Finite Group Databasebeta

← #27538684

Subgroups — #27538684

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811064C211yesmaximal, characteristic, complemented
2#20287818C211yesmaximal, characteristic, complemented
3#20284435C211yesmaximal, characteristic
4#20276847C214yesmaximal, complemented
5#20276838C214yesmaximal, complemented
6#20276509C211yesmaximal, characteristic, complemented
7#20276462C211yesmaximal, characteristic, complemented
8C2×#16810439C211yesmaximal, characteristic, complemented
9C2×C53×#473C211yesmaximal, characteristic, complemented
10#16811421—21no—
11C2×#13133239C2211yescharacteristic
12#16811064—22no—
13#16811064—22no—
14C2×#13133163C2212yescomplemented
15C2×#13133160C2212yescomplemented
16#16811040—22no—
17#16811040C2214yes—
18C2×#13133157—22no—
19C2×#13133157C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21#16810898C2212yes—
22#16810893C2212yes—
23#16810886—22no—
24C2×#13133153C2211yescharacteristic
25#16810605—21no—
26#16810452C2214yescomplemented
27#16810439C2214yescomplemented
28#16810438C2212yescomplemented
29#16810435C2212yescomplemented
30C2×#137×C53—21no—
31C22×Q16×C53C2211yescharacteristic
32C2×#131×C53C2211yescharacteristic
33C53×#473—22no—
34C53×#473C2214yes—
35C2×#108×C53C2211yescharacteristic
36C2×#101×C53—21no—
37C2×#11719945—21no—
38#13133248—41no—
39#13133247—22no—
40#13133243—41no—
41#13133242—22no—
42#13133239—21no—
43#13133239—21no—
44#13133239—21no—
45#13133239—21no—
46#13133237—42no—
47C2×Dic212C2311yescharacteristic
48C2×#11719913—21no—
49C2×D424C2311yescharacteristic
50C2×#11719917—21no—
51#13133163—24no—
52#13133160—24no—
53#13133157—21no—
54#13133157—24no—
55#13133157D412yes—
56#13133157D412yes—
57#13133157C2312yes—
58C2×#11719908—21no—
59C2×#11719908—24no—
60C2×#11719908C2312yes—
61#13133153—21no—
62#13133153C2312yes—
63#13133135—22no—
64#13133133—22no—
65C2×#41×C53D411yescharacteristic
66C22×Q8×C53—21no—
67#137×C53—24no—
68#137×C53—42no—
69C2×Q16×C53—21no—
70C2×Q16×C53—21no—
71C2×Q16×C53—21no—
72C2×Q16×C53—21no—
73C2×Q16×C53—21no—
74C2×Q16×C53—41no—
75C2×Q16×C53C2312yes—
76C2×SD16×C53—41no—
77#131×C53—21no—
78#131×C53—21no—
79#131×C53D412yes—
80#131×C53C2312yes—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53D411yescharacteristic
84C22×C8×C53—21no—
85C22×C8×C53C2311yescharacteristic
86#108×C53—21no—
87#108×C53C2312yes—
88#101×C53—24no—
89#41×C53—21no—
90#41×C53—21no—
91#41×C53D812yes—
92#41×C53C2×D412yes—
93C2×Q8×C53—22no—
94C2×Q8×C53—41no—
95C2×Q8×C53—41no—
96C2×Q8×C53—41no—
97C2×Q8×C53—41no—
98C2×Q8×C53—41no—
99C2×Q8×C53D811yescharacteristic
100C2×D4×C53—21no—
101C2×D4×C53D811yescharacteristic
102C22×C4×C53—21no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C2×C22⋊D53—41no—
106#11719945—41no—
107#11719945—41no—
108#11719945—41no—
109#11719945—41no—
110C2×D212—21no—
111C2×C4×D53—41no—
112C2×Dic106—21no—
113Q16×C53—22no—
114Q16×C53—22no—
115Q16×C53—44no—
116Q16×C53—41no—
117Q16×C53—41no—
118SD16×C53—44no—
119M16×C53—21no—
120M16×C53—22no—
121M16×C53—22no—
122M16×C53—22no—
123M16×C53—41no—
124M16×C53C2×D412yes—
125C2×C8×C53—21no—
126C2×C8×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53—41no—
129C2×C8×C53C2×D411yescharacteristic
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2×D411yescharacteristic
132C2×C8×C53#4111yescharacteristic
133C2×C8×C53#4111yescharacteristic
