Finite Group Databasebeta

← #27397975

Subgroups — #27397975

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285160C211yesmaximal, characteristic
2#20283427C211yesmaximal, characteristic, complemented
3#20283426C211yesmaximal, characteristic, complemented
4#20275547C211yesmaximal, characteristic, complemented
5#20275040C214yesmaximal, complemented
6#20275039C214yesmaximal, complemented
7#20274855C211yesmaximal, characteristic, complemented
8#20274467C211yesmaximal, characteristic, complemented
9C53×#1963C211yesmaximal, characteristic, complemented
10C2×#13133237C2214yes—
11C2×#13133225C2212yes—
12C2×#13133224C2212yes—
13#16811173C2211yescharacteristic
14#16811018C412yescomplemented
15#16811015C412yescomplemented
16#16811013C412yescomplemented
17#16811010C412yescomplemented
18#16810701—21no—
19#16810701—21no—
20#16810700—21no—
21#16810700—21no—
22D4×#88322—21no—
23D4×#88322—21no—
24#16810688—21no—
25#16810688—21no—
26#16810647C2211yescharacteristic
27C2×#13133145C2214yes—
28#16810591C2212yes—
29#16810590C2212yes—
30C2×#13133128C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810387C2214yes—
33#16810378C2212yes—
34#16810378C2212yes—
35C53×#532C2211yescharacteristic
36C53×#423C2211yescharacteristic
37C2×#104×C53C2214yes—
38C53×#376C2211yescharacteristic
39C2×#11719945C2312yes—
40#13133237—44no—
41#13133237—44no—
42C2×#11719925C2×C414yes—
43C2×#11719923C2×C414yes—
44C2×#11719924C2×C414yes—
45C2×#11719922C2×C414yes—
46#13133225—44no—
47#13133224—44no—
48C2×#11719917—22no—
49C2×#11719917C2312yes—
50C2×#11719912—22no—
51C22×#88322—22no—
52C22×#88322C2312yes—
53C22×#88322C2311yescharacteristic
54C22×#88322C2311yescharacteristic
55C2×C8×D53—22no—
56#13133217C2×C411yescharacteristic
57#13133214C2×C411yescharacteristic
58#13133205C2×C412yes—
59#13133191—21no—
60#13133191—21no—
61C4×D212—21no—
62C4×D212—21no—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65#13133145—44no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68#13133129—22no—
69#13133129—22no—
70#13133128C2×C412yes—
71#13133128C2×C412yes—
72C4×#88322D411yescharacteristic
73C4×#88322D411yescharacteristic
74C4×#88322D411yescharacteristic
75C4×#88322D411yescharacteristic
76C2×#41×C53C2312yes—
77C2×M16×C53C2312yes—
78C4⋊Q8×C53C2×C411yescharacteristic
79#127×C53C2×C411yescharacteristic
80#124×C53C2×C412yes—
81#121×C53—21no—
82#121×C53—21no—
83C4×D4×C53—21no—
84C4×D4×C53—21no—
85C2×C42×C53C2312yes—
86C2×C42×C53C2311yescharacteristic
87C4⋊C8×C53—21no—
88C4⋊C8×C53—21no—
89#104×C53—44no—
90#95×C53—21no—
91#95×C53—21no—
92#41×C53—24no—
93#41×C53—24no—
94C2×Q8×C53C22×C412yes—
95C2×D4×C53—22no—
96C2×D4×C53—21no—
97C2×D4×C53—21no—
98C2×D4×C53C22×C412yes—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53#4111yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×C4×C53C2411yescharacteristic
104C2×C22⋊D53—22no—
105#11719945—24no—
106#11719945—24no—
107C2×D212—21no—
108C2×D212—21no—
109C2×D212C22×C412yes—
110C2×C4×D53—22no—
111C2×Dic106C22×C412yes—
112M16×C53—24no—
113C2×C8×C53—22no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C42×C53—22no—
119C42×C53—22no—
120C42×C53C22×C411yescharacteristic
