Finite Group Databasebeta

← #18004227

Subgroups — #18004227

Order 4736.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 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×#11924569C211yesmaximal, characteristic, complemented
2C2×M16×D37C211yesmaximal, characteristic, complemented
3#14558045C211yesmaximal, characteristic
4C2×#11924516C211yesmaximal, characteristic, complemented
5#14557792C212yesmaximal, complemented
6#14557780C212yesmaximal, complemented
7#14557775C212yesmaximal, complemented
8#14557763C212yesmaximal, complemented
9#14557541C211yesmaximal, characteristic, complemented
10#14557532C211yesmaximal, characteristic, complemented
11C37×#428C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×Dic74C2211yescharacteristic
14C22×C4×D37C2211yescharacteristic
15#11924569—24no—
16C2×Dic148C2212yescomplemented
17C2×#10596930C2212yescomplemented
18C2×#10596934C2211yescharacteristic
19C2×#10596929C2212yes—
20M16×D37—24no—
21C2×C8×D37C2212yes—
22#11924520—21no—
23#11924516—21no—
24#11924516—21no—
25#11924516C2212yes—
26#11924516C2211yescharacteristic
27#11924516C2212yes—
28#11924516C2211yescharacteristic
29#11924511C2211yescharacteristic
30#11924511C2211yescharacteristic
31#11924506—21no—
32C4×D148—21no—
33C4×D148—21no—
34C2×C4⋊Dic37—21no—
35C2×#10596938—21no—
36#11924459C2212yescomplemented
37#11924459C2212yescomplemented
38#11924459C2212yescomplemented
39#11924456C2212yescomplemented
40C2×M16×C37C2211yescharacteristic
41#117×C37C2211yescharacteristic
42C2×C4⋊C4×C37C2211yescharacteristic
43#107×C37C2212yescomplemented
44#106×C37C2212yescomplemented
45C23×D37D411yescharacteristic
46C22×C4×C37—21no—
47C22×C4×C37C2311yescharacteristic
48C2×C22⋊D37—21no—
49C22×Dic37—21no—
50C22×Dic37D411yescharacteristic
51#10596962—22no—
52#10596962—22no—
53C2×D148—21no—
54C2×D148—21no—
55C2×D148C2311yescharacteristic
56C2×C4×D37—22no—
57C2×C4×D37—22no—
58C2×C4×D37—21no—
59C2×C4×D37D411yescharacteristic
60C2×C4×D37D411yescharacteristic
61C2×C4×D37C2311yescharacteristic
62C2×C4×D37C2311yescharacteristic
63C2×Dic74—22no—
64C2×Dic74—22no—
65C2×Dic74—21no—
66C2×Dic74C2312yes—
67C2×Dic74C2311yescharacteristic
68M16×C37D414yescomplemented
69C2×C8×C37C2312yes—
70C4⋊C4×C37—21no—
71C4⋊C4×C37C2311yescharacteristic
72C4⋊C4×C37C2311yescharacteristic
73C4⋊C4×C37C2312yes—
74C22⋊C4×C37—21no—
75C42×C37—21no—
76#10596938—22no—
77#10596938—22no—
78#10596938—21no—
79#10596938—21no—
80#10596938—21no—
81#10596938—21no—
82C4⋊Dic37—22no—
83C4⋊Dic37—22no—
84C4⋊Dic37—21no—
85C4⋊Dic37—21no—
86#10596934—22no—
87C2×#64436C2312yes—
88Dic148—24no—
89#10596930—24no—
90#10596929—24no—
91C8×D37—24no—
92C23×C37C2×D411yescharacteristic
93C22×D37—22no—
94C22×D37—22no—
95C22×D37—21no—
96C22×D37—41no—
97C22×D37C2×D411yescharacteristic
98C22×D37C2×D411yescharacteristic
99C2×C4×C37—22no—
100C2×C4×C37—21no—
101C2×C4×C37—21no—
102C2×C4×C37—42no—
103C2×C4×C37C2×D412yes—
104C2×C4×C37C2×D411yescharacteristic
105C2×C4×C37#4112yes—
106C2×C4×C37C2411yescharacteristic
107C22⋊D37—42no—
108C2×Dic37—22no—
109C2×Dic37—22no—
110C2×Dic37—21no—
111C2×Dic37—42no—
112C2×Dic37C2×D411yescharacteristic
113C2×Dic37C2×D411yescharacteristic
114D148—21no—
115D148—21no—
116D148—42no—
117D148—42no—
118C4×D37—22no—
119C4×D37—22no—
120C4×D37—21no—
121C4×D37—21no—
122C4×D37—21no—
123C4×D37—21no—
124C4×D37—42no—
125Dic74—22no—
126Dic74—21no—
127Dic74—22no—
128Dic74—21no—
129Dic74—42no—
130C8×C37C2×D414yes—
131#64436—22no—
132C22×C37—21no—
133C22×C37—21no—
134C22×C37#12112yes—
135C22×C37C22×D411yescharacteristic
136C2×D37—21no—
137C2×D37—21no—
138C2×D37—21no—
139C2×D37—21no—
140C2×D37—21no—
141C2×D37—44no—
142C2×D37—41no—
143C2×D37—41no—
144C2×D37—41no—
145C2×D37#13712yescomplemented
146C4×C37—41no—
147C4×C37—41no—
148C4×C37—42no—
149C4×C37#12112yes—
150C4×C37C22×D411yescharacteristic
151C4×C37C2×#4111yescharacteristic
152Dic37—21no—
153Dic37—21no—
154Dic37—42no—
155Dic37—41no—
156#3056—371nomaximal, Sylow
157C2×C37—22no—
158C2×C37C2×#12111yescharacteristic
159C2×C37C2×#13711yescharacteristic
160C2×C37#57511yescharacteristic
