Finite Group Databasebeta

← #18004061

Subgroups — #18004061

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×#11924563C212yesmaximal, complemented
2#14558084C211yesmaximal, characteristic, complemented
3C4×D4×D37C211yesmaximal, characteristic, complemented
4#14557751C212yesmaximal, complemented
5#14557750C212yesmaximal, complemented
6#14557735C211yesmaximal, characteristic, complemented
7#14557729C211yesmaximal, characteristic, complemented
8#14557728C211yesmaximal, characteristic, complemented
9#14557555C211yesmaximal, characteristic, complemented
10#14557533C212yesmaximal, complemented
11C37×#461C211yesmaximal, characteristic, complemented
12C2×#10596966C2212yes—
13C2×D4×D37C2211yescharacteristic
14C22×C4×D37—21no—
15#11924563—28no—
16C2×#10596941C2212yes—
17C2×D296C2212yes—
18C2×#10596939C2212yes—
19C2×#10596929C2212yes—
20#11924533C2211yescharacteristic, complemented
21#11924525C2212yescomplemented
22#11924511—21no—
23#11924511—21no—
24C4×D148C2211yescharacteristic
25D4×Dic37C2211yescharacteristic
26#11924498—21no—
27#11924498—21no—
28#11924498C2211yescharacteristic
29C4×C22⋊D37—21no—
30#11924496—21no—
31C4⋊C4×D37C2211yescharacteristic
32C22⋊C4×D37—21no—
33C42×D37C2211yescharacteristic
34C4⋊#64436C2211yescharacteristic, complemented
35#11924465C2212yescomplemented
36#11924460C2212yescomplemented
37#11924458C2212yescomplemented
38#11924456C2212yescomplemented
39C2×SD16×C37C2212yes—
40C4⋊Q8×C37C2211yescharacteristic, complemented
41C4×D4×C37C2211yescharacteristic
42C4⋊C8×C37C2211yescharacteristic, complemented
43#103×C37C2212yescomplemented
44C23×D37—21no—
45C2×Q8×C37C2312yes—
46C2×D4×C37C2311yescharacteristic
47C22×C4×C37—21no—
48C2×C22⋊D37—21no—
49C22×Dic37—21no—
50#10596966—24no—
51#10596966—24no—
52D4×D37—22no—
53D4×D37D414yescomplemented
54C2×D148—22no—
55C2×D148—21no—
56C2×D148—21no—
57C2×D148C2312yes—
58C2×D148C2311yescharacteristic
59C2×C4×D37—21no—
60C2×C4×D37—21no—
61C2×C4×D37—21no—
62C2×C4×D37—22no—
63C2×C4×D37—44no—
64C2×C4×D37D411yescharacteristic
65C2×C4×D37D411yescharacteristic
66C2×C4×D37C2311yescharacteristic
67SD16×C37—24no—
68C2×C8×C37C2312yes—
69C4⋊C4×C37—21no—
70C4⋊C4×C37—21no—
71C4⋊C4×C37C2311yescharacteristic
72C22⋊C4×C37—21no—
73C42×C37C2311yescharacteristic
74C22⋊Dic37—21no—
75#10596941—24no—
76#10596939—24no—
77#10596938—21no—
78#10596938—21no—
79#10596938—22no—
80#10596938—22no—
81C4⋊Dic37C2311yescharacteristic
82#10596936—21no—
83C4×Dic37D411yescharacteristic
84C4×Dic37D411yescharacteristic
85C4×Dic37C2311yescharacteristic
86C2×#64436C2312yes—
87D296—24no—
88#10596929—24no—
89C23×C37—21no—
90C22×D37—21no—
91C22×D37—24no—
92C22×D37—22no—
93C22×D37—42no—
94C22×D37—42no—
95C22×D37C2×D411yescharacteristic
96Q8×C37—22no—
97Q8×C37—22no—
98D4×C37—21no—
99D4×C37C2×D412yes—
100C2×C4×C37—21no—
101C2×C4×C37—22no—
102C2×C4×C37—42no—
103C2×C4×C37C2×D411yescharacteristic
104C2×C4×C37C2×D411yescharacteristic
105C2×C4×C37C2411yescharacteristic
106C22⋊D37—24no—
107C2×Dic37—21no—
108C2×Dic37—42no—
109C2×Dic37C2×D411yescharacteristic
110C2×Dic37C2×D411yescharacteristic
111C2×Dic37C2×D411yescharacteristic
112D148—22no—
113D148—21no—
114D148—22no—
115D148—42no—
116D148—44no—
117D148—42no—
118D148C2×D412yes—
119C4×D37—22no—
120C4×D37—41no—
121C4×D37—42no—
122C4×D37—42no—
123C4×D37—42no—
124C4×D37—44no—
125C4×D37C2×D412yes—
126C4×D37#4112yes—
127C8×C37—22no—
128#64436—22no—
129C22×C37—22no—
130C22×C37—41no—
131C22×C37—41no—
132C22×C37C22×D411yescharacteristic
133C2×D37—21no—
134C2×D37—21no—
135C2×D37—22no—
136C2×D37—48no—
137C2×D37—41no—
138C2×D37—41no—
139C2×D37—42no—
140C2×D37—42no—
141C2×D37—42no—
142C2×D37#12112yes—
143C4×C37—21no—
144C4×C37—42no—
145C4×C37—41no—
146C4×C37AGL(1,ℤ/8)12yescomplemented
147C4×C37C22×D411yescharacteristic