121C42×C53C22×C411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124#11719925—44no—
125#11719924—44no—
126#11719923—44no—
127#11719922—44no—
128#11719921—22no—
129#11719921—22no—
130C4⋊Dic53—21no—
131C4⋊Dic53—21no—
132#11719917—24no—
133#11719917—44no—
134C2×#88322—22no—
135C2×#88322—22no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—22no—
139C2×#88322—44no—
140C2×#88322C22×C414yes—
141C2×#88322C2×D411yescharacteristic
142C2×#88322C2×D411yescharacteristic
143C2×#88322C2×D411yescharacteristic
144C2×#88322C2×D411yescharacteristic
145#11719912—44no—
146C8×D53—44no—
147C23×C53—22no—
148C23×C53C2×#4111yescharacteristic
149C22×D53—22no—
150Q8×C53—24no—
151D4×C53—24no—
152D4×C53—42no—
153D4×C53—42no—
154D4×C53—44no—
155C2×C4×C53—22no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53C4×D411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53C23×C411yescharacteristic
168C2×C4×C53C22×D411yescharacteristic
169C2×C4×C53H(5,𝔽2)12yes—
170C2×C4×C53H(5,𝔽2)12yes—
171C22⋊D53—44no—
172C2×Dic53—22no—
173D212—24no—
174D212—42no—
175D212—42no—
176C4×D53—42no—
177C4×D53—42no—
178Dic106—24no—
179C8×C53—42no—
180#88322—24no—
181#88322—21no—
182#88322—21no—
183#88322—21no—
184#88322—21no—
185#88322—42no—
186#22281—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—42no—
190C22×C53—42no—
191C22×C53—42no—
192C22×C53C2×C4×D411yescharacteristic
193C22×C53#54012yes—
194C2×D53—42no—
195C2×D53—42no—
196C2×D53—42no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—42no—
202C4×C53—82no—
203C4×C53#51812yes—
204C4×C53#53511yescharacteristic
205C4×C53#53511yescharacteristic
206Dic53—42no—
207#3150—531no—
208#2931—531no—
209#2931—531no—
210#1963—531no—
211#1963D5311yescharacteristic, complemented
212#1876—534no—
213#1876—534no—
214#1838—531no—
215#1746—531no—
216C2×C53—22no—
217C2×C53—82no—
218C2×C53#232811yescharacteristic
219C2×C53#295511yescharacteristic
220C2×C53#295511yescharacteristic
221D53—82no—
222C2×#135—534no—
223C2×#131—532no—
224C2×#131—532no—
225#532—531no—
226#532C2×D5311yescharacteristic
227#497—532no—
228#497—532no—
229#496—532no—
230#496—532no—
231#436—1061no—
232#436—1061no—
233#436—1061no—
234#436—1061no—
235C8×D4—1061no—
236C8×D4—1061no—
237C8×D4—1061no—
238C8×D4—1061no—
239#423—531no—
240#423C2×D5311yescharacteristic
241C2×#104—534no—
242C2×#104C2×D5314yes—
243#408—532no—
244#408—532no—
245C2×#97—531no—
246C2×C4×C8—531no—
247#376C2×D5311yescharacteristic
248#341—534no—
249#336—532no—
250#336—532no—
251C53#2228111yescharacteristic, complemented, Sylow
252C2×#41—532no—
253C2×#41C22×D5312yes—
254#135—2124no—
255#135—2124no—
256C2×Q16—534no—
257C2×SD16—534no—
258C2×SD16—534no—
259C2×D8—534no—
260#131—2124no—
261#131—2124no—
262C2×M16—532no—
263C2×M16—1062no—
264C2×M16—1062no—
265C2×M16C22×D5312yes—
266C22×C8—531no—
267C22×C8—531no—
268C22×C8—532no—
269C22×C8—1062no—
270C22×C8—1062no—
271C4⋊Q8—531no—
272C4⋊Q8C4×D5311yescharacteristic
273#127—531no—
274#127C4×D5311yescharacteristic
275#124—532no—
276#124C4×D5312yes—
277#121—21no—
278#121—21no—
279#121—1061no—
280#121—1061no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—1061no—
284C4×D4—1061no—
285C2×C42C22×D5312yes—
286C2×C42C22×D5311yescharacteristic
