Finite Group Databasebeta

← #27538689

Subgroups — #27538689

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×#16811063C211yesmaximal, characteristic, complemented
2#20287818C211yesmaximal, characteristic, complemented
3C2×#16810898C211yesmaximal, characteristic, complemented
4#20276843C214yesmaximal, complemented
5#20276838C214yesmaximal, complemented
6#20276509C211yesmaximal, characteristic, complemented
7#20276464C211yesmaximal, characteristic, complemented
8C2×#16810438C211yesmaximal, characteristic, complemented
9C53×#3067C211yesmaximal, characteristic
10C2×#13133247—21no—
11C22×Dic212C2211yescharacteristic
12#16811063—22no—
13#16811063—22no—
14C2×#13133163C2212yescomplemented
15C2×#13133161C2212yescomplemented
16#16811040—22no—
17#16811040C2214yes—
18C2×#13133157—22no—
19C2×#13133157C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21#16810898—22no—
22#16810898C2214yes—
23C2×#13133153C2211yescharacteristic
24C2×#13133135—21no—
25#16810452C2214yescomplemented
26#16810439C2212yescomplemented
27#16810438C2214yescomplemented
28#16810436C2212yescomplemented
29C53×#575—21no—
30C2×#135×C53C2211yescharacteristic
31C2×#131×C53C2211yescharacteristic
32C53×#473C2212yes—
33C53×#472C2212yes—
34C53×#471—22no—
35C2×#108×C53C2211yescharacteristic
36C53×#413—21no—
37C22×Dic106—21no—
38#13133247—24no—
39#13133247—42no—
40C2×Dic212—21no—
41C2×Dic212—21no—
42C2×Dic212—21no—
43C2×Dic212—21no—
44C2×Dic212—21no—
45C2×Dic212C2312yes—
46C2×#11719925—41no—
47C2×#11719923—41no—
48C2×#11719917—21no—
49#13133163—24no—
50#13133161—24no—
51#13133157—21no—
52#13133157—24no—
53#13133157D412yes—
54#13133157D412yes—
55#13133157C2312yes—
56C2×#11719908—21no—
57C2×#11719908—24no—
58C2×#11719908C2312yes—
59#13133153—21no—
60#13133153C2312yes—
61#13133135—24no—
62C2×#41×C53—21no—
63C2×#41×C53D411yescharacteristic
64#137×C53—22no—
65#137×C53—41no—
66AGL(1,ℤ/8)×C53—22no—
67AGL(1,ℤ/8)×C53—41no—
68#135×C53—21no—
69#135×C53—21no—
70#135×C53—21no—
71#135×C53—21no—
72#135×C53—42no—
73C2×Q16×C53C2311yescharacteristic
74C2×SD16×C53—21no—
75C2×D8×C53C2311yescharacteristic
76#131×C53—21no—
77#131×C53—21no—
78#131×C53D412yes—
79#131×C53C2312yes—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53D411yescharacteristic
83C22×C8×C53—21no—
84C22×C8×C53C2311yescharacteristic
85#108×C53—21no—
86#108×C53C2312yes—
87#101×C53—22no—
88#100×C53—22no—
89#41×C53—21no—
90#41×C53—21no—
91#41×C53—41no—
92#41×C53—41no—
93#41×C53—41no—
94#41×C53—41no—
95#41×C53D812yes—
96#41×C53C2×D412yes—
97C2×Q8×C53—21no—
98C2×Q8×C53D811yescharacteristic
99C2×D4×C53—21no—
100C2×D4×C53—21no—
101C2×D4×C53—41no—
102C2×D4×C53D811yescharacteristic
103C22×C4×C53—21no—
104C22×C4×C53—41no—
105C22×C4×C53C2×D411yescharacteristic
106C22×Dic53—41no—
107C2×Dic106—22no—
108C2×Dic106—41no—
109C2×Dic106—41no—
110C2×Dic106—41no—
111C2×Dic106—41no—
112C2×Dic106—41no—
113Q16×C53—21no—
114Q16×C53—21no—
115Q16×C53—42no—
116SD16×C53—42no—
117SD16×C53—42no—
118SD16×C53—41no—
119SD16×C53—41no—
120D8×C53—21no—
121D8×C53—21no—
122D8×C53—42no—
123M16×C53—21no—
124M16×C53—22no—
125M16×C53—22no—
126M16×C53—22no—
127M16×C53—41no—
128M16×C53C2×D412yes—
129C2×C8×C53—21no—
130C2×C8×C53—22no—
131C2×C8×C53—22no—
132C2×C8×C53—41no—
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53#4111yescharacteristic
137C2×C8×C53#4111yescharacteristic
138C2×C8×C53C2411yescharacteristic
139#11719925—44no—
140#11719923—44no—
141#11719917—22no—
142#11719917—41no—
143C2×#88322—41no—
144Dic212—22no—
145Dic212—22no—
146Dic212—41no—
147Dic212—41no—
148#11719908—22no—
149#11719908C2×D414yes—
150C23×C53—21no—
151C23×C53—41no—
152C23×C53#13511yescharacteristic
153Q8×C53—21no—
154Q8×C53—41no—
155Q8×C53—41no—
156Q8×C53C2×D812yes—
157D4×C53—22no—
158D4×C53—22no—
159D4×C53—21no—
160D4×C53—41no—
161D4×C53—41no—
162D4×C53—81no—
163D4×C53—81no—
164D4×C53C2×D812yes—
165C2×C4×C53—21no—
166C2×C4×C53—22no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53#13511yescharacteristic
176C2×C4×C53C22×D411yescharacteristic
