Finite Group Databasebeta

← #27278751

Subgroups — #27278751

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283795C211yesmaximal, characteristic, complemented
2C4×#13133205C211yesmaximal, characteristic
3#20273107C211yesmaximal, characteristic
4#20273105C211yesmaximal, characteristic
5#20273038C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C53×#1276C211yesmaximal, characteristic, complemented
8C2×#13133205C411yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11C8×Dic106—22no—
12#16810747—22no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810673—22no—
16#16810529C411yescharacteristic
17#16810525C411yescharacteristic
18#16810514—22no—
19#16810489C411yescharacteristic
20C4×#11719921—22no—
21#16810352—22no—
22#16810350—22no—
23#16810348—22no—
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C43×C53C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×D212C2×C411yescharacteristic
29C22×C4×D53—21no—
30#13133205—22no—
31#13133205C814yescomplemented
32C4×Dic106—21no—
33C4×Dic106—21no—
34C4×Dic106—21no—
35C4×Dic106—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×#11719919D412yes—
43C2×#11719921—21no—
44C2×#11719921—22no—
45C2×#11719921—22no—
46C2×C4×Dic53D412yes—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49#13133146—24no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C4×#88322—21no—
53C4×#88322—21no—
54C8×Dic53—24no—
55#13133124—22no—
56#13133124—22no—
57C2×C42×C53—21no—
58C2×C42×C53—21no—
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53C2×C411yescharacteristic
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53D412yes—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C23×D53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53C22⋊C412yes—
76C22×Dic53C22⋊C412yes—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212C2×C812yes—
83C2×C4×D53—44no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×Dic106—21no—
87C2×Dic106—21no—
88C2×Dic106—21no—
89C2×Dic106—21no—
90C2×Dic106C2×C812yes—
91C2×Dic106M1612yes—
92C2×C8×C53—22no—
93C2×C8×C53—41no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—42no—
98C42×C53—42no—
99C42×C53C2×C812yes—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104#11719921—28no—
105#11719921—44no—
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112#11719919—22no—
113#11719919—22no—
114C4×Dic53—22no—
115C4×Dic53—22no—
116C2×#88322—22no—
117C2×#88322—41no—
118C23×C53C2×C22⋊C411yescharacteristic
119C22×D53—22no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—41no—
123C22×D53—41no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53C22×C811yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53#13111yescharacteristic
141C2×C4×C53#13111yescharacteristic
142C2×Dic53—22no—
143C2×Dic53—22no—
144C2×Dic53—22no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53C22⋊C814yes—
149D212—22no—
150D212—22no—
151D212—41no—
152D212—41no—
153C4×D53—82no—
154C4×D53—82no—
155Dic106—22no—
156Dic106—22no—
157Dic106—22no—
158Dic106—22no—
159C8×C53—42no—
160C8×C53—81no—
161#88322—42no—
162#88322—81no—
163#8960—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53#38611yescharacteristic
167C22×C53C2×C22⋊C811yescharacteristic
168C22×C53#40811yescharacteristic
169C2×D53—44no—
170C2×D53—42no—
171C2×D53—82no—
172C2×D53—81no—
173C2×D53—81no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—42no—
178C4×C53—42no—
179C4×C53C8×D411yescharacteristic
180C4×C53C8×D411yescharacteristic
181C4×C53#43511yescharacteristic
182C4×C53#43511yescharacteristic
183C4×C53#45412yes—
184Dic53—24no—
185Dic53—42no—
186Dic53—81no—
187#2982—531no—
188C4×#124—531no—
189#1478—531no—
190#1478—531no—
191#1461—531no—
192#1276—531no—
193#1276D5311yescharacteristic, complemented
194C2×C53—21no—
195C2×C53—21no—
196C2×C53#186911yescharacteristic
197C2×C53#188411yescharacteristic
198C2×C53#188811yescharacteristic
199D53—82no—
200C2×#124—531no—
201C2×C4×Q8—531no—
202C2×C4×D4—531no—
203C8×Q8—1062no—
204C8×Q8—1062no—
205#431—531no—