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—1061no—
290C4⋊C8—1061no—
291#104—44no—
292#104—2124no—
293C22⋊C8—1062no—
294C22⋊C8—1062no—
295C22⋊C8—1062no—
296C22⋊C8—1062no—
297#97—532no—
298#97—532no—
299C4×C8—531no—
300C4×C8—531no—
301C4×C8—531no—
302C4×C8—531no—
303#95—21no—
304#95—21no—
305#41—24no—
306#41—24no—
307#41—1064no—
308#41—1064no—
309C2×Q8—532no—
310C2×Q8C2×C4×D5312yes—
311C2×D4—22no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—532no—
315C2×D4—1062no—
316C2×D4—1061no—
317C2×D4—1061no—
318C2×D4C2×C4×D5312yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—1062no—
322C22×C4#1171994711yescharacteristic
323C22×C4#1171994711yescharacteristic
324C22×C4C23×D5311yescharacteristic
325Q16—2124no—
326SD16—2124no—
327SD16—2124no—
328D8—2124no—
329M16—24no—
330M16—1064no—
331M16—2124no—
332M16—2124no—
333C2×C8—22no—
334C2×C8—534no—
335C2×C8—531no—
336C2×C8—531no—
337C2×C8—531no—
338C2×C8—531no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—1062no—
342C2×C8—1062no—
343C2×C8—1062no—
344C2×C8—2124no—
345C2×C8—2124no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—1062no—
353C22⋊C4—1062no—
354C42—22no—
355C42—22no—
356C42C2×C4×D5311yescharacteristic
357C42C2×C4×D5311yescharacteristic
358C42D4×D5311yescharacteristic
359C42D4×D5311yescharacteristic
360C23—22no—
361C23—1062no—
362C23C2×#1171994711yescharacteristic
363Q8—24no—
364Q8—1064no—
365D4—24no—
366D4—42no—
367D4—42no—
368D4—44no—
369D4—1064no—
370D4—2122no—
371D4—2122no—
372D4—2124no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—1062no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4#1313317711yescharacteristic
385C2×C4#1313317711yescharacteristic
386C2×C4#1313317711yescharacteristic
387C2×C4#1313317711yescharacteristic
388C2×C4C22×C4×D5311yescharacteristic
389C2×C4C2×D4×D5311yescharacteristic
390C2×C4#1313326212yes—
391C2×C4#1313326312yes—
392C8—42no—
393C8—1064no—
394C8—1061no—
395C8—1061no—
396C8—1061no—
397C8—1061no—
398C8—2122no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—2122no—
405C22—2122no—
406C22—2122no—
407C22C2×#1313317711yescharacteristic
408C22#1681122612yes—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—82no—
415C4—2122no—
416C4#1681109512yes—
417C4#1681118611yescharacteristic
418C4#1681119111yescharacteristic
419C2—22no—
420C2—82no—
421C2—4242no—
422C2#2027743911yescharacteristic
423C2#2028361411yescharacteristic
424C2#2028361411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1963C2
socleC22×C53C2×C4×D4
prime radical#20285160C2
cosocleC22×C4×C53C24
hypercenter#1963D53
hypocenterC53#22281
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #20283427C2semidirect
3. #20283426C2semidirect
4. #20275547C2semidirect
5. #20275040C2semidirect
6. #20275039C2semidirect
7. #20274855C2semidirect
8. #20274467C2semidirect
9. C53×#1963C2semidirect
14. #16811018C4semidirect
15. #16811015C4semidirect
16. #16811013C4semidirect
17. #16811010C4semidirect
211. #1963D53semidirect
251. C53#22281semidirect

Maximal subgroups

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

subgroupindex
#202851602
#202834272
#202834262
#202755472
#202750402
#202750392
#202748552
#202744672
C53×#19632
#2228153