Finite Group Databasebeta

← #18004220

Subgroups — #18004220

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×M16×D37C211yesmaximal, characteristic, complemented
2#14558040C211yesmaximal, characteristic
3C2×#11924512C211yesmaximal, characteristic, complemented
4#14557788C212yesmaximal, complemented
5#14557776C212yesmaximal, complemented
6#14557771C212yesmaximal, complemented
7#14557759C212yesmaximal, complemented
8#14557575C211yesmaximal, characteristic, complemented
9#14557542C211yesmaximal, characteristic, complemented
10#14557534C211yesmaximal, characteristic, complemented
11C2×#137×C37C211yesmaximal, characteristic, complemented
12C22×C4×D37C2211yescharacteristic
13C2×#10596934C2211yescharacteristic
14C2×#10596929C2212yes—
15M16×D37—24no—
16C2×C8×D37C2212yes—
17#11924517—21no—
18#11924512—21no—
19#11924512—21no—
20#11924512C2212yes—
21#11924512C2211yescharacteristic
22#11924512C2212yes—
23#11924512C2211yescharacteristic
24#11924507C2211yescharacteristic
25#11924507C2211yescharacteristic
26#11924504—21no—
27C4×C22⋊D37—21no—
28C4×C22⋊D37—21no—
29#11924495C2211yescharacteristic
30C2×C4⋊Dic37C2211yescharacteristic
31C2×#10596936—21no—
32C2×#10596938—21no—
33#11924470C2212yescomplemented
34#11924468C2212yescomplemented
35#11924460C2212yescomplemented
36#11924460C2212yescomplemented
37#11924460C2212yescomplemented
38#11924457C2212yescomplemented
39C2×#41×C37C2211yescharacteristic
40C22×Q8×C37C2211yescharacteristic
41#137×C37—24no—
42C2×Q16×C37C2212yescomplemented
43C2×SD16×C37C2212yescomplemented
44C2×M16×C37C2211yescharacteristic
45C23×D37D411yescharacteristic
46#41×C37—22no—
47#41×C37—22no—
48C2×Q8×C37—21no—
49C2×Q8×C37—22no—
50C2×Q8×C37—22no—
51C2×Q8×C37C2312yes—
52C2×Q8×C37C2311yescharacteristic
53C2×D4×C37—21no—
54C2×D4×C37C2311yescharacteristic
55C22×C4×C37—21no—
56C22×C4×C37—21no—
57C22×C4×C37C2311yescharacteristic
58C2×C22⋊D37—21no—
59C2×C22⋊D37—21no—
60C22×Dic37—21no—
61C22×Dic37D411yescharacteristic
62C2×C4×D37—22no—
63C2×C4×D37—22no—
64C2×C4×D37D411yescharacteristic
65C2×C4×D37D411yescharacteristic
66C2×C4×D37C2311yescharacteristic
67C2×C4×D37C2311yescharacteristic
68Q16×C37—24no—
69SD16×C37—24no—
70M16×C37D414yescomplemented
71C2×C8×C37C2312yes—
72C22⋊Dic37—21no—
73C22⋊Dic37—21no—
74C22⋊Dic37—21no—
75#10596938—22no—
76#10596938—22no—
77#10596938—21no—
78#10596938—21no—
79C4⋊Dic37—21no—
80C4⋊Dic37C2312yes—
81C4⋊Dic37C2311yescharacteristic
82C4⋊Dic37C2311yescharacteristic
83#10596936—22no—
84#10596936—22no—
85#10596936—21no—
86#10596936—21no—
87C4×Dic37—21no—
88#10596934—22no—
89C2×#64436C2312yes—
90#10596929—24no—
91C8×D37—24no—
92C23×C37—21no—
93C23×C37C2×D411yescharacteristic
94C22×D37—22no—
95C22×D37—22no—
96C22×D37—41no—
97C22×D37C2×D411yescharacteristic
98C22×D37C2×D411yescharacteristic
99Q8×C37—22no—
100Q8×C37—21no—
101Q8×C37—22no—
102Q8×C37—21no—
103Q8×C37—42no—
104D4×C37—21no—
105D4×C37—21no—
106D4×C37—42no—
107C2×C4×C37—21no—
108C2×C4×C37—22no—
109C2×C4×C37—42no—
110C2×C4×C37—42no—
111C2×C4×C37C2×D411yescharacteristic
112C2×C4×C37C2×D412yes—
113C2×C4×C37#4112yes—
114C2×C4×C37C2411yescharacteristic
115C22⋊D37—42no—
116C22⋊D37—42no—
117C2×Dic37—22no—
118C2×Dic37—21no—
119C2×Dic37—21no—
120C2×Dic37—22no—
121C2×Dic37—42no—
122C2×Dic37C2×D411yescharacteristic
123C2×Dic37C2×D411yescharacteristic
124C4×D37—22no—
125C4×D37—22no—
126C4×D37—21no—
127C4×D37—21no—
128C4×D37—21no—
129C4×D37—21no—
130C8×C37C2×D414yes—
131#64436—22no—
132C22×C37—21no—
133C22×C37—21no—
134C22×C37—41no—
135C22×C37—41no—
136C22×C37—41no—
137C22×C37#12112yes—
138C22×C37C22×D411yescharacteristic
139C2×D37—21no—
140C2×D37—21no—
141C2×D37—21no—
142C2×D37—21no—
143C2×D37—21no—
144C2×D37—44no—
145C2×D37#13712yescomplemented
146C4×C37—41no—
147C4×C37—42no—
148C4×C37#12112yes—
149C4×C37C22×D411yescharacteristic
150C4×C37C2×#4111yescharacteristic
