Finite Group Databasebeta

← #27369389

Subgroups — #27369389

Order 13568.

740 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic
2#20284183C211yesmaximal, characteristic, complemented
3C2×#16810824C211yesmaximal, characteristic, complemented
4#20283795C211yesmaximal, characteristic, complemented
5#20274162C214yesmaximal, complemented
6#20274074C214yesmaximal, complemented
7#20273489C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic, complemented
9C53×#1540C211yesmaximal, characteristic, complemented
10C2×#13133247—21no—
11C22×#11719913—21no—
12C2×#13133214C2211yescharacteristic
13C2×C4×Dic106C2211yescharacteristic
14#16810999C2212yes—
15#16810971C2212yes—
16#16810842—22no—
17#16810830C2212yescomplemented
18#16810824—22no—
19#16810824C2214yes—
20#16810820C2212yescomplemented
21#16810818—22no—
22#16810756—22no—
23C8×Dic106—22no—
24#16810747C2214yescomplemented
25#16810676C2211yescharacteristic
26#16810673—22no—
27#16810647C2211yescharacteristic
28#16810624—21no—
29C2×#13133139—21no—
30#16810375C2214yescomplemented
31#16810366C2214yes—
32C53×#532C2211yescharacteristic
33C53×#431C2211yescharacteristic
34C2×C4⋊C8×C53C2211yescharacteristic
35C53×#332C2212yescomplemented
36C53×#331C2212yescomplemented
37C22×Dic106D411yescharacteristic
38C22×D212—21no—
39C22×D212—21no—
40C22×D212—21no—
41#13133247—44no—
42C2×#11719925—22no—
43C2×#11719913—24no—
44C2×#11719913—44no—
45C2×#11719923—22no—
46C2×#11719922—24no—
47C2×#11719917—21no—
48#13133214—21no—
49#13133214—21no—
50#13133214—21no—
51#13133214C2312yes—
52C4×Dic106—21no—
53C4×Dic106—21no—
54C4×Dic106D412yes—
55C4×Dic106C2312yes—
56C2×C4⋊Dic53D411yescharacteristic
57C2×#11719919—21no—
58C2×C4×Dic53—21no—
59C4⋊#88322—21no—
60C4⋊#88322D412yes—
61C4⋊#88322C2312yes—
62#13133146—24no—
63#13133141—22no—
64#13133139—24no—
65#13133138—22no—
66#13133137—24no—
67C22⋊#88322D412yes—
68C4×#88322C2312yes—
69C8×Dic53—24no—
70C2×#41×C53—21no—
71C22×C8×C53—21no—
72C4⋊Q8×C53C2311yescharacteristic
73#127×C53C2311yescharacteristic
74#124×C53—21no—
75#121×C53—21no—
76#121×C53—21no—
77C4×D4×C53—21no—
78C4×D4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C4⋊C8×C53C2312yes—
82C4⋊C8×C53C2312yes—
83C22⋊C8×C53—21no—
84C4×C8×C53—21no—
85C23×D53—41no—
86#41×C53—42no—
87#41×C53—42no—
88C2×Q8×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×Dic53—21no—
98C22×Dic53—21no—
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—42no—
103C2×D212—42no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—82no—
108C2×D212—82no—
109C2×Dic106—21no—
110C2×Dic106—22no—
111C2×Dic106D814yescomplemented
112C2×Dic106C2×D412yes—
113C2×C8×C53—22no—
114C2×C8×C53—22no—
115C2×C8×C53—42no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53C2411yescharacteristic
124#11719925—44no—
125#11719923—44no—
126#11719922—84no—
127C4⋊Dic53—21no—
128C4⋊Dic53C2×D412yes—
129#11719919—22no—
130#11719919—22no—
131C4×Dic53—22no—
132C4×Dic53—22no—
133#11719917—42no—
134C2×#88322—22no—
135C2×#88322C2×D414yes—
136#11719913—48no—
137C23×C53—41no—
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143Q8×C53—41no—
144Q8×C53—41no—
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—42no—
148D4×C53—81no—
149D4×C53—82no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—82no—
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53C2×D812yes—
158C2×C4×C53#13512yes—
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C2×#4111yescharacteristic
162C2×Dic53—22no—
163C2×Dic53—24no—
164C2×Dic53—22no—
165C2×Dic53—81no—
166D212—42no—
167D212—42no—
168D212—44no—
169D212—82no—
170D212—82no—
171D212—82no—
172D212—82no—
173Dic106—22no—
174Dic106—24no—
175Dic106C2×D814yes—
176C8×C53—42no—
177C8×C53—42no—
178#88322—24no—
179#88322—42no—
180#19398—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53#46612yes—
187C22×C53C2×#12111yescharacteristic
188C2×D53—82no—
189C2×D53—82no—
190C2×D53—82no—
191C2×D53—164no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—81no—
196C4×C53#46612yes—
197C4×C53C22×D811yescharacteristic
198C4×C53C2×#13511yescharacteristic
199C4×C53#57712yes—
200Dic53—24no—
201Dic53—42no—
202#3205—531no—
203#3037—531no—
204C2×#460—531no—
205#2982—531no—
206#1682—534no—
207#1668—534no—
208#1571—531no—
209#1545—531no—
210#1540D5311yescharacteristic, complemented
211C2×C53—22no—
212C2×C53—161no—
213C2×C53#304311yescharacteristic
214C2×C53C2×#46611yescharacteristic
215C2×C53#306811yescharacteristic
216D53—162no—
217C2×#137—1061no—
218C22×SD16—1061no—
219#532C2×D5311yescharacteristic
220C2×#127—531no—
