Finite Group Databasebeta

← #27301154

Subgroups — #27301154

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133160C211yesmaximal, characteristic
2C2×Q16×Dic53C211yesmaximal, characteristic, complemented
3C2×#16810425C211yesmaximal, characteristic
4#20275579C211yesmaximal, characteristic
5#20272806C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C2×C53×#358C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C2×Q8×Dic53—21no—
10C2×#13133160—22no—
11C2×#13133160—22no—
12C2×#13133160C414yescomplemented
13C22×#11719908C2211yescharacteristic
14Q16×Dic53—22no—
15Q16×Dic53—22no—
16C2×#13133151C2211yescharacteristic
17C2×#13133141—21no—
18C2×#13133137—21no—
19C2×C8×Dic53C2211yescharacteristic
20#16810524—21no—
21#16810425—22no—
22C22×Q16×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#358—22no—
26C22×D212—21no—
27C2×D424—21no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424C2×C412yes—
33C22×#88322D411yescharacteristic
34Q8×Dic53—24no—
35Q8×Dic53—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—41no—
38C2×#11719921—41no—
39C2×#11719921—41no—
40C2×C4×Dic53—41no—
41C2×C4×Dic53D411yescharacteristic
42#13133160—28no—
43C2×#11719908—21no—
44C2×#11719908—21no—
45C2×#11719908C2×C412yes—
46#13133151—21no—
47#13133151D412yes—
48#13133141—24no—
49#13133137—42no—
50C8×Dic53—21no—
51C8×Dic53—21no—
52C22×Q8×C53—21no—
53C2×Q16×C53—21no—
54C2×Q16×C53—21no—
55C2×Q16×C53—21no—
56C2×Q16×C53C2×C412yes—
57C2×Q16×C53D412yes—
58C2×Q16×C53D412yes—
59C22×C8×C53C2311yescharacteristic
60C2×C4⋊C4×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53—21no—
63C2×C16×C53—21no—
64#107×C53—21no—
65#107×C53—21no—
66C23×D53—41no—
67C2×Q8×C53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53—22no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—41no—
77C22×Dic53D811yescharacteristic
78C22×Dic53D811yescharacteristic
79C2×D212—22no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85Q16×C53—22no—
86Q16×C53—22no—
87Q16×C53—22no—
88Q16×C53C22⋊C414yescomplemented
89C2×C8×C53C22×C411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C2×C8×C53#4111yescharacteristic
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97#11719921—82no—
98#11719921—82no—
99C4⋊Dic53—22no—
100C4⋊Dic53—44no—
101C4⋊Dic53—41no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—44no—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322—21no—
108D424—22no—
109D424—22no—
110D424—41no—
111D424—41no—
112C16×C53—41no—
113#11719908—22no—
114C23×C53C2×D811yescharacteristic
115C22×D53—42no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—81no—
122Q8×C53—24no—
123Q8×C53—42no—
124Q8×C53—42no—
125Q8×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—42no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53SD3212yescomplemented
147D212—42no—
148D212—42no—
149D212—81no—
150D212—81no—
151C8×C53C2×C22⋊C411yescharacteristic
152C8×C53C4×D411yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12311yescharacteristic
155#88322—41no—
156#13013—531nomaximal, Sylow
157C22×C53C4×D811yescharacteristic
158C22×C53#44711yescharacteristic
159C22×C53#45911yescharacteristic
160C22×C53C2×SD3211yescharacteristic
161C22×C53#50811yescharacteristic
162C22×C53#50911yescharacteristic
163C22×C53#51011yescharacteristic
164C2×D53—82no—
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—161no—
168C2×D53—161no—
169C2×D53—161no—
170C2×D53—161no—
171C4×C53—44no—
172C4×C53—161no—
173C4×C53#38611yescharacteristic
174C4×C53#44211yescharacteristic
175C4×C53#44911yescharacteristic
176C4×C53#46311yescharacteristic
177Dic53—22no—
178Dic53—41no—
179Dic53—82no—
180C22×SD32—531no—
181C2×C4×Q16—531no—
182C2×#358D5311yescharacteristic, complemented
183C2×#357—531no—
184#1970—531no—
185#1380—531no—
186#1374—531no—
187C2×C53#187911yescharacteristic
188C2×C53C4×SD3211yescharacteristic
189C2×C53#217511yescharacteristic
190C2×C53#218511yescharacteristic
191C2×C53#218611yescharacteristic
192C2×C53#220711yescharacteristic
193C2×C53#221111yescharacteristic
194D53—162no—
195C22×Q16C2×D5311yescharacteristic
