Finite Group Databasebeta

← #27279983

Subgroups — #27279983

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284195C211yesmaximal, characteristic
2#20283283C211yesmaximal, characteristic, complemented
3#20273382C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272590C211yesmaximal, characteristic, complemented
7C53×#1477C211yesmaximal, characteristic, complemented
8C2×#13133242—21no—
9C2×#13133224—21no—
10C2×#13133214C2211yescharacteristic
11#16810823—21no—
12#16810823C412yes—
13#16810818—21no—
14#16810818C412yes—
15#16810650—22no—
16#16810650—21no—
17#16810647—21no—
18#16810647C2211yescharacteristic
19#16810647C2211yescharacteristic
20C2×C4⋊#88322—21no—
21C2×C4⋊#88322C2211yescharacteristic
22#16810627—21no—
23#16810366—21no—
24#16810366—21no—
25#16810366—22no—
26C4×#41×C53C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C53×Q8⋊C8—21no—
30C53×D4⋊C8—21no—
31C22×D212—21no—
32C22×D212—21no—
33C22×D212—21no—
34#13133242—42no—
35#13133242—42no—
36C2×#11719924—22no—
37C2×#11719922—22no—
38#13133224—42no—
39#13133224—42no—
40C2×#11719917—22no—
41C2×#11719917—21no—
42C2×#11719917—21no—
43C22×#88322—22no—
44C22×#88322—21no—
45#13133214—21no—
46#13133214—21no—
47#13133214—21no—
48#13133214C2×C412yes—
49C4⋊#88322—22no—
50C4⋊#88322—22no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322C2×C412yes—
55C4⋊#88322D411yescharacteristic
56C4⋊#88322D411yescharacteristic
57C4⋊#88322D412yes—
58#13133137—22no—
59#13133137—22no—
60C2×#41×C53D411yescharacteristic
61C22×C8×C53—21no—
62C4×Q8×C53C2×C411yescharacteristic
63C4×Q8×C53D411yescharacteristic
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53C2×C411yescharacteristic
67C4×D4×C53D411yescharacteristic
68#117×C53—21no—
69#117×C53D411yescharacteristic
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73C4⋊C8×C53—21no—
74C4⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76C4×C8×C53—21no—
77C23×D53—41no—
78#41×C53—21no—
79#41×C53—21no—
80#41×C53D812yes—
81#41×C53SD1612yes—
82C2×Q8×C53C22⋊C411yescharacteristic
83C2×D4×C53—21no—
84C2×D4×C53C22⋊C411yescharacteristic
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—82no—
102C2×D212—81no—
103C2×D212—81no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—81no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53C22⋊C411yescharacteristic
110C4⋊C4×C53C22⋊C411yescharacteristic
111C22⋊C4×C53—21no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—21no—
114C42×C53—21no—
115C42×C53—21no—
116C42×C53—21no—
117C42×C53C22×C411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121#11719924—44no—
122#11719922—44no—
123#11719917—44no—
124#11719917—41no—
125#11719917—41no—
126#11719917—41no—
127#11719917—41no—
128C2×#88322—22no—
129C2×#88322—21no—
130C2×#88322—22no—
131C2×#88322—21no—
132C2×#88322—44no—
133C2×#88322—41no—
134C2×#88322—41no—
135C23×C53—21no—
136C23×C53#12311yescharacteristic
137C22×D53—42no—
138C22×D53—82no—
139C22×D53—82no—
140C22×D53—81no—
141C22×D53—81no—
142Q8×C53—21no—
143Q8×C53#10212yes—
144D4×C53—22no—
145D4×C53—22no—
146D4×C53—21no—
147D4×C53#10212yes—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53C2×D811yescharacteristic
166C2×C4×C53C2×SD1611yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53#13711yescharacteristic
169D212—42no—
170D212—42no—
171D212—82no—
172D212—82no—
173D212—82no—
174D212—81no—
175D212—81no—
176D212—81no—
177D212—81no—
178C8×C53—81no—
179C8×C53—81no—
180#88322—42no—
181#88322—41no—
182#88322—42no—
183#88322—41no—
184#9173—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—22no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53#38611yescharacteristic
191C22×C53#48011yescharacteristic
192C22×C53#48111yescharacteristic
193C2×D53—82no—
194C2×D53—82no—
195C2×D53—82no—
196C2×D53—162no—
197C2×D53—162no—
198C4×C53—22no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53C2×#10211yescharacteristic
205C4×C53#41711yescharacteristic
206C4×C53#44711yescharacteristic
207C4×C53#44811yescharacteristic
208C4×C53#47111yescharacteristic
209C4×C53#47211yescharacteristic
210#3038—531no—
211#2906—531no—
212#1545—531no—
213C2×#331—531no—
214#1477D5311yescharacteristic, complemented
215#1301—531no—
216#1292—531no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—42no—
220C2×C53#187411yescharacteristic
221C2×C53#190311yescharacteristic
222C2×C53#190711yescharacteristic
223D53—162no—
224C2×AGL(1,ℤ/8)—1061no—
225C2×#131—1061no—
226C2×#127—531no—
227C4×#41C2×D5311yescharacteristic
