Finite Group Databasebeta

← #27279563

Subgroups — #27279563

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283671C211yesmaximal, characteristic
2C4×#13133248C211yesmaximal, characteristic
3#20273103C211yesmaximal, characteristic
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5#20272564C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1461C211yesmaximal, characteristic, complemented
8C2×#13133248C411yescharacteristic
9C4×#11719945C2211yescharacteristic
10#16810747—21no—
11#16810728—21no—
12#16810721—21no—
13C4×#11719925—21no—
14C4×#11719924—21no—
15#16810688—21no—
16#16810686—22no—
17#16810674—21no—
18#16810663C411yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20#16810632C411yescharacteristic
21#16810622C411yescharacteristic
22C4×#11719917C2211yescharacteristic
23C2×C4×#88322C2211yescharacteristic
24C4×C8×D53—21no—
25#16810355—21no—
26Q8⋊#88322—21no—
27Q8⋊#88322—21no—
28#16810352—21no—
29C2×C4×Q8×C53C2211yescharacteristic
30C53×#431C2211yescharacteristic
31C53×#431C2211yescharacteristic
32C53×#324—22no—
33C2×#11719945C2×C411yescharacteristic
34#13133248—22no—
35#13133248C814yescomplemented
36C2×#11719925—21no—
37C2×#11719924—21no—
38C2×#11719917C2×C411yescharacteristic
39C2×#11719917C2×C411yescharacteristic
40C22×#88322—21no—
41C22×#88322—21no—
42C2×C8×D53—22no—
43C4×Dic106D411yescharacteristic
44C4×Dic106D411yescharacteristic
45C4×D212D411yescharacteristic
46C4×D212D411yescharacteristic
47C4×C22⋊D53—21no—
48C4×C22⋊D53—21no—
49#13133176C2×C411yescharacteristic
50#13133174—21no—
51C42×D53—21no—
52#13133150—21no—
53#13133148—21no—
54C4⋊#88322—21no—
55C4⋊#88322—21no—
56#13133146—22no—
57#13133141—21no—
58#13133141—21no—
59#13133140—21no—
60#13133137—21no—
61#13133129—22no—
62#13133128—21no—
63C4×#88322—21no—
64C4×#88322—21no—
65C4×#88322D411yescharacteristic
66C4×#88322D411yescharacteristic
67C8×Dic53—22no—
68C22×Q8×C53C2×C411yescharacteristic
69C4×Q8×C53—21no—
70C4×Q8×C53—21no—
71C4×Q8×C53—21no—
72C4×Q8×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53C2×C411yescharacteristic
75C2×C42×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53—21no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53D412yes—
81C4×C8×C53—21no—
82C4×C8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53C2×C812yescomplemented
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C2×C22⋊D53—21no—
95#11719945—21no—
96#11719945—21no—
97#11719945C2×C812yes—
98#11719945M1612yes—
99C2×D212C22⋊C411yescharacteristic
100C2×C4×D53—21no—
101C2×C4×D53—21no—
102C2×C4×D53—21no—
103C2×Dic106C22⋊C411yescharacteristic
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C42×C53—41no—
111C42×C53—41no—
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53#4111yescharacteristic
116C22⋊Dic53—21no—
117#11719925—24no—
118#11719924—24no—
119#11719921—21no—
120#11719921—21no—
121C4⋊Dic53C22⋊C411yescharacteristic
122C4⋊Dic53C22⋊C411yescharacteristic
123#11719919—21no—
124#11719919—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128#11719917—21no—
129#11719917—21no—
130#11719917—21no—
131#11719917C2×C812yescomplemented
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—21no—
136C2×#88322—42no—
137C2×#88322—41no—
138C2×#88322—81no—
139C8×D53—44no—
140C23×C53C4×D411yescharacteristic
141C22×D53—21no—
142Q8×C53—22no—
143Q8×C53—22no—
144Q8×C53—41no—
145Q8×C53—41no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—81no—
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×C4×C53C22×C811yescharacteristic
161C2×C4×C53C2×M1611yescharacteristic
162C2×C4×C53#13111yescharacteristic
163C2×C4×C53#13111yescharacteristic
164C22⋊D53—22no—
165C22⋊D53—22no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169D212—21no—
170D212C22⋊C812yes—
171C4×D53—22no—
172C4×D53—22no—
173C4×D53—42no—
174C4×D53—41no—
175C4×D53—41no—
176Dic106—21no—
177Dic106C22⋊C812yes—
178C8×C53—42no—
179C8×C53—81no—
180#88322—22no—
181#88322—22no—
182#88322—41no—
183#88322—41no—
184#88322—81no—
185#9096—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53#38611yescharacteristic
189C22×C53C8×D411yescharacteristic
190C22×C53#43511yescharacteristic
191C2×D53—22no—
192C2×D53—41no—
193C2×D53—41no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—42no—
198C4×C53—81no—
199C4×C53C2×C22⋊C811yescharacteristic
200C4×C53#40811yescharacteristic
201C4×C53C8×D411yescharacteristic
202C4×C53#43511yescharacteristic
203C4×C53#47111yescharacteristic
204C4×C53#47311yescharacteristic
205Dic53—22no—
206Dic53—41no—
207Dic53—41no—
208C8×#41—531no—
209C4×#137—531no—
210#1477—531no—
211C2×Q8⋊C8—531no—
212#1461D5311yescharacteristic, complemented
213#1283—531no—
