Finite Group Databasebeta

← #27309360

Subgroups — #27309360

Order 13568.

2058 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133201C211yesmaximal, characteristic
2C2×C22⋊C4×Dic53C212yesmaximal, complemented
3#20272964C211yesmaximal, characteristic, complemented
4#20272954C212yesmaximal
5#20272951C214yesmaximal, complemented
6#20272949C214yesmaximal, complemented
7C53×#1440C211yesmaximal, characteristic, complemented
8C23×C22⋊D53C2211yescharacteristic
9C24×Dic53C2211yescharacteristic
10C2×#13133201—28no—
11C2×#13133201C418yescomplemented
12C22×C4⋊Dic53C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×#11719921C2214yes—
15C22×#11719921C2212yes—
16C22×C4×Dic53C2214yes—
17C22⋊C4×Dic53—28no—
18C2×#13133124C2214yes—
19C2×#13133124C2214yes—
20C2×#13133124C2214yes—
21C22×C22⋊C4×C53C2212yes—
22C22×C22⋊C4×C53C2211yescharacteristic
23C2×#95×C53C2214yes—
24C24×D53C2311yescharacteristic
25C22×C22⋊D53—24no—
26C22×C22⋊D53—44no—
27C22×C22⋊D53C2×C414yes—
28C23×Dic53—44no—
29C23×Dic53C2×C414yes—
30C23×Dic53D418yes—
31C23×Dic53C2311yescharacteristic
32C23×Dic53C2314yes—
33C23×Dic53C2312yes—
34#13133201—232no—
35C2×C4⋊Dic53—24no—
36C2×C4⋊Dic53—24no—
37C2×C4⋊Dic53C2×C418yes—
38C2×C22⋊Dic53—24no—
39C2×C22⋊Dic53—28no—
40C2×#11719921—28no—
41C2×#11719921—24no—
42C2×#11719921—24no—
43C2×#11719921—28no—
44C2×#11719921—44no—
45C2×#11719921C2×C418yes—
46C2×C4×Dic53—28no—
47C2×C4×Dic53—44no—
48C2×C4×Dic53—44no—
49#13133124—28no—
50#13133124—28no—
51#13133124—44no—
52C25×C53C2311yescharacteristic
53C23×C4×C53C2312yes—
54C23×C4×C53C2314yes—
55C2×C22⋊C4×C53—24no—
56C2×C22⋊C4×C53—24no—
57C2×C22⋊C4×C53—28no—
58C2×C22⋊C4×C53C2×C414yes—
59#95×C53—28no—
60C24×C53—24no—
61C24×C53—24no—
62C24×C53—24no—
63C24×C53C22×C412yes—
64C24×C53C2×D414yes—
65C24×C53C2411yescharacteristic
66C23×D53—22no—
67C23×D53—24no—
68C23×D53—24no—
69C23×D53—24no—
70C23×D53C22×C412yes—
71C22×C4×C53—24no—
72C22×C4×C53—22no—
73C22×C4×C53—22no—
74C22×C4×C53—24no—
75C22×C4×C53—24no—
76C22×C4×C53—24no—
77C22×C4×C53—24no—
78C22×C4×C53—24no—
79C22×C4×C53—24no—
80C22×C4×C53—24no—
81C22×C4×C53C22×C414yes—
82C22×C4×C53#4118yes—
83C2×C22⋊D53—28no—
84C2×C22⋊D53—48no—
85C2×C22⋊D53—416no—
86C22×Dic53—24no—
87C22×Dic53—24no—
88C22×Dic53—24no—
89C22×Dic53—22no—
90C22×Dic53—24no—
91C22×Dic53—24no—
92C22×Dic53—24no—
93C22×Dic53—24no—
94C22×Dic53—24no—
95C22×Dic53—24no—
96C22×Dic53—48no—
97C22×Dic53—416no—
98C22×Dic53C22⋊C4116yescomplemented
99C22×Dic53C22×C412yes—
100C22×Dic53C22×C414yes—
101C22×Dic53C2×D414yes—
102C22×Dic53C2×D414yes—
103C22×Dic53#4118yes—
104C22⋊C4×C53—216no—
105C22⋊C4×C53—28no—
106C22⋊Dic53—216no—
107#11719921—216no—
108#11719921—416no—
109C4⋊Dic53—216no—
110C4×Dic53—416no—
111C23×C53—28no—
112C23×C53—24no—
113C23×C53—24no—
114C23×C53—24no—
115C23×C53—28no—
116C23×C53—28no—
117C23×C53—44no—