206#431—531no—
207#431—1062no—
208#431C2×D5311yescharacteristic
209#431C2×D5311yescharacteristic
210#391—531no—
211#390—531no—
212#388—1062no—
213#382—531no—
214C4×C22⋊C4—1062no—
215C43C2×D5311yescharacteristic
216D4⋊C8—1062no—
217D4⋊C8—1062no—
218#324—1062no—
219C53#896011yescharacteristic, complemented, Sylow
220C22×Q8—531no—
221C22×D4—531no—
222C23×C4—1061no—
223#124—534no—
224#124—1062no—
225C4×Q8—1061no—
226C4×Q8—1061no—
227C4×Q8—1061no—
228C4×Q8—1061no—
229C4×D4—1061no—
230C4×D4—1061no—
231C4×D4—1061no—
232C4×D4—1061no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—532no—
235C2×C4⋊C4—531no—
236C2×C22⋊C4—1062no—
237C2×C22⋊C4—1062no—
238C2×C22⋊C4—1061no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—532no—
242C2×C42C4×D5311yescharacteristic
243C2×C42C4×D5311yescharacteristic
244C2×C42C22×D5311yescharacteristic
245C4⋊C8—21no—
246C4⋊C8—532no—
247C4⋊C8—1064no—
248C4⋊C8—1061no—
249C4⋊C8D21212yes—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—1061no—
253C22⋊C8—1061no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—1064no—
257C4×C8—1061no—
258C4×C8—1061no—
259#95—1062no—
260#95—1062no—
261C24—1061no—
262C2×Q8—532no—
263C2×Q8—532no—
264C2×Q8—1061no—
265C2×Q8—1061no—
266C2×Q8—1061no—
267C2×Q8—1061no—
268C2×D4—532no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—532no—
277C22×C4—532no—
278C22×C4—1061no—
279C22×C4—2124no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4C2×C4×D5311yescharacteristic
283C22×C4D4×D5311yescharacteristic
284C22×C4D4×D5311yescharacteristic
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—1062no—
288C2×C8—2121no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4—1062no—
292C4⋊C4—1062no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C22⋊C4—1068no—
296C22⋊C4—2124no—
297C22⋊C4—2121no—
298C22⋊C4—2121no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—42no—
303C42—42no—
304C42—1062no—
305C42—1062no—
306C42C8×D5312yes—
307C42C2×D21211yescharacteristic
308C42#1171994511yescharacteristic
309C42D4×D5311yescharacteristic
310C42#1171994911yescharacteristic
311C23—1062no—
312C23—2122no—
313C23—2122no—
314C23—2121no—
315C23—2121no—
316C23C22⋊C4×D5311yescharacteristic
317Q8—1062no—
318Q8—1062no—
319Q8—1062no—
320Q8—1062no—
321D4—1062no—
322D4—1062no—
323D4—2121no—
324D4—2121no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—534no—
334C2×C4—1062no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—4242no—
338C2×C4—4242no—
339C2×C4—2122no—
340C2×C4—2121no—
341C2×C4—2122no—
342C2×C4#1313317911yescharacteristic
343C2×C4C4×D21211yescharacteristic
344C2×C4#1313318711yescharacteristic
345C2×C4#1313319011yescharacteristic
346C2×C4#1313319211yescharacteristic
347C2×C4#1313320211yescharacteristic
348C2×C4C2×C8×D5311yescharacteristic
349C2×C4M16×D5311yescharacteristic
350C2×C4#1313322311yescharacteristic
351C2×C4#1313322511yescharacteristic
352C8—42no—
353C8—81no—
354C8—4241no—
355C8—2122no—
356C22—21no—
357C22—21no—
358C22—4242no—
359C22—4241no—
360C22—4241no—
361C22—2124no—
362C22—2122no—
363C22#1681050111yescharacteristic
364C22C22⋊C8×D5311yescharacteristic
365C22#1681058811yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—1064no—
372C4—4241no—
373C4—2122no—
374C4#1681068811yescharacteristic
375C4C8×D21211yescharacteristic
376C4#1681069311yescharacteristic
377C4#1681069611yescharacteristic
378C4#1681079712yes—
379C2—21no—
380C2—21no—
381C2—4242no—
382C2#2027500611yescharacteristic
383C2#2027509711yescharacteristic
384C2#2027511711yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1276D53
hypocenterC53#8960
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #20283795C2semidirect
5. #20273038C2semidirect
6. #20272548C2semidirect
7. C53×#1276C2semidirect
31. #13133205C8semidirect
193. #1276D53semidirect
219. C53#8960semidirect

Maximal subgroups

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

subgroupindex
#202837952
C4×#131332052
#202731072
#202731052
#202730382
#202725482
C53×#12762
#896053