Finite Group Databasebeta

← #27484045

Subgroups — #27484045

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285227C211yesmaximal, characteristic, complemented
2C2×#16810740C211yesmaximal, characteristic
3#20275889C212yesmaximal, complemented
4#20275885C212yesmaximal, complemented
5#20275614C211yesmaximal, characteristic
6#20274839C211yesmaximal, characteristic, complemented
7#20274838C211yesmaximal, characteristic, complemented
8#20274767C212yesmaximal, complemented
9#20274759C212yesmaximal, complemented
10#20274595C211yesmaximal, characteristic, complemented
11C53×#1976C211yesmaximal, characteristic
12C2×#13133224C2212yes—
13#16811176C2212yes—
14#16811027—21no—
15#16811024—21no—
16#16811023—21no—
17#16811020—21no—
18#16810740—21no—
19#16810740—21no—
20#16810740—21no—
21#16810740—21no—
22C2×#13133153C2211yescharacteristic
23C2×#13133152C2211yescharacteristic
24#16810661C2211yescharacteristic
25#16810659C2212yes—
26#16810659C2211yescharacteristic
27#16810640C2212yes—
28C2×#13133144C2212yes—
29#16810597C2212yes—
30#16810578C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32#16810394C2212yes—
33#16810393C2212yes—
34#16810384C2211yescharacteristic
35#16810382C2211yescharacteristic
36#16810379C2212yes—
37#16810379C2212yes—
38C2×#135×C53C2211yescharacteristic
39C53×#427C2211yescharacteristic
40C53×#421C2212yes—
41C53×#410C2212yes—
42C53×#405C2211yescharacteristic
43#13133224—22no—
44#13133224—22no—
45#13133224—22no—
46#13133224—22no—
47C2×#11719917—22no—
48C2×#11719917C2311yescharacteristic
49C2×#11719917C2312yes—
50C22×#88322—22no—
51C22×#88322C2311yescharacteristic
52#13133216—22no—
53#13133208—22no—
54#13133194—42no—
55C4×Dic106—42no—
56#13133176C2311yescharacteristic
57#13133176C2312yes—
58C2×C4×Dic53C2311yescharacteristic
59#13133153—21no—
60#13133153—21no—
61#13133152—21no—
62#13133152—21no—
63#13133151—21no—
64#13133150—22no—
65#13133150—21no—
66#13133149—21no—
67#13133148—22no—
68#13133148—21no—
69#13133144—22no—
70#13133144—22no—
71#13133144—42no—
72#13133132—42no—
73#13133127—21no—
74C8×Dic53—21no—
75C8×Dic53D411yescharacteristic
76C8×Dic53D411yescharacteristic
77C8×Dic53D411yescharacteristic
78C8×Dic53D411yescharacteristic
79C2×#41×C53C2312yes—
80#135×C53—21no—
81#135×C53—21no—
82#135×C53—21no—
83#135×C53—21no—
84C2×Q16×C53Q811yescharacteristic, complemented
85C2×SD16×C53Q812yescomplemented
86C2×D8×C53Q811yescharacteristic, complemented
87C2×M16×C53C2311yescharacteristic
88C22×C8×C53C2311yescharacteristic
89#117×C53C2311yescharacteristic
90#117×C53C2312yes—
91#107×C53—21no—
92#106×C53—21no—
93#104×C53—42no—
94#99×C53—42no—
95#97×C53—21no—
96C4×C8×C53—21no—
97#41×C53—22no—
98#41×C53—22no—
99#41×C53—22no—
100#41×C53—22no—
101C2×Q8×C53C2×Q812yes—
102C2×D4×C53—22no—
103C2×D4×C53C2×Q812yes—
104C22×C4×C53—22no—
105C22×C4×C53C2411yescharacteristic
106C22×Dic53—21no—
107C2×Dic106—42no—
108Q16×C53—21no—