134C2×C8×C53C2411yescharacteristic
135#11719925—42no—
136#11719924—42no—
137#11719923—42no—
138#11719922—42no—
139#11719917—22no—
140#11719917—41no—
141C2×#88322—41no—
142Dic212—21no—
143Dic212—21no—
144D424—21no—
145D424—21no—
146#11719913—41no—
147#11719913—41no—
148#11719908—22no—
149#11719908C2×D414yes—
150C23×C53—21no—
151C23×C53#13511yescharacteristic
152C22×D53—41no—
153Q8×C53—21no—
154Q8×C53—42no—
155Q8×C53—42no—
156Q8×C53—81no—
157Q8×C53—81no—
158Q8×C53C2×D812yes—
159D4×C53—22no—
160D4×C53—22no—
161D4×C53—21no—
162D4×C53C2×D812yes—
163C2×C4×C53—21no—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—42no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53C2×D811yescharacteristic
172C2×C4×C53C2×D811yescharacteristic
173C2×C4×C53#13511yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C22⋊D53—81no—
176C22⋊D53—81no—
177C2×Dic53—41no—
178D212—41no—
179D212—41no—
180C4×D53—81no—
181C4×D53—81no—
182Dic106—41no—
183Dic106—41no—
184C8×C53—22no—
185C8×C53—42no—
186C8×C53#12111yescharacteristic
187C8×C53#12111yescharacteristic
188C8×C53C22×D411yescharacteristic
189C8×C53C2×#4111yescharacteristic
190#88322—42no—
191#33619—531nomaximal, Sylow
192C22×C53—22no—
193C22×C53—41no—
194C22×C53—81no—
195C22×C53#46611yescharacteristic
196C22×C53#46611yescharacteristic
197C22×C53C2×#13511yescharacteristic
198C2×D53—81no—
199C2×D53—161no—
200C4×C53—22no—
201C4×C53—82no—
202C4×C53#46611yescharacteristic
203C4×C53#46611yescharacteristic
204C4×C53C2×#12111yescharacteristic
205C4×C53C22×D811yescharacteristic
206Dic53—81no—
207C2×#510—531no—
208#3407—531no—
209#3067—531no—
210C2×#473D5311yescharacteristic, complemented
211#2223—534no—
212#2222—534no—
213#2161—531no—
214#2148—531no—
215C2×#362—531no—
216C2×C53—21no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—42no—
220C2×C53C2×#46611yescharacteristic
221D53—161no—
222#575—1061no—
223C2×#137—21no—
224C2×#135—531no—
225C22×Q16C2×D5311yescharacteristic
226C2×#131C2×D5311yescharacteristic
227#510—1062no—
228#510—1062no—
229C2×Q32—532no—
230C2×SD32—532no—
231#504—534no—
232#504—1062no—
233C2×M32—531no—
234C2×M32—531no—
235C2×M32—1062no—
236#473—22no—
237#473—532no—
238#473C2×D5314yes—
239#472—532no—
240#471—1062no—
241C2×#108—531no—
242C2×#108C2×D5311yescharacteristic
243#413—1061no—
244C2×#101—21no—
245#368—534no—
246#362—534no—
247#362—532no—
248#361—532no—
249C53#3361911yescharacteristic, complemented, Sylow
250C2×#41—1061no—
251C2×#41C22⋊D5311yescharacteristic
252C22×Q8—21no—
253#137—24no—
254#137—42no—
255#137—1062no—
256#137—2121no—
257AGL(1,ℤ/8)—1062no—
258AGL(1,ℤ/8)—2121no—
259#135—1061no—
260#135—1061no—
261#135—1061no—
262#135—1061no—
263#135—2122no—
264C2×Q16—21no—
265C2×Q16—21no—
266C2×Q16—21no—
267C2×Q16—21no—
268C2×Q16—21no—
269C2×Q16—41no—
270C2×Q16—531no—
271C2×Q16C22×D5312yes—
272C2×SD16—41no—
273C2×SD16—1061no—
274C2×D8—531no—
275#131—21no—
276#131—21no—
277#131C22⋊D5312yes—
278#131C22×D5312yes—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—1061no—
282C2×M16C22⋊D5311yescharacteristic
283C22×C8—21no—
284C22×C8C22×D5311yescharacteristic
285Q32—1064no—
286SD32—1064no—
287M32—532no—
288M32—532no—
289M32—532no—
290M32—1064no—
291M32—1061no—
292C2×C16—532no—
293C2×C16—1064no—
294C2×C16—1061no—
295#108—21no—
296#108—532no—
297#108—1061no—
298#108C22×D5312yes—
299#101—24no—
