Finite Group Databasebeta

← #28313366

Subgroups — #28313366

Order 13568.

1782 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811565C212yesmaximal
2#20289901C218yesmaximal, complemented
3#20289889C214yesmaximal
4C2×#95×D53C211yesmaximal, characteristic
5C22×C4⋊#6157C2214yes—
6C22×C22⋊#6157C2214yes—
7#16811565—28no—
8C2×(C2×#6157)⋊C4C2214yes—
9C2×(C2×#6157)⋊C4C2218yes—
10C2×(C2×#6157)⋊C4C2218yes—
11C23×C4×D53C2212yes—
12C23×C4×D53C2211yescharacteristic
13C2×#13133124C412yescomplemented
14C2×#13133124C411yescharacteristic, complemented
15#95×D53C414yescomplemented
16#95×D53C2214yes—
17C2×#95×C53C411yescharacteristic, complemented
18C23×#6157C2318yes—
19C2×C4⋊#6157—28no—
20C2×C4⋊#6157—48no—
21C2×C22⋊#6157—28no—
22C2×C22⋊#6157—216no—
23(C2×#6157)⋊C4—28no—
24(C2×#6157)⋊C4—48no—
25(C2×#6157)⋊C4—48no—
26C24×D53C2311yescharacteristic
27C23×Dic53C2×C412yes—
28C23×Dic53C2×C411yescharacteristic
29C22×C4×D53—28no—
30C22×C4×D53—44no—
31C22×C4×D53—44no—
32C22×C4×D53—44no—
33C22×C4×D53C2×C414yes—
34C22×C4×D53C2×C412yes—
35C22×C4×D53D414yes—
36C22×C4×D53Q814yes—
37C22×C4×D53C2314yes—
38C22×C4×D53C2312yes—
39#13133124C2×C414yes—
40#13133124C2×C418yes—
41C23×C4×C53C2×C412yes—
42C23×C4×C53C2×C411yescharacteristic
43#95×C53C2×C414yes—
44C24×C53C22×C411yescharacteristic
45C23×D53—24no—
46C23×D53—22no—
47C23×D53—22no—
48C23×D53C22×C411yescharacteristic
49C23×D53C2×D412yes—
50C23×D53C2×Q812yes—
51C23×D53#4118yes—
52C23×D53C2411yescharacteristic
53C22×#6157—28no—
54C22×#6157—216no—
55C22×#6157—28no—
56C22×#6157—216no—
57C22×#6157—28no—
58C22×C4×C53—24no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C4⋊C414yes—
63C22×C4×C53C22×C414yes—
64C22×C4×C53C22×C412yes—
65C22×Dic53—24no—
66C22×Dic53—42no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C22×Dic53C4⋊C414yes—
70C22×Dic53C22×C414yes—
71C22×Dic53C22×C412yes—
72C2×C4×D53—28no—
73C2×C4×D53—28no—
74C2×C4×D53—28no—
75C2×C4×D53—28no—
76C2×C4×D53—28no—
77C2×C4×D53—48no—
78C2×C4×D53—48no—
79C2×C4×D53—44no—
80C2×C4×D53—44no—
81C2×C4×D53—44no—
82C2×C4×D53—432no—
83C2×C4×D53C4⋊C418yes—
84C2×C4×D53C2×D414yes—
85C2×C4×D53C2×Q814yes—
86C22⋊#6157—416no—
87C4⋊#6157—416no—
88C23×C53—21no—
89C23×C53—22no—
90C23×C53—21no—
91C23×C53C2×C4⋊C412yes—
92C23×C53#11714yes—
93C23×C53C23×C411yescharacteristic
94C22×D53—21no—
95C22×D53—22no—
96C22×D53—21no—
97C22×D53—24no—
98C22×D53—24no—
99C22×D53—216no—
100C22×D53—216no—
101C22×D53—24no—
102C22×D53—24no—
103C22×D53C2×C4⋊C412yes—
104C22×D53#11714yes—
105C22×D53#12418yes—
106C22×D53#12518yes—
107C22×D53C22×D411yescharacteristic
108C22×D53C22×Q811yescharacteristic
109C22×D53C2×#4114yes—
110C22×D53H(5,𝔽2)14yes—
111C22×D53H(5,𝔽2)12yes—
112C22×D532−1+412yes—
113C2×#6157—48no—
114C2×#6157—48no—
115C2×#6157—48no—
116C2×#6157—416no—
117C2×#6157—416no—
118C2×C4×C53—24no—
119C2×C4×C53—24no—
120C2×C4×C53—24no—
121C2×C4×C53—24no—
122C2×C4×C53—44no—
123C2×C4×C53—48no—
124C2×C4×C53C2×C4⋊C414yes—
125C2×Dic53—24no—
126C2×Dic53—24no—
127C2×Dic53—24no—
128C2×Dic53—24no—
129C2×Dic53—44no—
130C2×Dic53—48no—
131C2×Dic53C2×C4⋊C414yes—
132C4×D53—28no—
133C4×D53—28no—
134C4×D53—48no—
135C4×D53—416no—
136C4×D53—416no—
137#14347—531nomaximal, Sylow
138C22×C53—24no—
139C22×C53—24no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53#38214yes—
143C22×C53C22×C4⋊C411yescharacteristic
144C22×C53C2×#11712yes—
145C22×C53#51811yescharacteristic
146C22×C53#51812yes—
147C22×C53#51911yescharacteristic
148C2×D53—24no—
149C2×D53—24no—
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—216no—
153C2×D53—216no—
154C2×D53#38214yes—
155C2×D53C2×#12412yes—
156C2×D53C2×#12512yes—
157C2×D53#53412yes—
158C2×D53#53712yes—
159C2×D53#53818yes—
160C2×D53#53918yes—
161C2×D53#54312yes—
162C2×D53#54412yes—
163#6157—88no—
164C4×C53—24no—
165C4×C53—44no—
166Dic53—24no—
167Dic53—44no—
168C2×#380—532no—
