Finite Group Databasebeta

← #16811296

Subgroups — #16811296

Order 3392.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#13133212C211yesmaximal, characteristic, complemented
3#13133203C211yesmaximal, characteristic, complemented
4#13133201C211yesmaximal, characteristic, complemented
5#13133199C211yesmaximal, characteristic, complemented
6#13133197C211yesmaximal, characteristic, complemented
7#13133194C211yesmaximal, characteristic, complemented
8#13133185C211yesmaximal, characteristic
9D4×Dic53C211yesmaximal, characteristic, complemented
10#13133177C211yesmaximal, characteristic, complemented
11#13133176C211yesmaximal, characteristic, complemented
12#13133175C211yesmaximal, characteristic, complemented
13C2×#11719921C211yesmaximal, characteristic, complemented
14C22⋊C4×D53C211yesmaximal, characteristic, complemented
15#123×C53C211yesmaximal, characteristic, complemented
16C23×D53C2211yescharacteristic
17C2×D4×C53C2211yescharacteristic
18C22×C4×C53C2211yescharacteristic
19C2×C22⋊D53C2211yescharacteristic
20C2×C22⋊D53C2211yescharacteristic
21C22×Dic53C2211yescharacteristic
22C22×Dic53C2211yescharacteristic, complemented
23C2×C4×D53C2211yescharacteristic
24C2×C4×D53C2211yescharacteristic
25C2×Dic106C2211yescharacteristic, complemented
26C4⋊C4×C53C2211yescharacteristic, complemented
27C4⋊C4×C53C2211yescharacteristic, complemented
28C22⋊C4×C53C2211yescharacteristic
29C22⋊C4×C53C2211yescharacteristic, complemented
30C22⋊C4×C53C2211yescharacteristic, complemented
31C22⋊Dic53C2211yescharacteristic
32C22⋊Dic53C2211yescharacteristic
33C22⋊Dic53C2211yescharacteristic, complemented
34#11719921C2211yescharacteristic
35#11719921C2211yescharacteristic
36#11719921C2211yescharacteristic
37#11719921C2211yescharacteristic
38#11719921C2211yescharacteristic
39#11719921C2211yescharacteristic
40#11719921C2211yescharacteristic
41#11719921C2211yescharacteristic
42C4⋊Dic53C2211yescharacteristic, complemented
43C4⋊Dic53C2211yescharacteristic, complemented
44C4⋊Dic53C2211yescharacteristic, complemented
45C4⋊Dic53C2211yescharacteristic, complemented
46#11719919C2211yescharacteristic, complemented
47#11719919C2211yescharacteristic, complemented
48C4×Dic53C2211yescharacteristic, complemented
49C4×Dic53C2211yescharacteristic, complemented
50C4×Dic53C2211yescharacteristic, complemented
51C23×C53C2311yescharacteristic
52C23×C53C2311yescharacteristic
53C22×D53—22no—
54C22×D53—22no—
55C22×D53—41no—
56C22×D53C2311yescharacteristic
57C22×D53C2311yescharacteristic
58D4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53C2311yescharacteristic
61C2×C4×C53C2311yescharacteristic
62C2×C4×C53C2311yescharacteristic
63C2×C4×C53C2311yescharacteristic
64C2×C4×C53C2311yescharacteristic
65C22⋊D53—22no—
66C22⋊D53—41no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53C2311yescharacteristic
70C2×Dic53C2311yescharacteristic
71C2×Dic53C2311yescharacteristic
72C2×Dic53C2311yescharacteristic
73C2×Dic53C2311yescharacteristic
74C2×Dic53C2311yescharacteristic
75C4×D53—22no—
76C4×D53—41no—
77Dic106—41no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—21no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53#4112yes—
84C22×C53C2411yescharacteristic
85C2×D53—21no—
86C2×D53—21no—
87C2×D53—21no—
88C2×D53—21no—
89C2×D53—21no—
90C2×D53—44no—
91C2×D53#4112yes—
92C4×C53—21no—
93C4×C53—21no—
94C4×C53—21no—
95C4×C53—21no—
96C4×C53—21no—
97Dic53—21no—
98Dic53—21no—
99Dic53—21no—
100Dic53—21no—
101Dic53—21no—
102Dic53—21no—
103C2×C53—22no—
104C2×C53—41no—
105C2×C53C2×#4111yescharacteristic
106C2×C53C2×#4111yescharacteristic
107C2×C53H(5,𝔽2)11yescharacteristic
