Finite Group Databasebeta

← #28329161

Subgroups — #28329161

Order 13568.

1800 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291261C213yesmaximal
2#20291249C2112yesmaximal
3#20291215C2112yesmaximal
4#125×#6157C213yesmaximal
5#564×D53C211yesmaximal, characteristic
6#16811574C2216yes—
7#16811574C22112yes—
8#16811574C22112yes—
9#16811572C22112yes—
10#16811572C22112yes—
11#16811572C22112yes—
12#16811572C22112yes—
13#16811572C22112yes—
14#16811568C2214yes—
15#16811567C2213yes—
16C4⋊C4×#6157C2216yes—
17C4⋊C4×#6157C22112yes—
18C4⋊C4×#6157C22112yes—
19C4×C4⋊#6157C22112yes—
20C42×#6157C2211yescharacteristic
21C53⋊#564C413yescomplemented
22C53⋊#564C4112yescomplemented
23#125×D53C2213yes—
24#125×D53C22112yes—
25C53×#564C411yescharacteristic, complemented
26C2×C4⋊#6157C23112yes—
27C2×C4⋊#6157C23112yes—
28C2×C4⋊#6157C2316yes—
29C2×C4×#6157C2313yes—
30C2×C4×#6157C2313yes—
31C2×C4×#6157C23112yes—
32C2×C4×#6157C23112yes—
33(C2×#6157)⋊C4C23112yes—
34(C2×#6157)⋊C4C23112yes—
35(C2×#6157)⋊C4C23112yes—
36(C2×#6157)⋊C4C23112yes—
37(C2×#6157)⋊C4C23112yes—
38#13133209C2×C413yes—
39#13133209C2×C4112yes—
40#13133208C2×C416yes—
41#13133208C2×C4112yes—
42#13133208C2×C4112yes—
43#13133207C2×C4112yes—
44#13133207C2×C4112yes—
45#13133207C2×C4112yes—
46#13133207C2×C4112yes—
47#13133207C2×C4112yes—
48C4⋊C4×D53C2316yes—
49C4⋊C4×D53C23112yes—
50C4⋊C4×D53C23112yes—
51C42×D53C2311yescharacteristic
52C42×D53C2314yes—
53#125×C53C2×C413yes—
54#125×C53C2×C4112yes—
55C22×#6157C2411yescharacteristic
56C22×#6157C24112yes—
57C22×#6157C2413yes—
58C2×C4×D53C2413yes—
59C2×C4×D53C24112yes—
60C4⋊#6157—224no—
61C4⋊#6157—412no—
62C4⋊#6157—46no—
63C4×#6157—26no—
64C4×#6157—412no—
65C4×#6157—412no—
66C4×#6157—43no—
67C4⋊C4×C53C22×C416yes—
68C4⋊C4×C53C22×C4112yes—
69C4⋊C4×C53C22×C4112yes—
70C42×C53C22×C411yescharacteristic
71C42×C53C22×C414yes—
72C4⋊Dic53C22×C416yes—
73C4⋊Dic53C22×C4112yes—
74C4⋊Dic53C22×C4112yes—
75#11719919C22×C4112yes—
76#11719919C22×C4112yes—
77#11719919C22×C4112yes—
78#11719919C22×C4112yes—
79#11719919C22×C4112yes—
80C4×Dic53C22×C413yes—
81C4×Dic53C22×C4112yes—
82C22×D53C2511yescharacteristic
83C2×#6157—26no—
84C2×#6157—23no—
85C2×#6157—212no—
86C2×#6157—212no—
87C2×#6157—212no—
88C2×#61572−1+416yes—
89C2×C4×C53C23×C413yes—
90C2×C4×C53C23×C4112yes—
91C2×Dic53C23×C413yes—
92C2×Dic53C23×C4112yes—
93C4×D53—23no—
94C4×D53—212no—
95C4×D53—212no—
96C4×D53—23no—
97#34526—531nomaximal, Sylow
98C22×C53C24×C411yescharacteristic
99C2×D53C2×2−1+413yes—
100C2×D53#58513yes—
101#6157—26no—
102#6157—412no—
103#6157—41no—
104C4×C53—23no—
105C4×C53—212no—
106Dic53—23no—
107Dic53—212no—
