Finite Group Databasebeta

← #28329165

Subgroups — #28329165

Order 13568.

1800 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291261C213yesmaximal
2#20291250C216yesmaximal
3#20291249C216yesmaximal
4#20291215C216yesmaximal
5#20291209C216yesmaximal
6#125×#6157C213yesmaximal
7#564×D53C211yesmaximal, characteristic
8#16811574C2213yes—
9#16811574C2213yes—
10#16811574C2216yes—
11#16811574C2216yes—
12#16811574C2216yes—
13#16811574C2216yes—
14#16811572C2216yes—
15#16811572C2216yes—
16#16811572C2216yes—
17#16811572C2216yes—
18#16811572C2216yes—
19#16811572C2216yes—
20#16811572C2216yes—
21#16811572C2216yes—
22#16811572C2216yes—
23#16811572C2216yes—
24#16811569C2212yes—
25#16811568C2212yes—
26#16811567C2213yes—
27C4⋊C4×#6157C2216yes—
28C4⋊C4×#6157C2213yes—
29C4⋊C4×#6157C2216yes—
30C4⋊C4×#6157C2213yes—
31C4⋊C4×#6157C2216yes—
32C4⋊C4×#6157C2216yes—
33C4×C4⋊#6157C2216yes—
34#16811559C2216yes—
35C42×#6157C2211yescharacteristic
36C53⋊#564C413yescomplemented
37C53⋊#564C416yescomplemented
38C53⋊#564C416yescomplemented
39#125×D53C2213yes—
40#125×D53C2216yes—
41#125×D53C2216yes—
42C53×#564C411yescharacteristic, complemented
43C2×C4⋊#6157C2316yes—
44C2×C4⋊#6157C2316yes—
45C2×C4⋊#6157C2313yes—
46C2×C4⋊#6157C2313yes—
47C2×C4⋊#6157C2316yes—
48C2×C4⋊#6157C2316yes—
49C2×C4×#6157C2313yes—
50C2×C4×#6157C2316yes—
51C2×C4×#6157C2313yes—
52C2×C4×#6157C2316yes—
53C2×C4×#6157C2316yes—
54C2×C4×#6157C2316yes—
55(C2×#6157)⋊C4C2316yes—
56(C2×#6157)⋊C4C2316yes—
57(C2×#6157)⋊C4C2316yes—
58(C2×#6157)⋊C4C2316yes—
59(C2×#6157)⋊C4C2316yes—
60(C2×#6157)⋊C4C2316yes—
61(C2×#6157)⋊C4C2316yes—
62(C2×#6157)⋊C4C2316yes—
63(C2×#6157)⋊C4C2316yes—
64(C2×#6157)⋊C4C2316yes—
65#13133209C2×C413yes—
66#13133209C2×C416yes—
67#13133209C2×C416yes—
68#13133208C2×C413yes—
69#13133208C2×C413yes—
70#13133208C2×C416yes—
71#13133208C2×C416yes—
72#13133208C2×C416yes—
73#13133208C2×C416yes—
74#13133207C2×C416yes—
75#13133207C2×C416yes—
76#13133207C2×C416yes—
77#13133207C2×C416yes—
78#13133207C2×C416yes—
79#13133207C2×C416yes—
80#13133207C2×C416yes—
81#13133207C2×C416yes—
82#13133207C2×C416yes—
83#13133207C2×C416yes—
84C4⋊C4×D53C2316yes—
85C4⋊C4×D53C2313yes—
86C4⋊C4×D53C2316yes—
87C4⋊C4×D53C2313yes—
88C4⋊C4×D53C2316yes—
89C4⋊C4×D53C2316yes—
90C42×D53C2311yescharacteristic
91C42×D53C2312yes—
92C42×D53C2312yes—
93#125×C53C2×C413yes—
94#125×C53C2×C416yes—
95#125×C53C2×C416yes—
96C22×#6157C2416yes—
97C22×#6157C2416yes—
98C22×#6157C2411yescharacteristic
99C22×#6157C2413yes—
100C2×C4×D53C2413yes—
101C2×C4×D53C2416yes—
102C2×C4×D53C2416yes—
103C4⋊#6157—212no—
104C4⋊#6157—46no—
105C4⋊#6157—43no—
106C4⋊#6157—46no—
107C4⋊#6157—46no—
108C4⋊#6157—43no—
109C4×#6157—26no—
