Finite Group Databasebeta

← #27271215

Subgroups — #27271215

Order 13568.

464 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810349C211yesmaximal, characteristic
2#20272968C211yesmaximal, characteristic
3C2×C53×#343C211yesmaximal, characteristic
4C22×C4⋊Dic53C411yescharacteristic
5C2×C22⋊#88322—21no—
6C2×#13133124—21no—
7C2×#13133124—21no—
8C2×#13133124C411yescharacteristic
9#16810349—21no—
10#16810349C412yescomplemented
11C22×M16×C53—21no—
12C22×C4⋊C4×C53C2211yescharacteristic
13C53×#343—21no—
14C53×#343C412yescomplemented
15C23×Dic53—21no—
16C23×Dic53—21no—
17C22×#88322—41no—
18C2×C4⋊Dic53—42no—
19C2×C4⋊Dic53—41no—
20C22⋊#88322—22no—
21C22⋊#88322—41no—
22#13133124—42no—
23#13133124—41no—
24#13133124—41no—
25#13133124—41no—
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C2×M16×C53—22no—
30C2×M16×C53—22no—
31C2×M16×C53—22no—
32C2×M16×C53—41no—
33C2×M16×C53—41no—
34C2×M16×C53—41no—
35C22×C8×C53—41no—
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53Q811yescharacteristic
42C24×C53C22⋊C411yescharacteristic
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53C4211yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C4⋊C411yescharacteristic
59C22×C4×C53C4⋊C411yescharacteristic
60C22×C4×C53C4⋊C411yescharacteristic
61C22×C4×C53D811yescharacteristic
62C22×C4×C53SD1611yescharacteristic
63C22×C4×C53SD1611yescharacteristic
64C22×C4×C53Q1611yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53—41no—
70C22×Dic53—41no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79M16×C53—24no—
80M16×C53—42no—
81M16×C53—42no—
82M16×C53—42no—
83C2×C8×C53—42no—
84C2×C8×C53—42no—
85C2×C8×C53—42no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊Dic53—42no—
91C4⋊Dic53—42no—
92C2×#88322—42no—
93C2×#88322—81no—
94C2×#88322—81no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53#9511yescharacteristic
100C23×C53#9911yescharacteristic
101C23×C53#9911yescharacteristic
102C23×C53#10211yescharacteristic
103C23×C53#10311yescharacteristic
104C23×C53#10411yescharacteristic
105C23×C53#10411yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—81no—
132C2×C4×C53#10211yescharacteristic
133C2×C4×C53#10311yescharacteristic
134C2×C4×C53#10611yescharacteristic
135C2×C4×C53#10711yescharacteristic
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150C8×C53—44no—
151#88322—82no—
152#7251—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53#32711yescharacteristic
166C22×C53#32711yescharacteristic
167C22×C53#32811yescharacteristic
168C22×C53#32811yescharacteristic
169C22×C53#33911yescharacteristic
170C22×C53#34011yescharacteristic
171C22×C53#34211yescharacteristic
172C22×C53AΣL(1,𝔽16)11yescharacteristic
173C22×C53#35211yescharacteristic
174C22×C53#35511yescharacteristic
175C22×C53#35611yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184Dic53—82no—
185Dic53—82no—
186C2×#343D5311yescharacteristic
187C2×#324—531no—
188#1441—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53#129411yescharacteristic
194C2×C53#134511yescharacteristic
195C2×C53#134611yescharacteristic
196C2×C53#135311yescharacteristic
197C2×C53#135411yescharacteristic
198C2×C53#139011yescharacteristic
199C2×C53#139411yescharacteristic
200C22×M16—21no—
201C22×C4⋊C4—531no—
202C22×C4⋊C4C2×D5311yescharacteristic
203C2×C22⋊C8—1061no—
204C2×#95—531no—
205C2×#95—1061no—
206C2×#95—1061no—
207#343—21no—
208#343Dic5312yescomplemented
209#324—532no—
210#324—1061no—
211C53#725111yescharacteristic, complemented, Sylow
212C23×C4—1061no—
213C23×C4—1061no—
214C23×C4C4×D5311yescharacteristic
215C23×C4D21211yescharacteristic
216C23×C4C22⋊D5311yescharacteristic
217C2×M16—22no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—41no—
221C2×M16—41no—
222C2×M16—41no—
223C22×C8—41no—
224C22×C8—2121no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—2121no—
228C2×C4⋊C4—2122no—
229C2×C4⋊C4Dic10611yescharacteristic
230C2×C4⋊C4C4×D5311yescharacteristic
231C2×C4⋊C4C2×Dic5311yescharacteristic
232C2×C4⋊C4C22⋊D5311yescharacteristic
233C22⋊C8—1062no—
234C22⋊C8—2121no—
235#95—2121no—
236#95—2122no—
237#95—2121no—
238#95—2121no—
239C24#1171992111yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—1062no—
253C22×C4—1062no—
254C22×C4—2121no—
255C22×C4—2121no—
256C22×C4—2121no—
257C22×C4—2121no—
258C22×C4—2121no—
259C22×C4—2121no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4C4×Dic5311yescharacteristic
267C22×C4#1171991911yescharacteristic
268C22×C4#1171991911yescharacteristic
269C22×C4C4⋊Dic5311yescharacteristic
270C22×C4#1171992111yescharacteristic
271C22×C4#1171992211yescharacteristic
272C22×C4#1171992311yescharacteristic
273C22×C4#1171992411yescharacteristic
274C22×C4#1171992511yescharacteristic
275C22×C4C22⋊Dic5311yescharacteristic
276M16—24no—
277M16—42no—
278M16—42no—
279M16—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—4241no—
284C2×C8—4241no—
285C2×C8—2122no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—2122no—
291C4⋊C4—2122no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23#1313312411yescharacteristic
297C23#1313313111yescharacteristic
298C23#1313313111yescharacteristic
299C23#1313313711yescharacteristic
300C23#1313314011yescharacteristic
301C23#1313314511yescharacteristic
302C23#1313314511yescharacteristic
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4—4241no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4#1313313811yescharacteristic
344C2×C4#1313314111yescharacteristic
345C2×C4#1313314811yescharacteristic
346C2×C4#1313315011yescharacteristic
347C8—44no—
348C8—4242no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#1681035611yescharacteristic
362C22#1681035611yescharacteristic
363C22#1681035911yescharacteristic
364C22#1681035911yescharacteristic
365C22#1681038311yescharacteristic
366C22#1681038511yescharacteristic
367C22#1681038911yescharacteristic
368C22#1681041111yescharacteristic
369C22#1681041411yescharacteristic
370C22#1681041911yescharacteristic
371C22#1681042111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—4242no—
381C4—4242no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2#2027259411yescharacteristic
387C2#2027271711yescharacteristic
388C2#2027272011yescharacteristic
389C2#2027274111yescharacteristic
390C2#2027274411yescharacteristic
391C2#2027283411yescharacteristic
392C2#2027284111yescharacteristic
subgroupG/H
centerC23#13133131
commutatorC22×C4×C53C42
FrattiniC22×C4⋊C4C2×D53
FittingC2×C53×#343C2
socleC23×C53#99
prime radicalC24×C53C22⋊C4
cosocleC22×C4⋊C4×C53C22
hypercenterC2×#343D53
hypocenterC53#7251
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
10. #16810349C4semidirect
14. C53×#343C4semidirect
208. #343Dic53semidirect
211. C53#7251semidirect

Maximal subgroups

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

subgroupindex
C2×#168103492
#202729682
C2×C53×#3432
#725153