Finite Group Databasebeta

← #27279311

Subgroups — #27279311

Order 13568.

538 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285290C211yesmaximal, characteristic
2#20283459C211yesmaximal, characteristic, complemented
3#20275633C211yesmaximal, characteristic, complemented
4#20273115C211yesmaximal, characteristic, complemented
5#20272556C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1502C211yesmaximal, characteristic, complemented
8C2×#13133224—21no—
9#16811173C2211yescharacteristic
10#16811026C411yescharacteristic, complemented
11#16810999C411yescharacteristic, complemented
12#16810971C411yescharacteristic, complemented
13#16810969C411yescharacteristic, complemented
14#16810857—22no—
15C4×#11719922—22no—
16#16810701—21no—
17#16810695—22no—
18D4×#88322—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133145—21no—
21C2×#13133145—21no—
22#16810590—21no—
23C4×#11719917C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25#16810352—21no—
26#16810352—21no—
27C53×#532C2211yescharacteristic
28C53×#431C2211yescharacteristic
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#327—22no—
31C2×#11719945—21no—
32C2×#11719922—22no—
33#13133224—42no—
34#13133224—42no—
35C2×#11719917—22no—
36C2×#11719917D411yescharacteristic
37C2×#11719917D411yescharacteristic
38C22×#88322—22no—
39C22×#88322—21no—
40C22×#88322—21no—
41#13133217C2×C411yescharacteristic
42#13133214C2×C411yescharacteristic
43#13133205—21no—
44#13133191—21no—
45#13133191—21no—
46C4×D212—21no—
47C4×D212—21no—
48#13133150—22no—
49C4⋊#88322—21no—
50C4⋊#88322D411yescharacteristic
51C4⋊#88322D411yescharacteristic
52#13133145—42no—
53#13133145—42no—
54#13133138—22no—
55#13133137—22no—
56C22⋊#88322—22no—
57C22⋊#88322—22no—
58#13133128—21no—
59C4×#88322—21no—
60C4×#88322—21no—
61C4×#88322C2×C411yescharacteristic
62C4×#88322C2×C411yescharacteristic
63C2×#41×C53—21no—
64C4⋊Q8×C53C2×C411yescharacteristic
65#127×C53C2×C411yescharacteristic
66#124×C53—21no—
67#121×C53—21no—
68#121×C53D412yes—
69C4×D4×C53—21no—
70C4×D4×C53D412yes—
71C2×C4⋊C4×C53—21no—
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74C4⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76C4×C8×C53—21no—
77#95×C53—21no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82C2×Q8×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53C22⋊C412yes—
87C22×C4×C53—22no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C2×C22⋊D53—41no—
93#11719945—41no—
94#11719945—41no—
95#11719945—41no—
96#11719945—41no—
97C2×D212—21no—
98C2×D212—21no—
99C2×D212—21no—
100C2×C4×D53—41no—
101C2×Dic106—21no—
102C2×C8×C53—41no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53C22⋊C412yes—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C42×C53—41no—
110C42×C53—41no—
111C42×C53C22×C411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115#11719922—44no—
116#11719921—41no—
117#11719921—41no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120#11719917—21no—
121#11719917—21no—
122#11719917—21no—
123#11719917—44no—
124#11719917D812yescomplemented
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—42no—
132C2×#88322—44no—
133C2×#88322—41no—
134C2×#88322—41no—
135C23×C53—22no—
136C23×C53#12111yescharacteristic
137C22×D53—41no—
138Q8×C53—41no—
139Q8×C53—41no—
140D4×C53—42no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—41no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—81no—
155C2×C4×C53—81no—
156C2×C4×C53C2×C22⋊C411yescharacteristic
157C2×C4×C53C4×D411yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12311yescharacteristic
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C22⋊D53—81no—
167C22⋊D53—81no—
168C2×Dic53—41no—
169D212—42no—
170D212—41no—
171D212—41no—
172D212—41no—
173D212—41no—
174C4×D53—81no—
175C4×D53—81no—
176Dic106—41no—
177Dic106—41no—
178C8×C53—81no—
179#88322—22no—
180#88322—22no—
181#88322—41no—
182#88322—41no—
183#88322—41no—
184#88322—41no—
185#9053—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—42no—
189C22×C53—42no—
190C22×C53—42no—
191C22×C53#38611yescharacteristic
192C22×C53#45911yescharacteristic
193C22×C53#46311yescharacteristic
194C2×D53—81no—
195C2×D53—81no—
196C2×D53—81no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—42no—
201C4×C53—82no—
202C4×C53C4×D811yescharacteristic
203C4×C53#44211yescharacteristic
204C4×C53#44311yescharacteristic
205C4×C53#44411yescharacteristic
206C4×C53#44711yescharacteristic
207C4×C53#44911yescharacteristic
208Dic53—81no—
209#3169—531no—
210#2935—531no—
211#1980—531no—
212#1502D5311yescharacteristic, complemented
213#1480—531no—
214#1279—531no—
215#1278—531no—
216C2×C53—21no—
