Finite Group Databasebeta

← #18003867

Subgroups — #18003867

Order 4736.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11924582C211yesmaximal, characteristic, complemented
2C2×#11924561C211yesmaximal, characteristic, complemented
3C22×D296C211yesmaximal, characteristic, complemented
4#14558102C211yesmaximal, characteristic, complemented
5#14557681C214yesmaximal, complemented
6#14557668C214yesmaximal, complemented
7#14557537C211yesmaximal, characteristic, complemented
8#14557499C211yesmaximal, characteristic, complemented
9C37×#416C211yesmaximal, characteristic, complemented
10#11924582—24no—
11#11924582—24no—
12C2×#10596966C2212yescomplemented
13C2×#10596962—21no—
14C2×#10596962C2211yescharacteristic
15C22×D148—21no—
16C22×D148C2211yescharacteristic
17C22×D148C2211yescharacteristic, complemented
18C2×D4×D37—22no—
19C2×D4×D37C2212yescomplemented
20#11924561—24no—
21C2×#10596941C2212yescomplemented
22C2×D296—24no—
23C2×D296C2214yescomplemented
24C2×#10596939C2212yescomplemented
25C2×#10596934C2211yescharacteristic, complemented
26#11924533—21no—
27#11924524—21no—
28#11924511C2214yescomplemented
29#11924506—22no—
30#11924456C2214yescomplemented
31#11924448C2214yescomplemented
32C2×#41×C37C2211yescharacteristic
33C22×C8×C37C2211yescharacteristic, complemented
34#117×C37C2211yescharacteristic, complemented
35#103×C37C2212yescomplemented
36#102×C37C2212yescomplemented
37C23×D37—21no—
38C23×D37—22no—
39C23×D37—21no—
40#41×C37—22no—
41#41×C37D414yescomplemented
42C2×Q8×C37C2311yescharacteristic
43C2×D4×C37—21no—
44C2×D4×C37C2311yescharacteristic
45C22×C4×C37—21no—
46C22×C4×C37C2311yescharacteristic
47C2×C22⋊D37—22no—
48C2×C22⋊D37D412yes—
49#10596966—24no—
50#10596966—24no—
51D4×D37—216no—
52D4×D37—24no—
53D4×D37—24no—
54#10596962—28no—
55#10596962—22no—
56#10596962—22no—
57C2×D148—28no—
58C2×D148—22no—
59C2×D148—21no—
60C2×D148—22no—
61C2×D148—22no—
62C2×D148—22no—
63C2×D148—21no—
64C2×D148—24no—
65C2×D148D411yescharacteristic
66C2×D148D414yescomplemented
67C2×D148C2312yes—
68C2×D148C2312yes—
69C2×C4×D37—22no—
70C2×C4×D37C2312yes—
71C2×Dic74D411yescharacteristic
72C2×C8×C37—22no—
73C2×C8×C37C2312yes—
74C4⋊C4×C37C2312yes—
75C22⋊C4×C37—21no—
76C42×C37—21no—
77#10596941—24no—
78#10596939—24no—
79#10596938—24no—
80#10596934—22no—
81C2×#64436C2312yes—
82D296—28no—
83C23×C37—21no—
84C23×C37C2×D411yescharacteristic
85C22×D37—22no—
86C22×D37—28no—
87C22×D37—24no—
88C22×D37—22no—
89C22×D37—42no—
90C22×D37—84no—
91C22×D37—82no—
92C22×D37—82no—
93C22×D37C2×D412yes—
94Q8×C37—21no—
95Q8×C37C2×D412yes—
96D4×C37—24no—
97D4×C37—21no—
98D4×C37C2×D412yes—
99C2×C4×C37—21no—
100C2×C4×C37—24no—
101C2×C4×C37—22no—
102C2×C4×C37C2×D412yes—
103C2×C4×C37C2×D411yescharacteristic
104C2×C4×C37C2×D412yes—
105C2×C4×C37C2411yescharacteristic
106C22⋊D37—28no—
107C22⋊D37—24no—
108C2×Dic37C2×D412yes—
109D148—28no—
110D148—22no—
111D148—24no—
112D148—22no—
113D148—22no—
114D148—22no—
115D148—42no—
116D148—84no—
117D148C2×D414yes—
118C4×D37—28no—
119C4×D37—22no—
120C4×D37—22no—
121Dic74—22no—
122C8×C37—22no—
123#64436—22no—
124C22×C37—22no—
125C22×C37—22no—
126C22×C37—42no—
127C22×C37#12012yes—
128C22×C37C22×D411yescharacteristic
129C2×D37—22no—
130C2×D37—24no—
131C2×D37—24no—
132C2×D37—44no—
133C2×D37—42no—
134C2×D37—44no—
135C2×D37—84no—
136C2×D37—88no—
137C2×D37—84no—
138C4×C37—22no—
139C4×C37—42no—
140C4×C37#12012yes—
141C4×C37C22×D411yescharacteristic
142C4×C37C22×D411yescharacteristic
143Dic37—22no—
144#3008—371nomaximal, Sylow
145C2×C37—22no—
146C2×C37—42no—
147C2×C37C2×#12011yescharacteristic
148C2×C37#57612yes—
149D37—42no—
150D37—44no—
151D37—82no—
152C2×H(5,𝔽2)—371no—
153C2×AGL(1,ℤ/8)—371no—
154C22×D8—371no—
155#535—371no—
156#449—374no—
157#447—374no—
158#418—371no—
159#416D3711yescharacteristic, complemented