109Q16×C53—21no—
110SD16×C53—22no—
111SD16×C53—22no—
112D8×C53—21no—
113D8×C53—21no—
114M16×C53—21no—
115M16×C53—21no—
116C2×C8×C53—21no—
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2×Q811yescharacteristic
120C2×C8×C53C2×Q811yescharacteristic
121C2×C8×C53#4111yescharacteristic
122C2×C8×C53#4111yescharacteristic
123C4⋊C4×C53—22no—
124C4⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—22no—
127C42×C53—21no—
128C42×C53—22no—
129C22⋊Dic53—21no—
130C22⋊Dic53—22no—
131C4⋊Dic53—22no—
132C4⋊Dic53—21no—
133C4⋊Dic53—42no—
134#11719919—42no—
135#11719919—42no—
136C4×Dic53—21no—
137C4×Dic53—21no—
138C4×Dic53—22no—
139C4×Dic53C2×D411yescharacteristic
140C4×Dic53C2×D411yescharacteristic
141#11719917—22no—
142#11719917—21no—
143#11719917—22no—
144#11719917—21no—
145#11719917—42no—
146#11719917—42no—
147C2×#88322—21no—
148C2×#88322—22no—
149C2×#88322—42no—
150C2×#88322—42no—
151C2×#88322C2×D411yescharacteristic
152C2×#88322C2×D411yescharacteristic
153C2×#88322#4111yescharacteristic
154C2×#88322#4111yescharacteristic
155C2×#88322#4111yescharacteristic
156C2×#88322#4111yescharacteristic
157C23×C53—22no—
158C23×C53H(5,𝔽2)11yescharacteristic
159Q8×C53—22no—
160Q8×C53—22no—
161D4×C53—22no—
162D4×C53—22no—
163D4×C53—42no—
164D4×C53—42no—
165C2×C4×C53—22no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—41no—
170C2×C4×C53C22×D411yescharacteristic
171C2×C4×C53C22×Q811yescharacteristic
172C2×C4×C53C2×#4111yescharacteristic
173C2×C4×C53C2×#4111yescharacteristic
174C2×C4×C53C2×#4111yescharacteristic
175C2×C4×C53H(5,𝔽2)11yescharacteristic
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—41no—
181Dic106—84no—
182C8×C53—41no—
183C8×C53C22⋊Q811yescharacteristic
184C8×C53C22⋊Q811yescharacteristic
185C8×C53C22⋊Q811yescharacteristic
186C8×C53C22⋊Q811yescharacteristic
187#88322—42no—
188#88322—41no—
189#88322#12411yescharacteristic
190#88322#12411yescharacteristic
191#88322#12411yescharacteristic
192#88322#12411yescharacteristic
193#29154—531nomaximal, Sylow
194C22×C53—41no—
195C22×C53—42no—
196C22×C53—82no—
197C22×C53#53511yescharacteristic
198C22×C53#55411yescharacteristic
199C22×C53#55511yescharacteristic
200C4×C53—42no—
201C4×C53—82no—
202C4×C53—81no—
203C4×C53C2×C22⋊Q811yescharacteristic
204C4×C53C2×#12411yescharacteristic
205C4×C53#55411yescharacteristic
206C4×C53#55511yescharacteristic
207Dic53—22no—
208Dic53—41no—
209Dic53—82no—
210Dic53—81no—
211#3160—531no—
212C2×#443—531no—
213#2031—532no—
214#2031—532no—
215#1976—531no—
216#1976D5311yescharacteristic
217#1833—531no—
218#1833—531no—
219#1814—532no—
220#1812—532no—
221#1772—531no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—82no—
226C2×C53#256311yescharacteristic
227C2×#135C2×D5311yescharacteristic
228C2×#131—532no—
229#533—532no—
230#500—1061no—
231#499—1061no—
232#499—1061no—
233#498—1061no—
234#443—1061no—
235#443—1061no—
236#443—1061no—
237#443—1061no—
238C2×#108—531no—