217C2×C53—21no—
218C2×C53—82no—
219C2×C53#187611yescharacteristic
220C2×C53#187911yescharacteristic
221C2×C53#188411yescharacteristic
222D53—161no—
223C2×#131—1061no—
224#532—531no—
225#532C2×D5311yescharacteristic
226#500—531no—
227#493—531no—
228#486—531no—
229#486—531no—
230#466—1062no—
231C4×D8—1062no—
232#436—1061no—
233#435—1062no—
234C8×D4—1061no—
235#431C2×D5311yescharacteristic
236C2×C4⋊C8—531no—
237C2×#104—1061no—
238C2×#104—1061no—
239#408—1061no—
240C4×M16—531no—
241C2×C4×C8—531no—
242C4×C4⋊C4C2×D5311yescharacteristic
243#327—22no—
244D4⋊C8—1061no—
245D4⋊C8—1061no—
246C53#905311yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41—1061no—
249C2×D8—1062no—
250#131—2122no—
251#131—2122no—
252C2×M16—531no—
253C2×M16—531no—
254C2×M16—1062no—
255C22×C8—1062no—
256C22×C8—1061no—
257C22×C8—1061no—
258C4⋊Q8—531no—
259C4⋊Q8C4×D5311yescharacteristic
260#127—531no—
261#127C4×D5311yescharacteristic
262#124—21no—
263#124—1061no—
264#121—21no—
265#121—1061no—
266#121—1061no—
267#121C22⋊D5312yes—
268C4×D4—21no—
269C4×D4—1061no—
270C4×D4—1061no—
271C4×D4D21212yes—
272C2×C4⋊C4—21no—
273C2×C42—21no—
274C2×C42C22×D5311yescharacteristic
275#107—1062no—
276C4⋊C8—21no—
277C4⋊C8—531no—
278C4⋊C8—531no—
279C4⋊C8—1061no—
280#104—2122no—
281#104—2122no—
282#102—1062no—
283#102—1062no—
284C22⋊C8—21no—
285C22⋊C8—1062no—
286C22⋊C8—1062no—
287#97—1061no—
288C4×C8—21no—
289C4×C8—531no—
290C4×C8—531no—
291C4×C8—1061no—
292C4×C8—1061no—
293#95—21no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—2121no—
299#41—2121no—
300#41—2121no—
301#41—2121no—
302C2×Q8—21no—
303C2×Q8—1061no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—1061no—
308C2×D4—1061no—
309C2×D4—1061no—
310C2×D4—2121no—
311C2×D4#1171992112yes—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—2121no—
315C22×C4D4×D5311yescharacteristic
316C22×C4#1171994711yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318D8—2124no—
319M16—532no—
320M16—1061no—
321M16—1061no—
322M16—1061no—
323M16—2124no—
324C2×C8—41no—
325C2×C8—1061no—
326C2×C8—1061no—
327C2×C8—1061no—
328C2×C8—1061no—
329C2×C8—1061no—
330C2×C8—1061no—
331C2×C8—2124no—
332C2×C8—2122no—
333C2×C8—2121no—
334C2×C8—2121no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—1061no—
339C4⋊C4—1061no—
340C4⋊C4#1171992112yes—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C42—41no—
346C42—41no—
347C42C2×C4×D5311yescharacteristic
348C42C2×D21211yescharacteristic
349C42D4×D5311yescharacteristic
350C42#1171994711yescharacteristic
351C23—22no—
352C23—2121no—
353C23#1313318911yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—2121no—
357Q8—2121no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—4241no—
366D4—4241no—
367D4—2122no—
368D4—2121no—
369D4—2121no—
370D4—2121no—
371D4—2121no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—2121no—
384C2×C4C2×#1171992111yescharacteristic
385C2×C4#1313317711yescharacteristic
386C2×C4#1313317711yescharacteristic
387C2×C4#1313318711yescharacteristic
388C2×C4#1313318911yescharacteristic
389C2×C4#1313320111yescharacteristic
390C2×C4D8×D5311yescharacteristic
391C2×C4#1313323811yescharacteristic
392C2×C4#1313324111yescharacteristic
393C2×C4#1313324411yescharacteristic
394C8—81no—
395C8—1062no—
396C8—1062no—
397C8—2121no—
398C8—2121no—
399C8—2121no—
400C8—2121no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—4241no—
407C22—4241no—
408C22—4241no—
409C22#1681050011yescharacteristic
410C22#1681081611yescharacteristic
411C22#1681084011yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—82no—
417C4—4241no—
418C4#1681070311yescharacteristic
419C4#1681073211yescharacteristic
420C4#1681073711yescharacteristic
421C4#1681074211yescharacteristic
422C4#1681075811yescharacteristic
423C4#1681076911yescharacteristic
424C2—21no—
425C2—21no—
426C2—82no—
427C2—8481no—
428C2#2027504011yescharacteristic
429C2#2027506111yescharacteristic
430C2#2027509611yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1502C2
socleC22×C53#386
prime radical#20285290C2
cosocleC2×C42×C53C23
hypercenter#1502D53
hypocenterC53#9053
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #20283459C2semidirect
3. #20275633C2semidirect
4. #20273115C2semidirect
5. #20272556C2semidirect
6. #20272551C2semidirect
7. C53×#1502C2semidirect
10. #16811026C4semidirect
11. #16810999C4semidirect
12. #16810971C4semidirect
13. #16810969C4semidirect
124. #11719917D8semidirect
212. #1502D53semidirect
246. C53#9053semidirect

Maximal subgroups

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

subgroupindex
#202852902
#202834592
#202756332
#202731152
#202725562
#202725512
C53×#15022
#905353