228#460—532no—
229#460—1061no—
230#459—532no—
231#459—1061no—
232#431C2×D5311yescharacteristic
233#424—1062no—
234#424—1061no—
235#423—531no—
236#423—531no—
237#423—1061no—
238C2×C4⋊C8—531no—
239C2×C4⋊C8—1061no—
240C2×C4⋊C8C2×D5311yescharacteristic
241#418—1061no—
242#331—1061no—
243#331—1061no—
244#331—1062no—
245Q8⋊C8—21no—
246D4⋊C8—21no—
247C53#917311yescharacteristic, complemented, Sylow
248C2×#41D21211yescharacteristic
249C22×D4—1061no—
250C22×D4—1061no—
251C22×D4—1061no—
252AGL(1,ℤ/8)—2122no—
253AGL(1,ℤ/8)—2122no—
254C2×SD16—1062no—
255C2×D8—1062no—
256#131—2122no—
257#131—2122no—
258C2×M16—1062no—
259C2×M16—1061no—
260C2×M16—1061no—
261C22×C8—21no—
262C22×C8—1062no—
263C22×C8—1061no—
264#127—532no—
265#127—1061no—
266#127—1061no—
267#127—1061no—
268C4×Q8C4×D5311yescharacteristic
269C4×Q8C22⋊D5311yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D5311yescharacteristic
273C4×D4C22⋊D5311yescharacteristic
274#117—21no—
275#117D21211yescharacteristic
276C2×C42—21no—
277C2×C42C22×D5311yescharacteristic
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—532no—
282C4⋊C8—531no—
283C4⋊C8—531no—
284C4⋊C8—532no—
285C4⋊C8—1062no—
286C4⋊C8—1062no—
287C4⋊C8—1061no—
288C4⋊C8—1061no—
289C4⋊C8—1061no—
290#102—1062no—
291#102—1062no—
292C22⋊C8—21no—
293C4×C8—21no—
294C24—2121no—
295#41—21no—
296#41—21no—
297#41#1171991312yes—
298#41D42412yes—
299C2×Q8#1171992111yescharacteristic
300C2×D4—21no—
301C2×D4—1062no—
302C2×D4—1062no—
303C2×D4—1062no—
304C2×D4—4241no—
305C2×D4—4241no—
306C2×D4—4242no—
307C2×D4—2121no—
308C2×D4—2121no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4#1171992111yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4C2×D21211yescharacteristic
319C22×C4#1171994711yescharacteristic
320C22×C4#1171994711yescharacteristic
321SD16—2124no—
322D8—2124no—
323M16—2124no—
324M16—2121no—
325M16—2121no—
326M16—2121no—
327M16—2121no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—81no—
331C2×C8—1062no—
332C2×C8—1061no—
333C2×C8—1061no—
334C2×C8—1062no—
335C2×C8—2124no—
336C2×C8—2121no—
337C2×C8—2121no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4#1171992111yescharacteristic
341C4⋊C4#1171992111yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42C2×C4×D5311yescharacteristic
349C42D4×D5311yescharacteristic
350C42D4×D5311yescharacteristic
351C42C2×C22⋊D5311yescharacteristic
352C23—21no—
353C23—4242no—
354C23—4242no—
355C23—4241no—
356C23—4241no—
357C23—2122no—
358C23#1313320111yescharacteristic
359Q8—21no—
360Q8#1313313912yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—4242no—
365D4—4242no—
366D4—4242no—
367D4—4241no—
368D4—4241no—
369D4—4241no—
370D4—4241no—
371D4—2122no—
372D4—2122no—
373D4#1313313912yes—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×#1171992111yescharacteristic
386C2×C4#1313317711yescharacteristic
387C2×C4#1313317711yescharacteristic
388C2×C4#1313318711yescharacteristic
389C2×C4#1313318911yescharacteristic
390C2×C4#1313318911yescharacteristic
391C2×C4C2×D42411yescharacteristic
392C2×C4C2×#1171991311yescharacteristic
393C2×C4#1313324511yescharacteristic
394C2×C4#1313325011yescharacteristic
395C8—81no—
396C8—81no—
397C8—2121no—
398C8—2122no—
399C8—2121no—
400C8—2122no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—8482no—
407C22—8482no—
408C22—4242no—
409C22—4242no—
410C22—4242no—
411C22#1681050011yescharacteristic
412C22#1681093711yescharacteristic
413C22#1681094211yescharacteristic
414C4—21no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4C2×#1313313911yescharacteristic
421C4#1681062511yescharacteristic
422C4#1681075911yescharacteristic
423C4#1681076411yescharacteristic
424C4#1681088811yescharacteristic
425C4#1681089111yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—8482no—
430C2#2027503011yescharacteristic
431C2#2027519311yescharacteristic
432C2#2027522111yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1477C2
socleC22×C53#386
prime radical#20284195C2
cosocleC2×C42×C53C23
hypercenter#1477D53
hypocenterC53#9173
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #20283283C2semidirect
3. #20273382C2semidirect
4. C2×#16810366C2semidirect
5. #20272611C2semidirect
6. #20272590C2semidirect
7. C53×#1477C2semidirect
214. #1477D53semidirect
247. C53#9173semidirect

Maximal subgroups

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

subgroupindex
#202841952
#202832832
#202733822
C2×#168103662
#202726112
#202725902
C53×#14772
#917353