Finite Group Databasebeta

← #27278412

Subgroups — #27278412

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#20274732C211yesmaximal, characteristic
3#20273470C211yesmaximal, characteristic, complemented
4#20273095C211yesmaximal, characteristic
5#20272617C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#1571C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810683—22no—
10#16810679—21no—
11#16810675—21no—
12#16810647C2211yescharacteristic
13C2×#13133143—21no—
14#16810589—21no—
15#16810578—21no—
16C2×C4×#88322C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810384—21no—
19#16810373—22no—
20#16810353C411yescharacteristic, complemented
21#16810353C411yescharacteristic, complemented
22#16810350C411yescharacteristic, complemented
23#16810350C411yescharacteristic, complemented
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C53×#335—21no—
28C53×#335—21no—
29C2×#11719945D411yescharacteristic
30C2×#11719917—21no—
31C22×#88322—21no—
32C4×Dic106C2×C411yescharacteristic
33C4×Dic106Q811yescharacteristic
34C4×D212C2×C411yescharacteristic
35C4×D212Q811yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53—21no—
38#13133176D411yescharacteristic
39#13133174—21no—
40C2×C4⋊Dic53—21no—
41C2×C4×Dic53—21no—
42C42×D53—21no—
43C4⋊#88322—21no—
44C4⋊#88322D412yes—
45#13133143—42no—
46#13133129—22no—
47#13133129—22no—
48#13133127—22no—
49C4×#88322—21no—
50C4×#88322C2×C411yescharacteristic
51C4×#88322C2×C411yescharacteristic
52C8×Dic53—22no—
53#13133124—21no—
54C2×M16×C53—21no—
55C22×C8×C53—21no—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53C2×C411yescharacteristic
61C4⋊C8×C53C2×C411yescharacteristic
62C4⋊C8×C53D411yescharacteristic
63C4⋊C8×C53D411yescharacteristic
64C22⋊C8×C53—21no—
65C4×C8×C53—21no—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4111yescharacteristic
68C22×C4×C53#4111yescharacteristic
69C2×C22⋊D53—21no—
70C22×Dic53—41no—
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
80M16×C53—41no—
81M16×C53—41no—
82C2×C8×C53—21no—
83C2×C8×C53—21no—
84C2×C8×C53—21no—
85C2×C8×C53—21no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C42×C53C22×C411yescharacteristic
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53C2×Q811yescharacteristic
93C22⋊Dic53—21no—
94#11719921—21no—
95#11719921—21no—
96C4⋊Dic53—22no—
97C4⋊Dic53—41no—
98C4⋊Dic53C4⋊C411yescharacteristic
99C4⋊Dic53C4⋊C411yescharacteristic
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107#11719917—41no—
108#11719917—41no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—22no—
112C2×#88322—42no—
113C23×C53#12311yescharacteristic
114C22×D53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53C2×C4⋊C411yescharacteristic
119C2×C4×C53C4×D411yescharacteristic
120C2×C4×C53C4×D411yescharacteristic
121C2×C4×C53#12011yescharacteristic
122C2×C4×C53C22⋊Q811yescharacteristic
123C2×C4×C53C22⋊Q811yescharacteristic
124C2×C4×C53#13511yescharacteristic
125C2×C4×C53#13511yescharacteristic
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C53#13711yescharacteristic
128C22⋊D53—41no—
129C22⋊D53—41no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—42no—
134C2×Dic53—81no—
135C2×Dic53—81no—
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—41no—
146C8×C53—41no—
147C8×C53—41no—
148C8×C53—41no—
149C8×C53—81no—
150#88322—22no—
151#88322—41no—
152#88322—42no—
153#8905—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#42711yescharacteristic
166C4×C53#42811yescharacteristic
167C4×C53#44311yescharacteristic
168C4×C53#44411yescharacteristic
169C4×C53#44911yescharacteristic
170C4×C53#45211yescharacteristic
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174Dic53—82no—
175#2925—531no—
176#1806—531no—
177#1571D5311yescharacteristic, complemented
178#1566—531no—
179#1476—531no—
180#1303—531no—
181#1279—531no—
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—1061no—
