Finite Group Databasebeta

← #16811361

Subgroups — #16811361

Order 3392.

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#13133213C211yesmaximal, characteristic, complemented
2#13133212C211yesmaximal, characteristic, complemented
3#13133210C211yesmaximal, characteristic, complemented
4#13133202C211yesmaximal, characteristic, complemented
5#13133201C211yesmaximal, characteristic, complemented
6#13133200C211yesmaximal, characteristic, complemented
7#13133191C211yesmaximal, characteristic, complemented
8#13133190C211yesmaximal, characteristic, complemented
9#13133189C211yesmaximal, characteristic, complemented
10#13133189C211yesmaximal, characteristic, complemented
11#13133188C211yesmaximal, characteristic, complemented
12#13133187C211yesmaximal, characteristic, complemented
13#13133185C211yesmaximal, characteristic, complemented
14#13133185C211yesmaximal, characteristic, complemented
15#121×C53C211yesmaximal, characteristic, complemented
16C23×D53C2211yescharacteristic
17C2×D4×C53C2211yescharacteristic
18C2×D4×C53C2211yescharacteristic
19C2×D4×C53C2211yescharacteristic, complemented
20C22×C4×C53C2211yescharacteristic, complemented
21C2×C22⋊D53C2211yescharacteristic
22C2×C22⋊D53C2211yescharacteristic
23C2×C22⋊D53C2211yescharacteristic
24C2×C22⋊D53C2211yescharacteristic
25C2×C22⋊D53C2211yescharacteristic
26C2×C22⋊D53C2211yescharacteristic, complemented
27C2×C22⋊D53C2211yescharacteristic, complemented
28C22×Dic53C2211yescharacteristic, complemented
29C2×D212C2211yescharacteristic
30C2×D212C2211yescharacteristic
31C2×C4×D53C2211yescharacteristic, complemented
32C4⋊C4×C53C2211yescharacteristic, complemented
33C22⋊C4×C53C2211yescharacteristic
34C22⋊C4×C53C2211yescharacteristic, complemented
35C22⋊Dic53C2211yescharacteristic
36C22⋊Dic53C2211yescharacteristic
37C22⋊Dic53C2211yescharacteristic, complemented
38#11719921C2211yescharacteristic
39#11719921C2211yescharacteristic
40#11719921C2211yescharacteristic
41#11719921C2211yescharacteristic
42#11719921C2211yescharacteristic
43#11719921C2211yescharacteristic
44#11719921C2211yescharacteristic, complemented
45C4⋊Dic53C2211yescharacteristic, complemented
46C4⋊Dic53C2211yescharacteristic, complemented
47#11719919C2211yescharacteristic, complemented
48#11719919C2211yescharacteristic, complemented
49#11719919C2211yescharacteristic, complemented
50C4×Dic53C2211yescharacteristic, complemented
51C23×C53C2311yescharacteristic
52C23×C53C2311yescharacteristic
53C23×C53C2311yescharacteristic
54C22×D53—41no—
55C22×D53—41no—
56C22×D53—41no—
57C22×D53C2311yescharacteristic
58C22×D53C2311yescharacteristic
59C22×D53C2311yescharacteristic
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53C2311yescharacteristic
65C2×C4×C53C2311yescharacteristic
66C2×C4×C53C2311yescharacteristic
67C2×C4×C53C2311yescharacteristic
68C22⋊D53—41no—
69C22⋊D53—41no—
70C22⋊D53—41no—
71C22⋊D53—41no—
72C22⋊D53—41no—
73C22⋊D53—41no—
74C22⋊D53—41no—
75C2×Dic53—41no—
76C2×Dic53C2311yescharacteristic
77C2×Dic53C2311yescharacteristic
78C2×Dic53C2311yescharacteristic
79C2×Dic53C2311yescharacteristic
80C2×Dic53C2311yescharacteristic
81D212—41no—
82D212—41no—
83C4×D53—41no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53C2411yescharacteristic
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—21no—
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—44no—
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—21no—
108Dic53—21no—
109Dic53—21no—
110Dic53—21no—
111Dic53—21no—
112Dic53—21no—
113C2×C53—41no—
114C2×C53—41no—
115C2×C53—41no—
116C2×C53H(5,𝔽2)11yescharacteristic
117C2×C53H(5,𝔽2)11yescharacteristic
118C2×C53H(5,𝔽2)11yescharacteristic