118C23×C53—44no—
119C23×C53—44no—
120C23×C53—44no—
121C23×C53C2×C22⋊C414yes—
122C23×C53C23×C411yescharacteristic
123C23×C53C22×D412yes—
124C23×C53C2×#4114yes—
125C23×C53C2×#4114yes—
126C23×C53H(5,𝔽2)14yes—
127C22×D53—22no—
128C22×D53—28no—
129C22×D53—24no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—44no—
133C22×D53—44no—
134C22×D53—44no—
135C22×D53—44no—
136C22×D53—44no—
137C22×D53—44no—
138C2×C4×C53—24no—
139C2×C4×C53—28no—
140C2×C4×C53—28no—
141C2×C4×C53—28no—
142C2×C4×C53—28no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—44no—
146C2×C4×C53—44no—
147C2×C4×C53—44no—
148C2×C4×C53—44no—
149C2×C4×C53—44no—
150C22⋊D53—416no—
151C2×Dic53—24no—
152C2×Dic53—24no—
153C2×Dic53—28no—
154C2×Dic53—28no—
155C2×Dic53—22no—
156C2×Dic53—24no—
157C2×Dic53—28no—
158C2×Dic53—28no—
159C2×Dic53—28no—
160C2×Dic53—416no—
161C2×Dic53—44no—
162C2×Dic53—44no—
163C2×Dic53—44no—
164C2×Dic53—44no—
165C2×Dic53—44no—
166C2×Dic53C2×C22⋊C418yes—
167#14167—531nomaximal, Sylow
168C22×C53—24no—
169C22×C53—28no—
170C22×C53—48no—
171C22×C53—48no—
172C22×C53—48no—
173C22×C53C22×C22⋊C411yescharacteristic
174C22×C53C4×#4114yes—
175C22×C53#51812yes—
176C22×C53#52512yes—
177C22×C53#52512yes—
178C22×C53#53214yes—
179C22×C53#53414yes—
180C22×C53#55114yes—
181C22×C53#55314yes—
182C22×C53#55314yes—
183C22×C53#55914yes—
184C2×D53—42no—
185C2×D53—48no—
186C2×D53—44no—
187C2×D53—84no—
188C2×D53—84no—
189C4×C53—48no—
190C4×C53—44no—
191Dic53—28no—
192Dic53—42no—
193Dic53—48no—
194C22×#123—531no—
195C2×C4×C22⋊C4—532no—
196#1440—531no—
197#1440D5311yescharacteristic, complemented
198#1438—532no—
199#1437—534no—
200#1436—534no—
201C2×C53—44no—
202C2×C53#229712yes—
203C2×C53#230811yescharacteristic
204C2×C53#237414yes—
205C2×C53#260214yes—
206C2×C53#260214yes—
207D53—82no—
208C23×D4—531no—
209C24×C4—531no—
210C2×#123—538no—
211C2×#123—1068no—
212C22×C4⋊C4—532no—
213C22×C22⋊C4—532no—
214C22×C22⋊C4—532no—
215C22×C22⋊C4—534no—
216C22×C22⋊C4C2×D5312yes—
217C22×C22⋊C4C2×D5311yescharacteristic
218C22×C42—534no—
219C4×C22⋊C4—1068no—
220C2×#95—534no—
221C2×#95—534no—
222C2×#95—534no—
223C2×#95C2×D5314yes—
224C53#1416711yescharacteristic, complemented, Sylow
225C25—531no—
226C25C22×D5311yescharacteristic
227C22×D4—534no—
228C22×D4—1064no—
229C22×D4—2124no—
230C23×C4—532no—
231C23×C4—534no—
232C23×C4—538no—
233C23×C4—534no—
234C23×C4—531no—
235C23×C4—2124no—
236C23×C4C22×D5312yes—
237C23×C4C22×D5314yes—
238#123—10632no—
239C2×C4⋊C4—538no—
240C2×C4⋊C4—1064no—
241C2×C4⋊C4—1064no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—538no—
246C2×C22⋊C4—1064no—
247C2×C22⋊C4—1064no—
248C2×C22⋊C4—1068no—
249C2×C22⋊C4—1064no—
250C2×C22⋊C4—1068no—
251C2×C22⋊C4—1068no—
252C2×C22⋊C4—2124no—
253C2×C22⋊C4C4×D5314yes—
254C2×C42—1068no—
255C2×C42—2124no—
256C2×C42—2124no—
257#95—28no—
258#95—1068no—
259#95—1068no—