151Dic37—21no—
152Dic37—21no—
153Dic37—41no—
154Dic37—42no—
155Dic37—41no—
156#3056—371nomaximal, Sylow
157C2×C37—22no—
158C2×C37—81no—
159C2×C37C2×#12111yescharacteristic
160C2×C37C2×#13711yescharacteristic
161C2×C37#57511yescharacteristic
162D37—22no—
163D37—41no—
164C2×#137D3711yescharacteristic, complemented
165C22×M16—371no—
166#525—371no—
167C2×C22⋊Q8—371no—
168#470—372no—
169#468—372no—
170#467—372no—
171#465—372no—
172#428—371no—
173#419—371no—
174#417—371no—
175C37#305611yescharacteristic, complemented, Sylow
176C2×#41C2×D3711yescharacteristic
177C22×Q8C2×D3711yescharacteristic
178C23×C4—371no—
179#137—24no—
180C2×Q16C2×D3712yescomplemented
181C2×SD16C2×D3712yescomplemented
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#117—371no—
200C2×C4⋊C4—371no—
201C2×C4⋊C4—741no—
202C2×C22⋊C4—741no—
203#107—372no—
204#106—372no—
205#103—372no—
206#103—372no—
207#103—372no—
208#102—372no—
209C24—371no—
210#41—22no—
211#41—22no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C22×D3712yes—
216C2×Q8C22×D3711yescharacteristic
217C2×D4—21no—
218C2×D4—741no—
219C2×D4—741no—
220C2×D4C22×D3711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—371no—
224C22×C4—371no—
225C22×C4—371no—
226C22×C4—371no—
227C22×C4—371no—
228C22×C4—742no—
229C22×C4—742no—
230C22×C4—741no—
231C22×C4C22×D3711yescharacteristic
232Q16—24no—
233SD16—24no—
234M16—742no—
235M16—744no—
236M16C22⋊D3714yescomplemented
237C2×C8—372no—
238C2×C8—744no—
239C2×C8C22×D3712yes—
240C4⋊C4—372no—
241C4⋊C4—371no—
242C4⋊C4—371no—
243C4⋊C4—742no—
244C4⋊C4—742no—
245C4⋊C4—741no—
246C4⋊C4—741no—
247C4⋊C4—741no—
248C22⋊C4—742no—
249C22⋊C4—742no—
250C22⋊C4—741no—
251C22⋊C4—741no—
252C22⋊C4—741no—
253C22⋊C4—741no—
254C22⋊C4—741no—
255C42—741no—
256C23—21no—
257C23—371no—
258C23—371no—
259C23—742no—
260C23—742no—
261C23—1481no—
262C23D4×D3711yescharacteristic
263Q8—22no—
264Q8—21no—
265Q8—22no—
266Q8—21no—
267Q8—42no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—1482no—
272D4—1482no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
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—741no—
288C2×C4—742no—
289C2×C4—1482no—
290C2×C4D4×D3711yescharacteristic
291C2×C4#1059696412yes—
292C2×C4C2×C22⋊D3712yes—
293C2×C4C23×D3711yescharacteristic
294C8—742no—
295C8C2×C22⋊D3714yes—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—372no—
302C22—741no—
303C22—741no—
304C22—741no—
305C22—741no—
306C22—741no—
307C22—1484no—
308C22#1192450712yes—
309C22C2×D4×D3711yescharacteristic
310C4—41no—
311C4—42no—
312C4—741no—
313C4—741no—
314C4—1481no—
315C4—1482no—
316C4—1481no—
317C4#1192450712yes—
318C4C22×C22⋊D3711yescharacteristic
319C4C2×#1059696411yescharacteristic
320C2—22no—
321C2—81no—
322C2—742no—
323C2—1481no—
324C2C2×#1192450711yescharacteristic
325C2#137×D3711yescharacteristic
326C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×#137×C37C2
socleC22×C37C22×D4
prime radical#14558040C2
hypercenterC2×#137D37
hypocenterC37#3056
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×M16×D37C2semidirect
3. C2×#11924512C2semidirect
4. #14557788C2semidirect
5. #14557776C2semidirect
6. #14557771C2semidirect
7. #14557759C2semidirect
8. #14557575C2semidirect
9. #14557542C2semidirect
10. #14557534C2semidirect
11. C2×#137×C37C2semidirect
33. #11924470C22semidirect
34. #11924468C22semidirect
35. #11924460C22semidirect
36. #11924460C22semidirect
37. #11924460C22semidirect
38. #11924457C22semidirect
42. C2×Q16×C37C22semidirect
43. C2×SD16×C37C22semidirect
70. M16×C37D4semidirect
145. C2×D37#137semidirect
164. C2×#137D37semidirect
175. C37#3056semidirect
180. C2×Q16C2×D37semidirect
181. C2×SD16C2×D37semidirect
236. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D372
#145580402
C2×#119245122
#145577882
#145577762
#145577712
#145577592
#145575752
#145575422
#145575342
C2×#137×C372
#305637