Finite Group Databasebeta

← #16811206

Subgroups — #16811206

Order 3392.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#13133212C211yesmaximal, characteristic, complemented
2#13133211C211yesmaximal, characteristic, complemented
3#13133205C211yesmaximal, characteristic, complemented
4#13133203C211yesmaximal, characteristic, complemented
5#13133202C211yesmaximal, characteristic, complemented
6#13133200C211yesmaximal, characteristic, complemented
7#13133194C211yesmaximal, characteristic, complemented
8#13133191C211yesmaximal, characteristic, complemented
9#13133190C211yesmaximal, characteristic, complemented
10#13133189C211yesmaximal, characteristic, complemented
11#13133187C211yesmaximal, characteristic, complemented
12#13133185C211yesmaximal, characteristic, complemented
13C4×C22⋊D53C211yesmaximal, characteristic, complemented
14C2×#11719921C211yesmaximal, characteristic, complemented
15C4×D4×C53C211yesmaximal, characteristic, complemented
16C23×D53C2211yescharacteristic
17C2×D4×C53C2211yescharacteristic
18C22×C4×C53C2211yescharacteristic
19C22×C4×C53C2211yescharacteristic, complemented
20C2×C22⋊D53C2211yescharacteristic
21C2×C22⋊D53C2211yescharacteristic
22C2×C22⋊D53C2211yescharacteristic
23C2×C22⋊D53C2211yescharacteristic, complemented
24C22×Dic53C2211yescharacteristic
25C2×D212C2211yescharacteristic
26C2×D212C2211yescharacteristic
27C2×C4×D53C2211yescharacteristic, complemented
28C2×Dic106C2211yescharacteristic, complemented
29C4⋊C4×C53C2211yescharacteristic, complemented
30C22⋊C4×C53C2211yescharacteristic
31C22⋊C4×C53C2211yescharacteristic, complemented
32C42×C53C2211yescharacteristic, complemented
33C22⋊Dic53C2211yescharacteristic
34C22⋊Dic53C2211yescharacteristic, complemented
35#11719921C2211yescharacteristic
36#11719921C2211yescharacteristic
37#11719921C2211yescharacteristic
38#11719921C2211yescharacteristic
39#11719921C2211yescharacteristic
40#11719921C2211yescharacteristic
41#11719921C2211yescharacteristic
42#11719921C2211yescharacteristic
43#11719921C2211yescharacteristic, complemented
44#11719921C2211yescharacteristic, complemented
45C4⋊Dic53C2211yescharacteristic, complemented
46#11719919C2211yescharacteristic, complemented
47#11719919C2211yescharacteristic, complemented
48#11719919C2211yescharacteristic, complemented
49#11719919C2211yescharacteristic, complemented
50C4×Dic53C2211yescharacteristic, complemented
51C23×C53C2311yescharacteristic
52C23×C53C2311yescharacteristic
53C22×D53—22no—
54C22×D53—41no—
55C22×D53—41no—
56C22×D53C2311yescharacteristic
57C22×D53C2311yescharacteristic
58C22×D53C2311yescharacteristic
59D4×C53—22no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53C2311yescharacteristic
63C2×C4×C53C2311yescharacteristic
64C2×C4×C53C2311yescharacteristic
65C2×C4×C53C2311yescharacteristic
66C2×C4×C53C2311yescharacteristic
67C22⋊D53—22no—
68C22⋊D53—41no—
69C22⋊D53—41no—
70C22⋊D53—41no—
71C2×Dic53—41no—
72C2×Dic53C2311yescharacteristic
73C2×Dic53C2311yescharacteristic
74C2×Dic53C2311yescharacteristic
75C2×Dic53C2311yescharacteristic
76C2×Dic53C2311yescharacteristic
77D212—41no—
78D212—41no—
79C4×D53—41no—
80Dic106—41no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53#4112yes—
87C22×C53C2411yescharacteristic
88C2×D53—21no—
89C2×D53—21no—
90C2×D53—21no—
91C2×D53—21no—
92C2×D53—21no—
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—44no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—21no—
103Dic53—21no—
104Dic53—21no—
105Dic53—21no—
106Dic53—21no—
107Dic53—21no—
108C2×C53—22no—
109C2×C53—41no—
110C2×C53C2×#4111yescharacteristic
111C2×C53H(5,𝔽2)11yescharacteristic
112C2×C53H(5,𝔽2)11yescharacteristic
113D53—41no—
114D53—41no—
