Finite Group Databasebeta

← #27288666

Subgroups — #27288666

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275641C211yesmaximal, characteristic
2#20275520C211yesmaximal, characteristic
3#20275476C211yesmaximal, characteristic, complemented
4#20275431C211yesmaximal, characteristic, complemented
5C2×#16810373C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic
7C53×#1296C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133148—21no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133146—21no—
12C2×#13133139—21no—
13C2×#13133139—21no—
14C2×#13133137—21no—
15C2×#13133137—21no—
16C2×C4×#88322C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810385—21no—
19#16810377—21no—
20#16810373—22no—
21#16810366—22no—
22C2×C4⋊C8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C22×D212—21no—
26C22×D212—21no—
27C22×D212—21no—
28C22×#88322—21no—
29C22×#88322D411yescharacteristic
30C22×#88322D411yescharacteristic
31#13133214—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214Q812yes—
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4×Dic53—21no—
38#13133148—42no—
39C4⋊#88322—21no—
40C4⋊#88322D412yes—
41#13133146—42no—
42#13133139—42no—
43#13133139—42no—
44#13133137—42no—
45#13133137—42no—
46C4×#88322—21no—
47C4×#88322—21no—
48#13133124—21no—
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C42×C53C2311yescharacteristic
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C23×D53—41no—
60C22×C4×C53—41no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53#4111yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C2×D212—22no—
67C2×D212—22no—
68C2×D212—22no—
69C2×D212—41no—
70C2×D212—41no—
71C2×D212—41no—
72C2×D212—41no—
73C2×D212—41no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—81no—
81C2×D212—81no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—81no—
93C42×C53C2×D411yescharacteristic
94C42×C53C2×Q811yescharacteristic
95C42×C53#4111yescharacteristic
96C42×C53#4111yescharacteristic
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53SD1612yes—
101C4×Dic53—41no—
102C4×Dic53—41no—
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322—21no—
106C2×#88322—22no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322SD1612yes—
112C23×C53#12111yescharacteristic
113C22×D53—42no—
114C22×D53—81no—
115C22×D53—81no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C22×D53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53#12111yescharacteristic
126C2×C4×C53#12111yescharacteristic
127C2×C4×C53C22⋊Q811yescharacteristic
128C2×C4×C53C22⋊Q811yescharacteristic
129C2×C4×C53C22⋊Q811yescharacteristic
130C2×C4×C53#12611yescharacteristic
131C2×C4×C53C2×SD1611yescharacteristic
132C2×C4×C53C2×SD1611yescharacteristic
133C2×C4×C53#13511yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×Dic53—22no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145D212—42no—
146D212—42no—
147D212—42no—
148D212—42no—
149D212—82no—
150D212—82no—
151D212—81no—
152D212—81no—
153D212—81no—
154D212—81no—
155C8×C53—81no—
156C8×C53—81no—
157#88322—22no—
158#88322—42no—
159#88322—41no—
160#10595—531nomaximal, Sylow
161C22×C53#39611yescharacteristic
162C22×C53#46011yescharacteristic
163C22×C53#46311yescharacteristic
164C22×C53#46511yescharacteristic
165C22×C53#46911yescharacteristic
166C22×C53#47111yescharacteristic
167C22×C53#47211yescharacteristic
168C2×D53—82no—
169C2×D53—82no—
170C2×D53—82no—
171C2×D53—161no—
172C2×D53—161no—
173C2×D53—161no—
174C2×D53—161no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—161no—
178C4×C53#46011yescharacteristic
179C4×C53#46311yescharacteristic
180C4×C53#46511yescharacteristic
181C4×C53#46911yescharacteristic
182C4×C53#47611yescharacteristic
183C4×C53#47611yescharacteristic
184C4×C53#47811yescharacteristic
185C4×C53#47811yescharacteristic
186Dic53—42no—
187Dic53—81no—
188#1982—531no—
189#1957—531no—
190#1948—531no—
191#1939—531no—
192C2×#334—531no—
193C2×#331—531no—
194#1296D5311yescharacteristic, complemented
195C2×C53#166611yescharacteristic
196C2×C53#167711yescharacteristic
197C2×C53#168811yescharacteristic
198C2×C53#170111yescharacteristic
199C2×C53#206311yescharacteristic
200C2×C53#206511yescharacteristic
201C2×C53#206511yescharacteristic
202D53—162no—
