Finite Group Databasebeta

← #27537128

Subgroups — #27537128

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D53C211yesmaximal, characteristic
2C2×#16810660C211yesmaximal, characteristic
3#427×D53C211yesmaximal, characteristic
4#20276504C211yesmaximal, characteristic, complemented
5#20276503C211yesmaximal, characteristic, complemented
6#20276502C212yesmaximal, complemented
7#20276501C211yesmaximal, characteristic, complemented
8#20276500C212yesmaximal, complemented
9#20276498C212yesmaximal, complemented
10#20276496C212yesmaximal, complemented
11C53×#2159C211yesmaximal, characteristic, complemented
12C22×C8×D53C2211yescharacteristic
13C2×#13133175—21no—
14#117×D53—21no—
15C2×#13133157C2211yescharacteristic
16C2×#13133156C2212yes—
17M32×D53C418yescomplemented
18C2×C16×D53C2212yes—
19#16810661C2211yescharacteristic
20#16810660—21no—
21#16810660—21no—
22#16810660C2211yescharacteristic
23#16810660C2212yes—
24#16810660C2211yescharacteristic
25#16810660C2212yes—
26#16810659—21no—
27#16810658—21no—
28C2×#13133151C2211yescharacteristic
29#107×D53C2211yescharacteristic
30#107×D53C2211yescharacteristic
31C2×#13133148—21no—
32#106×D53—21no—
33#16810451C2212yescomplemented
34#16810450C2212yescomplemented
35#16810450C2212yescomplemented
36#16810449C2212yescomplemented
37#16810448C2212yescomplemented
38#16810448C2212yescomplemented
39C2×M32×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C2×#107×C53C2211yescharacteristic
42C53×#367C2212yescomplemented
43C53×#366C2212yescomplemented
44C22×C4×D53D411yescharacteristic
45C22×#88322D411yescharacteristic
46C2×C8×D53C2×C414yes—
47C2×C8×D53D411yescharacteristic
48C2×C8×D53D411yescharacteristic
49C2×C8×D53Q814yes—
50C2×C8×D53C2311yescharacteristic
51C2×C8×D53C2311yescharacteristic
52#13133176—21no—
53#13133175—21no—
54#13133175—21no—
55#13133175—22no—
56#13133175—22no—
57#13133175—41no—
58#13133175—41no—
59#13133174—41no—
60#13133173—41no—
61C2×C4⋊Dic53—21no—
62C4⋊C4×D53—21no—
63C4⋊C4×D53—21no—
64C2×#11719921—41no—
65C22⋊C4×D53—41no—
66C2×C4×Dic53—41no—
67C42×D53—41no—
68#13133157C2×C414yescomplemented
69#13133156C2×C418yescomplemented
70C2×#11719908C2312yes—
71C16×D53C2×C418yescomplemented
72#13133151—21no—
73#13133151C2311yescharacteristic
74#13133151C2311yescharacteristic
75#13133151C2312yes—
76#13133150—21no—
77#13133150—21no—
78#13133149—21no—
79#13133148—21no—
80#13133148—21no—
81#13133148—22no—
82#13133148—22no—
83C22×C8×C53C2311yescharacteristic
84#117×C53—21no—
85C2×C4⋊C4×C53—21no—
86M32×C53C2×C414yescomplemented
87C2×C16×C53C2312yes—
88#107×C53—21no—
89#107×C53C2311yescharacteristic
90#107×C53C2312yes—
91#107×C53C2311yescharacteristic
92#106×C53—21no—
93C23×D53D811yescharacteristic
94C22×C4×C53—41no—
95C22×C4×C53C2×D411yescharacteristic
96C22×Dic53—41no—
97C22×Dic53D811yescharacteristic
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53C4⋊C414yes—
101C2×C4×D53D811yescharacteristic
102C2×C4×D53D811yescharacteristic
103C2×C4×D53Q1614yes—
104C2×C4×D53C2×D411yescharacteristic
105C2×C4×D53C2×D411yescharacteristic
106C2×C8×C53C22×C412yes—
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×Q812yes—
109C2×C8×C53C2411yescharacteristic
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53—41no—
116C22⋊Dic53—41no—
117#11719921—41no—
118#11719921—41no—
119#11719921—42no—
120#11719921—42no—
121C4⋊Dic53—22no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—41no—