110C4×#6157—212no—
111C4×#6157—46no—
112C4×#6157—46no—
113C4×#6157—43no—
114C4×#6157—46no—
115C4⋊C4×C53C22×C413yes—
116C4⋊C4×C53C22×C416yes—
117C4⋊C4×C53C22×C416yes—
118C4⋊C4×C53C22×C413yes—
119C4⋊C4×C53C22×C416yes—
120C4⋊C4×C53C22×C416yes—
121C42×C53C22×C411yescharacteristic
122C42×C53C22×C412yes—
123C42×C53C22×C412yes—
124C4⋊Dic53C22×C416yes—
125C4⋊Dic53C22×C413yes—
126C4⋊Dic53C22×C416yes—
127C4⋊Dic53C22×C413yes—
128C4⋊Dic53C22×C416yes—
129C4⋊Dic53C22×C416yes—
130#11719919C22×C416yes—
131#11719919C22×C416yes—
132#11719919C22×C416yes—
133#11719919C22×C416yes—
134#11719919C22×C416yes—
135#11719919C22×C416yes—
136#11719919C22×C416yes—
137#11719919C22×C416yes—
138#11719919C22×C416yes—
139#11719919C22×C416yes—
140C4×Dic53C22×C413yes—
141C4×Dic53C22×C416yes—
142C4×Dic53C22×C416yes—
143C22×D53C2511yescharacteristic
144C2×#6157—23no—
145C2×#6157—23no—
146C2×#6157—26no—
147C2×#6157—26no—
148C2×#6157—26no—
149C2×#6157—26no—
150C2×#6157—26no—
151C2×#6157—23no—
152C2×#6157—26no—
153C2×#61572−1+416yes—
154C2×C4×C53C23×C413yes—
155C2×C4×C53C23×C416yes—
156C2×C4×C53C23×C416yes—
157C2×Dic53C23×C413yes—
158C2×Dic53C23×C416yes—
159C2×Dic53C23×C416yes—
160C4×D53—23no—
161C4×D53—26no—
162C4×D53—26no—
163C4×D53—26no—
164C4×D53—23no—
165C4×D53—26no—
166#34530—531nomaximal, Sylow
167C22×C53C24×C411yescharacteristic
168C2×D53C2×2−1+413yes—
169C2×D53#58513yes—
170#6157—26no—
171#6157—46no—
172#6157—46no—
173#6157—41no—
174C4×C53—23no—
175C4×C53—26no—
176C4×C53—26no—
177Dic53—23no—
178Dic53—26no—
179Dic53—26no—
180C2×#564—531no—
181#2382—533no—
182#2371—536no—
183#2370—536no—
184#2336—536no—
185#2330—536no—
186C4×#125—533no—
187C2×C53C4×2−1+413yes—
188D53#356313yes—
189D53#356711yescharacteristic
190#564—536no—
191#564—536no—
192#564—533no—
193#564#615711yescharacteristic, complemented
194C2×#125—533no—
195C2×#125—536no—
196C2×#125—536no—
197#389—533no—
198#389—533no—
199#389—536no—
200#389—536no—
201#389—536no—
202#389—536no—
203#387—536no—
204#387—536no—
205#387—536no—
206#387—536no—
207#387—536no—
208#387—536no—
209#387—536no—
210#387—536no—
211#387—536no—
212#387—536no—
213#384—532no—
214#383—532no—
215#382—533no—
216C4×C4⋊C4—536no—
217C4×C4⋊C4—533no—
218C4×C4⋊C4—536no—
219C4×C4⋊C4—533no—
220C4×C4⋊C4—536no—
221C4×C4⋊C4—536no—
222C4×C4⋊C4—536no—
223#376—536no—
224C43—531no—
225C53#3453011yescharacteristic, complemented, Sylow
226#125—536no—
227#125—536no—
228#125—536no—
229#125—536no—
230#125—536no—
231#125—536no—
232#125—536no—
233#125—536no—
234#125—536no—
235#125—536no—
236#125—533no—
237#125—533no—
238#125—536no—
239#125—536no—
240#125—536no—
241#125—536no—
242#125—533no—
243#125—536no—
244#125—536no—
245#125C2×#615713yes—