260#95—2124no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—532no—
265C24—1062no—
266C24—1064no—
267C24—1064no—
268C24—1064no—
269C24C2×C4×D5312yes—
270C24D4×D5314yes—
271C24C23×D5311yescharacteristic
272C2×D4—1068no—
273C2×D4—2128no—
274C2×D4—21216no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—534no—
286C22×C4—534no—
287C22×C4—534no—
288C22×C4—5316no—
289C22×C4—532no—
290C22×C4—538no—
291C22×C4—1064no—
292C22×C4—1064no—
293C22×C4—1064no—
294C22×C4—1062no—
295C22×C4—1064no—
296C22×C4—1064no—
297C22×C4—1064no—
298C22×C4—1064no—
299C22×C4—1064no—
300C22×C4—1064no—
301C22×C4—2128no—
302C22×C4—21216no—
303C22×C4C2×C4×D5314yes—
304C22×C4#1171994718yes—
305C4⋊C4—10616no—
306C22⋊C4—216no—
307C22⋊C4—28no—
308C22⋊C4—10616no—
309C22⋊C4—10616no—
310C22⋊C4—21216no—
311C42—21216no—
312C23—28no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—28no—
317C23—28no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—1062no—
323C23—1068no—
324C23—1064no—
325C23—2124no—
326C23—2124no—
327C23—2124no—
328C23—2124no—
329C23—2122no—
330C23—2122no—
331C23—2124no—
332C23—2124no—
333C23C22⋊C4×D5314yes—
334C23C22×C4×D5311yescharacteristic
335C23C2×D4×D5312yes—
336C23#41×D5314yes—
337C23C2×#1171994714yes—
338C23#1313326314yes—
339D4—21216no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—538no—
353C2×C4—1064no—
354C2×C4—1064no—
355C2×C4—1068no—
356C2×C4—1068no—
357C2×C4—1068no—
358C2×C4—1068no—
359C2×C4—1062no—
360C2×C4—1064no—
361C2×C4—1068no—
362C2×C4—21216no—
363C2×C4—2124no—
364C2×C4—2124no—
365C2×C4—2124no—
366C2×C4—2124no—
367C2×C4—2124no—
368C22—24no—
369C22—28no—
370C22—48no—
371C22—48no—
372C22—48no—
373C22—4244no—
374C22—4244no—
375C22—2122no—
376C22—2128no—
377C22—2124no—
378C22C2×C22⋊C4×D5311yescharacteristic
379C22C4×#1171994714yes—
380C22#1681109712yes—
381C22#1681112912yes—
382C22#1681113112yes—
383C22#1681116814yes—
384C22#1681118414yes—
385C22#1681129614yes—
386C22#1681132414yes—
387C22#1681132514yes—
388C22#1681135514yes—
389C4—48no—
390C4—44no—
391C4—1068no—
392C4—2128no—
393C4—2122no—
394C2—44no—
395C2—4242no—
396C2#2027720012yes—
397C2#2027729211yescharacteristic
398C2#2027784514yes—
399C2#2028032214yes—
400C2#2028032814yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1440C2
socle, cosocleC24×C53C24
prime radicalC23×C22⋊D53C22
hypercenter#1440D53
hypocenterC53#14167
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×Dic53C2semidirect
3. #20272964C2semidirect
5. #20272951C2semidirect
6. #20272949C2semidirect
7. C53×#1440C2semidirect
11. C2×#13133201C4semidirect
98. C22×Dic53C22⋊C4semidirect
197. #1440D53semidirect
224. C53#14167semidirect

Maximal subgroups

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

subgroupindex
C22×#131332012
C2×C22⋊C4×Dic532
#202729642
#202729542
#202729512
#202729492
C53×#14402
#1416753