Finite Group Databasebeta

← #27278402

Subgroups — #27278402

Order 13568.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283380C211yesmaximal, characteristic
2#20274736C211yesmaximal, characteristic
3#20273485C211yesmaximal, characteristic, complemented
4#20273474C211yesmaximal, characteristic, complemented
5#20273098C211yesmaximal, characteristic
6#20272618C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810683—22no—
10#16810679—21no—
11#16810676C2211yescharacteristic
12#16810675—21no—
13#16810647C2211yescharacteristic
14#16810647C2211yescharacteristic
15C2×#13133145—21no—
16#16810589—21no—
17#16810578—21no—
18#16810383—21no—
19#16810376—22no—
20#16810373—21no—
21#16810373—21no—
22#16810355C411yescharacteristic, complemented
23#16810355C411yescharacteristic, complemented
24#16810352C411yescharacteristic, complemented
25#16810352C411yescharacteristic, complemented
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C2×#11719945D411yescharacteristic
30C2×#11719917—21no—
31C2×#11719917—21no—
32C4×Dic106C2×C411yescharacteristic
33C4×Dic106Q811yescharacteristic
34C4×D212C2×C411yescharacteristic
35C4×D212Q811yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53—21no—
38#13133176D411yescharacteristic
39#13133174—21no—
40C42×D53—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322C2×C411yescharacteristic
45C4⋊#88322C2×C411yescharacteristic
46C4⋊#88322D412yes—
47#13133145—42no—
48C22⋊#88322—21no—
49#13133129—22no—
50#13133129—22no—
51#13133127—22no—
52C4×#88322—21no—
53C8×Dic53—22no—
54C22×C8×C53—21no—
55C22×C8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C42×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53D411yescharacteristic
61C4⋊C8×C53D411yescharacteristic
62C4×C8×C53—21no—
63C4×C8×C53C2×C411yescharacteristic
64C4×C8×C53C2×C411yescharacteristic
65#95×C53—21no—
66C22×C4×C53—41no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53#4111yescharacteristic
69C22×C4×C53#4111yescharacteristic
70C2×C22⋊D53—21no—
71#11719945—21no—
72#11719945—21no—
73#11719945—21no—
74#11719945—21no—
75C2×D212C4⋊C411yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×Dic106C4⋊C411yescharacteristic
80C2×C8×C53—21no—
81C2×C8×C53—21no—
82C2×C8×C53—21no—
83C2×C8×C53—21no—
84C2×C8×C53—42no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C42×C53—41no—
90C42×C53—41no—
91C42×C53C22×C411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53C2×Q811yescharacteristic
95C22⋊Dic53—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53C4⋊C411yescharacteristic
99C4⋊Dic53C4⋊C411yescharacteristic
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105#11719917—41no—
106#11719917—41no—
107#11719917—41no—
108#11719917—41no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—22no—
112C2×#88322—41no—
113C23×C53#12311yescharacteristic
114C22×D53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53C2×C4⋊C411yescharacteristic
122C2×C4×C53C4×D411yescharacteristic
123C2×C4×C53C4×D411yescharacteristic
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53C22⋊Q811yescharacteristic
126C2×C4×C53C22⋊Q811yescharacteristic
127C2×C4×C53#13511yescharacteristic
128C2×C4×C53#13511yescharacteristic
129C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C53#13711yescharacteristic
131C22⋊D53—41no—
132C22⋊D53—41no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136D212—21no—
137D212—21no—
138C4×D53—41no—
139C4×D53—41no—
140C4×D53—41no—
141C4×D53—41no—
142C4×D53—81no—
143Dic106—21no—
144Dic106—21no—
145C8×C53—22no—
146C8×C53—41no—
147C8×C53—41no—
148C8×C53—41no—
149#88322—41no—
150#88322—41no—
151#88322—42no—
152#88322—81no—
153#8903—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53#38511yescharacteristic
157C22×C53#48211yescharacteristic
158C22×C53#48511yescharacteristic
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—82no—
166C4×C53#42711yescharacteristic
167C4×C53#42811yescharacteristic
168C4×C53#44311yescharacteristic
169C4×C53#44411yescharacteristic
170C4×C53#44911yescharacteristic
171C4×C53#45211yescharacteristic
172Dic53—41no—
173Dic53—41no—
174Dic53—41no—
175#2925—531no—
