Finite Group Databasebeta

← #27282940

Subgroups — #27282940

Order 13568.

554 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
1C2×C4×#11719924C211yesmaximal, characteristic
2#20275652C211yesmaximal, characteristic, complemented
3#20275508C211yesmaximal, characteristic
4C2×#16810373C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6C2×C8×#88322C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719924C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C4×#11719924—22no—
11C4×#11719924—22no—
12C2×#13133148C411yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133141C411yescharacteristic
15C2×#13133137C411yescharacteristic
16C2×#13133129—22no—
17C2×C4×#88322C2211yescharacteristic
18C2×C4×#88322C2211yescharacteristic
19#16810378—22no—
20#16810373—22no—
21#16810352—22no—
22C8×#88322—22no—
23C2×C4×Q8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719924—22no—
30C2×#11719924—22no—
31C2×#11719924C814yescomplemented
32C22×#88322—21no—
33C22×#88322C2×C411yescharacteristic
34C22×#88322C2×C411yescharacteristic
35C22×#88322D412yes—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719921—21no—
42#13133148—22no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133141—22no—
46#13133137—22no—
47#13133129—44no—
48C4×#88322—21no—
49C4×#88322—21no—
50C4×#88322D412yes—
51C4×#88322D412yes—
52C22×Q8×C53C2×C411yescharacteristic
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×Q8×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C42×C53—21no—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4×C8×C53—21no—
66C4×C8×C53D412yes—
67C23×D53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53C2×C812yes—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×Dic53—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212C2×C812yes—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—42no—
89C2×C4×D53—42no—
90C2×C8×C53—24no—
91C2×C8×C53—22no—
92C2×C8×C53—41no—
93C2×C8×C53—42no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C42×C53—41no—
99C42×C53—41no—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104#11719924—28no—
105#11719921—41no—
106#11719921—41no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C2×#88322—22no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—22no—
113C2×#88322—41no—
114C2×#88322—42no—
115C2×#88322C22⋊C412yes—
116C2×#88322C22⋊C412yes—
117C2×#88322C2×C812yes—
118C2×#88322M1612yes—
119C2×#88322SD1614yescomplemented
120C23×C53C4×D411yescharacteristic
121C22×D53—22no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—42no—
125C22×D53—42no—
126Q8×C53—22no—
127Q8×C53—22no—
128Q8×C53—41no—
129Q8×C53—41no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12411yescharacteristic
141C2×C4×C53#12711yescharacteristic
142C2×C4×C53C22×C811yescharacteristic
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53#13111yescharacteristic
145C2×C4×C53#13111yescharacteristic
146C2×C4×C53C2×SD1612yes—
147C2×C4×C53#13512yes—
148C2×Dic53—41no—
149C2×Dic53—42no—
150D212—22no—
151D212—22no—
152D212—41no—
153D212—41no—
154C4×D53—82no—
155C4×D53—81no—
156C4×D53—81no—
157C8×C53—24no—
158C8×C53—81no—
159#88322—22no—
160#88322—24no—
161#88322—81no—
162#88322C22⋊C814yescomplemented
163#9655—531nomaximal, Sylow
164C22×C53#39011yescharacteristic
165C22×C53C8×D411yescharacteristic
166C22×C53#43611yescharacteristic
167C22×C53C4×SD1612yes—
168C22×C53#44312yes—
169C2×D53—44no—
170C2×D53—42no—
171C2×D53—82no—
172C2×D53—81no—
173C2×D53—81no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—42no—
177C4×C53—81no—
178C4×C53C2×C22⋊C811yescharacteristic
179C4×C53#40811yescharacteristic
180C4×C53C8×D411yescharacteristic
181C4×C53#43611yescharacteristic
182C4×C53#46512yes—
183C4×C53#49211yescharacteristic
184C4×C53#49511yescharacteristic
185Dic53—81no—
186C2×C4×SD16—531no—
187#1984—531no—
188#1954—531no—
189C2×#334—531no—
190C2×Q8⋊C8D5311yescharacteristic, complemented
191C2×D4⋊C8—531no—
192C2×C82—531no—
193C2×C53C8×SD1612yes—
194C2×C53#159012yes—
195C2×C53#195411yescharacteristic
196C2×C53#196811yescharacteristic
197C2×C53#197411yescharacteristic
198D53—82no—