196C22×D8—531no—
197C2×C4×Q8—1061no—
198C2×SD32—534no—
199C2×SD32—1062no—
200C2×SD32—1062no—
201C22×C16—531no—
202C22×C16C2×D5311yescharacteristic
203C4×Q16—1062no—
204C4×Q16—1062no—
205C2×#107—531no—
206C2×#107C2×D5311yescharacteristic
207C2×#103—1061no—
208C2×#102—1061no—
209C2×C4×C8—531no—
210#390—1061no—
211#358—22no—
212#357—1062no—
213C53#1301311yescharacteristic, complemented, Sylow
214C22×Q8—21no—
215C22×D4—1061no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D5312yes—
220C2×Q16D21212yes—
221C2×Q16C22⋊D5312yes—
222C2×D8—532no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C22×C8—531no—
229C22×C8C22×D5311yescharacteristic
230C4×Q8—1064no—
231C4×Q8—2122no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—2121no—
235C2×C22⋊C4—2121no—
236C2×C22⋊C4—2121no—
237C2×C42—531no—
238C2×C42—2121no—
239SD32—1068no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—532no—
244C2×C16—1061no—
245C2×C16—1061no—
246#107—21no—
247#107—21no—
248#107—532no—
249#107—1061no—
250#103—1064no—
251#102—2122no—
252C4×C8—1061no—
253C4×C8—1061no—
254C24—2121no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—1062no—
262C2×D4—2121no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—2121no—
272C22×C4D4×D5311yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16#1171992114yescomplemented
277D8—1062no—
278D8—1062no—
279D8—2121no—
280D8—2121no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8C2×C4×D5311yescharacteristic
285C2×C8C2×D21211yescharacteristic
286C2×C8D4×D5311yescharacteristic
287C2×C8#1171994711yescharacteristic
288C2×C8#1171994711yescharacteristic
289C2×C8C2×C22⋊D5311yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—1062no—
293C4⋊C4—2124no—
294C4⋊C4—2121no—
295C22⋊C4—4242no—
296C22⋊C4—4242no—
297C42—1061no—
298C42—1061no—
299C42—2124no—
300C16—41no—
301C16—1062no—
302C23—4241no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—4241no—
307C23—4241no—
308C23—2122no—
309C23D8×D5311yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—4241no—
315D4—4241no—
316D4—2122no—
317D4—2122no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—532no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4—2122no—
333C2×C4#1313317711yescharacteristic
334C2×C4#1313318711yescharacteristic
335C2×C4#1313318911yescharacteristic
336C2×C4#1313323811yescharacteristic
337C2×C4#1313324111yescharacteristic
338C2×C4#1313324411yescharacteristic
339C8—2121no—
340C8C2×#1171992111yescharacteristic
341C8#1313317711yescharacteristic
342C8#1313318911yescharacteristic
343C8#1313320111yescharacteristic
344C22—8481no—
345C22—8481no—
346C22—8481no—
347C22—8481no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22#1681070311yescharacteristic
352C22#1681075811yescharacteristic
353C22#1681081611yescharacteristic
354C22SD32×D5311yescharacteristic
355C22#1681105611yescharacteristic
356C22#1681106011yescharacteristic
357C22#1681106511yescharacteristic
358C4—44no—
359C4—161no—
360C4—1062no—
361C4—4242no—
362C4—2121no—
363C4#1681050011yescharacteristic
364C4#1681073211yescharacteristic
365C4#1681076911yescharacteristic
366C4#1681084011yescharacteristic
367C2—8482no—
368C2#2027506111yescharacteristic
369C2#2027656411yescharacteristic
370C2#2027658211yescharacteristic
371C2#2027663311yescharacteristic
372C2#2027664111yescharacteristic
373C2#2027675311yescharacteristic
374C2#2027677911yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#358C2
socleC23×C53C2×D8
prime radicalC22×D424C22
cosocleC22×C8×C53C23
hypercenterC2×#358D53
hypocenterC53#13013
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic53C2semidirect
5. #20272806C2semidirect
6. #20272792C2semidirect
7. C2×C53×#358C2semidirect
12. C2×#13133160C4semidirect
88. Q16×C53C22⋊C4semidirect
146. C2×Dic53SD32semidirect
182. C2×#358D53semidirect
213. C53#13013semidirect
276. Q16#11719921semidirect

Maximal subgroups

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

subgroupindex
C22×#131331602
C2×Q16×Dic532
C2×#168104252
#202755792
#202728062
#202727922
C2×C53×#3582
#1301353