148C4×C37C2×#4111yescharacteristic
149Dic37—21no—
150Dic37—21no—
151Dic37—41no—
152Dic37#12112yes—
153#3036—371nomaximal, Sylow
154C2×C37—42no—
155C2×C37C2×#12111yescharacteristic
156C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
157C2×C37#57511yescharacteristic
158D37—22no—
159D37—42no—
160D37—82no—
161C2×AGL(1,ℤ/8)—372no—
162#532—371no—
163C2×C4×D4—371no—
164#463—372no—
165#463—372no—
166#461—371no—
167#461D3711yescharacteristic, complemented
168#459—371no—
169#459—371no—
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×SD16C2×D3712yes—
179C2×D8—372no—
180C2×D8—372no—
181C2×M16—372no—
182C4⋊Q8C2×D3711yescharacteristic, complemented
183#127—371no—
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#103C2×D3712yescomplemented
203#102—372no—
204#102—372no—
205C24—741no—
206#41—744no—
207#41—744no—
208C2×Q8C22×D3712yes—
209C2×D4—374no—
210C2×D4—371no—
211C2×D4—372no—
212C2×D4—742no—
213C2×D4—742no—
214C2×D4—741no—
215C2×D4—741no—
216C2×D4—741no—
217C2×D4C22×D3711yescharacteristic
218C22×C4—21no—
219C22×C4—371no—
220C22×C4—371no—
221C22×C4—371no—
222C22×C4—741no—
223C22×C4—742no—
224C22×C4—741no—
225C22×C4—741no—
226C22×C4—741no—
227C22×C4—1484no—
228SD16—24no—
229SD16—744no—
230D8—744no—
231D8—744no—
232M16—744no—
233C2×C8—372no—
234C2×C8C22×D3712yes—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—371no—
238C4⋊C4—741no—
239C4⋊C4C22×D3711yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—741no—
242C22⋊C4—742no—
243C22⋊C4—742no—
244C22⋊C4—741no—
245C22⋊C4—741no—
246C42—371no—
247C42—371no—
248C42—371no—
249C42C22×D3711yescharacteristic
250C23—21no—
251C23—371no—
252C23—744no—
253C23—741no—
254C23—742no—
255C23—1482no—
256C23—1482no—
257Q8—22no—
258Q8—22no—
259D4—21no—
260D4—372no—
261D4—744no—
262D4—742no—
263D4—741no—
264D4—742no—
265D4—1482no—
266D4—1484no—
267D4—1482no—
268D4D4×D3712yes—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—371no—
273C2×C4—372no—
274C2×C4—372no—
275C2×C4—371no—
276C2×C4—371no—
277C2×C4—742no—
278C2×C4—741no—
279C2×C4—1481no—
280C2×C4—1484no—
281C2×C4—1482no—
282C2×C4—1482no—
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—372no—
294C22—741no—
295C22—741no—
296C22—742no—
297C22—1488no—
298C22—1482no—
299C22—1482no—
300C22—1482no—
301C22—1481no—
302C22—1481no—
303C22C2×D4×D3711yescharacteristic
304C4—21no—
305C4—42no—
306C4—41no—
307C4—372no—
308C4—741no—
309C4—741no—
310C4—1481no—
311C4#1192456312yescomplemented
312C4C2×D4×D3711yescharacteristic
313C4#41×D3711yescharacteristic
314C2—42no—
315C2—2962no—
316C2—742no—
317C2—1482no—
318C2#121×D3711yescharacteristic
319C2C2×#1192456311yescharacteristic
320C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#461C2
socleC22×C37C22×D4
hypercenter#461D37
hypocenterC37#3036
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#11924563C2semidirect
2. #14558084C2semidirect
3. C4×D4×D37C2semidirect
4. #14557751C2semidirect
5. #14557750C2semidirect
6. #14557735C2semidirect
7. #14557729C2semidirect
8. #14557728C2semidirect
9. #14557555C2semidirect
10. #14557533C2semidirect
11. C37×#461C2semidirect
20. #11924533C22semidirect
21. #11924525C22semidirect
34. C4⋊#64436C22semidirect
35. #11924465C22semidirect
36. #11924460C22semidirect
37. #11924458C22semidirect
38. #11924456C22semidirect
40. C4⋊Q8×C37C22semidirect
42. C4⋊C8×C37C22semidirect
43. #103×C37C22semidirect
53. D4×D37D4semidirect
146. C4×C37AGL(1,ℤ/8)semidirect
167. #461D37semidirect
172. C37#3036semidirect
182. C4⋊Q8C2×D37semidirect
200. C4⋊C8C2×D37semidirect
202. #103C2×D37semidirect
311. C4#11924563semidirect

Maximal subgroups

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

subgroupindex
C2×#119245632
#145580842
C4×D4×D372
#145577512
#145577502
#145577352
#145577292
#145577282
#145575552
#145575332
C37×#4612
#303637