dic37—22no—
111C2×Dic37—42no—
112C2×Dic37—44no—
113C2×Dic37C2×D411yescharacteristic
114C2×Dic37C2×D411yescharacteristic
115C2×Dic37C2×D411yescharacteristic
116D148—21no—
117D148C2×D412yes—
118C4×D37—22no—
119C4×D37—41no—
120C4×D37—42no—
121C4×D37—42no—
122C4×D37—42no—
123C4×D37C2×D412yes—
124C4×D37#4112yes—
125Dic74—22no—
126Dic74—22no—
127C8×C37—22no—
128#64436—22no—
129C22×C37—22no—
130C22×C37—41no—
131C22×C37—41no—
132C22×C37—42no—
133C22×C37—42no—
134C22×C37—42no—
135C22×C37C22×D411yescharacteristic
136C2×D37—21no—
137C2×D37—21no—
138C2×D37—22no—
139C2×D37—48no—
140C2×D37—41no—
141C2×D37—41no—
142C2×D37#12112yes—
143C4×C37—21no—
144C4×C37—41no—
145C4×C37AGL(1,ℤ/8)12yescomplemented
146C4×C37C22×D411yescharacteristic
147C4×C37C2×#4111yescharacteristic
148Dic37—21no—
149Dic37—21no—
150Dic37—42no—
151Dic37—41no—
152Dic37#12112yes—
153#3036—371nomaximal, Sylow
154C2×C37—42no—
155C2×C37—82no—
156C2×C37C2×#12111yescharacteristic
157C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C37#57511yescharacteristic
159D37—22no—
160D37—42no—
161C2×AGL(1,ℤ/8)—372no—
162#532—371no—
163C2×C4×D4—371no—
164#463—372no—
165#463—372no—
166#461—371no—
167#461—371no—
168#459—371no—
169#459D3711yescharacteristic, complemented
170#423—371no—
171#417—372no—
172C37#303611yescharacteristic, complemented, Sylow
173C2×#41—372no—
174C22×D4—371no—
175C23×C4—741no—
176AGL(1,ℤ/8)—748no—
177C2×SD16—372no—
178C2×SD16—372no—
179C2×D8—372no—
180C2×D8C2×D3712yes—
181C2×M16—372no—
182C4⋊Q8—371no—
183#127C2×D3711yescharacteristic, complemented
184#124—372no—
185#121—741no—
186#121—741no—
187C4×D4—371no—
188C4×D4—371no—
189C4×D4—371no—
190C4×D4—741no—
191C4×D4—741no—
192C4×D4—741no—
193C4×D4—741no—
194C4×D4C2×D3711yescharacteristic
195C2×C4⋊C4—371no—
196C2×C22⋊C4—741no—
197C2×C42—371no—
198C4⋊C8—372no—
199C4⋊C8—371no—
200C4⋊C8C2×D3711yescharacteristic, complemented
201#103—372no—
202#103—372no—
203#102—372no—
204#102C2×D3712yescomplemented
205C24—741no—
206#41—744no—
207#41—744no—
208C2×Q8—372no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—374no—
212C2×D4—371no—
213C2×D4—742no—
214C2×D4—741no—
215C2×D4—742no—
216C2×D4C22×D3712yes—
217C2×D4C22×D3711yescharacteristic
218C22×C4—21no—
219C22×C4—371no—
220C22×C4—371no—
221C22×C4—371no—
222C22×C4—741no—
223C22×C4—741no—
224C22×C4—741no—
225C22×C4—742no—
226C22×C4—741no—
227C22×C4—1484no—
228SD16—744no—
229SD16—744no—
230D8—24no—
231D8—744no—
232M16—744no—
233C2×C8—372no—
234C2×C8C22×D3712yes—
235C4⋊C4—371no—
236C4⋊C4—741no—
237C4⋊C4—741no—
238C4⋊C4—741no—
239C4⋊C4C22×D3711yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—741no—
242C22⋊C4—741no—
243C22⋊C4—742no—
244C22⋊C4—742no—
245C22⋊C4—741no—
246C42—371no—
247C42—371no—
248C42—371no—
249C42C22×D3711yescharacteristic
250C23—21no—
251C23—22no—
252C23—371no—
253C23—744no—
254C23—741no—
255C23—1482no—
256C23—1482no—
257Q8—742no—
258Q8—742no—
259D4—22no—
260D4—21no—
261D4—22no—
262D4—42no—
263D4—42no—
264D4—372no—
265D4—744no—
266D4—741no—
267D4—1484no—
268D4D4×D3712yes—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—371no—
272C2×C4—372no—
273C2×C4—372no—
274C2×C4—371no—
275C2×C4—371no—
276C2×C4—742no—
277C2×C4—741no—
278C2×C4—742no—
279C2×C4—1481no—
280C2×C4—1482no—
281C2×C4—1482no—
282C2×C4—1484no—
283C2×C4—1482no—
284C2×C4—1482no—
285C2×C4D4×D3711yescharacteristic
286C2×C4D4×D3711yescharacteristic
287C2×C4C23×D3711yescharacteristic
288C8—22no—
289C8—742no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—372no—
297C22—741no—
298C22—741no—
299C22—742no—
300C22—1488no—
301C22—1481no—
302C22—1481no—
303C22C2×D4×D3711yescharacteristic
304C4—21no—
305C4—41no—
306C4—372no—
307C4—741no—
308C4—741no—
309C4—1482no—
310C4—1481no—
311C4#1192456012yescomplemented
312C4C2×D4×D3711yescharacteristic
313C4#41×D3711yescharacteristic
314C2—42no—
315C2—82no—
316C2—742no—
317C2—1482no—
318C2#121×D3711yescharacteristic
319C2C2×#1192456011yescharacteristic
320C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#459C2
socleC22×C37C22×D4
hypercenter#459D37
hypocenterC37#3036
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#11924560C2semidirect
2. #14558081C2semidirect
3. C4×D4×D37C2semidirect
4. #14557747C2semidirect
5. #14557746C2semidirect
6. #14557739C2semidirect
7. #14557738C2semidirect
8. #14557725C2semidirect
9. #14557555C2semidirect
10. #14557531C2semidirect
11. C37×#459C2semidirect
20. #11924536C22semidirect
21. #11924523C22semidirect
34. C4⋊#64436C22semidirect
35. #11924465C22semidirect
36. #11924461C22semidirect
37. #11924459C22semidirect
38. #11924457C22semidirect
40. #127×C37C22semidirect
42. C4⋊C8×C37C22semidirect
43. #102×C37C22semidirect
57. D4×D37D4semidirect
145. C4×C37AGL(1,ℤ/8)semidirect
169. #459D37semidirect
172. C37#3036semidirect
183. #127C2×D37semidirect
200. C4⋊C8C2×D37semidirect
204. #102C2×D37semidirect
311. C4#11924560semidirect

Maximal subgroups

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

subgroupindex
C2×#119245602
#145580812
C4×D4×D372
#145577472
#145577462
#145577392
#145577382
#145577252
#145575552
#145575312
C37×#4592
#303637