221C2×C4×Q8—531no—
222#493—532no—
223#486—532no—
224#464—1062no—
225#462—532no—
226#460—532no—
227#460—534no—
228#460—1062no—
229#459—1062no—
230#447—1062no—
231C8×Q8—534no—
232C8×Q8—1062no—
233#431—531no—
234#431—1062no—
235#431C2×D5311yescharacteristic
236#423—531no—
237C2×C4⋊C8C2×D5311yescharacteristic
238#417—1061no—
239C2×#102—1061no—
240#335—534no—
241#332C2×D5312yescomplemented
242#331—534no—
243#331C2×D5312yescomplemented
244C53#1939811yescharacteristic, complemented, Sylow
245C2×#41—21no—
246C22×Q8—531no—
247C22×D4—1061no—
248C22×D4—1061no—
249C22×D4—1061no—
250#137—2124no—
251C2×Q16—1062no—
252C2×SD16—1064no—
253C2×SD16—1062no—
254C2×SD16—2124no—
255C2×D8—1064no—
256C2×M16—1061no—
257C22×C8—21no—
258C4⋊Q8C22×D5311yescharacteristic
259#127—532no—
260#127—1061no—
261#127—1061no—
262#127—1061no—
263#127C22×D5311yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—532no—
268C4×Q8—532no—
269C4×Q8—1061no—
270C4×Q8—1061no—
271C4×D4—21no—
272C4×D4—21no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4—1061no—
275C2×C42—1061no—
276C2×C42C22×D5311yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—532no—
279C4⋊C8—532no—
280C4⋊C8—1064no—
281C4⋊C8—1061no—
282C4⋊C8C22×D5312yes—
283C4⋊C8C22×D5312yes—
284#103—1062no—
285#102—1064no—
286#102—1064no—
287#102—1062no—
288C22⋊C8—21no—
289C22⋊C8—532no—
290C4×C8—21no—
291C4×C8—532no—
292C4×C8—1064no—
293C24—2121no—
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×Q8—534no—
298C2×Q8—532no—
299C2×Q8—1061no—
300C2×Q8—1062no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—1062no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—4242no—
309C2×D4—4242no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2122no—
314C2×D4—2122no—
315C22×C4—41no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4#1171994511yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C22×C4D4×D5311yescharacteristic
321Q16—2124no—
322SD16—2128no—
323SD16—2124no—
324D8—4244no—
325M16—2122no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—42no—
329C2×C8—534no—
330C2×C8—1062no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—532no—
334C4⋊C4—1061no—
335C4⋊C4—1062no—
336C4⋊C4—1062no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—1062no—
340C42—1062no—
341C42#1171994511yescharacteristic
342C42D4×D5311yescharacteristic
343C42D4×D5311yescharacteristic
344C42C23×D5311yescharacteristic
345C23—41no—
346C23—4242no—
347C23—4242no—
348C23—4242no—
349C23—2122no—
350C23#1313319011yescharacteristic
351Q8—41no—
352Q8—41no—
353Q8—534no—
354Q8—1064no—
355Q8—1062no—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—81no—
360D4—82no—
361D4—4242no—
362D4—4242no—
363D4—4242no—
364D4—4242no—
365D4—2122no—
366D4—2122no—
367D4—2124no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—1064no—
373C2×C4—1062no—
374C2×C4—1062no—
375C2×C4—4241no—
376C2×C4#1313319011yescharacteristic
377C2×C4#1313319011yescharacteristic
378C2×C4#1313319011yescharacteristic
379C2×C4D8×D5312yes—
380C2×C4#1313323812yes—
381C2×C4C2×D4×D5311yescharacteristic
382C2×C4C2×D4×D5311yescharacteristic
383C2×C4C2×#1171994511yescharacteristic
384C8—42no—
385C8—42no—
386C8—1064no—
387C8—2122no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—8484no—
394C22—4242no—
395C22—4242no—
396C22—4242no—
397C22#1681085812yes—
398C22C2×#1313319011yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—1064no—
404C4—2122no—
405C4#1681085812yes—
406C4C2×D8×D5311yescharacteristic
407C4C2×#1313323811yescharacteristic
408C4#1681143012yes—
409C2—22no—
410C2—161no—
411C2—8482no—
412C2#2028423911yescharacteristic
413C2C2×#1681085811yescharacteristic
414C2#2028444611yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1540C2
socleC22×C53C2×#121
prime radical#20285630C2
hypercenter#1540D53
hypocenterC53#19398
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
2. #20284183C2semidirect
3. C2×#16810824C2semidirect
4. #20283795C2semidirect
5. #20274162C2semidirect
6. #20274074C2semidirect
7. #20273489C2semidirect
8. #20273382C2semidirect
9. C53×#1540C2semidirect
17. #16810830C22semidirect
20. #16810820C22semidirect
24. #16810747C22semidirect
30. #16810375C22semidirect
35. C53×#332C22semidirect
36. C53×#331C22semidirect
111. C2×Dic106D8semidirect
210. #1540D53semidirect
241. #332C2×D53semidirect
243. #331C2×D53semidirect
244. C53#19398semidirect

Maximal subgroups

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

subgroupindex
#202856302
#202841832
C2×#168108242
#202837952
#202741622
#202740742
#202734892
#202733822
C53×#15402
#1939853