125#11719919—41no—
126#11719919—41no—
127C4×Dic53—41no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130C4×Dic53—42no—
131C4×Dic53—42no—
132C2×#88322C4⋊C414yes—
133C2×#88322C2×D411yescharacteristic
134C2×#88322C2×D411yescharacteristic
135C8×D53C4⋊C414yes—
136C8×D53C4⋊C414yes—
137C8×D53C22×C412yes—
138C8×D53C22×C412yes—
139C8×D53C2×Q812yes—
140C8×D53C2×Q812yes—
141C16×C53C22×C414yes—
142#11719908C22×C414yes—
143C23×C53C2×D811yescharacteristic
144C22×D53—41no—
145C22×D53—41no—
146C22×D53#10714yes—
147C22×D53C2×D811yescharacteristic
148C22×D53C2×D811yescharacteristic
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—82no—
153C2×C4×C53C2×C4⋊C412yes—
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×Q1612yes—
156C2×C4×C53C22×D411yescharacteristic
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—42no—
160C2×Dic53—82no—
161C2×Dic53#10714yes—
162C2×Dic53C2×D811yescharacteristic
163C2×Dic53C2×D811yescharacteristic
164C4×D53—82no—
165C4×D53—82no—
166C4×D53#10714yes—
167C4×D53#10714yes—
168C4×D53C2×C4⋊C412yes—
169C4×D53C2×C4⋊C412yes—
170C4×D53C2×Q1612yes—
171C4×D53C2×Q1612yes—
172C8×C53C2×C4⋊C412yes—
173C8×C53C23×C411yescharacteristic
174C8×C53C22×Q811yescharacteristic
175#88322C2×C4⋊C412yes—
176#88322C2×C4⋊C412yes—
177#33409—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53C2×#10712yes—
181C22×C53C22×D811yescharacteristic
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C2×D53—21no—
186C2×D53—44no—
187C2×D53C2×#10712yes—
188C2×D53C2×#10712yes—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53—82no—
192C4×C53C2×#10712yes—
193C4×C53C22×C4⋊C411yescharacteristic
194C4×C53C22×Q1611yescharacteristic
195Dic53—81no—
196Dic53—81no—
197Dic53—82no—
198Dic53C2×#10712yes—
199Dic53C2×#10712yes—
200C22×M32—531no—
201C2×#427—531no—
202C2×#427—531no—
203#2159—531no—
204#2159—531no—
205#2159—531no—
206#2159—532no—
207#2159—532no—
208#2159D5311yescharacteristic, complemented
209#2158—532no—
210#2158—532no—
211C2×C53—22no—
212C2×C53C22×#10711yescharacteristic
213C2×C53#341411yescharacteristic
214C2×C53#341411yescharacteristic
215D53—22no—
216D53—22no—
217C23×C8—531no—
218C2×#117—1061no—
219C2×#117—1061no—
220C2×M32—531no—
221C2×M32—538no—
222C2×M32—532no—
223C2×M32C2×D5311yescharacteristic
224C22×C16—532no—
225#427—531no—
226#427—531no—
227#427—532no—
228#427—531no—
229#427—532no—
230#427—1061no—
231#427—1061no—
232#427—1061no—
233#427—1061no—
234#427C2×D5311yescharacteristic
235C2×#107—531no—
236C2×#107—531no—
237C2×#107—531no—
238C2×#107C2×D5311yescharacteristic
239C2×#106—1061no—
240C2×#106—1061no—
241#367—532no—
242#367—532no—
243#367—532no—
244#367C2×D5312yescomplemented
245#366—532no—
246#366—532no—
247#366—532no—
248#366C2×D5312yescomplemented
249C53#3340911yescharacteristic, complemented, Sylow
250C23×C4—531no—
251C22×C8—531no—
252C22×C8—534no—
253C22×C8—531no—
254C22×C8—531no—
255C22×C8—534no—
256C22×C8—531no—
257C22×C8—531no—
258C22×C8C22×D5311yescharacteristic
259#117—21no—
260#117—1061no—
261#117—1061no—
262#117—1062no—
263#117—1062no—
264#117—1061no—
265#117—2121no—
266#117—2121no—
267#117—2121no—
268#117—2121no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C22⋊C4—2121no—
274C2×C22⋊C4—2121no—
275C2×C42—2121no—
276C2×C42—2121no—
277M32—534no—
278M32—538no—
279M32C4×D5314yescomplemented
280C2×C16—532no—
281C2×C16—538no—
