Finite Group Databasebeta

← #27301037

Subgroups — #27301037

Order 13568.

574 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×#16810861C211yesmaximal, characteristic
2C2×#107×D53C211yesmaximal, characteristic
3C2×#16810427C211yesmaximal, characteristic, complemented
4C2×#16810402C211yesmaximal, characteristic
5#20272807C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C2×C53×#358C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×#13133193—21no—
10C2×C4⋊C4×D53—21no—
11C22×#11719908C2211yescharacteristic
12#16810861—22no—
13#16810861C414yescomplemented
14C2×#13133151C2211yescharacteristic
15C2×#13133151C2211yescharacteristic
16C2×#13133150—21no—
17#107×D53—22no—
18#107×D53—22no—
19C2×#13133141—21no—
20#16810427—22no—
21#16810402—22no—
22C22×Q16×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#358—22no—
26C22×C4×D53—41no—
27C22×C4×D53D411yescharacteristic
28C22×#88322D411yescharacteristic
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53C2×C412yes—
34C2×C8×D53D412yes—
35#13133193—24no—
36#13133193—42no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×#11719919—41no—
40C2×#11719919—41no—
41C4⋊C4×D53—24no—
42C4⋊C4×D53—42no—
43C2×#11719921—41no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47#13133151—21no—
48#13133151—21no—
49#13133151C2×C412yescomplemented
50#13133151D412yes—
51#13133150—24no—
52#13133141—24no—
53C22×Q8×C53—21no—
54C2×Q16×C53—21no—
55C2×Q16×C53—21no—
56C2×Q16×C53—21no—
57C2×Q16×C53—21no—
58C2×Q16×C53—21no—
59C2×Q16×C53C2×C412yescomplemented
60C22×C8×C53C2311yescharacteristic
61C2×C4⋊C4×C53—21no—
62C2×C16×C53—21no—
63C2×C16×C53—21no—
64C2×C16×C53—21no—
65#107×C53—21no—
66#107×C53D412yes—
67C23×D53D811yescharacteristic
68C2×Q8×C53—22no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53D811yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—44no—
85C2×C4×D53—82no—
86C2×C4×D53—82no—
87C2×C4×D53C22⋊C412yes—
88C2×C4×D53SD1612yes—
89Q16×C53—22no—
90Q16×C53—22no—
91Q16×C53—41no—
92Q16×C53—41no—
93C2×C8×C53C22×C411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C2×C8×C53#4111yescharacteristic
99C4⋊C4×C53—22no—
100C4⋊C4×C53—41no—
101#11719921—44no—
102C4⋊Dic53—22no—
103C4⋊Dic53—22no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106#11719919—44no—
107#11719919—44no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322C22⋊C412yes—
111C8×D53—22no—
112C8×D53—22no—
113C8×D53—22no—
114C8×D53—22no—
115C16×C53—41no—
116#11719908—41no—
117C23×C53C2×D811yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53#10212yes—
123C22×D53SD3212yes—
124C22×D53Q3212yes—
125Q8×C53—42no—
126Q8×C53—42no—
127Q8×C53—81no—
128Q8×C53—81no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53#10212yes—
150C4×D53—22no—
151C4×D53—22no—
152C4×D53—22no—
153C4×D53—22no—
154C4×D53—88no—
155C8×C53C4×D411yescharacteristic
156C8×C53C4×D411yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12111yescharacteristic
159#88322—22no—
160#12993—531nomaximal, Sylow
161C22×C53C2×#10211yescharacteristic
162C22×C53#44711yescharacteristic
163C22×C53#48011yescharacteristic
164C22×C53C2×SD3211yescharacteristic
165C22×C53C2×Q3211yescharacteristic
166C22×C53#50911yescharacteristic
167C22×C53#51011yescharacteristic
168C2×D53—22no—
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C2×D53#35814yescomplemented
175C4×C53—82no—
176C4×C53—82no—
177C4×C53#38611yescharacteristic
178C4×C53#41711yescharacteristic
179C4×C53#44811yescharacteristic
180C4×C53#48111yescharacteristic
181Dic53—22no—
182Dic53—82no—
183Dic53—82no—
184C2×#467—531no—
185C22×#107—531no—
186C2×#358—531no—
187C2×#358D5311yescharacteristic, complemented
188C2×C22⋊C16—531no—
189#1380—531no—
190#1380—531no—
191C2×C53#187411yescharacteristic
192C2×C53C2×#35811yescharacteristic
193C2×C53#213911yescharacteristic
194C2×C53#218711yescharacteristic
195C2×C53#218811yescharacteristic
196C2×C53#223511yescharacteristic
197C2×C53#223611yescharacteristic
198D53—24no—
199C22×Q16C2×D5311yescharacteristic
200C23×C8—531no—
201C2×C22⋊Q8—1061no—
202C22×C4⋊C4—1061no—
203C22×C16—531no—
204C22×C16C2×D5311yescharacteristic