108D53—22no—
109D53—41no—
110#551—531nomaximal, Sylow
111C53#55111yescharacteristic, complemented, Sylow
112#126—531no—
113#126—531no—
114#124—531no—
115#123—531no—
116#123—531no—
117#123D5311yescharacteristic, complemented
118C22⋊Q8—531no—
119C22⋊Q8—531no—
120#120—531no—
121C4×D4—531no—
122C4×D4—531no—
123#117—531no—
124#117—531no—
125C2×C22⋊C4—531no—
126C2×C22⋊C4—531no—
127C24—531no—
128C2×Q8—531no—
129C2×D4—531no—
130C2×D4—531no—
131C2×D4C2×D5311yescharacteristic
132C22×C4—531no—
133C22×C4—531no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4C2×D5311yescharacteristic
137C4⋊C4—531no—
138C4⋊C4—531no—
139C4⋊C4—531no—
140C4⋊C4—531no—
141C4⋊C4—531no—
142C4⋊C4—531no—
143C4⋊C4C2×D5311yescharacteristic, complemented
144C4⋊C4C2×D5311yescharacteristic, complemented
145C22⋊C4—531no—
146C22⋊C4—531no—
147C22⋊C4—531no—
148C22⋊C4—531no—
149C22⋊C4—531no—
150C22⋊C4—531no—
151C22⋊C4—531no—
152C22⋊C4—531no—
153C22⋊C4—531no—
154C22⋊C4—531no—
155C22⋊C4—531no—
156C22⋊C4C2×D5311yescharacteristic
157C22⋊C4C2×D5311yescharacteristic, complemented
158C22⋊C4C2×D5311yescharacteristic, complemented
159C42—531no—
160C42—531no—
161C42—531no—
162C23—531no—
163C23—531no—
164C23—1062no—
165C23—1062no—
166C23—2121no—
167C23C22×D5311yescharacteristic
168C23C22×D5311yescharacteristic
169Q8—2121no—
170D4—22no—
171D4—1062no—
172D4—2121no—
173C2×C4—22no—
174C2×C4—531no—
175C2×C4—531no—
176C2×C4—531no—
177C2×C4—531no—
178C2×C4—531no—
179C2×C4—531no—
180C2×C4—1062no—
181C2×C4—2121no—
182C2×C4—2121no—
183C2×C4—2121no—
184C2×C4C22×D5311yescharacteristic
185C2×C4C22×D5311yescharacteristic
186C2×C4C22×D5311yescharacteristic
187C2×C4C22×D5311yescharacteristic
188C2×C4C22×D5311yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—532no—
195C22—1061no—
196C22—1061no—
197C22—1061no—
198C22—1061no—
199C22—1061no—
200C22—2124no—
201C22#1171994712yes—
202C22C23×D5311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—1061no—
209C4—1061no—
210C4—1061no—
211C4—1061no—
212C4—1061no—
213C4—1061no—
214C2—22no—
215C2—41no—
216C2—1062no—
217C2—2121no—
218C2#41×D5311yescharacteristic
219C2C2×#1171994711yescharacteristic
220C2#1313326311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D53
commutator, socle, cosocleC22×C53C24
Fitting#123×C53C2
prime radical#13133185C2
hypercenter#123D53
hypocenterC53#551
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #13133212C2semidirect
2. #13133212C2semidirect
3. #13133203C2semidirect
4. #13133201C2semidirect
5. #13133199C2semidirect
6. #13133197C2semidirect
7. #13133194C2semidirect
9. D4×Dic53C2semidirect
10. #13133177C2semidirect
11. #13133176C2semidirect
12. #13133175C2semidirect
13. C2×#11719921C2semidirect
14. C22⋊C4×D53C2semidirect
15. #123×C53C2semidirect
22. C22×Dic53C22semidirect
25. C2×Dic106C22semidirect
26. C4⋊C4×C53C22semidirect
27. C4⋊C4×C53C22semidirect
29. C22⋊C4×C53C22semidirect
30. C22⋊C4×C53C22semidirect
33. C22⋊Dic53C22semidirect
42. C4⋊Dic53C22semidirect
43. C4⋊Dic53C22semidirect
44. C4⋊Dic53C22semidirect
45. C4⋊Dic53C22semidirect
46. #11719919C22semidirect
47. #11719919C22semidirect
48. C4×Dic53C22semidirect
49. C4×Dic53C22semidirect
50. C4×Dic53C22semidirect
111. C53#551semidirect
117. #123D53semidirect
143. C4⋊C4C2×D53semidirect
144. C4⋊C4C2×D53semidirect
157. C22⋊C4C2×D53semidirect
158. C22⋊C4C2×D53semidirect

Maximal subgroups

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

subgroupindex
#131332122
#131332032
#131332012
#131331992
#131331972
#131331942
#131331852
D4×Dic532
#131331772
#131331762
#131331752
C2×#117199212
C22⋊C4×D532
#123×C532
#55153