Finite Group Databasebeta

← #27301044

Subgroups — #27301044

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810865C211yesmaximal, characteristic
2C2×#16810660C211yesmaximal, characteristic
3C2×#16810428C211yesmaximal, characteristic, complemented
4C2×#16810426C211yesmaximal, characteristic, complemented
5C2×#16810402C211yesmaximal, characteristic
6#20272806C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×#13133197—21no—
11C2×#13133175—21no—
12C22×#11719908C2211yescharacteristic
13#16810865—22no—
14#16810865C414yescomplemented
15#16810660—22no—
16#16810660—22no—
17C2×#13133151C2211yescharacteristic
18C2×#13133148—21no—
19C2×#13133140—21no—
20#16810428—22no—
21#16810426—22no—
22#16810402—22no—
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C22×Dic106—21no—
27C22×C4×D53D411yescharacteristic
28C2×Dic212—21no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×Dic212—21no—
32C2×Dic212—21no—
33C2×Dic212C2×C412yes—
34C22×#88322D411yescharacteristic
35C2×C8×D53—21no—
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53C2×C412yes—
40C2×C8×D53D412yes—
41#13133197—24no—
42#13133197—42no—
43#13133175—24no—
44#13133175—42no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—41no—
47C2×#11719921—41no—
48C2×#11719921—41no—
49C2×C4×Dic53—41no—
50C2×#11719908—21no—
51C2×#11719908—21no—
52C2×#11719908—21no—
53#13133151—21no—
54#13133151D412yes—
55#13133148—24no—
56#13133140—24no—
57C22×C8×C53C2311yescharacteristic
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53—21no—
63#107×C53—21no—
64#107×C53—21no—
65#107×C53C2×C412yes—
66#107×C53D412yes—
67C23×D53D811yescharacteristic
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53D811yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53C22⋊C412yes—
79C2×C4×D53SD1612yes—
80C2×Dic106—22no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×Dic106—41no—
85C2×Dic106—41no—
86C2×C8×C53C22×C411yescharacteristic
87C2×C8×C53C2×D411yescharacteristic
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53#4111yescharacteristic
91C2×C8×C53#4111yescharacteristic
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96#11719921—44no—
97#11719921—44no—
98C4⋊Dic53—22no—
99C4⋊Dic53—41no—
100C4⋊Dic53—44no—
101C4×Dic53—44no—
102C2×#88322—21no—
103C2×#88322—21no—
104C2×#88322C22⋊C412yes—
105Dic212—22no—
106Dic212—22no—
107Dic212—41no—
108Dic212—41no—
109C8×D53—22no—
110C8×D53—22no—
111C8×D53—22no—
112C8×D53—22no—
113C16×C53—41no—
114#11719908—41no—
115C23×C53C2×D811yescharacteristic
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53#10212yes—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12311yescharacteristic
132C2×C4×C53C2×SD1611yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C53#13711yescharacteristic
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142C2×Dic53—81no—
143C2×Dic53#10212yes—
144C4×D53—22no—
145C4×D53—22no—
146C4×D53—22no—
147C4×D53—22no—
148Dic106—42no—
149Dic106—42no—
150Dic106—81no—
151Dic106—81no—
152C8×C53C4×D411yescharacteristic
153C8×C53C4×D411yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12111yescharacteristic
156#88322—22no—
157#12994—531nomaximal, Sylow
158C22×C53C2×#10211yescharacteristic
159C22×C53#44711yescharacteristic
160C22×C53#48011yescharacteristic
161C22×C53#50811yescharacteristic
162C22×C53#50811yescharacteristic
163C22×C53#51011yescharacteristic
164C22×C53#51011yescharacteristic
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—22no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C4×C53—82no—
173C4×C53—82no—
174C4×C53#38611yescharacteristic
175C4×C53#41711yescharacteristic
176C4×C53#44811yescharacteristic
177C4×C53#48111yescharacteristic
178Dic53—22no—
179Dic53—82no—
180Dic53—82no—
181C2×#467—531no—
182C2×#427—531no—
183C2×#358—531no—
184C2×#358—531no—
185C2×C22⋊C16—531no—
186#1380—531no—
187#1380D5311yescharacteristic, complemented
188C2×C53#187411yescharacteristic
189C2×C53#213811yescharacteristic
190C2×C53#214111yescharacteristic
191C2×C53#218911yescharacteristic
192C2×C53#218911yescharacteristic
