Finite Group Databasebeta

← #27164756

Subgroups — #27164756

Order 12736.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#20263951C211yesmaximal, characteristic, complemented
2#20263950C211yesmaximal, characteristic, complemented
3#20263948C211yesmaximal, characteristic, complemented
4#20263940C211yesmaximal, characteristic, complemented
5#20263939C211yesmaximal, characteristic, complemented
6#20263938C211yesmaximal, characteristic, complemented
7#20263929C211yesmaximal, characteristic, complemented
8#20263928C211yesmaximal, characteristic, complemented
9#20263927C211yesmaximal, characteristic, complemented
10#20263927C211yesmaximal, characteristic, complemented
11#20263926C211yesmaximal, characteristic, complemented
12#20263925C211yesmaximal, characteristic, complemented
13#20263923C211yesmaximal, characteristic, complemented
14#20263923C211yesmaximal, characteristic, complemented
15#121×C199C211yesmaximal, characteristic, complemented
16C23×D199C2211yescharacteristic
17C2×D796C2211yescharacteristic
18C2×D796C2211yescharacteristic
19C2×C22⋊D199C2211yescharacteristic
20C2×C22⋊D199C2211yescharacteristic
21C2×C22⋊D199C2211yescharacteristic
22C2×C22⋊D199C2211yescharacteristic
23C2×C22⋊D199C2211yescharacteristic
24C2×C22⋊D199C2211yescharacteristic, complemented
25C2×C22⋊D199C2211yescharacteristic, complemented
26C22×Dic199C2211yescharacteristic, complemented
27C2×C4×D199C2211yescharacteristic, complemented
28C4⋊Dic199C2211yescharacteristic, complemented
29C4⋊Dic199C2211yescharacteristic, complemented
30Dic199⋊C4C2211yescharacteristic, complemented
31Dic199⋊C4C2211yescharacteristic, complemented
32Dic199⋊C4C2211yescharacteristic, complemented
33C22⋊Dic199C2211yescharacteristic
34C22⋊Dic199C2211yescharacteristic
35C22⋊Dic199C2211yescharacteristic, complemented
36(C2×D199)⋊C4C2211yescharacteristic
37(C2×D199)⋊C4C2211yescharacteristic
38(C2×D199)⋊C4C2211yescharacteristic
39(C2×D199)⋊C4C2211yescharacteristic
40(C2×D199)⋊C4C2211yescharacteristic
41(C2×D199)⋊C4C2211yescharacteristic
42(C2×D199)⋊C4C2211yescharacteristic, complemented
43C4×Dic199C2211yescharacteristic, complemented
44C2×D4×C199C2211yescharacteristic
45C2×D4×C199C2211yescharacteristic
46C2×D4×C199C2211yescharacteristic, complemented
47C22×C4×C199C2211yescharacteristic, complemented
48C4⋊C4×C199C2211yescharacteristic, complemented
49C22⋊C4×C199C2211yescharacteristic
50C22⋊C4×C199C2211yescharacteristic, complemented
51C22×D199—41no—
52C22×D199—41no—
53C22×D199—41no—
54C22×D199C2311yescharacteristic
55C22×D199C2311yescharacteristic
56C22×D199C2311yescharacteristic
57D796—41no—
58D796—41no—
59C22⋊D199—41no—
60C22⋊D199—41no—
61C22⋊D199—41no—
62C22⋊D199—41no—
63C22⋊D199—41no—
64C22⋊D199—41no—
65C22⋊D199—41no—
66C2×Dic199—41no—
67C2×Dic199C2311yescharacteristic
68C2×Dic199C2311yescharacteristic
69C2×Dic199C2311yescharacteristic
70C2×Dic199C2311yescharacteristic
71C2×Dic199C2311yescharacteristic
72C4×D199—41no—
73C23×C199C2311yescharacteristic
74C23×C199C2311yescharacteristic
75C23×C199C2311yescharacteristic
76D4×C199—41no—
77D4×C199—41no—
78D4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199C2311yescharacteristic
81C2×C4×C199C2311yescharacteristic
82C2×C4×C199C2311yescharacteristic
83C2×C4×C199C2311yescharacteristic
84C22×C199—21no—
85C22×C199—21no—
86C22×C199—21no—
87C22×C199—21no—
88C22×C199—21no—
89C22×C199—21no—
90C22×C199—21no—
91C22×C199—21no—
92C22×C199—21no—
93C22×C199C2411yescharacteristic
94C2×D199—21no—
95C2×D199—21no—
96C2×D199—21no—
97C2×D199—21no—
98C2×D199—21no—
99C2×D199—21no—
100C2×D199—21no—
101C2×D199—21no—
102C2×D199—21no—
103C2×D199—44no—
104C4×C199—21no—
105C4×C199—21no—
106C4×C199—21no—
107C4×C199—21no—
108Dic199—21no—
109Dic199—21no—
110Dic199—21no—
111Dic199—21no—
112Dic199—21no—
113C2×C199—41no—