205#467—534no—
206#467—1062no—
207C2×#107—531no—
208C2×#107—531no—
209C2×#107—1062no—
210C2×#107—1061no—
211C2×#107—1062no—
212C2×#107C2×D5311yescharacteristic
213C2×#103—1061no—
214#358—22no—
215#358—1062no—
216C22⋊C16—1062no—
217C53#1299311yescharacteristic, complemented, Sylow
218C22×Q8—21no—
219C23×C4—531no—
220C23×C4—2121no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16C4×D5312yescomplemented
227C22×C8—531no—
228C22×C8—532no—
229C22×C8—532no—
230C22×C8—1061no—
231C22×C8—1061no—
232C22×C8—1061no—
233C22×C8—1061no—
234C22×C8C22×D5311yescharacteristic
235C22⋊Q8—1064no—
236C22⋊Q8—2122no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1064no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—2122no—
242C2×C4⋊C4—2121no—
243C2×C4⋊C4—2121no—
244C2×C22⋊C4—2121no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—1061no—
249C2×C16—1061no—
250C2×C16—1061no—
251#107—21no—
252#107—532no—
253#107—532no—
254#107—1064no—
255#107—1061no—
256#107—1061no—
257#107C22⋊D5312yes—
258#103—1064no—
259C24—531no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—531no—
269C22×C4—532no—
270C22×C4—532no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—4242no—
276C22×C4—4242no—
277C22×C4—2124no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4D4×D5311yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—41no—
284Q16—41no—
285C2×C8—532no—
286C2×C8—1061no—
287C2×C8—1061no—
288C2×C8—1062no—
289C2×C8—1062no—
290C2×C8—1062no—
291C2×C8—1062no—
292C2×C8C2×C4×D5311yescharacteristic
293C2×C8#1171994511yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8C2×C22⋊D5311yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—1062no—
301C4⋊C4—1062no—
302C4⋊C4—2124no—
303C4⋊C4—2124no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—2124no—
307C16—41no—
308C16—2121no—
309C23—532no—
310C23—532no—
311C23—532no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23D8×D5311yescharacteristic
317Q8—42no—
318Q8—42no—
319Q8—81no—
320Q8—81no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—532no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1062no—
331C2×C4—1062no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4—4248no—
335C2×C4—4241no—
336C2×C4—4241no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4C22⋊C4×D5311yescharacteristic
342C2×C4#1313318511yescharacteristic
343C2×C4#1313319911yescharacteristic
344C2×C4SD16×D5311yescharacteristic
345C2×C4#1313324111yescharacteristic
346C2×C4#1313324611yescharacteristic
347C8—1062no—
348C8#1313317711yescharacteristic
349C8C4×C22⋊D5311yescharacteristic
350C8#1313318811yescharacteristic
351C8#1313319011yescharacteristic
352C22—534no—
353C22—1062no—
354C22—1062no—
355C22—1062no—
356C22—1062no—
357C22—1062no—
358C22—1062no—
359C22#102×D5311yescharacteristic
360C22#1681075511yescharacteristic
361C22#1681093511yescharacteristic
362C22SD32×D5311yescharacteristic
363C22Q32×D5311yescharacteristic
364C22#1681106011yescharacteristic
365C22#1681106511yescharacteristic
366C4—82no—
367C4—82no—
368C4—1062no—
369C4—4242no—
370C4—4242no—
371C4#1681049911yescharacteristic
372C4#1681062111yescharacteristic
373C4#1681076011yescharacteristic
374C4#1681094011yescharacteristic
375C2—1064no—
376C2#2027502911yescharacteristic
377C2#358×D5311yescharacteristic
378C2#2027642611yescharacteristic
379C2#2027664211yescharacteristic
380C2#2027664711yescharacteristic
381C2#2027691111yescharacteristic
382C2#2027691611yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#358C2
socleC23×C53C2×D8
prime radicalC23×D53D8
cosocleC22×C8×C53C23
hypercenterC2×#358D53
hypocenterC53#12993
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
3. C2×#16810427C2semidirect
5. #20272807C2semidirect
6. #20272806C2semidirect
7. C2×C53×#358C2semidirect
13. #16810861C4semidirect
49. #13133151C2×C4semidirect
59. C2×Q16×C53C2×C4semidirect
174. C2×D53#358semidirect
187. C2×#358D53semidirect
217. C53#12993semidirect
226. C2×Q16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108612
C2×#107×D532
C2×#168104272
C2×#168104022
#202728072
#202728062
C2×C53×#3582
#1299353