Finite Group Databasebeta

← #27302945

Subgroups — #27302945

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133163C211yesmaximal, characteristic, complemented
2C22×#13133160C211yesmaximal, characteristic
3C2×#16811007C211yesmaximal, characteristic
4C2×#16810426C211yesmaximal, characteristic, complemented
5C2×#16810423C211yesmaximal, characteristic
6#20272793C211yesmaximal, characteristic, complemented
7C53×#1969C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×#11719913—21no—
10C22×D424C2211yescharacteristic
11C2×#13133205—21no—
12C2×#13133163—22no—
13C2×#13133163—22no—
14C2×#13133163—22no—
15C2×#13133160—22no—
16C2×#13133160—22no—
17C2×#13133160—22no—
18C22×#11719908C2211yescharacteristic
19C22×#11719908C2211yescharacteristic
20#16811007—22no—
21#16811007—22no—
22#16810426—22no—
23#16810423—22no—
24C22×Q16×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C2×#103×C53—21no—
27C2×C4×C8×C53C2211yescharacteristic
28C53×#391—21no—
29C22×Dic106—21no—
30C22×D212—21no—
31C2×Dic212—21no—
32C2×Dic212—21no—
33C2×Dic212—21no—
34C2×Dic212—21no—
35C2×Dic212—21no—
36C2×Dic212D412yes—
37C2×#11719913—24no—
38C2×#11719913—42no—
39C2×#11719913—42no—
40C2×D424—21no—
41C2×D424—21no—
42C2×D424—21no—
43C2×D424—21no—
44C2×D424—21no—
45C2×D424D412yes—
46#13133205—24no—
47#13133205—42no—
48C2×#11719921—42no—
49#13133163—28no—
50#13133160—28no—
51C2×#11719908—21no—
52C2×#11719908—21no—
53C2×#11719908—21no—
54C2×#11719908—21no—
55C2×#11719908D412yes—
56C2×#11719908D412yes—
57C22×Q8×C53—21no—
58C2×Q16×C53—21no—
59C2×Q16×C53—21no—
60C2×Q16×C53—21no—
61C2×Q16×C53—21no—
62C2×Q16×C53—21no—
63C2×Q16×C53D412yes—
64C22×C8×C53D411yescharacteristic
65C22×C8×C53C2311yescharacteristic
66C2×C4⋊C4×C53—21no—
67C2×C42×C53D411yescharacteristic
68#107×C53—21no—
69#107×C53—21no—
70#103×C53—42no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73#95×C53—42no—
74C23×D53—41no—
75C2×Q8×C53—22no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53D812yes—
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—41no—
86C2×D212—22no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×Dic106—22no—
93C2×Dic106—41no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×Dic106—41no—
97C2×Dic106—41no—
98Q16×C53—22no—
99Q16×C53—22no—
100Q16×C53—41no—
101Q16×C53—41no—
102C2×C8×C53—21no—
103C2×C8×C53—21no—
104C2×C8×C53D812yes—
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C42×C53—21no—
114C42×C53—21no—
115#11719921—48no—
116Dic212—22no—
117Dic212—22no—
118Dic212—41no—
119Dic212—41no—
120D424—22no—
121D424—22no—
122D424—41no—
123D424—41no—
124#11719913—48no—
125#11719908—22no—
126#11719908—22no—
127C23×C53C2×D811yescharacteristic
128C22×D53—42no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53—82no—
132C22×D53—82no—
133Q8×C53—42no—
134Q8×C53—42no—
135Q8×C53—81no—
136Q8×C53—81no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—42no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53—82no—
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12711yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×Dic53—42no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154D212—42no—
155D212—42no—
156D212—81no—
157D212—81no—
158Dic106—42no—
159Dic106—42no—
160Dic106—81no—
161Dic106—81no—
162C8×C53—22no—
163C8×C53#12011yescharacteristic
164C8×C53#12011yescharacteristic
165C8×C53#12111yescharacteristic
166C8×C53#12111yescharacteristic
167#13329—531nomaximal, Sylow
168C22×C53#44711yescharacteristic
169C22×C53#45911yescharacteristic
170C22×C53#49311yescharacteristic
171C22×C53#50812yes—
172C22×C53#51012yes—
173C2×D53—84no—
174C2×D53—82no—
175C2×D53—162no—
176C2×D53—161no—
177C2×D53—161no—
178C4×C53—42no—
179C4×C53—82no—
180C4×C53—161no—
181C4×C53#39211yescharacteristic
182C4×C53#44711yescharacteristic
183C4×C53#45911yescharacteristic
184C4×C53#49611yescharacteristic
185Dic53—82no—
186C22×Q32—531no—
187C22×SD32—531no—
188C2×#495—531no—
189C2×#358—531no—
190C2×#357—531no—
191#1969D5311yescharacteristic, complemented
192#1374—531no—
193C2×C53#201111yescharacteristic
194C2×C53#218912yes—
195C2×C53#221112yes—
196C2×C53#224911yescharacteristic
197C2×C53#225111yescharacteristic
198D53—162no—
