Finite Group Databasebeta

← #27276744

Subgroups — #27276744

Order 13568.

842 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×#13133212C211yesmaximal, characteristic
2#20274793C212yesmaximal, complemented
3#20273045C211yesmaximal, characteristic, complemented
4#20273041C212yesmaximal
5C53×#1463C211yesmaximal, characteristic, complemented
6C2×#13133212—24no—
7C2×#13133212—24no—
8C2×#13133212—22no—
9C2×#13133212—22no—
10C22×C4⋊Dic53C2211yescharacteristic
11C22×#11719919C411yescharacteristic
12C22×#11719921C411yescharacteristic
13C22×#11719921C411yescharacteristic
14C22×#11719921C2211yescharacteristic
15C22×C4×Dic53C411yescharacteristic
16#16810587C2212yes—
17#16810391—24no—
18#16810349—22no—
19#16810346—24no—
20C22×C4⋊C4×C53C2211yescharacteristic
21C53×#407C2212yes—
22C53×#324—22no—
23C24×D53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C2×#11719917—22no—
28#13133212—216no—
29#13133212—216no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D414yes—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921—22no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×#11719921—22no—
45C2×#11719921—22no—
46C2×#11719921D414yes—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—22no—
49C2×C4×Dic53—22no—
50C22⋊#88322—22no—
51C22⋊#88322—22no—
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C2×M16×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53D412yes—
59C2×C4⋊C4×C53D412yes—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C24×C53C22×C411yescharacteristic
63C23×D53—21no—
64C23×D53—21no—
65C23×D53—42no—
66C23×D53—42no—
67C23×D53—42no—
68C23×D53C22⋊C412yes—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C2×D412yes—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—42no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53C22⋊C412yes—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C22⋊C412yes—
94M16×C53—44no—
95C2×C8×C53—42no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—24no—
98#11719921—28no—
99#11719921—28no—
100#11719921—24no—
101#11719921—24no—
102C4⋊Dic53—24no—
103C4⋊Dic53—24no—
104#11719919—28no—
105C4×Dic53—28no—
106#11719917—44no—
107C2×#88322—42no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53C2×C22⋊C411yescharacteristic
113C23×C53C2×C22⋊C411yescharacteristic
114C22×D53—22no—
115C22×D53—24no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—42no—
124C22×D53—88no—
125C2×C4×C53—24no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—24no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53#12012yes—
138C2×C4×C53#12012yes—
139C2×Dic53—22no—
140C2×Dic53—22no—
141C2×Dic53—22no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C8×C53—82no—
152#88322—82no—
153#8556—531nomaximal, Sylow
154C22×C53—28no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53#37911yescharacteristic
162C22×C53#41012yes—
163C22×C53#41312yes—
164C22×C53#41312yes—
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—42no—
168C2×D53—84no—
169C2×D53—84no—
170C2×D53—84no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—42no—
174C4×C53—42no—
175Dic53—42no—
176Dic53—42no—
177Dic53—42no—
178C22×#126—531no—
179#1821—532no—
180#1463—531no—
181#1463D5311yescharacteristic, complemented
182#1462—532no—
183C2×C53—22no—
184C2×C53—22no—
185C2×C53—22no—
186C2×C53#178211yescharacteristic
187C2×C53#178511yescharacteristic
188C2×C53#178511yescharacteristic
189D53—82no—
190C2×#126—1064no—
191C2×#126—1064no—
192C2×#126—1062no—
193C2×#126—1062no—
194C22×C4⋊C4—531no—
195C22×C4⋊C4—531no—
196C22×C4⋊C4C2×D5311yescharacteristic
197C22×C22⋊C4—531no—
198C22×C22⋊C4—531no—
199C22×C22⋊C4—531no—
200C22×C42—531no—
201#407—532no—
202#407C2×D5312yes—
203#343—1064no—
204#324—22no—
205#324—1062no—
206#323—1064no—
207C53#855611yescharacteristic, complemented, Sylow
208C25—531no—
209C23×C4—531no—
210C23×C4—531no—
211C23×C4—531no—
212C23×C4C4×D5311yescharacteristic
213C23×C4C4×D5311yescharacteristic
214C23×C4C22×D5311yescharacteristic
215C2×M16—22no—
216C2×M16—1062no—
217#126—10616no—
218#126—10616no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—534no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4D21212yes—
229C2×C4⋊C4C22⋊D5312yes—
230C2×C22⋊C4—534no—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4—1062no—
233C2×C22⋊C4—1062no—
234C2×C22⋊C4—1062no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1062no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1062no—
239C2×C22⋊C4—1062no—
240C2×C42—1062no—
241C2×C42—1062no—
242C2×C42—1062no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—1062no—
246C22⋊C8—1062no—
247C24—532no—
248C24—1061no—
249C24—1061no—
250C24—2122no—
251C24—2122no—
252C24—2122no—
253C24C2×C4×D5311yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—532no—
262C22×C4—532no—
263C22×C4—532no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—2122no—
271C22×C4—2122no—
272C22×C4—2122no—
273C22×C4#1171992112yes—
274C22×C4#1171992112yes—
275C22×C4C2×D21211yescharacteristic
276C22×C4D4×D5312yes—
277C22×C4D4×D5312yes—
278C22×C4C2×C22⋊D5311yescharacteristic
279M16—44no—
280M16—2124no—
281C2×C8—42no—
282C2×C8—2122no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—1068no—
286C4⋊C4—1064no—
287C4⋊C4—1064no—
288C22⋊C4—1068no—
289C22⋊C4—1068no—
290C22⋊C4—1064no—
291C22⋊C4—1064no—
292C42—1068no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—1064no—
297C23—1062no—
298C23—4248no—
299C23—2122no—
300C23—2122no—
301C23—2122no—
302C23—2122no—
303C23—2122no—
304C23—2122no—
305C23—2122no—
306C23—2122no—
307C23C22⋊C4×D5311yescharacteristic
308C23C22⋊C4×D5311yescharacteristic
309C23C2×#1171992111yescharacteristic
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—1062no—
323C2×C4—1062no—
324C2×C4—1062no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4#1313318512yes—
335C2×C4#1313318712yes—
336C8—82no—
337C8—4242no—
338C22—28no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—4244no—
346C22—4244no—
347C22—4244no—
348C22—2122no—
349C22—2122no—
350C22—2122no—
351C22#1681047811yescharacteristic
352C22#1681059512yes—
353C22#1681060512yes—
354C22#1681060612yes—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—42no—
359C4—2122no—
360C4—2122no—
361C4—2122no—
362C2—22no—
363C2—22no—
364C2—22no—
365C2—4242no—
366C2#2027463111yescharacteristic
367C2#2027464111yescharacteristic
368C2#2027464211yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1463C2
socleC22×C53#379
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1463D53
hypocenterC53#8556
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #20274793C2semidirect
3. #20273045C2semidirect
5. C53×#1463C2semidirect
181. #1463D53semidirect
207. C53#8556semidirect

Maximal subgroups

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

subgroupindex
C22×#131332122
#202747932
#202730452
#202730412
C53×#14632
#855653