300#101—1062no—
301#100—1062no—
302#41—21no—
303#41—21no—
304#41—2121no—
305#41—2121no—
306#41—2121no—
307#41—2121no—
308#41#1171992212yes—
309#41C2×C22⋊D5312yes—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—1061no—
317C2×Q8#1171992211yescharacteristic
318C2×D4—21no—
319C2×D4—1061no—
320C2×D4—2121no—
321C2×D4#1171992211yescharacteristic
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—2121no—
325C22×C4C2×C22⋊D5311yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—44no—
329Q16—41no—
330Q16—41no—
331Q16—1061no—
332Q16—1061no—
333Q16—2122no—
334SD16—44no—
335SD16—2122no—
336SD16—2122no—
337SD16—2121no—
338SD16—2121no—
339D8—1061no—
340D8—1061no—
341D8—2122no—
342M16—21no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—41no—
347M16—1062no—
348M16—2121no—
349M16C2×C22⋊D5312yes—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—2121no—
355C2×C8D4×D5311yescharacteristic
356C2×C8D4×D5311yescharacteristic
357C2×C8#1171994711yescharacteristic
358C2×C8#1171994711yescharacteristic
359C2×C8C2×C22⋊D5311yescharacteristic
360C2×C8C23×D5311yescharacteristic
361C16—534no—
362C16—1062no—
363C23—21no—
364C23—2121no—
365C23#1313323711yescharacteristic
366Q8—21no—
367Q8—42no—
368Q8—42no—
369Q8—81no—
370Q8—81no—
371Q8—2121no—
372Q8—2121no—
373Q8C2×#1171992212yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—4241no—
378D4—4241no—
379D4—2121no—
380D4—2121no—
381D4C2×#1171992212yes—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—4241no—
389C2×C4—4241no—
390C2×C4—2121no—
391C2×C4#1313318911yescharacteristic
392C2×C4#1313318911yescharacteristic
393C2×C4C2×#1171992211yescharacteristic
394C2×C4C2×#1171992211yescharacteristic
395C2×C4#1313323711yescharacteristic
396C2×C4C22×C22⋊D5311yescharacteristic
397C8—22no—
398C8—42no—
399C8—2122no—
400C8#1313318911yescharacteristic
401C8#1313318911yescharacteristic
402C8C2×D4×D5311yescharacteristic
403C8C2×#1171994711yescharacteristic
404C22—22no—
405C22—41no—
406C22—81no—
407C22—8481no—
408C22—4241no—
409C22#1681085711yescharacteristic
410C22#1681085711yescharacteristic
411C22C2×#1313323711yescharacteristic
412C4—22no—
413C4—82no—
414C4—4241no—
415C4#1681085711yescharacteristic
416C4#1681085711yescharacteristic
417C4C2×#1313318911yescharacteristic
418C4C22×#1171992211yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—8481no—
424C2C2×#1681085711yescharacteristic
subgroupG/H
centerC2C2×#16810857
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#473C2
socleC2×C53C2×#466
prime radical#20284435C2
hypercenterC2×#473D53
hypocenterC53#33619
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#473

Complements

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

NH ≅ G/Ntype
1. C2×#16811064C2semidirect
2. #20287818C2semidirect
4. #20276847C2semidirect
5. #20276838C2semidirect
6. #20276509C2semidirect
7. #20276462C2semidirect
8. C2×#16810439C2semidirect
9. C2×C53×#473C2semidirect
14. C2×#13133163C22semidirect
15. C2×#13133160C22semidirect
26. #16810452C22semidirect
27. #16810439C22semidirect
28. #16810438C22semidirect
29. #16810435C22semidirect
210. C2×#473D53semidirect
249. C53#33619semidirect

Maximal subgroups

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

subgroupindex
C2×#168110642
#202878182
#202844352
#202768472
#202768382
#202765092
#202764622
C2×#168104392
C2×C53×#4732
#3361953