282C2×C16C22×D5312yes—
283#107—21no—
284#107—531no—
285#107—531no—
286#107—532no—
287#107—1061no—
288#107—1061no—
289#107—1061no—
290#107C22×D5311yescharacteristic
291#107C22×D5312yes—
292#107C22×D5311yescharacteristic
293#106—21no—
294#106—1061no—
295#106—1061no—
296#106—1062no—
297#106—1062no—
298#106—1061no—
299C24—531no—
300C22×C4—41no—
301C22×C4—531no—
302C22×C4—534no—
303C22×C4—531no—
304C22×C4—531no—
305C22×C4—534no—
306C22×C4—531no—
307C22×C4—531no—
308C22×C4—2121no—
309C22×C4—2121no—
310C22×C4—2121no—
311C22×C4D4×D5311yescharacteristic
312C2×C8—534no—
313C2×C8—531no—
314C2×C8—531no—
315C2×C8—534no—
316C2×C8—534no—
317C2×C8—532no—
318C2×C8—532no—
319C2×C8—532no—
320C2×C8—532no—
321C2×C8C2×C4×D5312yes—
322C2×C8D4×D5311yescharacteristic
323C2×C8Q8×D5312yes—
324C2×C8C23×D5311yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—1062no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C4⋊C4—2121no—
335C22⋊C4—41no—
336C22⋊C4—2121no—
337C22⋊C4—2121no—
338C22⋊C4—2122no—
339C22⋊C4—2122no—
340C22⋊C4—2121no—
341C42—41no—
342C42—2121no—
343C42—2121no—
344C42—2122no—
345C42—2122no—
346C42—2121no—
347C16—534no—
348C16C2×C4×D5314yes—
349C23—531no—
350C23—531no—
351C23—534no—
352C23—2121no—
353C23—2121no—
354C23D8×D5311yescharacteristic
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—531no—
360C2×C4—531no—
361C2×C4—534no—
362C2×C4—532no—
363C2×C4—532no—
364C2×C4—532no—
365C2×C4—532no—
366C2×C4—534no—
367C2×C4—534no—
368C2×C4—4242no—
369C2×C4—4242no—
370C2×C4—4242no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2122no—
374C2×C4C4⋊C4×D5312yes—
375C2×C4D8×D5311yescharacteristic
376C2×C4Q16×D5312yes—
377C2×C4C2×D4×D5311yescharacteristic
378C8—532no—
379C8—532no—
380C8C4⋊C4×D5312yes—
381C8C22×C4×D5311yescharacteristic
382C8C2×Q8×D5311yescharacteristic
383C22—21no—
384C22—21no—
385C22—532no—
386C22—532no—
387C22—1061no—
388C22—1061no—
389C22—1061no—
390C22—1061no—
391C22—2124no—
392C22#107×D5312yes—
393C22C2×D8×D5311yescharacteristic
394C4—82no—
395C4—81no—
396C4—81no—
397C4—532no—
398C4—532no—
399C4—4241no—
400C4—4241no—
401C4—4242no—
402C4#107×D5312yes—
403C4C2×C4⋊C4×D5311yescharacteristic
404C4C2×Q16×D5311yescharacteristic
405C2—22no—
406C2—1062no—
407C2—1062no—
408C2C2×#107×D5311yescharacteristic
409C2#2028784711yescharacteristic
410C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2159C2
socleC22×C53C22×D8
prime radicalC23×D53D8
cosocleC2×C8×C53C24
hypercenter#2159D53
hypocenterC53#33409
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #20276504C2semidirect
5. #20276503C2semidirect
6. #20276502C2semidirect
7. #20276501C2semidirect
8. #20276500C2semidirect
9. #20276498C2semidirect
10. #20276496C2semidirect
11. C53×#2159C2semidirect
17. M32×D53C4semidirect
33. #16810451C22semidirect
34. #16810450C22semidirect
35. #16810450C22semidirect
36. #16810449C22semidirect
37. #16810448C22semidirect
38. #16810448C22semidirect
42. C53×#367C22semidirect
43. C53×#366C22semidirect
68. #13133157C2×C4semidirect
69. #13133156C2×C4semidirect
71. C16×D53C2×C4semidirect
86. M32×C53C2×C4semidirect
208. #2159D53semidirect
244. #367C2×D53semidirect
248. #366C2×D53semidirect
249. C53#33409semidirect
279. M32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D532
C2×#168106602
#427×D532
#202765042
#202765032
#202765022
#202765012
#202765002
#202764982
#202764962
C53×#21592
#3340953