193C2×C53#223711yescharacteristic
194C2×C53#223711yescharacteristic
195D53—42no—
196C22×Q16—531no—
197C23×C8—531no—
198C2×C22⋊Q8—1061no—
199C2×#117—1061no—
200C22×C16—531no—
201C22×C16C2×D5311yescharacteristic
202#467—534no—
203#467—1062no—
204#427—1062no—
205#427—1062no—
206C2×#107—531no—
207C2×#107C2×D5311yescharacteristic
208C2×#107C2×D5311yescharacteristic
209C2×#106—1061no—
210C2×#103—1061no—
211#358—1062no—
212#358—1062no—
213C22⋊C16—1062no—
214C53#1299411yescharacteristic, complemented, Sylow
215C22×Q8—1061no—
216C23×C4—531no—
217C2×Q16—532no—
218C2×Q16—1061no—
219C2×Q16—1061no—
220C2×Q16—1061no—
221C2×Q16—1061no—
222C2×Q16—1061no—
223C22×C8—531no—
224C22×C8—532no—
225C22×C8—532no—
226C22×C8—1061no—
227C22×C8—1061no—
228C22×C8—1061no—
229C22×C8—1061no—
230C22×C8C22×D5311yescharacteristic
231C22⋊Q8—1064no—
232C22⋊Q8—2122no—
233#117—1064no—
234#117—2122no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4—2121no—
239C2×C22⋊C4—2121no—
240C2×C22⋊C4—2121no—
241C2×C42—2121no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1061no—
246C2×C16—1061no—
247C2×C16—1061no—
248#107—21no—
249#107—21no—
250#107—532no—
251#107—1061no—
252#107C4×D5312yes—
253#107D21212yes—
254#106—1064no—
255#103—1064no—
256C24—531no—
257C2×Q8—1062no—
258C2×Q8—2121no—
259C2×Q8—2121no—
260C2×Q8—2121no—
261C2×Q8—2121no—
262C2×Q8—2121no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—531no—
266C22×C4—532no—
267C22×C4—532no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4D4×D5311yescharacteristic
275Q16—1062no—
276Q16—1062no—
277Q16—2121no—
278Q16—2121no—
279C2×C8—532no—
280C2×C8—1061no—
281C2×C8—1061no—
282C2×C8—1062no—
283C2×C8—1062no—
284C2×C8—1062no—
285C2×C8—1062no—
286C2×C8C2×C4×D5311yescharacteristic
287C2×C8C2×D21211yescharacteristic
288C2×C8#1171994511yescharacteristic
289C2×C8D4×D5311yescharacteristic
290C2×C8D4×D5311yescharacteristic
291C2×C8#1171994911yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—1062no—
297C4⋊C4—2124no—
298C4⋊C4—2121no—
299C22⋊C4—2124no—
300C22⋊C4—2124no—
301C42—2124no—
302C16—41no—
303C16—2121no—
304C23—532no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23—1061no—
311C23D8×D5311yescharacteristic
312Q8—4241no—
313Q8—4241no—
314Q8—2122no—
315Q8—2122no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—532no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4C22⋊C4×D5311yescharacteristic
336C2×C4#1313318711yescharacteristic
337C2×C4#1313320211yescharacteristic
338C2×C4SD16×D5311yescharacteristic
339C2×C4#1313324411yescharacteristic
340C2×C4#1313324911yescharacteristic
341C8—1062no—
342C8#1313317911yescharacteristic
343C8C4×D21211yescharacteristic
344C8#1313319011yescharacteristic
345C8#1313319211yescharacteristic
346C22—1062no—
347C22—1062no—
348C22—1062no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22#102×D5311yescharacteristic
354C22#1681075811yescharacteristic
355C22#1681093811yescharacteristic
356C22#1681105211yescharacteristic
357C22#1681105411yescharacteristic
358C22#1681106211yescharacteristic
359C22#1681106511yescharacteristic
360C4—82no—
361C4—82no—
362C4—1062no—
363C4—4242no—
364C4—4242no—
365C4#1681050111yescharacteristic
366C4#1681062311yescharacteristic
367C4#1681076311yescharacteristic
368C4#1681094311yescharacteristic
369C2—2122no—
370C2#2027503111yescharacteristic
371C2#2027641911yescharacteristic
372C2#2027643311yescharacteristic
373C2#2027665611yescharacteristic
374C2#2027665811yescharacteristic
375C2#2027692511yescharacteristic
376C2#2027692611yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC23×D53D8
cosocleC22×C8×C53C23
hypercenter#1380D53
hypocenterC53#12994
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#16810428C2semidirect
4. C2×#16810426C2semidirect
6. #20272806C2semidirect
7. C53×#1380C2semidirect
14. #16810865C4semidirect
187. #1380D53semidirect
214. C53#12994semidirect

Maximal subgroups

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

subgroupindex
C2×#168108652
C2×#168106602
C2×#168104282
C2×#168104262
C2×#168104022
#202728062
C53×#13802
#1299453