119D53—41no—
120D53—41no—
121D53—41no—
122#560—531nomaximal, Sylow
123C53#56011yescharacteristic, complemented, Sylow
124#127—531no—
125#126—531no—
126#126—531no—
127#123—531no—
128#123—531no—
129#123—531no—
130#121—531no—
131#121—531no—
132#121—531no—
133#121—531no—
134#121—531no—
135#121D5311yescharacteristic, complemented
136#120—531no—
137#120—531no—
138#120—531no—
139C24—531no—
140C2×D4—531no—
141C2×D4—531no—
142C2×D4—531no—
143C2×D4—531no—
144C2×D4—531no—
145C2×D4—531no—
146C2×D4—531no—
147C2×D4—531no—
148C2×D4—531no—
149C2×D4C2×D5311yescharacteristic
150C2×D4C2×D5311yescharacteristic
151C2×D4C2×D5311yescharacteristic, complemented
152C22×C4—531no—
153C22×C4—531no—
154C22×C4C2×D5311yescharacteristic, complemented
155C4⋊C4—531no—
156C4⋊C4—531no—
157C4⋊C4—531no—
158C4⋊C4—531no—
159C4⋊C4—531no—
160C4⋊C4C2×D5311yescharacteristic, complemented
161C22⋊C4—531no—
162C22⋊C4—531no—
163C22⋊C4—531no—
164C22⋊C4—531no—
165C22⋊C4—531no—
166C22⋊C4—531no—
167C22⋊C4—531no—
168C22⋊C4—531no—
169C22⋊C4—531no—
170C22⋊C4—531no—
171C22⋊C4C2×D5311yescharacteristic
172C22⋊C4C2×D5311yescharacteristic, complemented
173C42—531no—
174C23—531no—
175C23—531no—
176C23—531no—
177C23—2121no—
178C23—2121no—
179C23—2121no—
180C23C22×D5311yescharacteristic
181C23C22×D5311yescharacteristic
182C23C22×D5311yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—2121no—
187D4—2121no—
188D4—2121no—
189D4—2121no—
190D4—2121no—
191D4—2121no—
192D4—2121no—
193D4—2121no—
194D4—2121no—
195C2×C4—41no—
196C2×C4—531no—
197C2×C4—531no—
198C2×C4—531no—
199C2×C4—531no—
200C2×C4—531no—
201C2×C4—2121no—
202C2×C4—2121no—
203C2×C4C22×D5311yescharacteristic
204C2×C4C22×D5311yescharacteristic
205C2×C4C22×D5311yescharacteristic
206C2×C4C22×D5311yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—1061no—
217C22—1061no—
218C22—1061no—
219C22—1061no—
220C22—1061no—
221C22—1061no—
222C22—1061no—
223C22—1061no—
224C22—1061no—
225C22—2124no—
226C22C23×D5311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—1061no—
232C4—1061no—
233C4—1061no—
234C4—1061no—
235C4—1061no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—2121no—
240C2—2121no—
241C2—2121no—
242C2#1313326211yescharacteristic
243C2#1313326211yescharacteristic
244C2#1313326311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D53
commutator, socle, cosocleC22×C53C24
Fitting#121×C53C2
hypercenter#121D53
hypocenterC53#560
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #13133213C2semidirect
2. #13133212C2semidirect
3. #13133210C2semidirect
4. #13133202C2semidirect
5. #13133201C2semidirect
6. #13133200C2semidirect
7. #13133191C2semidirect
8. #13133190C2semidirect
9. #13133189C2semidirect
10. #13133189C2semidirect
11. #13133188C2semidirect
12. #13133187C2semidirect
13. #13133185C2semidirect
14. #13133185C2semidirect
15. #121×C53C2semidirect
19. C2×D4×C53C22semidirect
20. C22×C4×C53C22semidirect
26. C2×C22⋊D53C22semidirect
27. C2×C22⋊D53C22semidirect
28. C22×Dic53C22semidirect
31. C2×C4×D53C22semidirect
32. C4⋊C4×C53C22semidirect
34. C22⋊C4×C53C22semidirect
37. C22⋊Dic53C22semidirect
44. #11719921C22semidirect
45. C4⋊Dic53C22semidirect
46. C4⋊Dic53C22semidirect
47. #11719919C22semidirect
48. #11719919C22semidirect
49. #11719919C22semidirect
50. C4×Dic53C22semidirect
123. C53#560semidirect
135. #121D53semidirect
151. C2×D4C2×D53semidirect
154. C22×C4C2×D53semidirect
160. C4⋊C4C2×D53semidirect
172. C22⋊C4C2×D53semidirect

Maximal subgroups

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

subgroupindex
#131332132
#131332122
#131332102
#131332022
#131332012
#131332002
#131331912
#131331902
#131331892
#131331882
#131331872
#131331852
#121×C532
#56053