169C22×#95—531no—
170#1442—538no—
171C22⋊#95—534no—
172C2×C53—24no—
173C2×C53C2×#38211yescharacteristic
174C2×C53#231212yes—
175C2×C53#231312yes—
176C2×C53#231511yescharacteristic
177C2×C53#231611yescharacteristic
178D53—24no—
179D53#263914yes—
180D53#264614yes—
181C24×C4—531no—
182C24×C4—532no—
183C22×C4⋊C4—534no—
184C22×C22⋊C4—534no—
185#380—1068no—
186C2×#95—532no—
187C2×#95—531no—
188C2×#95—534no—
189C2×#95—534no—
190C2×#95—534no—
191C2×#95—538no—
192C2×#95—538no—
193C2×#95#615711yescharacteristic, complemented
194C53#1434711yescharacteristic, complemented, Sylow
195C25—531no—
196C23×C4—531no—
197C23×C4—532no—
198C23×C4—534no—
199C23×C4—534no—
200C23×C4—532no—
201C23×C4—532no—
202C23×C4—534no—
203C23×C4—534no—
204C23×C4—538no—
205C23×C4—1068no—
206C23×C4—2124no—
207C23×C4—2124no—
208C23×C4—2124no—
209C23×C4C2×#615712yes—
210C23×C4C2×#615711yescharacteristic
211C2×C4⋊C4—1068no—
212C2×C4⋊C4—2128no—
213C2×C22⋊C4—1068no—
214C2×C22⋊C4—10616no—
215#95—538no—
216#95—534no—
217#95—1068no—
218#95—2128no—
219#95—2128no—
220#95C2×#615714yes—
221C24—531no—
222C24—532no—
223C24—532no—
224C24—538no—
225C24—531no—
226C24—1062no—
227C24—1064no—
228C24—1062no—
229C24C22×#615711yescharacteristic
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—532no—
235C22×C4—534no—
236C22×C4—534no—
237C22×C4—534no—
238C22×C4—538no—
239C22×C4—534no—
240C22×C4—1068no—
241C22×C4—1068no—
242C22×C4—1068no—
243C22×C4—1068no—
244C22×C4—1068no—
245C22×C4—1064no—
246C22×C4—1068no—
247C22×C4—1068no—
248C22×C4—10616no—
249C22×C4—10616no—
250C22×C4—1068no—
251C22×C4—2124no—
252C22×C4—2124no—
253C22×C4—2124no—
254C22×C4—2128no—
255C22×C4—2128no—
256C22×C4—21232no—
257C22×C4—2122no—
258C22×C4—2122no—
259C22×C4—2122no—
260C22×C4C4⋊#615714yes—
261C22×C4C22×#615714yes—
262C22×C4C22×#615712yes—
263C4⋊C4—21216no—
264C22⋊C4—21216no—
265C23—21no—
266C23—22no—
267C23—21no—
268C23—531no—
269C23—531no—
270C23—534no—
271C23—532no—
272C23—534no—
273C23—532no—
274C23—538no—
275C23—538no—
276C23—532no—
277C23—534no—
278C23—10616no—
279C23—1064no—
280C23—1064no—
281C23—10616no—
282C23—1064no—
283C23—1064no—
284C23—1061no—
285C23—1062no—
286C23—1061no—
287C23C2×C4⋊#615712yes—
288C23#1313329614yes—
289C23C23×#615711yescharacteristic
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—534no—
297C2×C4—1068no—
298C2×C4—1068no—
299C2×C4—1064no—
300C2×C4—1064no—
301C2×C4—1064no—
302C2×C4—1064no—
303C2×C4—2128no—
304C2×C4—21216no—
305C2×C4—21216no—
306C2×C4—2124no—
307C2×C4—2128no—
308C2×C4—2128no—
309C2×C4—21216no—
310C2×C4—21216no—
311C2×C4—2128no—
312C2×C4—2128no—
313C2×C4C2×C4⋊#615714yes—
314C22—24no—
315C22—24no—
316C22—22no—
317C22—22no—
318C22—532no—
319C22—532no—
320C22—538no—
321C22—538no—
322C22—532no—
323C22—532no—
324C22—532no—
325C22—532no—
326C22—534no—
327C22—10616no—
328C22—10616no—
329C22—1064no—
330C22—1064no—
331C22—1062no—
332C22—1062no—
333C22#1681156714yes—
334C22C22×C4⋊#615711yescharacteristic
335C22C2×#1313329612yes—
336C22#1681169011yescharacteristic
337C22#1681169012yes—
338C22#1681169111yescharacteristic
339C4—24no—
340C4—44no—
341C4—1064no—
342C4—4248no—
343C4—2124no—
344C2—24no—
345C2—534no—
346C2—534no—
347C2—1064no—
348C2C2×#1681156711yescharacteristic
349C2#2029119112yes—
350C2#2029119212yes—
351C2#2029119411yescharacteristic
352C2#2029119511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC2×#95×C53C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC2×#95#6157
hypocenterC53#14347
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #20289901C2semidirect
13. C2×#13133124C4semidirect
14. C2×#13133124C4semidirect
15. #95×D53C4semidirect
17. C2×#95×C53C4semidirect
193. C2×#95#6157semidirect
194. C53#14347semidirect

Maximal subgroups

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

subgroupindex
C2×#168115652
#202899012
#202898892
C2×#95×D532
#1434753