176#1806—531no—
177#1570—531no—
178#1567—531no—
179#1476—531no—
180#1303—531no—
181#1279D5311yescharacteristic, complemented
182C2×C53—21no—
183C2×C53—21no—
184C2×C53#185411yescharacteristic
185C2×C53#186111yescharacteristic
186C2×C53#186611yescharacteristic
187D53—81no—
188C4×#41—531no—
189#433—1062no—
190#432—1061no—
191#431—531no—
192#431—1061no—
193#423—531no—
194#423—531no—
195C2×C4⋊C8C2×D5311yescharacteristic
196C2×#104—1061no—
197#408—1061no—
198#405—1061no—
199C2×C4×C8C2×D5311yescharacteristic
200C4×C4⋊C4C2×D5311yescharacteristic
201#339—1061no—
202#335—1062no—
203#334—1061no—
204#334—1061no—
205Q8⋊C8—531no—
206Q8⋊C8—531no—
207D4⋊C8—531no—
208D4⋊C8—531no—
209C53#890311yescharacteristic, complemented, Sylow
210C2×#41—531no—
211C2×M16—1061no—
212C2×M16—1061no—
213C22×C8—21no—
214C22×C8—21no—
215C4×Q8—531no—
216C4×Q8—531no—
217C4×D4—531no—
218C4×D4—531no—
219C4×D4—1061no—
220C4×D4—1061no—
221#117—531no—
222#117—1061no—
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—1061no—
226C2×C42C22×D5311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—531no—
229C4⋊C8—531no—
230C4⋊C8—532no—
231C4⋊C8—1061no—
232C4⋊C8—1061no—
233C4⋊C8—1061no—
234C4⋊C8D21211yescharacteristic
235C4⋊C8D21211yescharacteristic
236#104—2122no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239C22⋊C8—1061no—
240#97—1062no—
241C4×C8—21no—
242C4×C8—1062no—
243C4×C8—1061no—
244C4×C8C4×D5311yescharacteristic
245C4×C8C4×D5311yescharacteristic
246#95—21no—
247#41—1061no—
248#41—1061no—
249#41—1061no—
250#41—1061no—
251C2×Q8—531no—
252C2×D4—531no—
253C2×D4—1061no—
254C22×C4—41no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4#1171994511yescharacteristic
259C22×C4D4×D5311yescharacteristic
260C22×C4#1171994711yescharacteristic
261M16—2121no—
262M16—2121no—
263M16—2121no—
264M16—2121no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—1061no—
273C2×C8—1061no—
274C2×C8—1062no—
275C2×C8—2121no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C42—41no—
286C42—41no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C42C2×C4×D5311yescharacteristic
291C42C2×D21211yescharacteristic
292C42D4×D5311yescharacteristic
293C42Q8×D5311yescharacteristic
294C23—1061no—
295C23#1313319911yescharacteristic
296Q8—1061no—
297Q8—1061no—
298D4—1061no—
299D4—1061no—
300D4—2121no—
301D4—2121no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—4241no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4C4⋊C4×D5311yescharacteristic
317C2×C4#1313317711yescharacteristic
318C2×C4C4×D21211yescharacteristic
319C2×C4#1313318711yescharacteristic
320C2×C4#1313319511yescharacteristic
321C2×C4#1313319711yescharacteristic
322C2×C4#1313323611yescharacteristic
323C2×C4#1313324011yescharacteristic
324C2×C4#1313324411yescharacteristic
325C2×C4#1313324611yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—4241no—
331C8—2122no—
332C8—2121no—
333C8—2121no—
334C22—21no—
335C22—21no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22#1681049711yescharacteristic
340C22#1681094511yescharacteristic
341C22#1681096311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—82no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4#1681066011yescharacteristic
350C4#1681066211yescharacteristic
351C4#1681074111yescharacteristic
352C4#1681074611yescharacteristic
353C4#1681077111yescharacteristic
354C4#1681078611yescharacteristic
355C2—21no—
356C2—21no—
357C2—4241no—
358C2#2027491711yescharacteristic
359C2#2027495611yescharacteristic
360C2#2027499311yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC22×C53#385
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8903
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #20273485C2semidirect
4. #20273474C2semidirect
6. #20272618C2semidirect
7. C53×#1279C2semidirect
22. #16810355C4semidirect
23. #16810355C4semidirect
24. #16810352C4semidirect
25. #16810352C4semidirect
181. #1279D53semidirect
209. C53#8903semidirect

Maximal subgroups

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

subgroupindex
#202833802
#202747362
#202734852
#202734742
#202730982
#202726182
C53×#12792
#890353