Finite Group Databasebeta

← #27273609

Subgroups — #27273609

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283348C211yesmaximal, characteristic
2C2×C4×#11719917C211yesmaximal, characteristic
3#20272553C214yesmaximal, complemented
4C2×C4×M16×C53C211yesmaximal, characteristic
5C22×#11719917—21no—
6#16810680—21no—
7#16810680C412yes—
8#16810676—21no—
9#16810676C2212yes—
10#16810647C412yes—
11#16810587—21no—
12C4×#11719917—21no—
13C4×#11719917—21no—
14C4×#11719917—21no—
15C4×#11719917C412yes—
16C4×#11719917C412yes—
17C2×#13133128—21no—
18C2×C4×#88322C2212yes—
19C22×M16×C53C412yes—
20C22×C42×C53C2211yescharacteristic
21C4×M16×C53—21no—
22C4×M16×C53—21no—
23C4×M16×C53C412yes—
24C4×M16×C53C2212yes—
25C2×#97×C53—21no—
26C2×C4×C8×C53—21no—
27C2×#11719917—28no—
28C2×#11719917—22no—
29C2×#11719917—22no—
30C2×#11719917—22no—
31C22×#88322—22no—
32C4⋊#88322—22no—
33C4⋊#88322C2×C414yes—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36#13133128—22no—
37#13133128—22no—
38#13133128—22no—
39C4×#88322—22no—
40C4×#88322—22no—
41C4×#88322C2×C414yes—
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53Q811yescharacteristic
45C2×M16×C53—28no—
46C2×M16×C53—22no—
47C2×M16×C53C2×C414yes—
48C22×C8×C53—22no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53Q811yescharacteristic
60C2×C42×C53C2311yescharacteristic
61#97×C53—22no—
62#97×C53—22no—
63C4×C8×C53—22no—
64C4×C8×C53—22no—
65C24×C53C4⋊C411yescharacteristic
66C22×C4×C53—24no—
67C22×C4×C53—24no—
68C22×C4×C53—42no—
69C22×C4×C53C4211yescharacteristic
70C22×C4×C53C4211yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C4⋊C411yescharacteristic
78C22×C4×C53C4⋊C411yescharacteristic
79C22×C4×C53C4⋊C411yescharacteristic
80C22×C4×C53C4⋊C411yescharacteristic
81C22×C4×C53C4⋊C411yescharacteristic
82C22×C4×C53C4⋊C411yescharacteristic
83C22×C4×C53C4⋊C411yescharacteristic
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×Q811yescharacteristic
87M16×C53—28no—
88M16×C53—24no—
89M16×C53M1618yescomplemented
90C2×C8×C53—28no—
91C2×C8×C53—24no—
92C42×C53C4211yescharacteristic
93C42×C53C4211yescharacteristic
94C42×C53C22⋊C411yescharacteristic
95C42×C53C22⋊C411yescharacteristic
96C42×C53C22⋊C411yescharacteristic
97C42×C53C22⋊C411yescharacteristic
98C42×C53C22⋊C411yescharacteristic
99C42×C53C22⋊C411yescharacteristic
100C42×C53C4⋊C411yescharacteristic
101C42×C53C4⋊C411yescharacteristic
102C42×C53C4⋊C411yescharacteristic
103C42×C53C4⋊C411yescharacteristic
104C42×C53C22×C411yescharacteristic
105C42×C53C22×C411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108#11719917—28no—
109#11719917—28no—
110#11719917—44no—
111C2×#88322—28no—
112C2×#88322—24no—
113C2×#88322—24no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—41no—
117C23×C53#9511yescharacteristic
118C23×C53#9511yescharacteristic
119C23×C53C2×C4⋊C411yescharacteristic
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53#9511yescharacteristic
137C2×C4×C53#9511yescharacteristic
138C2×C4×C53#9511yescharacteristic
139C2×C4×C53#9511yescharacteristic
140C2×C4×C53#9511yescharacteristic
141C2×C4×C53#9511yescharacteristic
142C2×C4×C53#10414yes—
143C2×C4×C53C2×C4211yescharacteristic
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C2×C4⋊C411yescharacteristic
148C2×C4×C53C2×C4⋊C411yescharacteristic
149C2×C4×C53C2×M1614yes—
150C8×C53—28no—
151#88322—28no—
152#88322—44no—
153#7887—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—44no—
162C22×C53#33912yes—
163C22×C53C2×#9511yescharacteristic
164C22×C53#42312yes—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53#33912yes—
173C4×C53C4×M1612yes—
174C4×C53#40712yes—
175C4×C53C2×#10412yes—
176C4×C53#43012yes—
177#2920—531no—
178C2×C4×M16—531no—
179C2×C4×M16D5311yescharacteristic
180#1278—534no—
181C2×C53—22no—
182C2×C53—22no—
183C2×C53—41no—
184C2×C53#172811yescharacteristic
185C2×C53C2×#33911yescharacteristic
186C2×C53#173811yescharacteristic
187C22×M16—1061no—
188C22×M16Dic5312yes—
189C22×C42C2×D5311yescharacteristic
190#432—532no—
191#432—1061no—
192#431—532no—
193#431—1061no—
194#423—532no—
195#407—1061no—
196C4×M16—21no—
197C4×M16—21no—
198C4×M16—532no—
199C4×M16—532no—
200C4×M16—1061no—
201C4×M16—1061no—
202C4×M16—1061no—
203C4×M16Dic5312yes—
204C4×M16C2×D5312yes—
205C2×#97—21no—
206C2×#97—1061no—
207C2×C4×C8—21no—
208C2×C4×C8—532no—