199C22×Q16—531no—
200C22×Q16C2×D5311yescharacteristic
201C22×SD16—1061no—
202C22×D8—531no—
203C2×#124—1061no—
204C2×Q32—1062no—
205C2×Q32—1062no—
206C2×Q32—1062no—
207C2×SD32—1062no—
208C2×SD32—1062no—
209C2×SD32—1062no—
210C22×C16—531no—
211C22×C16—531no—
212#495—1062no—
213#495—1062no—
214C2×#107C2×D5311yescharacteristic
215C2×#103—21no—
216C2×C4×C8C2×D5311yescharacteristic
217#391—21no—
218#358—1062no—
219#357—1062no—
220C53#1332911yescharacteristic, complemented, Sylow
221C22×Q8—21no—
222C22×Q8—1061no—
223C22×D4—1061no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16—21no—
227C2×Q16—21no—
228C2×Q16—21no—
229C2×Q16—532no—
230C2×Q16—1061no—
231C2×Q16—1061no—
232C2×Q16—1061no—
233C2×Q16—1061no—
234C2×Q16—1061no—
235C2×Q16D21212yes—
236C2×SD16—1064no—
237C2×SD16—2122no—
238C2×SD16—2122no—
239C2×D8—532no—
240C2×D8—1061no—
241C2×D8—1061no—
242C2×D8—1061no—
243C2×D8—1061no—
244C2×D8—1061no—
245C22×C8C22⋊D5311yescharacteristic
246C22×C8C22×D5311yescharacteristic
247#124—1064no—
248#124—2122no—
249C2×C4⋊C4—21no—
250C2×C22⋊C4—2122no—
251C2×C42C22⋊D5311yescharacteristic
252Q32—1068no—
253SD32—1068no—
254C2×C16—532no—
255C2×C16—532no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259C2×C16—1061no—
260#107—21no—
261#107—21no—
262#103—42no—
263C4×C8—21no—
264C4×C8—21no—
265#95—42no—
266C24—2121no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—1062no—
274C2×Q8—2121no—
275C2×Q8—2121no—
276C2×Q8—2121no—
277C2×Q8—2121no—
278C2×Q8—2121no—
279C2×D4—1062no—
280C2×D4—2121no—
281C2×D4—2121no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4—2121no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—2121no—
288C22×C4#1171992212yes—
289C22×C4C2×C22⋊D5311yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—41no—
293Q16—41no—
294Q16—1062no—
295Q16—1062no—
296Q16—2121no—
297Q16—2121no—
298SD16—2128no—
299D8—1062no—
300D8—1062no—
301D8—2121no—
302D8—2121no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8#1171992212yes—
306C2×C8C2×D21211yescharacteristic
307C2×C8#1171994511yescharacteristic
308C2×C8D4×D5311yescharacteristic
309C2×C8D4×D5311yescharacteristic
310C2×C8D4×D5311yescharacteristic
311C2×C8D4×D5311yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—41no—
314C22⋊C4—2128no—
315C42—21no—
316C42—21no—
317C16—1062no—
318C16—1062no—
319C23—4242no—
320C23—4242no—
321C23—4241no—
322C23—4241no—
323C23—2122no—
324C23C2×#1171992211yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329Q8—4241no—
330Q8—4241no—
331Q8—2122no—
332Q8—2122no—
333D4—4241no—
334D4—4241no—
335D4—2122no—
336D4—2122no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—2122no—
348C2×C4#1313318511yescharacteristic
349C2×C4#1313319111yescharacteristic
350C2×C4#1313321311yescharacteristic
351C2×C4C2×#1171992211yescharacteristic
352C2×C4#1313324211yescharacteristic
353C2×C4#1313324211yescharacteristic
354C8—22no—
355C8#1313318711yescharacteristic
356C8#1313318711yescharacteristic
357C8#1313319011yescharacteristic
358C8#1313319011yescharacteristic
359C22—8482no—
360C22—8481no—
361C22—8481no—
362C22—4242no—
363C22—4244no—
364C22#1681075611yescharacteristic
365C22#1681081811yescharacteristic
366C22#1681099911yescharacteristic
367C22#1681105312yes—
368C22#1681106412yes—
369C4—42no—
370C4—82no—
371C4—161no—
372C4—4242no—
373C4#1681053211yescharacteristic
374C4#1681075611yescharacteristic
375C4#1681081811yescharacteristic
376C4#1681101311yescharacteristic
377C2—8482no—
378C2#2027578511yescharacteristic
379C2#2027665312yes—
380C2#2027677712yes—
381C2#2027697411yescharacteristic
382C2#2027698411yescharacteristic
subgroupG/H
centerC23C2×#11719922
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1969C2
socleC23×C53C2×D8
prime radicalC22×D424C22
hypercenter#1969D53
hypocenterC53#13329
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
1. C22×#13133163C2semidirect
4. C2×#16810426C2semidirect
6. #20272793C2semidirect
7. C53×#1969C2semidirect
191. #1969D53semidirect
220. C53#13329semidirect

Maximal subgroups

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

subgroupindex
C22×#131331632
C22×#131331602
C2×#168110072
C2×#168104262
C2×#168104232
#202727932
C53×#19692
#1332953