Finite Group Databasebeta

← #18003864

Subgroups — #18003864

Order 4736.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11924561C211yesmaximal, characteristic, complemented
2C22×#10596939C211yesmaximal, characteristic, complemented
3#14558097C211yesmaximal, characteristic, complemented
4#14557678C214yesmaximal, complemented
5#14557667C214yesmaximal, complemented
6#14557538C211yesmaximal, characteristic, complemented
7#14557529C211yesmaximal, characteristic, complemented
8#14557500C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C37C211yesmaximal, characteristic, complemented
10C22×D148C2211yescharacteristic, complemented
11#11924561—24no—
12C2×#10596941C2212yescomplemented
13C2×#10596939—24no—
14C2×#10596939C2214yescomplemented
15C2×#10596939C2212yescomplemented
16C2×#10596934C2211yescharacteristic, complemented
17C22×#64436C2211yescharacteristic, complemented
18#11924532—21no—
19#11924525—21no—
20#11924510C2214yescomplemented
21#11924504—22no—
22#11924495C2211yescharacteristic, complemented
23#11924460C2212yescomplemented
24#11924457C2212yescomplemented
25#11924457C2214yescomplemented
26C22⋊#64436C2214yescomplemented
27H(5,𝔽2)×C37—24no—
28H(5,𝔽2)×C37—24no—
29C2×#41×C37—21no—
30C2×#41×C37C2211yescharacteristic
31C2×#41×C37C2211yescharacteristic
32C2×#41×C37C2212yescomplemented
33C22×D4×C37—21no—
34C22×D4×C37—22no—
35C22×D4×C37C2211yescharacteristic
36C22×D4×C37C2212yescomplemented
37C24×C37—21no—
38C24×C37—22no—
39C23×D37—21no—
40#41×C37—28no—
41#41×C37—24no—
42#41×C37—24no—
43#41×C37—22no—
44#41×C37—22no—
45#41×C37—22no—
46#41×C37D414yescomplemented
47C2×Q8×C37D411yescharacteristic
48C2×Q8×C37C2311yescharacteristic
49C2×D4×C37—216no—
50C2×D4×C37—28no—
51C2×D4×C37—21no—
52C2×D4×C37—22no—
53C2×D4×C37—24no—
54C2×D4×C37—24no—
55C2×D4×C37—21no—
56C2×D4×C37—22no—
57C2×D4×C37—22no—
58C2×D4×C37D412yes—
59C2×D4×C37D411yescharacteristic
60C2×D4×C37D414yescomplemented
61C2×D4×C37C2312yes—
62C2×D4×C37C2311yescharacteristic
63C22×C4×C37—21no—
64C22×C4×C37—22no—
65C22×C4×C37C2312yes—
66C22×C4×C37C2311yescharacteristic
67C2×C22⋊D37—24no—
68C2×D148—22no—
69C2×D148—22no—
70C2×D148—21no—
71C2×D148C2312yes—
72C22⋊Dic37—21no—
73#10596941—24no—
74#10596939—28no—
75#10596939—24no—
76#10596938—24no—
77C4⋊Dic37C2312yes—
78C4×Dic37—21no—
79#10596934—22no—
80C2×#64436—22no—
81C2×#64436C2312yes—
82C2×#64436C2312yes—
83C23×C37—28no—
84C23×C37—21no—
85C23×C37—22no—
86C23×C37—24no—
87C23×C37—82no—
88C23×C37—84no—
89C23×C37C2×D412yes—
90C23×C37C2×D411yescharacteristic
91C22×D37—22no—
92C22×D37—42no—
93C22×D37—82no—
94Q8×C37—22no—
95Q8×C37—21no—
96Q8×C37C2×D412yes—
97D4×C37—28no—
98D4×C37—28no—
99D4×C37—24no—
100D4×C37—24no—
101D4×C37—22no—
102D4×C37—24no—
103D4×C37—22no—
104D4×C37—21no—
105D4×C37C2×D412yes—
106D4×C37C2×D414yes—
107C2×C4×C37—28no—
108C2×C4×C37—21no—
109C2×C4×C37—24no—
110C2×C4×C37—22no—
111C2×C4×C37—22no—
112C2×C4×C37C2×D412yes—
113C2×C4×C37C2×D411yescharacteristic
114C2×C4×C37C2×D412yes—
115C2×C4×C37C2×D412yes—
116C2×C4×C37C2411yescharacteristic
117C22⋊D37—84no—
118C2×Dic37—22no—
119D148—22no—
120D148—22no—
121D148—42no—
122#64436—22no—
123#64436—22no—
124C22×C37—22no—
125C22×C37—22no—
126C22×C37—24no—
127C22×C37—22no—
128C22×C37—24no—
129C22×C37—44no—
130C22×C37—42no—
131C22×C37—84no—
132C22×C37—88no—
133C22×C37#12012yes—
134C22×C37C22×D411yescharacteristic
135C2×D37—42no—
136C2×D37—44no—
137C2×D37—84no—
138C4×C37—22no—
139C4×C37—22no—
140C4×C37#12012yes—
141C4×C37C22×D411yescharacteristic
142C4×C37C22×D411yescharacteristic
143Dic37—42no—
144#3008—371nomaximal, Sylow
145C2×C37—22no—
146C2×C37—42no—
147C2×C37—44no—
148C2×C37—42no—
149C2×C37C2×#12011yescharacteristic