209C53#788711yescharacteristic, complemented, Sylow
210C23×C4Dic10611yescharacteristic
211C23×C4D21211yescharacteristic
212C23×C4C2×Dic5311yescharacteristic
213C2×M16—22no—
214C2×M16—28no—
215C2×M16—1068no—
216C2×M16—1062no—
217C2×M16—1062no—
218C2×M16—1062no—
219C2×M16C2×Dic5314yes—
220C22×C8—22no—
221C22×C8—1062no—
222C2×C42Dic10611yescharacteristic
223C2×C42C4×D5311yescharacteristic
224C2×C42C4×D5311yescharacteristic
225C2×C42C4×D5311yescharacteristic
226C2×C42C4×D5311yescharacteristic
227C2×C42D21211yescharacteristic
228C2×C42C2×Dic5311yescharacteristic
229C2×C42C22⋊D5311yescharacteristic
230C2×C42C22⋊D5311yescharacteristic
231C2×C42C22⋊D5311yescharacteristic
232C2×C42C22⋊D5311yescharacteristic
233C2×C42C22×D5311yescharacteristic
234C4⋊C8—534no—
235C4⋊C8—1062no—
236C22⋊C8—1062no—
237C22⋊C8—1062no—
238#97—22no—
239#97—22no—
240#97—1062no—
241#97—1062no—
242#97—1062no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—534no—
246C4×C8—1062no—
247C4×C8—1062no—
248C24C4⋊Dic5311yescharacteristic
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4C4×Dic5311yescharacteristic
253C22×C4C4×Dic5311yescharacteristic
254C22×C4#1171991911yescharacteristic
255C22×C4#1171991911yescharacteristic
256C22×C4#1171991911yescharacteristic
257C22×C4#1171991911yescharacteristic
258C22×C4C4⋊Dic5311yescharacteristic
259C22×C4C4⋊Dic5311yescharacteristic
260C22×C4C4⋊Dic5311yescharacteristic
261C22×C4#1171992111yescharacteristic
262C22×C4#1171992111yescharacteristic
263C22×C4#1171992111yescharacteristic
264C22×C4#1171992111yescharacteristic
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4C22⋊Dic5311yescharacteristic
267C22×C4C2×Dic10611yescharacteristic
268C22×C4C2×D21211yescharacteristic
269C22×C4C22×Dic5311yescharacteristic
270M16—28no—
271M16—24no—
272M16—1068no—
273M16—1068no—
274M16—2124no—
275M16#1171991718yescomplemented
276C2×C8—28no—
277C2×C8—24no—
278C2×C8—1068no—
279C2×C8—1064no—
280C2×C8—1064no—
281C42C4×Dic5311yescharacteristic
282C42C4×Dic5311yescharacteristic
283C42#1171991911yescharacteristic
284C42#1171991911yescharacteristic
285C42#1171991911yescharacteristic
286C42#1171991911yescharacteristic
287C42#1171992111yescharacteristic
288C42#1171992111yescharacteristic
289C42#1171992111yescharacteristic
290C42#1171992111yescharacteristic
291C42C22⋊Dic5311yescharacteristic
292C42C22⋊Dic5311yescharacteristic
293C42C2×C4×D5311yescharacteristic
294C42C2×C4×D5311yescharacteristic
295C42C2×C22⋊D5311yescharacteristic
296C42C2×C22⋊D5311yescharacteristic
297C23—22no—
298C23—22no—
299C23—41no—
300C23#1313312411yescharacteristic
301C23#1313312411yescharacteristic
302C23C2×C4⋊Dic5311yescharacteristic
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4#1313312411yescharacteristic
320C2×C4#1313312411yescharacteristic
321C2×C4#1313312411yescharacteristic
322C2×C4#1313312411yescharacteristic
323C2×C4#1313312411yescharacteristic
324C2×C4#1313312411yescharacteristic
325C2×C4#1313314314yes—
326C2×C4C2×C4×Dic5311yescharacteristic
327C2×C4C2×#1171992111yescharacteristic
328C2×C4C2×#1171992111yescharacteristic
329C2×C4C2×C22⋊Dic5311yescharacteristic
330C2×C4C2×#1171991911yescharacteristic
331C2×C4C2×#1171991911yescharacteristic
332C2×C4C2×#1171991714yes—
333C8—28no—
334C8—1068no—
335C8—2124no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—44no—
344C22#1681038412yes—
345C22C2×#1313312411yescharacteristic
346C22#1681064712yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4#1681038412yes—
355C4C4×#1171991712yes—
356C4#1681058712yes—
357C4C2×#1313314312yes—
358C4#1681067112yes—
359C2—22no—
360C2—22no—
361C2—41no—
362C2#2027440911yescharacteristic
363C2C2×#1681038411yescharacteristic
364C2#2027444111yescharacteristic
subgroupG/H
centerC42C2×C22⋊D53
commutatorC2×C4×C53C2×C42
FrattiniC2×C42C22×D53
FittingC2×C4×M16×C53C2
socleC22×C53C2×#95
prime radicalC24×C53C4⋊C4
cosocleC2×C42×C53C23
hypercenterC2×C4×M16D53
hypocenterC53#7887
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C42 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
3. #20272553C2semidirect
89. M16×C53M16semidirect
209. C53#7887semidirect
275. M16#11719917semidirect

Maximal subgroups

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

subgroupindex
#202833482
C2×C4×#117199172
#202725532
C2×C4×M16×C532
#788753