192#431C2×D5311yescharacteristic
193#423—531no—
194#423C2×D5311yescharacteristic
195C2×C4⋊C8C2×D5311yescharacteristic
196C2×#104—1061no—
197#408—1061no—
198#405—1061no—
199C2×C4×C8—531no—
200C4×C4⋊C4—531no—
201#339—1061no—
202#335—21no—
203#335—21no—
204#334—1062no—
205Q8⋊C8—531no—
206Q8⋊C8—531no—
207D4⋊C8—531no—
208D4⋊C8—531no—
209C53#890511yescharacteristic, complemented, Sylow
210C2×#41—531no—
211C2×M16—21no—
212C2×M16—1061no—
213C22×C8—21no—
214C22×C8—1061no—
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—1061no—
224C2×C42—1061no—
225C2×C42—1061no—
226C2×C42C22×D5311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—532no—
231C4⋊C8—1061no—
232C4⋊C8C4×D5311yescharacteristic
233C4⋊C8C4×D5311yescharacteristic
234C4⋊C8C22⋊D5311yescharacteristic
235C4⋊C8C22⋊D5311yescharacteristic
236#104—2122no—
237C22⋊C8—21no—
238C22⋊C8—1062no—
239C22⋊C8—1062no—
240#97—1062no—
241C4×C8—21no—
242C4×C8—531no—
243C4×C8—531no—
244C4×C8—1062no—
245C4×C8—1061no—
246#95—1061no—
247#41—1061no—
248#41—1061no—
249#41—1061no—
250#41—1061no—
251C2×Q8—531no—
252C2×D4—531no—
253C2×D4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—2121no—
258C22×C4#1171994511yescharacteristic
259C22×C4D4×D5311yescharacteristic
260C22×C4#1171994911yescharacteristic
261M16—41no—
262M16—41no—
263M16—2121no—
264M16—2121no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—1061no—
273C2×C8—1061no—
274C2×C8—1062no—
275C2×C8—2122no—
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—1062no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—2121no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C42—1061no—
286C42—1061no—
287C42—1061no—
288C42—2121no—
289C42—2121no—
290C42C2×C4×D5311yescharacteristic
291C42D4×D5311yescharacteristic
292C42Q8×D5311yescharacteristic
293C42C2×C22⋊D5311yescharacteristic
294C23—1061no—
295C23#1313320211yescharacteristic
296Q8—1061no—
297Q8—1061no—
298D4—1061no—
299D4—1061no—
300D4—2121no—
301D4—2121no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—4241no—
309C2×C4—4241no—
310C2×C4—4241no—
311C2×C4—2122no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4C4⋊C4×D5311yescharacteristic
317C2×C4C4×C22⋊D5311yescharacteristic
318C2×C4#1313317911yescharacteristic
319C2×C4#1313318511yescharacteristic
320C2×C4#1313319311yescharacteristic
321C2×C4#1313319511yescharacteristic
322C2×C4#1313323611yescharacteristic
323C2×C4#1313324011yescharacteristic
324C2×C4#1313324111yescharacteristic
325C2×C4#1313324911yescharacteristic
326C8—41no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—81no—
331C8—1062no—
332C8—2121no—
333C8—2122no—
334C22—21no—
335C22—21no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22#1681049411yescharacteristic
340C22#1681095011yescharacteristic
341C22#1681096611yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—4242no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4#1681066011yescharacteristic
350C4#1681066411yescharacteristic
351C4#1681073811yescharacteristic
352C4#1681074311yescharacteristic
353C4#1681076511yescharacteristic
354C4#1681078411yescharacteristic
355C2—21no—
356C2—21no—
357C2—4241no—
358C2#2027491111yescharacteristic
359C2#2027495211yescharacteristic
360C2#2027498911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1571C2
socleC22×C53#385
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1571D53
hypocenterC53#8905
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
3. #20273470C2semidirect
5. #20272617C2semidirect
6. #20272554C2semidirect
7. C53×#1571C2semidirect
20. #16810353C4semidirect
21. #16810353C4semidirect
22. #16810350C4semidirect
23. #16810350C4semidirect
177. #1571D53semidirect
209. C53#8905semidirect

Maximal subgroups

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

subgroupindex
#202833802
#202747322
#202734702
#202730952
#202726172
#202725542
C53×#15712
#890553