Finite Group Databasebeta

← #27280819

Subgroups — #27280819

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285633C211yesmaximal, characteristic
2#20275547C211yesmaximal, characteristic, complemented
3#20273383C211yesmaximal, characteristic, complemented
4#20273274C211yesmaximal, characteristic, complemented
5#20272573C211yesmaximal, characteristic, complemented
6#20272544C211yesmaximal, characteristic, complemented
7C53×#2328C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810999C412yescomplemented
10#16810969C412yescomplemented
11#16810815—22no—
12#16810757—22no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×#13133145—21no—
17C2×#13133145—21no—
18#16810367—22no—
19#16810356—22no—
20C2×#127×C53C2211yescharacteristic
21C2×C4×D4×C53—21no—
22C53×#390—21no—
23C53×#386—22no—
24C53×#386—22no—
25C53×#382C2211yescharacteristic
26C53×#376C2211yescharacteristic
27C2×#11719945—21no—
28C2×#11719922—24no—
29C2×#11719917—21no—
30#13133217C2×C411yescharacteristic, complemented
31#13133214C2×C411yescharacteristic, complemented
32#13133205—21no—
33#13133191—21no—
34#13133191—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41#13133145—42no—
42#13133145—42no—
43#13133138—24no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C4×#88322—21no—
47C4×#88322C2×C412yescomplemented
48C22×D4×C53—21no—
49C22×D4×C53—21no—
50C22×D4×C53D412yes—
51C23×C4×C53—22no—
52#127×C53—21no—
53#127×C53—21no—
54#127×C53C2×C412yes—
55#127×C53D412yes—
56C4×D4×C53—42no—
57C4×D4×C53—42no—
58C2×C4⋊C4×C53—21no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53—22no—
61C2×C22⋊C4×C53—22no—
62C2×C42×C53D411yescharacteristic
63C2×C42×C53D411yescharacteristic
64C2×C42×C53C2311yescharacteristic
65#95×C53—21no—
66#95×C53—21no—
67#95×C53—21no—
68#95×C53D412yes—
69C24×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—22no—
74C2×D4×C53—22no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—82no—
80C2×D4×C53—82no—
81C2×D4×C53C22⋊C414yes—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—44no—
86C22×C4×C53—82no—
87C22×C4×C53—82no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C2×C22⋊D53—41no—
92#11719945—41no—
93#11719945—41no—
94#11719945—41no—
95#11719945—41no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212—21no—
99C2×C4×D53—41no—
100C2×Dic106—21no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C22⋊C4×C53—44no—
104C22⋊C4×C53—82no—
105C22⋊C4×C53—82no—
106C42×C53—21no—
107C42×C53—21no—
108C42×C53—21no—
109C42×C53D812yescomplemented
110C42×C53C22×C411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53#4111yescharacteristic
113C42×C53#4111yescharacteristic
114#11719922—84no—
115#11719921—41no—
116#11719921—41no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119#11719917—41no—
120#11719917—41no—
121C2×#88322—22no—
122C2×#88322—41no—
123C2×#88322—41no—
124C23×C53—24no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53#12011yescharacteristic
132C22×D53—41no—
133D4×C53—44no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—82no—
139D4×C53—82no—
140D4×C53—82no—
141C2×C4×C53—22no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—41no—
152C2×C4×C53—88no—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C22⋊D53—81no—
164C22⋊D53—81no—
165C2×Dic53—41no—
166D212—42no—
167D212—41no—
168D212—41no—
169D212—81no—
170C4×D53—81no—
171C4×D53—81no—
172Dic106—41no—
173Dic106—41no—
174#88322—42no—
175#88322—81no—
176#88322—81no—
177#9299—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—44no—
181C22×C53—44no—
182C22×C53—44no—
183C22×C53—82no—
184C22×C53—82no—
185C22×C53—82no—
186C22×C53—82no—
187C22×C53#38611yescharacteristic
188C22×C53#44711yescharacteristic
189C22×C53#44911yescharacteristic
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—42no—
198C4×C53—81no—
199C4×C53—82no—
200C4×C53C4×D811yescharacteristic
201C4×C53#44211yescharacteristic
202C4×C53#45911yescharacteristic
203C4×C53#46311yescharacteristic
204Dic53—81no—
205#3205—531no—
206#2328D5311yescharacteristic, complemented
207#1963—531no—
208#1545—531no—
209#1517—531no—
210#1286—531no—
211#1274—531no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—84no—
215C2×C53#187911yescharacteristic
216C2×C53#190411yescharacteristic
