Finite Group Databasebeta

← #27278968

Subgroups — #27278968

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284497C211yesmaximal, characteristic
2#20283280C211yesmaximal, characteristic
3#20273170C211yesmaximal, characteristic, complemented
4#20273095C211yesmaximal, characteristic
5#20272611C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C53×#1481C211yesmaximal, characteristic, complemented
8C2×#13133223—21no—
9C4×#11719945C2211yescharacteristic
10#16810927—21no—
11#16810920—21no—
12#16810915C411yescharacteristic
13#16810910C411yescharacteristic
14#16810905C411yescharacteristic
15#16810900C411yescharacteristic
16#16810659—21no—
17#16810649—21no—
18#16810648—22no—
19#16810647C2211yescharacteristic
20#16810645—21no—
21C2×C4⋊#88322C2211yescharacteristic
22C4⋊C8×D53—21no—
23#16810617—21no—
24C2×C4×#88322C2211yescharacteristic
25Q8⋊#88322—22no—
26#16810353—21no—
27#16810350—21no—
28C53×#533C2211yescharacteristic
29C53×#431C2211yescharacteristic
30C2×C4⋊C8×C53C2211yescharacteristic
31C53×Q8⋊C8—21no—
32C53×Q8⋊C8—21no—
33C2×#11719945D411yescharacteristic
34#13133223—42no—
35#13133223—42no—
36C2×#11719917—21no—
37C2×#11719912—22no—
38C22×#88322—21no—
39C22×#88322—21no—
40C2×C8×D53—22no—
41C4×Dic106C2×C411yescharacteristic
42C4×Dic106D411yescharacteristic
43C4×D212C2×C411yescharacteristic
44C4×D212D411yescharacteristic
45C4×C22⋊D53—21no—
46C4×C22⋊D53—21no—
47#13133176D411yescharacteristic
48#13133174—21no—
49C42×D53—21no—
50#13133150—21no—
51#13133150—21no—
52#13133148—21no—
53#13133148—21no—
54C4⋊#88322—21no—
55C4⋊#88322—21no—
56C4⋊#88322C2×C411yescharacteristic
57C4⋊#88322C2×C411yescharacteristic
58C4⋊#88322D412yes—
59#13133146—22no—
60#13133146—22no—
61#13133140—21no—
62#13133140—21no—
63#13133137—21no—
64#13133137—21no—
65C4×#88322—21no—
66C4×#88322—21no—
67C22×C8×C53—21no—
68C4⋊Q8×C53C2×C411yescharacteristic
69C4⋊Q8×C53C2×C411yescharacteristic
70#125×C53—21no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53—21no—
75#117×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53—21no—
79C4⋊C8×C53D411yescharacteristic
80C4⋊C8×C53D411yescharacteristic
81C22⋊C8×C53—21no—
82C4×C8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C2×C22⋊D53—21no—
89#11719945—21no—
90#11719945—21no—
91#11719945SD1612yes—
92#11719945Q1612yes—
93C2×D212C22⋊C411yescharacteristic
94C2×C4×D53—21no—
95C2×C4×D53—21no—
96C2×C4×D53—21no—
97C2×Dic106C22⋊C411yescharacteristic
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C42×C53—41no—
111C42×C53C22×C411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C22⋊Dic53—21no—
116#11719921—21no—
117#11719921—21no—
118C4⋊Dic53C22⋊C411yescharacteristic
119C4⋊Dic53C22⋊C411yescharacteristic
120#11719919—21no—
121#11719919—21no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125#11719917—41no—
126#11719917—41no—
127C2×#88322—22no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—22no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135#11719912—44no—
136C8×D53—44no—
137C23×C53#12311yescharacteristic
138C22×D53—21no—
139Q8×C53—42no—
140Q8×C53—41no—
141Q8×C53—41no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53C2×SD1611yescharacteristic
154C2×C4×C53C2×Q1611yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C22⋊D53—22no—
158C22⋊D53—22no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162D212—21no—
163D212#10312yes—
164C4×D53—22no—
165C4×D53—22no—
166C4×D53—41no—
167C4×D53—41no—
168C4×D53—81no—
169Dic106—21no—
170Dic106#10312yes—
171C8×C53—41no—
172C8×C53—41no—
173C8×C53—81no—
174#88322—42no—
175#88322—41no—
176#88322—41no—
177#88322—42no—
178#8993—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53#38611yescharacteristic
182C22×C53#48311yescharacteristic
183C22×C53#48411yescharacteristic
184C2×D53—22no—
185C2×D53—41no—
186C2×D53—41no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—81no—
191C4×C53—81no—
192C4×C53C2×#10311yescharacteristic
193C4×C53#41711yescharacteristic
194C4×C53#44311yescharacteristic
195C4×C53#44411yescharacteristic
196C4×C53#45011yescharacteristic
197C4×C53#45111yescharacteristic
198Dic53—22no—
199Dic53—41no—
200Dic53—41no—
201#3076—531no—
202#2906—531no—
203#1493—531no—
204#1481D5311yescharacteristic, complemented
205#1476—531no—
206#1301—531no—
207#1277—531no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53#187211yescharacteristic