203C2×#127—531no—
204C2×#106—1061no—
205C2×C4⋊C8—531no—
206C2×C4⋊C8—1061no—
207C2×C4⋊C8C2×D5311yescharacteristic
208C2×C4⋊C8C2×D5311yescharacteristic
209C2×#102—1061no—
210C2×#102—1061no—
211C2×#102—1061no—
212C2×#102—1061no—
213C2×C4×C8—531no—
214#384C2×D5311yescharacteristic
215C4×C4⋊C4—531no—
216#340—1061no—
217#336—1061no—
218#334—1062no—
219#331—1062no—
220C53#1059511yescharacteristic, complemented, Sylow
221C22×D4—1061no—
222C22×D4—1061no—
223C22×D4—1061no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—531no—
227C22×C8—531no—
228C22×C8—1061no—
229#127—532no—
230#127—1061no—
231#127—1061no—
232#127—1061no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C42—1061no—
239C2×C42C22×D5311yescharacteristic
240#106—2122no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—532no—
246C4⋊C8—1061no—
247C4⋊C8—2122no—
248#102—2122no—
249#102—2122no—
250#102—2122no—
251#102—2122no—
252C4×C8—1061no—
253C4×C8—1061no—
254#95—1061no—
255C24—2121no—
256C2×D4—1062no—
257C2×D4—1062no—
258C2×D4—1062no—
259C2×D4—4241no—
260C2×D4—4241no—
261C2×D4—2121no—
262C2×D4—2121no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C22×C4—41no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4#1171994511yescharacteristic
276C22×C4D4×D5311yescharacteristic
277C22×C4D4×D5311yescharacteristic
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—532no—
285C2×C8—1061no—
286C2×C8—1061no—
287C2×C8—1061no—
288C2×C8—1062no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—2121no—
292C2×C8—2121no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—81no—
298C4⋊C4—532no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C42—2121no—
303C42—2121no—
304C42#1171994511yescharacteristic
305C42#1171994511yescharacteristic
306C42D4×D5311yescharacteristic
307C42Q8×D5311yescharacteristic
308C23—4241no—
309C23—4241no—
310C23—4241no—
311C23—4241no—
312C23—4241no—
313C23—4241no—
314C23—2122no—
315C23#1313319011yescharacteristic
316D4—4242no—
317D4—4242no—
318D4—4241no—
319D4—4241no—
320D4—4241no—
321D4—4241no—
322D4—2122no—
323D4—2122no—
324D4—2122no—
325D4—2122no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—1062no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4#1313319011yescharacteristic
338C2×C4#1313319011yescharacteristic
339C2×C4#1313319511yescharacteristic
340C2×C4#1313319511yescharacteristic
341C2×C4#1313319511yescharacteristic
342C2×C4#1313321111yescharacteristic
343C2×C4SD16×D5311yescharacteristic
344C2×C4SD16×D5311yescharacteristic
345C2×C4#1313324011yescharacteristic
346C2×C4#1313324011yescharacteristic
347C2×C4#1313324111yescharacteristic
348C2×C4#1313324111yescharacteristic
349C2×C4#1313324411yescharacteristic
350C2×C4#1313324411yescharacteristic
351C8—81no—
352C8—81no—
353C8—1062no—
354C8—2122no—
355C8—2121no—
356C22—8481no—
357C22—8481no—
358C22—8481no—
359C22—8481no—
360C22—4242no—
361C22—4242no—
362C22—4242no—
363C22#1681054711yescharacteristic
364C22#1681082211yescharacteristic
365C22#1681084111yescharacteristic
366C22#1681085511yescharacteristic
367C22#1681087511yescharacteristic
368C22#1681088911yescharacteristic
369C22#1681089211yescharacteristic
370C4—21no—
371C4—21no—
372C4—161no—
373C4—4241no—
374C4—2122no—
375C4#1681082211yescharacteristic
376C4#1681084111yescharacteristic
377C4#1681085511yescharacteristic
378C4#1681087511yescharacteristic
379C4#1681091211yescharacteristic
380C4#1681091211yescharacteristic
381C4#1681092611yescharacteristic
382C4#1681092611yescharacteristic
383C2—8482no—
384C2#2027406611yescharacteristic
385C2#2027412711yescharacteristic
386C2#2027419811yescharacteristic
387C2#2027427911yescharacteristic
388C2#2027607611yescharacteristic
389C2#2027609011yescharacteristic
390C2#2027609011yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC2×#13133214C22
hypercenter#1296D53
hypocenterC53#10595
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #20275476C2semidirect
4. #20275431C2semidirect
5. C2×#16810373C2semidirect
7. C53×#1296C2semidirect
194. #1296D53semidirect
220. C53#10595semidirect

Maximal subgroups

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

subgroupindex
#202756412
#202755202
#202754762
#202754312
C2×#168103732
C2×#168103662
C53×#12962
#1059553