115D53—41no—
116#538—531nomaximal, Sylow
117C53#53811yescharacteristic, complemented, Sylow
118#126—531no—
119#126—531no—
120#124—531no—
121#124—531no—
122#123—531no—
123#123—531no—
124C22⋊Q8—531no—
125#121—531no—
126#121—531no—
127#121—531no—
128#120—531no—
129#120—531no—
130C4×D4—531no—
131C4×D4D5311yescharacteristic, complemented
132C2×C22⋊C4—531no—
133C24—531no—
134C2×Q8—531no—
135C2×D4—531no—
136C2×D4—531no—
137C2×D4—531no—
138C2×D4—531no—
139C2×D4—531no—
140C2×D4—531no—
141C2×D4C2×D5311yescharacteristic
142C22×C4—531no—
143C22×C4—531no—
144C22×C4C2×D5311yescharacteristic
145C22×C4C2×D5311yescharacteristic, complemented
146C4⋊C4—531no—
147C4⋊C4—531no—
148C4⋊C4—531no—
149C4⋊C4—531no—
150C4⋊C4—531no—
151C4⋊C4C2×D5311yescharacteristic, complemented
152C22⋊C4—531no—
153C22⋊C4—531no—
154C22⋊C4—531no—
155C22⋊C4—531no—
156C22⋊C4—531no—
157C22⋊C4—531no—
158C22⋊C4—531no—
159C22⋊C4—531no—
160C22⋊C4—531no—
161C22⋊C4—531no—
162C22⋊C4—531no—
163C22⋊C4—531no—
164C22⋊C4C2×D5311yescharacteristic
165C22⋊C4C2×D5311yescharacteristic, complemented
166C42—531no—
167C42C2×D5311yescharacteristic, complemented
168C23—531no—
169C23—531no—
170C23—531no—
171C23—1062no—
172C23—2121no—
173C23—2121no—
174C23C22×D5311yescharacteristic
175C23C22×D5311yescharacteristic
176Q8—2121no—
177D4—22no—
178D4—1062no—
179D4—2121no—
180D4—2121no—
181D4—2121no—
182D4—2121no—
183D4—2121no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—531no—
187C2×C4—531no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—531no—
191C2×C4—2121no—
192C2×C4—2121no—
193C2×C4C22×D5311yescharacteristic
194C2×C4C22×D5311yescharacteristic
195C2×C4C22×D5311yescharacteristic
196C2×C4C22×D5311yescharacteristic
197C2×C4C22×D5311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—1061no—
204C22—1061no—
205C22—1061no—
206C22—1061no—
207C22—1061no—
208C22—1061no—
209C22—1061no—
210C22—1061no—
211C22—1061no—
212C22—2124no—
213C22#1171994512yes—
214C22C23×D5311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—1061no—
221C4—1061no—
222C4—1061no—
223C4—1061no—
224C4—1061no—
225C2—22no—
226C2—41no—
227C2—2121no—
228C2—2121no—
229C2—2121no—
230C2C2×#1171994511yescharacteristic
231C2#1313326211yescharacteristic
232C2#1313326311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D53
commutator, socle, cosocleC22×C53C24
FittingC4×D4×C53C2
hypercenterC4×D4D53
hypocenterC53#538
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #13133212C2semidirect
2. #13133211C2semidirect
3. #13133205C2semidirect
4. #13133203C2semidirect
5. #13133202C2semidirect
6. #13133200C2semidirect
7. #13133194C2semidirect
8. #13133191C2semidirect
9. #13133190C2semidirect
10. #13133189C2semidirect
11. #13133187C2semidirect
12. #13133185C2semidirect
13. C4×C22⋊D53C2semidirect
14. C2×#11719921C2semidirect
15. C4×D4×C53C2semidirect
19. C22×C4×C53C22semidirect
23. C2×C22⋊D53C22semidirect
27. C2×C4×D53C22semidirect
28. C2×Dic106C22semidirect
29. C4⋊C4×C53C22semidirect
31. C22⋊C4×C53C22semidirect
32. C42×C53C22semidirect
34. C22⋊Dic53C22semidirect
43. #11719921C22semidirect
44. #11719921C22semidirect
45. C4⋊Dic53C22semidirect
46. #11719919C22semidirect
47. #11719919C22semidirect
48. #11719919C22semidirect
49. #11719919C22semidirect
50. C4×Dic53C22semidirect
117. C53#538semidirect
131. C4×D4D53semidirect
145. C22×C4C2×D53semidirect
151. C4⋊C4C2×D53semidirect
165. C22⋊C4C2×D53semidirect
167. C42C2×D53semidirect

Maximal subgroups

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

subgroupindex
#131332122
#131332112
#131332052
#131332032
#131332022
#131332002
#131331942
#131331912
#131331902
#131331892
#131331872
#131331852
C4×C22⋊D532
C2×#117199212
C4×D4×C532
#53853