214#1278—531no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53#186911yescharacteristic
218C2×C53#190611yescharacteristic
219C2×C53#190811yescharacteristic
220D53—42no—
221C2×#137—531no—
222C4×#41—531no—
223C2×C4×Q8C2×D5311yescharacteristic
224C8×Q8—1061no—
225#441—1061no—
226#440—1061no—
227C4×Q16—1061no—
228C4×SD16—1061no—
229C8×D4—1061no—
230C8×D4—1062no—
231#431—1061no—
232#431C2×D5311yescharacteristic
233#431C2×D5311yescharacteristic
234#428—531no—
235C2×C4⋊C8—531no—
236#419—531no—
237#417—531no—
238C4×M16—531no—
239C2×C4×C8—531no—
240C2×C4×C8—1061no—
241Q8⋊C8—1061no—
242Q8⋊C8—1061no—
243Q8⋊C8—1061no—
244D4⋊C8—1061no—
245#324—22no—
246C53#909611yescharacteristic, complemented, Sylow
247C2×#41—531no—
248C22×Q8C4×D5311yescharacteristic
249#137—534no—
250#137—1062no—
251C2×Q16—1061no—
252C2×SD16—1061no—
253C2×M16—531no—
254C2×M16—531no—
255C22×C8—1062no—
256C22×C8—1061no—
257C22×C8—1061no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—531no—
263C4×Q8—531no—
264C4×D4—531no—
265C4×D4—531no—
266C4×D4—1061no—
267C4×D4—1061no—
268#117—531no—
269#117—1061no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C4×D5311yescharacteristic
272C2×C42—21no—
273C2×C42—1061no—
274C2×C42C22×D5311yescharacteristic
275#107—1061no—
276#106—1061no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—1062no—
280C4⋊C8—1061no—
281C4⋊C8—1061no—
282#103—1061no—
283#103—1061no—
284#103—1061no—
285#102—1061no—
286C22⋊C8—21no—
287C22⋊C8—1062no—
288C22⋊C8C22⋊D5312yes—
289#97—1061no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—531no—
293C4×C8—531no—
294C4×C8—1062no—
295C4×C8—1061no—
296C4×C8—1061no—
297#41—532no—
298#41—532no—
299#41—1061no—
300#41—1061no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—531no—
307C2×Q8C8×D5312yescomplemented
308C2×D4—531no—
309C2×D4—1061no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4C2×C4×D5311yescharacteristic
316C22×C4#1171994511yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318Q16—1064no—
319SD16—1064no—
320M16—532no—
321M16—1061no—
322M16—1061no—
323M16—1061no—
324C2×C8—22no—
325C2×C8—41no—
326C2×C8—1061no—
327C2×C8—1061no—
328C2×C8—1061no—
329C2×C8—1061no—
330C2×C8—4241no—
331C2×C8—2124no—
332C2×C8—2122no—
333C2×C8—2121no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—531no—
339C4⋊C4—531no—
340C4⋊C4—1061no—
341C4⋊C4—1061no—
342C22⋊C4—1061no—
343C22⋊C4—1061no—
344C22⋊C4—1061no—
345C42—41no—
346C42—41no—
347C42—1061no—
348C42—1061no—
349C42—1061no—
350C42D4×D5311yescharacteristic
351C42D4×D5311yescharacteristic
352C42D4×D5311yescharacteristic
353C42#1171994711yescharacteristic
354C23—1061no—
355C23C4×C22⋊D5311yescharacteristic
356Q8—22no—
357Q8—22no—
358Q8—41no—
359Q8—41no—
360Q8—532no—
361Q8—1061no—
362D4—532no—
363D4—1062no—
364D4—1062no—
365D4—1061no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—1061no—
375C2×C4—1062no—
376C2×C4—1062no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—2122no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4C22⋊C4×D5311yescharacteristic
383C2×C4#1313317711yescharacteristic
384C2×C4#1313318511yescharacteristic
385C2×C4#1313318811yescharacteristic
386C2×C4#1313319011yescharacteristic
387C2×C4#1313319911yescharacteristic
388C2×C4C2×C8×D5311yescharacteristic
389C2×C4M16×D5311yescharacteristic
390C2×C4#1313322311yescharacteristic
391C2×C4#1313322511yescharacteristic
392C8—42no—
393C8—81no—
394C8—1062no—
395C8—1062no—
396C8—4241no—
397C8—2121no—
398C8—2121no—
399C22—21no—
400C22—21no—
401C22—1062no—
402C22—2121no—
403C22—2121no—
404C22#1681049911yescharacteristic
405C22C8×C22⋊D5311yescharacteristic
406C22#1681069211yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—1062no—
413C4—2121no—
414C4—2121no—
415C4C22⋊C8×D5311yescharacteristic
416C4#1681058811yescharacteristic
417C4#1681068611yescharacteristic
418C4#1681069711yescharacteristic
419C4#1681088911yescharacteristic
420C4#1681089711yescharacteristic
421C2—21no—
422C2—21no—
423C2—2122no—
424C2#2027500411yescharacteristic
425C2#2027521311yescharacteristic
426C2#2027522711yescharacteristic
subgroupG/H
centerC2×C4#13133190
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1461C2
socleC22×C53#386
prime radicalC2×#11719945C2×C4
cosocleC2×C42×C53C23
hypercenter#1461D53
hypocenterC53#9096
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊#88322C2semidirect
5. #20272564C2semidirect
6. #20272551C2semidirect
7. C53×#1461C2semidirect
35. #13133248C8semidirect
88. C2×Q8×C53C2×C8semidirect
131. #11719917C2×C8semidirect
212. #1461D53semidirect
246. C53#9096semidirect
307. C2×Q8C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202836712
C4×#131332482
#202731032
C2×Q8⋊#883222
#202725642
#202725512
C53×#14612
#909653