161D37—22no—
162D37—41no—
163D37—81no—
164C2×#137—371no—
165C22×M16—371no—
166#525—371no—
167C2×C22⋊Q8—371no—
168#470—372no—
169#468—372no—
170#467—372no—
171#465—372no—
172#428D3711yescharacteristic, complemented
173#419—371no—
174#417—371no—
175C37#305611yescharacteristic, complemented, Sylow
176C2×#41—371no—
177C22×Q8—371no—
178C23×C4—371no—
179#137—744no—
180C2×Q16—372no—
181C2×SD16—372no—
182C2×M16—371no—
183C2×M16—372no—
184C2×M16—744no—
185C2×M16C2×D3711yescharacteristic
186C22×C8—372no—
187#123—741no—
188C22⋊Q8—372no—
189C22⋊Q8—372no—
190C22⋊Q8—371no—
191C22⋊Q8—371no—
192C22⋊Q8—741no—
193C22⋊Q8—741no—
194#121—371no—
195#121—371no—
196#120—741no—
197C4×D4—741no—
198C4×D4—741no—
199#117C2×D3711yescharacteristic
200C2×C4⋊C4—741no—
201C2×C4⋊C4C2×D3711yescharacteristic
202C2×C22⋊C4—741no—
203#107C2×D3712yescomplemented
204#106C2×D3712yescomplemented
205#103—372no—
206#103—372no—
207#103—372no—
208#102—372no—
209C24—371no—
210#41—742no—
211#41—742no—
212C2×Q8—372no—
213C2×Q8—371no—
214C2×Q8—741no—
215C2×Q8—742no—
216C2×Q8—742no—
217C2×D4—371no—
218C2×D4—741no—
219C2×D4—741no—
220C2×D4—741no—
221C22×C4—21no—
222C22×C4—371no—
223C22×C4—371no—
224C22×C4—371no—
225C22×C4—371no—
226C22×C4—371no—
227C22×C4—742no—
228C22×C4—742no—
229C22×C4—741no—
230C22×C4—741no—
231C22×C4C22×D3711yescharacteristic
232Q16—744no—
233SD16—744no—
234M16—742no—
235M16—744no—
236M16D14814yescomplemented
237C2×C8—372no—
238C2×C8—744no—
239C2×C8C22×D3712yes—
240C4⋊C4—21no—
241C4⋊C4—742no—
242C4⋊C4—742no—
243C4⋊C4—741no—
244C4⋊C4—741no—
245C4⋊C4C22×D3712yes—
246C4⋊C4C22×D3711yescharacteristic
247C4⋊C4C22×D3711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—741no—
250C22⋊C4—741no—
251C22⋊C4—742no—
252C22⋊C4—742no—
253C22⋊C4—741no—
254C22⋊C4—741no—
255C42—21no—
256C23—371no—
257C23—371no—
258C23—742no—
259C23—742no—
260C23—741no—
261C23—1481no—
262C23D4×D3711yescharacteristic
263Q8—742no—
264Q8—741no—
265Q8—742no—
266Q8—741no—
267Q8—1482no—
268D4—741no—
269D4—741no—
270D4—1482no—
271D4—1482no—
272D4—1482no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—371no—
278C2×C4—371no—
279C2×C4—742no—
280C2×C4—742no—
281C2×C4—742no—
282C2×C4—741no—
283C2×C4—741no—
284C2×C4—741no—
285C2×C4—741no—
286C2×C4—741no—
287C2×C4—742no—
288C2×C4—1482no—
289C2×C4—1482no—
290C2×C4C2×D14812yes—
291C2×C4D4×D3711yescharacteristic
292C2×C4#1059696612yes—
293C2×C4C23×D3711yescharacteristic
294C8—742no—
295C8C2×D14814yes—
296C22—21no—
297C22—21no—
298C22—372no—
299C22—741no—
300C22—741no—
301C22—741no—
302C22—741no—
303C22—741no—
304C22—1484no—
305C22—1481no—
306C22—1481no—
307C22—1481no—
308C22#1192451112yes—
309C22C2×D4×D3711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—741no—
314C4—741no—
315C4—1481no—
316C4—1482no—
317C4#1192451112yes—
318C4C22×D14811yescharacteristic
319C4C2×#1059696611yescharacteristic
320C2—22no—
321C2—2961no—
322C2—742no—
323C2—1481no—
324C2C2×#1192451111yescharacteristic
325C2#137×D3711yescharacteristic
326C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#428C2
socleC22×C37C22×D4
prime radical#14558045C2
hypercenter#428D37
hypocenterC37#3056
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×#11924569C2semidirect
2. C2×M16×D37C2semidirect
4. C2×#11924516C2semidirect
5. #14557792C2semidirect
6. #14557780C2semidirect
7. #14557775C2semidirect
8. #14557763C2semidirect
9. #14557541C2semidirect
10. #14557532C2semidirect
11. C37×#428C2semidirect
16. C2×Dic148C22semidirect
17. C2×#10596930C22semidirect
36. #11924459C22semidirect
37. #11924459C22semidirect
38. #11924459C22semidirect
39. #11924456C22semidirect
43. #107×C37C22semidirect
44. #106×C37C22semidirect
68. M16×C37D4semidirect
145. C2×D37#137semidirect
172. #428D37semidirect
175. C37#3056semidirect
203. #107C2×D37semidirect
204. #106C2×D37semidirect
236. M16D148semidirect

Maximal subgroups

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

subgroupindex
C2×#119245692
C2×M16×D372
#145580452
C2×#119245162
#145577922
#145577802
#145577752
#145577632
#145575412
#145575322
C37×#4282
#305637