Finite Group Databasebeta

← #18003866

Subgroups — #18003866

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×#11924564C211yesmaximal, characteristic, complemented
3C22×D296C211yesmaximal, characteristic, complemented
4#14558097C211yesmaximal, characteristic, complemented
5#14557680C214yesmaximal, complemented
6#14557669C214yesmaximal, complemented
7#14557539C211yesmaximal, characteristic, complemented
8#14557530C211yesmaximal, characteristic, complemented
9C37×#408C211yesmaximal, characteristic, complemented
10#11924582—24no—
11#11924582—24no—
12C2×#10596966—21no—
13C2×#10596962C2211yescharacteristic
14C2×#10596962C2212yescomplemented
15C22×D148C2211yescharacteristic
16C22×D148C2211yescharacteristic, complemented
17C22×D148C2212yescomplemented
18C2×D4×D37—21no—
19C2×D4×D37—22no—
20#11924564—24no—
21C2×#10596930C2212yescomplemented
22C2×D296—24no—
23C2×D296C2214yescomplemented
24C2×D296C2212yescomplemented
25#11924532—21no—
26#11924525—21no—
27#11924510C2214yescomplemented
28#11924504—22no—
29#11924495C2211yescharacteristic, complemented
30#11924461C2212yescomplemented
31#11924458C2212yescomplemented
32#11924458C2214yescomplemented
33C2×#41×C37C2211yescharacteristic
34C2×M16×C37C2211yescharacteristic, complemented
35C22×C8×C37C2211yescharacteristic, complemented
36C22⋊C8×C37C2214yescomplemented
37C23×D37—21no—
38C23×D37—22no—
39C23×D37—21no—
40#41×C37—22no—
41#41×C37—22no—
42C2×Q8×C37D411yescharacteristic
43C2×D4×C37D412yes—
44C2×D4×C37D411yescharacteristic
45C22×C4×C37C2312yes—
46C22×C4×C37C2311yescharacteristic
47C2×C22⋊D37—21no—
48C2×C22⋊D37—22no—
49C2×C22⋊D37—24no—
50#10596966—28no—
51D4×D37—216no—
52D4×D37—28no—
53#10596962—22no—
54#10596962—24no—
55#10596962—24no—
56#10596962D414yescomplemented
57C2×D148—21no—
58C2×D148—22no—
59C2×D148—22no—
60C2×D148—24no—
61C2×D148—24no—
62C2×D148—22no—
63C2×D148—22no—
64C2×D148—21no—
65C2×D148D414yescomplemented
66C2×D148C2312yes—
67C2×D148C2312yes—
68C2×D148C2311yescharacteristic
69C2×C4×D37—21no—
70C2×C4×D37—22no—
71C2×Dic74C2311yescharacteristic
72M16×C37—22no—
73C2×C8×C37—22no—
74C2×C8×C37C2312yes—
75C2×C8×C37C2312yes—
76C22⋊Dic37—21no—
77#10596938—24no—
78C4⋊Dic37C2312yes—
79C4×Dic37—21no—
80D296—28no—
81D296—24no—
82#10596930—24no—
83C23×C37C2×D412yes—
84C23×C37C2×D411yescharacteristic
85C22×D37—21no—
86C22×D37—22no—
87C22×D37—28no—
88C22×D37—24no—
89C22×D37—22no—
90C22×D37—42no—
91C22×D37—82no—
92C22×D37—84no—
93C22×D37—82no—
94Q8×C37—22no—
95D4×C37—24no—
96D4×C37—22no—
97C2×C4×C37—22no—
98C2×C4×C37—22no—
99C2×C4×C37C2×D412yes—
100C2×C4×C37C2×D411yescharacteristic
101C2×C4×C37C2×D412yes—
102C2×C4×C37C2×D412yes—
103C2×C4×C37C2411yescharacteristic
104C22⋊D37—28no—
105C22⋊D37—24no—
106C22⋊D37—84no—
107C2×Dic37—21no—
108C2×Dic37—22no—
109D148—28no—
110D148—24no—
111D148—21no—
112D148—22no—
113D148—22no—
114D148—22no—
115D148—42no—
116D148C2×D412yes—
117D148C2×D414yes—
118C4×D37—28no—
119C4×D37—24no—
120Dic74—21no—
121Dic74C2×D412yes—
122C8×C37—22no—
123C8×C37—22no—
124C22×C37—22no—
125C22×C37—22no—
126C22×C37—24no—
127C22×C37#12012yes—
128C22×C37C22×D411yescharacteristic
129C2×D37—24no—
130C2×D37—22no—
131C2×D37—44no—
132C2×D37—42no—
133C2×D37—42no—
134C2×D37—44no—
135C2×D37—84no—
136C2×D37—88no—
137C2×D37—84no—
138C4×C37—22no—
139C4×C37#12012yes—
140C4×C37C22×D411yescharacteristic
141C4×C37C22×D411yescharacteristic
142Dic37—22no—
143Dic37—42no—
144#3008—371nomaximal, Sylow
145C2×C37—22no—
146C2×C37—42no—
147C2×C37C2×#12011yescharacteristic
148C2×C37#57612yes—
149D37—44no—
150D37—42no—