246#125C2×#615716yes—
247#125C2×#615716yes—
248C2×C4⋊C4—533no—
249C2×C4⋊C4—536no—
250C2×C4⋊C4—536no—
251C2×C4⋊C4—533no—
252C2×C4⋊C4—536no—
253C2×C4⋊C4—536no—
254C2×C4⋊C4—536no—
255C2×C4⋊C4—533no—
256C2×C4⋊C4—533no—
257C2×C4⋊C4—536no—
258C2×C4⋊C4—536no—
259C2×C4⋊C4—536no—
260C2×C42—531no—
261C2×C42—532no—
262C2×C42—532no—
263C2×C42—533no—
264C2×C42—536no—
265C2×C42—533no—
266C2×C42—536no—
267C2×C42—536no—
268C2×C42—536no—
269#95—536no—
270#95—536no—
271#95—536no—
272#95—536no—
273#95—536no—
274#95—536no—
275#95—536no—
276#95—536no—
277#95—536no—
278#95—536no—
279C22×C4—533no—
280C22×C4—536no—
281C22×C4—533no—
282C22×C4—536no—
283C22×C4—536no—
284C22×C4—536no—
285C22×C4—531no—
286C4⋊C4—536no—
287C4⋊C4—536no—
288C4⋊C4—536no—
289C4⋊C4—536no—
290C4⋊C4—536no—
291C4⋊C4—536no—
292C4⋊C4—536no—
293C4⋊C4—536no—
294C4⋊C4—536no—
295C4⋊C4—536no—
296C4⋊C4—533no—
297C4⋊C4—536no—
298C4⋊C4—536no—
299C4⋊C4—533no—
300C4⋊C4—536no—
301C4⋊C4—536no—
302C4⋊C4—10612no—
303C4⋊C4—2126no—
304C4⋊C4—2123no—
305C4⋊C4—2126no—
306C4⋊C4—2126no—
307C4⋊C4—2123no—
308C4⋊C4C22×#615713yes—
309C4⋊C4C22×#615716yes—
310C4⋊C4C22×#615716yes—
311C4⋊C4C22×#615713yes—
312C4⋊C4C22×#615716yes—
313C4⋊C4C22×#615716yes—
314C42—533no—
315C42—536no—
316C42—536no—
317C42—1066no—
318C42—10612no—
319C42—2126no—
320C42—2126no—
321C42—2123no—
322C42—2126no—
323C42C22×#615711yescharacteristic
324C42C22×#615712yes—
325C42C22×#615712yes—
326C23—531no—
327C2×C4—536no—
328C2×C4—533no—
329C2×C4—536no—
330C2×C4—536no—
331C2×C4—1063no—
332C2×C4—1066no—
333C2×C4—1066no—
334C2×C4—1063no—
335C2×C4—1066no—
336C2×C4—1066no—
337C2×C4—1063no—
338C2×C4—1063no—
339C2×C4—1066no—
340C2×C4—1066no—
341C2×C4—1066no—
342C2×C4—1066no—
343C2×C4—1063no—
344C2×C4—1066no—
345C2×C4—1066no—
346C2×C4C23×#615713yes—
347C2×C4C23×#615716yes—
348C2×C4C23×#615716yes—
349C22—533no—
350C22—533no—
351C22C24×#615711yescharacteristic
352C4—23no—
353C4—26no—
354C4—26no—
355C4—1066no—
356C4—1063no—
357C4—1066no—
358C4—1066no—
359C4—2126no—
360C4—2126no—
361C4—2121no—
362C2—533no—
363C2—531no—
364C22−1+4×#615713yes—
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×#564C4
prime radical, cosocleC22×D53C25
hypercenter#564#6157
hypocenterC53#34530
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #564

Complements

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

NH ≅ G/Ntype
36. C53⋊#564C4semidirect
37. C53⋊#564C4semidirect
38. C53⋊#564C4semidirect
42. C53×#564C4semidirect
193. #564#6157semidirect
225. C53#34530semidirect

Maximal subgroups

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

subgroupindex
#202912612
#202912502
#202912492
#202912152
#202912092
#125×#61572
#564×D532
#3453053