114C2×C199—41no—
115C2×C199—41no—
116C2×C199H(5,𝔽2)11yescharacteristic
117C2×C199H(5,𝔽2)11yescharacteristic
118C2×C199H(5,𝔽2)11yescharacteristic
119D199—41no—
120D199—41no—
121D199—41no—
122C199#56011yescharacteristic, complemented, Sylow
123#560—1991nomaximal, Sylow
124#127—1991no—
125#126—1991no—
126#126—1991no—
127#123—1991no—
128#123—1991no—
129#123—1991no—
130#121—1991no—
131#121—1991no—
132#121—1991no—
133#121—1991no—
134#121—1991no—
135#121D19911yescharacteristic, complemented
136#120—1991no—
137#120—1991no—
138#120—1991no—
139C24—1991no—
140C2×D4—1991no—
141C2×D4—1991no—
142C2×D4—1991no—
143C2×D4—1991no—
144C2×D4—1991no—
145C2×D4—1991no—
146C2×D4—1991no—
147C2×D4—1991no—
148C2×D4—1991no—
149C2×D4C2×D19911yescharacteristic
150C2×D4C2×D19911yescharacteristic
151C2×D4C2×D19911yescharacteristic, complemented
152C22×C4—1991no—
153C22×C4—1991no—
154C22×C4C2×D19911yescharacteristic, complemented
155C4⋊C4—1991no—
156C4⋊C4—1991no—
157C4⋊C4—1991no—
158C4⋊C4—1991no—
159C4⋊C4—1991no—
160C4⋊C4C2×D19911yescharacteristic, complemented
161C22⋊C4—1991no—
162C22⋊C4—1991no—
163C22⋊C4—1991no—
164C22⋊C4—1991no—
165C22⋊C4—1991no—
166C22⋊C4—1991no—
167C22⋊C4—1991no—
168C22⋊C4—1991no—
169C22⋊C4—1991no—
170C22⋊C4—1991no—
171C22⋊C4C2×D19911yescharacteristic
172C22⋊C4C2×D19911yescharacteristic, complemented
173C42—1991no—
174C23—7961no—
175C23—7961no—
176C23—7961no—
177C23—1991no—
178C23—1991no—
179C23—1991no—
180C23C22×D19911yescharacteristic
181C23C22×D19911yescharacteristic
182C23C22×D19911yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—7961no—
187D4—7961no—
188D4—7961no—
189D4—7961no—
190D4—7961no—
191D4—7961no—
192D4—7961no—
193D4—7961no—
194D4—7961no—
195C2×C4—41no—
196C2×C4—7961no—
197C2×C4—7961no—
198C2×C4—1991no—
199C2×C4—1991no—
200C2×C4—1991no—
201C2×C4—1991no—
202C2×C4—1991no—
203C2×C4C22×D19911yescharacteristic
204C2×C4C22×D19911yescharacteristic
205C2×C4C22×D19911yescharacteristic
206C2×C4C22×D19911yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—7964no—
217C22—3981no—
218C22—3981no—
219C22—3981no—
220C22—3981no—
221C22—3981no—
222C22—3981no—
223C22—3981no—
224C22—3981no—
225C22—3981no—
226C22C23×D19911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—3981no—
232C4—3981no—
233C4—3981no—
234C4—3981no—
235C4—3981no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—7961no—
240C2—7961no—
241C2—7961no—
242C2#2026400011yescharacteristic
243C2#2026400011yescharacteristic
244C2#2026400111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#121×C199C2
hypercenter#121D199
hypocenterC199#560
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20263951C2semidirect
2. #20263950C2semidirect
3. #20263948C2semidirect
4. #20263940C2semidirect
5. #20263939C2semidirect
6. #20263938C2semidirect
7. #20263929C2semidirect
8. #20263928C2semidirect
9. #20263927C2semidirect
10. #20263927C2semidirect
11. #20263926C2semidirect
12. #20263925C2semidirect
13. #20263923C2semidirect
14. #20263923C2semidirect
15. #121×C199C2semidirect
24. C2×C22⋊D199C22semidirect
25. C2×C22⋊D199C22semidirect
26. C22×Dic199C22semidirect
27. C2×C4×D199C22semidirect
28. C4⋊Dic199C22semidirect
29. C4⋊Dic199C22semidirect
30. Dic199⋊C4C22semidirect
31. Dic199⋊C4C22semidirect
32. Dic199⋊C4C22semidirect
35. C22⋊Dic199C22semidirect
42. (C2×D199)⋊C4C22semidirect
43. C4×Dic199C22semidirect
46. C2×D4×C199C22semidirect
47. C22×C4×C199C22semidirect
48. C4⋊C4×C199C22semidirect
50. C22⋊C4×C199C22semidirect
122. C199#560semidirect
135. #121D199semidirect
151. C2×D4C2×D199semidirect
154. C22×C4C2×D199semidirect
160. C4⋊C4C2×D199semidirect
172. C22⋊C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202639512
#202639502
#202639482
#202639402
#202639392
#202639382
#202639292
#202639282
#202639272
#202639262
#202639252
#202639232
#121×C1992
#560199