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#416—371no—
160#408D3711yescharacteristic, complemented
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—371no—
169C22×D4—371no—
170C22×D4—372no—
171C22×D4—742no—
172C22×D4—741no—
173AGL(1,ℤ/8)—744no—
174C2×SD16—372no—
175C2×D8—374no—
176C2×D8—372no—
177C2×D8—744no—
178C2×M16C2×D3711yescharacteristic, complemented
179C22×C8C2×D3711yescharacteristic, complemented
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⋊C8C2×D3714yescomplemented
189C24—742no—
190C24—741no—
191C24—741no—
192#41—22no—
193#41—22no—
194#41—374no—
195#41—744no—
196#41—744no—
197#41—748no—
198#41—742no—
199C2×Q8—371no—
200C2×Q8D14811yescharacteristic
201C2×D4—374no—
202C2×D4—372no—
203C2×D4—371no—
204C2×D4—372no—
205C2×D4—744no—
206C2×D4—744no—
207C2×D4—7416no—
208C2×D4—748no—
209C2×D4—744no—
210C2×D4—742no—
211C2×D4—741no—
212C2×D4—742no—
213C2×D4—741no—
214C2×D4—741no—
215C2×D4—742no—
216C2×D4—742no—
217C2×D4—742no—
218C2×D4D14812yes—
219C2×D4D14811yescharacteristic
220C22×C4—742no—
221C22×C4—741no—
222C22×C4C22×D3712yes—
223C22×C4C22×D3711yescharacteristic
224SD16—744no—
225D8—748no—
226D8—744no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D3712yes—
230C2×C8C22×D3712yes—
231C4⋊C4—372no—
232C22⋊C4—741no—
233C22⋊C4—744no—
234C42—741no—
235C23—2962no—
236C23—2962no—
237C23—2964no—
238C23—748no—
239C23—744no—
240C23—742no—
241C23—741no—
242C23—742no—
243C23—1482no—
244C23C2×D14812yes—
245C23C2×D14811yescharacteristic
246Q8—22no—
247Q8—372no—
248Q8—741no—
249D4—24no—
250D4—22no—
251D4—374no—
252D4—372no—
253D4—2964no—
254D4—748no—
255D4—748no—
256D4—744no—
257D4—744no—
258D4—741no—
259D4—742no—
260D4—742no—
261D4—742no—
262D4—1482no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—748no—
266C2×C4—741no—
267C2×C4—744no—
268C2×C4—742no—
269C2×C4C2×D14812yes—
270C2×C4C2×D14811yescharacteristic
271C2×C4D4×D3712yes—
272C2×C4D4×D3712yes—
273C2×C4C23×D3711yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—2964no—
280C22—2964no—
281C22—2968no—
282C22—744no—
283C22—742no—
284C22—1482no—
285C22—1484no—
286C22—1484no—
287C22—1482no—
288C22#1192450612yes—
289C22C22×D14811yescharacteristic
290C4—22no—
291C4—742no—
292C4—1482no—
293C4#1192450612yes—
294C4C2×D4×D3711yescharacteristic
295C4C2×D4×D3711yescharacteristic
296C2—22no—
297C2—42no—
298C2—2962no—
299C2—1484no—
300C2—1482no—
301C2C2×#1192450611yescharacteristic
302C2#1455833812yes—
subgroupG/H
centerC22C22×D148
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#408C2
socleC22×C37C22×D4
hypercenter#408D37
hypocenterC37#3008
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#11924582C2semidirect
2. C2×#11924564C2semidirect
3. C22×D296C2semidirect
4. #14558097C2semidirect
5. #14557680C2semidirect
6. #14557669C2semidirect
7. #14557539C2semidirect
8. #14557530C2semidirect
9. C37×#408C2semidirect
14. C2×#10596962C22semidirect
16. C22×D148C22semidirect
17. C22×D148C22semidirect
21. C2×#10596930C22semidirect
23. C2×D296C22semidirect
24. C2×D296C22semidirect
27. #11924510C22semidirect
29. #11924495C22semidirect
30. #11924461C22semidirect
31. #11924458C22semidirect
32. #11924458C22semidirect
34. C2×M16×C37C22semidirect
35. C22×C8×C37C22semidirect
36. C22⋊C8×C37C22semidirect
56. #10596962D4semidirect
65. C2×D148D4semidirect
160. #408D37semidirect
161. C37#3008semidirect
178. C2×M16C2×D37semidirect
179. C22×C8C2×D37semidirect
188. C22⋊C8C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245822
C2×#119245642
C22×D2962
#145580972
#145576802
#145576692
#145575392
#145575302
C37×#4082
#300837