Finite Group Databasebeta

← #27276743

Subgroups — #27276743

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×#13133210C211yesmaximal, characteristic
2#20274793C212yesmaximal, complemented
3#20273044C212yesmaximal, complemented
4#20273042C211yesmaximal, characteristic
5C53×#1462C211yesmaximal, characteristic
6C2×#13133210—24no—
7C2×#13133210—24no—
8C2×#13133210—22no—
9C2×#13133210—22no—
10C22×C4⋊Dic53C2211yescharacteristic
11C22×#11719919C411yescharacteristic
12C22×#11719919C2211yescharacteristic
13C22×C22⋊Dic53C411yescharacteristic
14C22×C22⋊Dic53C411yescharacteristic
15C22×C4×Dic53C411yescharacteristic
16#16810587C2212yes—
17#16810391—24no—
18#16810348—24no—
19#16810347—22no—
20C22×C22⋊C4×C53C2211yescharacteristic
21C53×#407C2212yes—
22C53×#323—22no—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C2×#11719917—22no—
28#13133210—216no—
29#13133210—216no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D414yes—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—22no—
39C2×#11719919—22no—
40C2×#11719919D414yes—
41C2×C22⋊Dic53—22no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—22no—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—22no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—22no—
49C2×C4×Dic53—22no—
50C22⋊#88322—22no—
51C22⋊#88322—22no—
52C25×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C2×M16×C53—22no—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—22no—
58C2×C22⋊C4×C53D412yes—
59C2×C22⋊C4×C53D412yes—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C24×C53—21no—
63C24×C53—21no—
64C24×C53—42no—
65C24×C53—42no—
66C24×C53—42no—
67C24×C53C22⋊C412yes—
68C24×C53C22×C411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53C2×D412yes—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D412yes—
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—42no—
87C22×Dic53—42no—
88C22×Dic53—42no—
89C22×Dic53—42no—
90C22×Dic53C22⋊C412yes—
91C22×Dic53C22⋊C412yes—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C22⋊C412yes—
94M16×C53—44no—
95C2×C8×C53—42no—
96C22⋊C4×C53—24no—
97C22⋊C4×C53—24no—
98C22⋊Dic53—28no—
99C22⋊Dic53—28no—
100C4⋊Dic53—24no—
101C4⋊Dic53—24no—
102#11719919—28no—
103#11719919—24no—
104#11719919—24no—
105C4×Dic53—28no—
106#11719917—44no—
107C2×#88322—42no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—24no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—88no—
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53C2×C22⋊C411yescharacteristic
124C23×C53C2×C22⋊C411yescharacteristic
125C2×C4×C53—24no—
126C2×C4×C53—22no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53#12012yes—
134C2×C4×C53#12012yes—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—22no—
139C2×Dic53—24no—
140C2×Dic53—42no—
141C2×Dic53—42no—
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—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—84no—
165C22×C53—84no—
166C22×C53—84no—
167C22×C53#37911yescharacteristic
168C22×C53#41012yes—
169C22×C53#41312yes—
170C22×C53#41312yes—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—42no—
174Dic53—42no—
175Dic53—42no—
176Dic53—42no—
177Dic53—42no—
178C22×#126—531no—
179#1821—532no—
180#1463—532no—
181#1462—531no—
182#1462D5311yescharacteristic
183C2×C53—22no—
184C2×C53—22no—
185C2×C53—22no—
186C2×C53—82no—
187C2×C53#178211yescharacteristic
188C2×C53#178511yescharacteristic
189C2×C53#178511yescharacteristic
190C2×#126—1064no—
191C2×#126—1064no—
192C2×#126—1062no—
193C2×#126—1062no—
194C22×C4⋊C4—531no—
195C22×C4⋊C4—531no—
196C22×C4⋊C4—531no—
197C22×C22⋊C4—531no—
198C22×C22⋊C4—531no—
199C22×C22⋊C4C2×D5311yescharacteristic
200C22×C42—531no—
201#407—532no—
202#407C2×D5312yes—
203#343—1064no—
204#324—1064no—
205#323—22no—
206#323—1062no—
207C53#855611yescharacteristic, complemented, Sylow
208C25C4×D5311yescharacteristic
209C23×C4—531no—
210C23×C4—531no—
211C23×C4—531no—
212C23×C4—531no—
213C23×C4C4×D5311yescharacteristic
214C23×C4C22×D5311yescharacteristic
215C2×M16—22no—
216C2×M16—1062no—
217#126—10616no—
218#126—10616no—
219C2×C4⋊C4—534no—
220C2×C4⋊C4—534no—
221C2×C4⋊C4—1062no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1062no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1062no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—1062no—
233C2×C22⋊C4—1062no—
234C2×C22⋊C4—1062no—
235C2×C22⋊C4—1062no—
236C2×C22⋊C4—1062no—
237C2×C22⋊C4—1062no—
238C2×C22⋊C4D21212yes—
239C2×C22⋊C4C22⋊D5312yes—
240C2×C42—1062no—
241C2×C42—1062no—
242C2×C42—1062no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—1062no—
246C22⋊C8—1062no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24#1171992112yes—
253C24C2×C4×D5311yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—532no—
261C22×C4—532no—
262C22×C4—1061no—
263C22×C4—1061no—
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—2122no—
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—1068no—
284C4⋊C4—1064no—
285C4⋊C4—1064no—
286C4⋊C4—1064no—
287C4⋊C4—1064no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—1068no—
291C22⋊C4—1068no—
292C42—1068no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—24no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—88no—
307C23C22⋊C4×D5311yescharacteristic
308C23C22⋊C4×D5311yescharacteristic
309C23C2×#1171992111yescharacteristic
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—1064no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—2122no—
324C2×C4—2122no—
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—42no—
346C22—42no—
347C22—42no—
348C22—84no—
349C22—84no—
350C22—84no—
351C22#1681047811yescharacteristic
352C22#1681059512yes—
353C22#1681060412yes—
354C22#1681060512yes—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—2122no—
359C4—2122no—
360C4—2122no—
361C4—2122no—
362C2—22no—
363C2—22no—
364C2—22no—
365C2—82no—
366C2#2027463111yescharacteristic
367C2#2027464011yescharacteristic
368C2#2027464111yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1462C2
socleC22×C53#379
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1462D53
hypocenterC53#8556
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #20274793C2semidirect
3. #20273044C2semidirect
207. C53#8556semidirect

Maximal subgroups

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

subgroupindex
C22×#131332102
#202747932
#202730442
#202730422
C53×#14622
#855653