108C2×#564—531no—
109#2382—533no—
110#2370—5312no—
111#2336—5312no—
112C4×#125—533no—
113C2×C53C4×2−1+413yes—
114D53#355313yes—
115D53#355611yescharacteristic
116#564—5312no—
117#564—533no—
118#564#615711yescharacteristic, complemented
119C2×#125—533no—
120C2×#125—5312no—
121#389—536no—
122#389—5312no—
123#389—5312no—
124#387—5312no—
125#387—5312no—
126#387—5312no—
127#387—5312no—
128#387—5312no—
129#383—534no—
130#382—533no—
131C4×C4⋊C4—536no—
132C4×C4⋊C4—5312no—
133C4×C4⋊C4—5312no—
134C4×C4⋊C4—5312no—
135C43—531no—
136C53#3452611yescharacteristic, complemented, Sylow
137#125—5312no—
138#125—5312no—
139#125—5312no—
140#125—5312no—
141#125—5312no—
142#125—536no—
143#125—5312no—
144#125—5312no—
145#125—533no—
146#125—5312no—
147#125C2×#615713yes—
148#125C2×#6157112yes—
149C2×C4⋊C4—536no—
150C2×C4⋊C4—5312no—
151C2×C4⋊C4—5312no—
152C2×C4⋊C4—5312no—
153C2×C4⋊C4—536no—
154C2×C4⋊C4—5312no—
155C2×C42—531no—
156C2×C42—534no—
157C2×C42—533no—
158C2×C42—533no—
159C2×C42—5312no—
160C2×C42—5312no—
161#95—5312no—
162#95—5312no—
163#95—5312no—
164#95—5312no—
165#95—5312no—
166C22×C4—533no—
167C22×C4—533no—
168C22×C4—5312no—
169C22×C4—531no—
170C22×C4—5312no—
171C4⋊C4—5312no—
172C4⋊C4—5312no—
173C4⋊C4—5312no—
174C4⋊C4—5312no—
175C4⋊C4—5312no—
176C4⋊C4—536no—
177C4⋊C4—5312no—
178C4⋊C4—5312no—
179C4⋊C4—10624no—
180C4⋊C4—21212no—
181C4⋊C4—2126no—
182C4⋊C4C22×#615716yes—
183C4⋊C4C22×#6157112yes—
184C4⋊C4C22×#6157112yes—
185C42—533no—
186C42—5312no—
187C42—1066no—
188C42—21212no—
189C42—21212no—
190C42—2123no—
191C42C22×#615711yescharacteristic
192C42C22×#615714yes—
193C23—531no—
194C2×C4—536no—
195C2×C4—533no—
196C2×C4—5312no—
197C2×C4—1063no—
198C2×C4—10612no—
199C2×C4—1063no—
200C2×C4—10612no—
201C2×C4—1066no—
202C2×C4—1063no—
203C2×C4—10612no—
204C2×C4—10612no—
205C2×C4—10612no—
206C2×C4C23×#615713yes—
207C2×C4C23×#6157112yes—
208C22—533no—
209C22—533no—
210C22C24×#615711yescharacteristic
211C4—23no—
212C4—212no—
213C4—1066no—
214C4—1063no—
215C4—10612no—
216C4—21212no—
217C4—2121no—
218C2—533no—
219C2—531no—
220C22−1+4×#615713yes—
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×#564C4
prime radical, cosocleC22×D53C25
hypercenter#564#6157
hypocenterC53#34526
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #564

Complements

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

NH ≅ G/Ntype
21. C53⋊#564C4semidirect
22. C53⋊#564C4semidirect
25. C53×#564C4semidirect
118. #564#6157semidirect
136. C53#34526semidirect

Maximal subgroups

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

subgroupindex
#202912612
#202912492
#202912152
#125×#61572
#564×D532
#3452653