199C22×SD16—531no—
200C2×C4×Q8C2×D5311yescharacteristic
201C2×C4×D4—531no—
202C4×SD16—1062no—
203C4×SD16—1062no—
204C2×#106—531no—
205C2×C4⋊C8—531no—
206C2×C4⋊C8C2×D5311yescharacteristic
207C2×#103—531no—
208C2×#102—531no—
209C2×C22⋊C8—1062no—
210C2×C4×C8—531no—
211C2×C4×C8—531no—
212C2×C4×C8C2×D5311yescharacteristic
213#336—1062no—
214#334—1062no—
215Q8⋊C8—22no—
216D4⋊C8—1062no—
217C82—1062no—
218C53#965511yescharacteristic, complemented, Sylow
219C22×Q8C4×D5311yescharacteristic
220C22×D4—531no—
221C23×C4—1061no—
222C2×SD16—534no—
223C2×SD16—1062no—
224C2×SD16—1062no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—531no—
228C22×C8—531no—
229C22×C8—532no—
230C22×C8—1061no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238C4×D4—1061no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4C4×D5311yescharacteristic
242C2×C22⋊C4—1061no—
243C2×C42—21no—
244C2×C42C22×D5311yescharacteristic
245#106—1062no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—1061no—
249C4⋊C8—1061no—
250#103—1062no—
251#102—1062no—
252C22⋊C8—2124no—
253C4×C8—21no—
254C4×C8—532no—
255C4×C8—532no—
256C4×C8—1061no—
257C4×C8—1061no—
258C4×C8C22⋊D5312yes—
259C24—1061no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8C8×D5312yes—
266C2×D4—532no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—1061no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4—2122no—
278C22×C4—2122no—
279C22×C4C2×C4×D5311yescharacteristic
280C22×C4D4×D5311yescharacteristic
281C22×C4#1171994711yescharacteristic
282SD16—1068no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—532no—
288C2×C8—532no—
289C2×C8—532no—
290C2×C8—532no—
291C2×C8—534no—
292C2×C8—1062no—
293C2×C8—1061no—
294C2×C8—1061no—
295C2×C8—1062no—
296C2×C8—2121no—
297C2×C8—2122no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C22⋊C4—2121no—
305C22⋊C4—2121no—
306C42—41no—
307C42—41no—
308C42#1171994511yescharacteristic
309C42D4×D5311yescharacteristic
310C42D4×D5311yescharacteristic
311C42C2×C22⋊D5311yescharacteristic
312C23—1062no—
313C23—2121no—
314C23—2121no—
315C23—2122no—
316C23—2122no—
317C23#1313317711yescharacteristic
318Q8—22no—
319Q8—22no—
320Q8—41no—
321Q8—41no—
322D4—1062no—
323D4—1062no—
324D4—2121no—
325D4—2121no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—4241no—
333C2×C4—4242no—
334C2×C4—4241no—
335C2×C4—2121no—
336C2×C4—2122no—
337C2×C4C22⋊C4×D5311yescharacteristic
338C2×C4C4×C22⋊D5311yescharacteristic
339C2×C4#1313318911yescharacteristic
340C2×C4#1313319011yescharacteristic
341C2×C4#1313320311yescharacteristic
342C2×C4#1313321311yescharacteristic
343C2×C4C2×C8×D5311yescharacteristic
344C2×C4M16×D5311yescharacteristic
345C2×C4#1313322311yescharacteristic
346C2×C4#1313322511yescharacteristic
347C2×C4SD16×D5312yes—
348C2×C4#1313324012yes—
349C8—24no—
350C8—81no—
351C8—534no—
352C8—1062no—
353C8—1064no—
354C8—4241no—
355C22—4241no—
356C22—4241no—
357C22—4242no—
358C22—2122no—
359C22—2124no—
360C22#1681052311yescharacteristic
361C22#1681068611yescharacteristic
362C22#1681069811yescharacteristic
363C22#1681070812yes—
364C22#1681073712yes—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—81no—
369C4—4241no—
370C4C22⋊C8×D5311yescharacteristic
371C4#1681058811yescharacteristic
372C4C8×C22⋊D5311yescharacteristic
373C4#1681069911yescharacteristic
374C4#1681085512yes—
375C4#1681099411yescharacteristic
376C4#1681100811yescharacteristic
377C2—4242no—
378C2#2027351212yes—
379C2#2027360212yes—
380C2#2027550711yescharacteristic
381C2#2027556611yescharacteristic
382C2#2027560011yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53C4×D4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#9655
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #20275652C2semidirect
4. C2×#16810373C2semidirect
6. C2×C8×#88322C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
31. C2×#11719924C8semidirect
119. C2×#88322SD16semidirect
162. #88322C22⋊C8semidirect
190. C2×Q8⋊C8D53semidirect
218. C53#9655semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199242
#202756522
#202755082
C2×#168103732
C2×#168103522
C2×C8×#883222
C2×C53×Q8⋊C82
#965553