217C2×C53#191111yescharacteristic
218D53—161no—
219#532—531no—
220C2×#127C2×D5311yescharacteristic
221C2×C4×D4—21no—
222#493—532no—
223#486—532no—
224#459—1062no—
225#447—1062no—
226#431—531no—
227#431—531no—
228#423—531no—
229C2×#104—1061no—
230C2×#104—1061no—
231#390—21no—
232#386—22no—
233#386—22no—
234#382C2×D5311yescharacteristic
235#376C2×D5311yescharacteristic
236#331—1062no—
237#327—1062no—
238C53#929911yescharacteristic, complemented, Sylow
239C2×#41—1061no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C22⋊D5312yes—
243C23×C4—22no—
244C2×D8—1064no—
245C2×M16—1061no—
246C4⋊Q8—531no—
247#127—21no—
248#127—21no—
249#127—531no—
250#127C4×D5312yes—
251#127D21212yes—
252#124—1061no—
253#121—1061no—
254#121—1061no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—1061no—
258C4×D4—1061no—
259C2×C4⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C42C22⋊D5311yescharacteristic
264C2×C42C22⋊D5311yescharacteristic
265C2×C42C22×D5311yescharacteristic
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270#104—2122no—
271#104—2122no—
272#102—1064no—
273C22⋊C8—1061no—
274C22⋊C8—1061no—
275C4×C8—532no—
276C4×C8—1061no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95C22⋊D5312yes—
281C24—22no—
282#41—2121no—
283#41—2121no—
284#41—2121no—
285#41—2121no—
286C2×Q8—1061no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—2121no—
302C2×D4#1171992114yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—44no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—2121no—
310C22×C4C2×C22⋊D5311yescharacteristic
311C22×C4C2×C22⋊D5311yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313D8—4244no—
314M16—2121no—
315M16—2121no—
316C2×C8—1062no—
317C2×C8—2121no—
318C2×C8—2121no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—1061no—
322C4⋊C4—1061no—
323C22⋊C4—44no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42#1171992212yescomplemented
332C42C2×C4×D5311yescharacteristic
333C42C2×D21211yescharacteristic
334C42#1171994511yescharacteristic
335C42#1171994511yescharacteristic
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—2121no—
344C23#1313318611yescharacteristic
345Q8—2121no—
346Q8—2121no—
347D4—44no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—82no—
353D4—82no—
354D4—82no—
355D4—4241no—
356D4—4241no—
357D4—4241no—
358D4—2122no—
359D4—2121no—
360D4—2121no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—88no—
373C2×C4—4241no—
374C2×C4—4241no—
375C2×C4—2121no—
376C2×C4C2×#1171992111yescharacteristic
377C2×C4C4×C22⋊D5311yescharacteristic
378C2×C4C4×C22⋊D5311yescharacteristic
379C2×C4#1313319111yescharacteristic
380C2×C4#1313319111yescharacteristic
381C2×C4#1313320011yescharacteristic
382C2×C4C2×#1171992211yescharacteristic
383C2×C4#1313323711yescharacteristic
384C2×C4#1313324211yescharacteristic
385C2×C4#1313324311yescharacteristic
386C8—4241no—
387C8—4241no—
388C8—2122no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—4241no—
399C22—4241no—
400C22—4241no—
401C22#1681050211yescharacteristic
402C22#1681075711yescharacteristic
403C22#1681076811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—82no—
410C4—81no—
411C4—4241no—
412C4C4×#1171992211yescharacteristic
413C4#1681073311yescharacteristic
414C4#1681081811yescharacteristic
415C4#1681083811yescharacteristic
416C2—21no—
417C2—21no—
418C2—84no—
419C2—8481no—
420C2#2027506311yescharacteristic
421C2#2027520211yescharacteristic
422C2#2027524711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2328C2
socleC22×C53#386
prime radical#20285633C2
cosocleC2×C42×C53C23
hypercenter#2328D53
hypocenterC53#9299
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #20275547C2semidirect
3. #20273383C2semidirect
4. #20273274C2semidirect
5. #20272573C2semidirect
6. #20272544C2semidirect
7. C53×#2328C2semidirect
9. #16810999C4semidirect
10. #16810969C4semidirect
30. #13133217C2×C4semidirect
31. #13133214C2×C4semidirect
47. C4×#88322C2×C4semidirect
109. C42×C53D8semidirect
206. #2328D53semidirect
238. C53#9299semidirect
331. C42#11719922semidirect

Maximal subgroups

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

subgroupindex
#202856332
#202755472
#202733832
#202732742
#202725732
#202725442
C53×#23282
#929953