211C2×C53#187611yescharacteristic
212C2×C53#188111yescharacteristic
213D53—42no—
214C2×#131—1061no—
215#533C2×D5311yescharacteristic
216C4×#41—531no—
217#479—1061no—
218#478—1061no—
219#477—531no—
220#476—531no—
221#475—531no—
222#474—531no—
223#431C2×D5311yescharacteristic
224#427—1061no—
225#424—1062no—
226#424—1061no—
227#423—531no—
228#423—1061no—
229C2×C4⋊C8—531no—
230C2×C4⋊C8—1061no—
231C2×C4⋊C8C2×D5311yescharacteristic
232#416—1061no—
233C2×C4×C8—531no—
234Q8⋊C8—21no—
235Q8⋊C8—21no—
236Q8⋊C8—1061no—
237Q8⋊C8—1062no—
238D4⋊C8—1061no—
239C53#899311yescharacteristic, complemented, Sylow
240C2×#41—531no—
241#131—2122no—
242#131—2122no—
243C2×M16—1062no—
244C2×M16—1061no—
245C22×C8—21no—
246C22×C8—1062no—
247C22×C8—1061no—
248C22×C8—1061no—
249C4⋊Q8C4×D5311yescharacteristic
250C4⋊Q8C4×D5311yescharacteristic
251#125—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—531no—
257C4×Q8—531no—
258C4×D4—531no—
259C4×D4—531no—
260C4×D4—1061no—
261C4×D4—1061no—
262#117—21no—
263#117—531no—
264#117—1061no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267#107—1061no—
268#107—1061no—
269#106—1061no—
270#106—1061no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—531no—
274C4⋊C8—531no—
275C4⋊C8—532no—
276C4⋊C8—1062no—
277C4⋊C8—1062no—
278C4⋊C8—1061no—
279C4⋊C8—1061no—
280C4⋊C8C22⋊D5311yescharacteristic
281C4⋊C8C22⋊D5311yescharacteristic
282#103—1061no—
283#103—1061no—
284#102—1061no—
285#102—1061no—
286C22⋊C8—21no—
287C4×C8—21no—
288C4×C8—1061no—
289C4×C8—1061no—
290#41—532no—
291#41—532no—
292#41—1061no—
293#41—1061no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—531no—
297C2×D4—531no—
298C2×D4—1061no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4#1171994511yescharacteristic
303C22×C4D4×D5311yescharacteristic
304C22×C4#1171994711yescharacteristic
305M16—2124no—
306M16—2121no—
307M16—2121no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—1062no—
314C2×C8—1061no—
315C2×C8—1061no—
316C2×C8—1062no—
317C2×C8—2124no—
318C2×C8—2121no—
319C2×C8—2121no—
320C2×C8—2121no—
321C2×C8—2121no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—531no—
329C4⋊C4—531no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C22⋊C4—41no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C42—41no—
337C42—1061no—
338C42—1061no—
339C42—1061no—
340C42C2×C4×D5311yescharacteristic
341C42D4×D5311yescharacteristic
342C42D4×D5311yescharacteristic
343C42C2×C22⋊D5311yescharacteristic
344C23—1061no—
345C23#1313319911yescharacteristic
346Q8—42no—
347Q8—41no—
348Q8—41no—
349Q8—532no—
350Q8—1061no—
351D4—532no—
352D4—1062no—
353D4—1062no—
354D4—1061no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—1061no—
361C2×C4—1062no—
362C2×C4—1062no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—4241no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4C22⋊C4×D5311yescharacteristic
369C2×C4#1313317711yescharacteristic
370C2×C4C4×C22⋊D5311yescharacteristic
371C2×C4#1313318511yescharacteristic
372C2×C4#1313318811yescharacteristic
373C2×C4#1313319011yescharacteristic
374C2×C4SD16×D5311yescharacteristic
375C2×C4Q16×D5311yescharacteristic
376C2×C4#1313324411yescharacteristic
377C2×C4#1313324911yescharacteristic
378C8—41no—
379C8—41no—
380C8—81no—
381C8—2122no—
382C8—2122no—
383C8—2121no—
384C8—2121no—
385C22—21no—
386C22—21no—
387C22—1062no—
388C22—2121no—
389C22—2121no—
390C22#1681049911yescharacteristic
391C22#1681095211yescharacteristic
392C22#1681095711yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—1062no—
399C4—2121no—
400C4—2121no—
401C4#103×D5311yescharacteristic
402C4#1681062311yescharacteristic
403C4#1681073811yescharacteristic
404C4#1681074311yescharacteristic
405C4#1681077211yescharacteristic
406C4#1681077711yescharacteristic
407C2—21no—
408C2—21no—
409C2—2122no—
410C2#2027501911yescharacteristic
411C2#2027504511yescharacteristic
412C2#2027507411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1481C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1481D53
hypocenterC53#8993
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. #20273170C2semidirect
5. #20272611C2semidirect
6. #20272549C2semidirect
7. C53×#1481C2semidirect
204. #1481D53semidirect
239. C53#8993semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202832802
#202731702
#202730952
#202726112
#202725492
C53×#14812
#899353