Finite Group Databasebeta

← #27294752

Subgroups — #27294752

Order 13568.

572 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#20274789C211yesmaximal, characteristic
2#20273042C211yesmaximal, characteristic
3#20272715C214yesmaximal, complemented
4C2×C53×#397C211yesmaximal, characteristic
5#16810587—21no—
6#16810479—21no—
7#16810390—21no—
8#16810390C2212yes—
9#16810347—21no—
10#16810347C2212yes—
11C22×C4⋊C4×C53C411yescharacteristic, complemented
12C22×C22⋊C4×C53—21no—
13C22×C22⋊C4×C53C2211yescharacteristic
14C53×#397—21no—
15C53×#397—21no—
16C53×#397C412yescomplemented
17C53×#397C2212yes—
18C2×#95×C53—21no—
19C2×#95×C53C411yescharacteristic, complemented
20C2×#11719917—41no—
21C2×C22⋊Dic53—22no—
22C2×C22⋊Dic53—41no—
23C2×C22⋊Dic53—41no—
24C22⋊#88322—22no—
25C22⋊#88322—41no—
26C25×C53D411yescharacteristic
27C23×C4×C53—21no—
28C23×C4×C53—21no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—41no—
33C2×C4⋊C4×C53—41no—
34C2×C4⋊C4×C53C2×C412yes—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—41no—
38C2×C22⋊C4×C53—41no—
39C2×C22⋊C4×C53—44no—
40C2×C22⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44#95×C53—22no—
45#95×C53—21no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C24×C53—22no—
49C24×C53—21no—
50C24×C53—21no—
51C24×C53—41no—
52C24×C53—42no—
53C24×C53—42no—
54C24×C53C22⋊C411yescharacteristic
55C24×C53C22⋊C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—44no—
84C22⋊C4×C53—44no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C22⋊Dic53—82no—
88C22⋊Dic53—82no—
89#11719917—82no—
90C2×#88322—42no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—48no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—81no—
110C23×C53—81no—
111C23×C53—84no—
112C23×C53#9911yescharacteristic
113C23×C53#9911yescharacteristic
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53AGL(1,ℤ/8)12yes—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—81no—
134C2×C4×C53—84no—
135C2×C4×C53—82no—
136C2×C4×C53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C2×Dic53—82no—
140#88322—82no—
141#11797—531nomaximal, Sylow
142C22×C53—24no—
143C22×C53—24no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—84no—
160C22×C53—84no—
161C22×C53—81no—
162C22×C53#35212yes—
163C22×C53#35412yes—
164C22×C53C2×#9911yescharacteristic
165C22×C53#41811yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—82no—
172C4×C53—82no—
173Dic53—162no—
174C2×#397D5311yescharacteristic
175#1819—531no—
176#1462—531no—
177#1344—534no—
178C2×C53—21no—
179C2×C53—22no—
180C2×C53—22no—
181C2×C53—21no—
182C2×C53—42no—
183C2×C53—81no—
184C2×C53#151011yescharacteristic
185C2×C53C2×#35211yescharacteristic
186C2×C53C2×#35411yescharacteristic
187C22×C4⋊C4Dic5311yescharacteristic, complemented
188C22×C22⋊C4—21no—
189C22×C22⋊C4C2×D5311yescharacteristic
190#407—1061no—
191#397—21no—
192#397—21no—
193#397Dic5312yescomplemented
194#397C2×D5312yes—
195#379—1061no—
196C2×#95—21no—
197C2×#95Dic5311yescharacteristic, complemented
198#342—532no—
199#342—1061no—
200#323—532no—
201#323—1061no—
202C53#1179711yescharacteristic, complemented, Sylow
203C25C22⋊D5311yescharacteristic
204C23×C4—21no—
205C23×C4—21no—
206C23×C4C2×Dic5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×M16—2121no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4C2×Dic5312yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—1062no—
219C2×C22⋊C4—2121no—
220C2×C22⋊C4—2121no—
221C2×C22⋊C4C2×Dic5311yescharacteristic
222C2×C22⋊C4C22⋊D5311yescharacteristic
223C2×C22⋊C4C22⋊D5311yescharacteristic
224C2×C22⋊C4C22×D5311yescharacteristic
225C22⋊C8—1062no—
226C22⋊C8—2121no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95C2×Dic5312yes—
231C24—22no—
232C24—21no—
233C24—21no—
234C24—41no—
235C24—42no—
236C24—42no—
237C24C22⋊Dic5311yescharacteristic
238C24C22⋊Dic5311yescharacteristic
239C24C2×C22⋊D5311yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—2122no—
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22×Dic5311yescharacteristic
259C22×C4C2×C22⋊D5311yescharacteristic
260M16—4242no—
261C2×C8—2122no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—4242no—
271C22⋊C4—4242no—
272C22⋊C4—2122no—
273C22⋊C4—2122no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—42no—
284C23—42no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—48no—
290C23—42no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23—84no—
295C23#1313313211yescharacteristic
296C23#1313313211yescharacteristic
297C23C2×C22⋊Dic5311yescharacteristic
298C23#1313324312yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—4242no—
321C2×C4—4242no—
322C2×C4—4242no—
323C8—4242no—
324C22—24no—
325C22—24no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—84no—
342C22—84no—
343C22—81no—
344C22#1681041312yes—
345C22#1681041812yes—
346C22C2×#1313313211yescharacteristic
347C22#1681062811yescharacteristic
348C22#1681064011yescharacteristic
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—82no—
354C4—82no—
355C4—8482no—
356C2—21no—
357C2—21no—
358C2—22no—
359C2—22no—
360C2—42no—
361C2—81no—
362C2#2027324411yescharacteristic
363C2C2×#1681041311yescharacteristic
364C2C2×#1681041811yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC2×C53×#397C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC2×#397D53
hypocenterC53#11797
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
3. #20272715C2semidirect
11. C22×C4⋊C4×C53C4semidirect
16. C53×#397C4semidirect
19. C2×#95×C53C4semidirect
187. C22×C4⋊C4Dic53semidirect
193. #397Dic53semidirect
197. C2×#95Dic53semidirect
202. C53#11797semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202730422
#202727152
C2×C53×#3972
#1179753