150C2×C37#57612yes—
151D37—82no—
152C2×H(5,𝔽2)D3711yescharacteristic, complemented
153C2×AGL(1,ℤ/8)—371no—
154C22×D8—371no—
155#535—371no—
156#449—374no—
157#447—374no—
158#418—371no—
159#416—371no—
160#408—371no—
161C37#300811yescharacteristic, complemented, Sylow
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D3711yescharacteristic
166C2×#41C2×D3711yescharacteristic
167C2×#41C2×D3712yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—371no—
171C22×D4C2×D3711yescharacteristic
172C22×D4C2×D3712yescomplemented
173AGL(1,ℤ/8)—744no—
174C2×SD16—372no—
175C2×D8—374no—
176C2×D8—372no—
177C2×D8—744no—
178C2×M16—371no—
179C22×C8—371no—
180#127—741no—
181#124—741no—
182#121—374no—
183#120—742no—
184#117—371no—
185#103—372no—
186#102—372no—
187#102—374no—
188C22⋊C8—374no—
189C24—21no—
190C24—22no—
191C24—741no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D3714yescomplemented
199C2×Q8C22⋊D3711yescharacteristic
200C2×Q8C22×D3711yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—372no—
211C2×D4—744no—
212C2×D4—742no—
213C2×D4—742no—
214C2×D4—741no—
215C2×D4C22⋊D3712yes—
216C2×D4C22⋊D3711yescharacteristic
217C2×D4C22⋊D3714yescomplemented
218C2×D4C22×D3712yes—
219C2×D4C22×D3711yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D3712yes—
223C22×C4C22×D3711yescharacteristic
224SD16—744no—
225D8—748no—
226D8—744no—
227M16—742no—
228C2×C8—372no—
229C2×C8—372no—
230C2×C8—742no—
231C4⋊C4—372no—
232C22⋊C4—741no—
233C22⋊C4—744no—
234C42—741no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—2962no—
242C23—742no—
243C23—1482no—
244C23C2×C22⋊D3712yes—
245C23C2×C22⋊D3711yescharacteristic
246Q8—22no—
247Q8—21no—
248Q8C2×C22⋊D3712yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—24no—
255D4—22no—
256D4—21no—
257D4—2964no—
258D4—742no—
259D4—742no—
260D4—1482no—
261D4C2×C22⋊D3712yes—
262D4C2×C22⋊D3714yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—742no—
269C2×C4C2×C22⋊D3712yes—
270C2×C4C2×C22⋊D3711yescharacteristic
271C2×C4C2×C22⋊D3712yes—
272C2×C4C2×C22⋊D3712yes—
273C2×C4C23×D3711yescharacteristic
274C8—742no—
275C8—742no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—2964no—
286C22—1482no—
287C22—1484no—
288C22#1192450512yes—
289C22C22×C22⋊D3711yescharacteristic
290C4—22no—
291C4—22no—
292C4—1482no—
293C4#1192450512yes—
294C4C22×C22⋊D3711yescharacteristic
295C4C22×C22⋊D3711yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—2962no—
301C2C2×#1192450511yescharacteristic
302C2#1455833612yes—
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×H(5,𝔽2)×C37C2
socleC22×C37C22×D4
hypercenterC2×H(5,𝔽2)D37
hypocenterC37#3008
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#11924561C2semidirect
2. C22×#10596939C2semidirect
3. #14558097C2semidirect
4. #14557678C2semidirect
5. #14557667C2semidirect
6. #14557538C2semidirect
7. #14557529C2semidirect
8. #14557500C2semidirect
9. C2×H(5,𝔽2)×C37C2semidirect
10. C22×D148C22semidirect
12. C2×#10596941C22semidirect
14. C2×#10596939C22semidirect
15. C2×#10596939C22semidirect
16. C2×#10596934C22semidirect
17. C22×#64436C22semidirect
20. #11924510C22semidirect
22. #11924495C22semidirect
23. #11924460C22semidirect
24. #11924457C22semidirect
25. #11924457C22semidirect
26. C22⋊#64436C22semidirect
32. C2×#41×C37C22semidirect
36. C22×D4×C37C22semidirect
46. #41×C37D4semidirect
60. C2×D4×C37D4semidirect
152. C2×H(5,𝔽2)D37semidirect
161. C37#3008semidirect
167. C2×#41C2×D37semidirect
172. C22×D4C2×D37semidirect
198. #41C22⋊D37semidirect
217. C2×D4C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245612
C22×#105969392
#145580972
#145576782
#145576672
#145575382
#145575292
#145575002
C2×H(5,𝔽2)×C372
#300837