160#408—371no—
161C37#300811yescharacteristic, complemented, Sylow
162H(5,𝔽2)—744no—
163H(5,𝔽2)—744no—
164C2×#41—371no—
165C2×#41—372no—
166C2×#41—741no—
167C2×#41C2×D3711yescharacteristic
168C22×D4—372no—
169C22×D4—371no—
170C22×D4—371no—
171C22×D4—741no—
172C22×D4—742no—
173AGL(1,ℤ/8)—744no—
174C2×SD16—372no—
175C2×D8—372no—
176C2×D8—374no—
177C2×D8—744no—
178C2×M16—371no—
179C22×C8C2×D3711yescharacteristic, complemented
180#127—741no—
181#124—741no—
182#121—374no—
183#120—742no—
184#117C2×D3711yescharacteristic, complemented
185#103C2×D3712yescomplemented
186#102—374no—
187#102C2×D3712yescomplemented
188C22⋊C8—374no—
189C24—742no—
190C24—741no—
191C24—741no—
192#41—22no—
193#41—748no—
194#41—744no—
195#41—744no—
196#41—742no—
197#41—742no—
198#41D14814yescomplemented
199C2×Q8—371no—
200C2×Q8C22×D3711yescharacteristic
201C2×D4—21no—
202C2×D4—372no—
203C2×D4—374no—
204C2×D4—371no—
205C2×D4—372no—
206C2×D4—372no—
207C2×D4—744no—
208C2×D4—742no—
209C2×D4—748no—
210C2×D4—742no—
211C2×D4—744no—
212C2×D4—7416no—
213C2×D4—744no—
214C2×D4—742no—
215C2×D4—742no—
216C2×D4—741no—
217C2×D4—742no—
218C2×D4—741no—
219C2×D4C22×D3711yescharacteristic
220C22×C4—21no—
221C22×C4—372no—
222C22×C4—742no—
223C22×C4C22×D3711yescharacteristic
224SD16—744no—
225D8—748no—
226D8—744no—
227M16—742no—
228C2×C8—22no—
229C2×C8—372no—
230C2×C8C22×D3712yes—
231C4⋊C4C22×D3712yes—
232C22⋊C4—21no—
233C22⋊C4—744no—
234C42—21no—
235C23—21no—
236C23—372no—
237C23—2962no—
238C23—2962no—
239C23—2964no—
240C23—748no—
241C23—744no—
242C23—742no—
243C23—742no—
244C23—1482no—
245C23D4×D3711yescharacteristic
246Q8—21no—
247Q8—742no—
248Q8C2×D14812yes—
249D4—24no—
250D4—21no—
251D4—374no—
252D4—2964no—
253D4—748no—
254D4—748no—
255D4—744no—
256D4—744no—
257D4—742no—
258D4—742no—
259D4—742no—
260D4—742no—
261D4—1482no—
262D4C2×D14812yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—372no—
267C2×C4—748no—
268C2×C4—742no—
269C2×C4—742no—
270C2×C4C2×D14812yes—
271C2×C4D4×D3711yescharacteristic
272C2×C4D4×D3712yes—
273C2×C4C23×D3711yescharacteristic
274C8—22no—
275C8—742no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—2964no—
280C22—2964no—
281C22—2968no—
282C22—744no—
283C22—742no—
284C22—744no—
285C22—1482no—
286C22—1484no—
287C22—1484no—
288C22#1192450612yes—
289C22C2×D4×D3711yescharacteristic
290C4—22no—
291C4—42no—
292C4—742no—
293C4#1192450612yes—
294C4C2×D4×D3711yescharacteristic
295C4C22×D14811yescharacteristic
296C2—22no—
297C2—42no—
298C2—2962no—
299C2—1482no—
300C2—1484no—
301C2C2×#1192450611yescharacteristic
302C2#1455833512yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#416C2
socleC22×C37C22×D4
hypercenter#416D37
hypocenterC37#3008
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#11924582C2semidirect
2. C2×#11924561C2semidirect
3. C22×D296C2semidirect
4. #14558102C2semidirect
5. #14557681C2semidirect
6. #14557668C2semidirect
7. #14557537C2semidirect
8. #14557499C2semidirect
9. C37×#416C2semidirect
12. C2×#10596966C22semidirect
17. C22×D148C22semidirect
19. C2×D4×D37C22semidirect
21. C2×#10596941C22semidirect
23. C2×D296C22semidirect
24. C2×#10596939C22semidirect
25. C2×#10596934C22semidirect
28. #11924511C22semidirect
30. #11924456C22semidirect
31. #11924448C22semidirect
33. C22×C8×C37C22semidirect
34. #117×C37C22semidirect
35. #103×C37C22semidirect
36. #102×C37C22semidirect
41. #41×C37D4semidirect
66. C2×D148D4semidirect
159. #416D37semidirect
161. C37#3008semidirect
179. C22×C8C2×D37semidirect
184. #117C2×D37semidirect
185. #103C2×D37semidirect
187. #102C2×D37semidirect
198. #41D148semidirect

Maximal subgroups

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

subgroupindex
C2×#119245822
C2×#119245612
C22×D2962
#145581022
#145576812
#145576682
#145575372
#145574992
C37×#4162
#300837