177C2×Dic53—42no—
178C2×Dic53—81no—
179C2×Dic53—81no—
180Dic106—42no—
181Dic106—42no—
182Dic106—81no—
183Dic106—81no—
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—81no—
196C22×C53—161no—
197C22×C53#46611yescharacteristic
198C22×C53#46611yescharacteristic
199C22×C53C2×#13511yescharacteristic
200C4×C53—22no—
201C4×C53—81no—
202C4×C53#46611yescharacteristic
203C4×C53#46611yescharacteristic
204C4×C53C2×#12111yescharacteristic
205C4×C53C22×D811yescharacteristic
206Dic53—82no—
207C2×#510—531no—
208#3407—531no—
209#3067D5311yescharacteristic
210C2×#473—531no—
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×C53—161no—
221C2×C53C2×#46611yescharacteristic
222#575—21no—
223C2×#137—1061no—
224C2×#135C2×D5311yescharacteristic
225C22×Q16—531no—
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—534no—
237#473—1062no—
238#473C2×D5312yes—
239#472C2×D5312yes—
240#471—22no—
241C2×#108—531no—
242C2×#108C2×D5311yescharacteristic
243#413—21no—
244C2×#101—1061no—
245#368—534no—
246#362—532no—
247#362—534no—
248#361—532no—
249C53#3361911yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C2×#41C22⋊D5311yescharacteristic
252C22×Q8—1061no—
253#137—22no—
254#137—41no—
255#137—1064no—
256#137—2122no—
257AGL(1,ℤ/8)—22no—
258AGL(1,ℤ/8)—41no—
259#135—21no—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—42no—
264C2×Q16—532no—
265C2×Q16—1061no—
266C2×Q16—1061no—
267C2×Q16—1061no—
268C2×Q16—1061no—
269C2×Q16—1061no—
270C2×Q16—2121no—
271C2×Q16C22×D5311yescharacteristic
272C2×SD16—21no—
273C2×SD16—2121no—
274C2×D8C22×D5311yescharacteristic
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—22no—
300#101—1064no—
301#100—22no—
302#41—21no—
303#41—21no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41#1171992212yes—
309#41C2×C22⋊D5312yes—
310C2×Q8—21no—
311C2×Q8—1062no—
312C2×Q8—2121no—
313C2×Q8—2121no—
314C2×Q8—2121no—
315C2×Q8—2121no—
316C2×Q8—2121no—
317C2×Q8#1171992211yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4#1171992211yescharacteristic
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—2121no—
325C22×C4C2×C22⋊D5311yescharacteristic
326Q16—21no—
327Q16—21no—
328Q16—42no—
329Q16—1062no—
330Q16—1062no—
331Q16—2124no—
332Q16—2121no—
333Q16—2121no—
334SD16—42no—
335SD16—42no—
336SD16—41no—
337SD16—41no—
338SD16—2124no—
339D8—21no—
340D8—21no—
341D8—42no—
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—41no—
365C23#1313323711yescharacteristic
366Q8—21no—
367Q8—41no—
368Q8—41no—
369Q8—4241no—
370Q8—4241no—
371Q8—2122no—
372Q8—2122no—
373Q8C2×#1171992212yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381D4C2×#1171992212yes—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—4241no—
389C2×C4—4241no—
390C2×C4—2122no—
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—81no—
408C22—161no—
409C22#1681085711yescharacteristic
410C22#1681085711yescharacteristic
411C22C2×#1313323711yescharacteristic
412C4—22no—
413C4—81no—
414C4—4242no—
415C4#1681085711yescharacteristic
416C4#1681085711yescharacteristic
417C4C2×#1313318911yescharacteristic
418C4C22×#1171992211yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—161no—
424C2C2×#1681085711yescharacteristic
subgroupG/H
centerC2C2×#16810857
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3067C2
socleC2×C53C2×#466
hypercenter#3067D53
hypocenterC53#33619
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#16811063C2semidirect
2. #20287818C2semidirect
3. C2×#16810898C2semidirect
4. #20276843C2semidirect
5. #20276838C2semidirect
6. #20276509C2semidirect
7. #20276464C2semidirect
8. C2×#16810438C2semidirect
14. C2×#13133163C22semidirect
15. C2×#13133161C22semidirect
25. #16810452C22semidirect
26. #16810439C22semidirect
27. #16810438C22semidirect
28. #16810436C22semidirect
249. C53#33619semidirect

Maximal subgroups

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

subgroupindex
C2×#168110632
#202878182
C2×#168108982
#202768432
#202768382
#202765092
#202764642
C2×#168104382
C53×#30672
#3361953