239C2×#108—531no—
240#427—531no—
241#427—531no—
242#427—532no—
243#427C2×D5311yescharacteristic
244#421—532no—
245#421C2×D5312yes—
246C2×#104—532no—
247#410—532no—
248#410C2×D5312yes—
249#405—531no—
250#405C2×D5311yescharacteristic
251C2×C4×C8—531no—
252#344—532no—
253#344—532no—
254#339—531no—
255#339—531no—
256H(3,ℤ/4)—532no—
257H(3,ℤ/4)—532no—
258C53#2915411yescharacteristic, complemented, Sylow
259C2×#41C22×D5312yes—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—21no—
264C2×Q16Dic10611yescharacteristic, complemented
265C2×SD16Dic10612yescomplemented
266C2×D8Dic10611yescharacteristic, complemented
267#131—1062no—
268#131—1062no—
269#131—1062no—
270#131—1062no—
271C2×M16—532no—
272C2×M16—531no—
273C2×M16—1062no—
274C2×M16C22×D5311yescharacteristic
275C22×C8—531no—
276C22×C8—1062no—
277C22×C8C22×D5311yescharacteristic
278C4⋊Q8—1062no—
279#125—1062no—
280C22⋊Q8—2122no—
281C4×Q8—2122no—
282#117—532no—
283#117—531no—
284#117C22×D5312yes—
285#117C22×D5311yescharacteristic
286C2×C42—531no—
287#108—1061no—
288#108—1061no—
289#108—1061no—
290#108—1061no—
291#107—21no—
292#107—1061no—
293#107—1062no—
294#107—1061no—
295#106—21no—
296#106—1061no—
297#106—1062no—
298#106—1061no—
299#104—42no—
300#104—1062no—
301#104—1062no—
302#104—2122no—
303#99—42no—
304#99—2122no—
305#97—21no—
306#97—1061no—
307C4×C8—21no—
308C4×C8—531no—
309C4×C8—531no—
310C4×C8—531no—
311C4×C8—531no—
312C4×C8—1061no—
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317C2×Q8—2122no—
318C2×Q8C2×Dic10612yes—
319C2×D4—22no—
320C2×D4C2×Dic10612yes—
321C22×C4—22no—
322C22×C4—1061no—
323C22×C4C23×D5311yescharacteristic
324Q16—21no—
325Q16—21no—
326SD16—22no—
327SD16—22no—
328D8—21no—
329D8—21no—
330M16—21no—
331M16—21no—
332M16—1062no—
333M16—1061no—
334M16—1062no—
335M16—1061no—
336M16—2122no—
337M16—2122no—
338C2×C8—21no—
339C2×C8—531no—
340C2×C8—531no—
341C2×C8—531no—
342C2×C8—531no—
343C2×C8—531no—
344C2×C8—531no—
345C2×C8—1062no—
346C2×C8—1061no—
347C2×C8—2122no—
348C2×C8—2122no—
349C2×C8C2×Dic10611yescharacteristic
350C2×C8C2×Dic10611yescharacteristic
351C2×C8D4×D5311yescharacteristic
352C2×C8D4×D5311yescharacteristic
353C2×C8#1171994711yescharacteristic
354C2×C8#1171994711yescharacteristic
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—1062no—
358C4⋊C4—1061no—
359C4⋊C4—2122no—
360C4⋊C4—2122no—
361C4⋊C4—2122no—
362C22⋊C4—21no—
363C22⋊C4—22no—
364C22⋊C4—1061no—
365C22⋊C4—1062no—
366C42—21no—
367C42—22no—
368C42—531no—
369C42—531no—
370C42—1061no—
371C42—1061no—
372C42—1062no—
373C23—22no—
374C23#1313326211yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—4244no—
378D4—22no—
379D4—22no—
380D4—42no—
381D4—42no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—2122no—
390C2×C4—2121no—
391C2×C4—2122no—
392C2×C4C2×D4×D5311yescharacteristic
393C2×C4C22×Dic10611yescharacteristic
394C2×C4#41×D5311yescharacteristic
395C2×C4#41×D5311yescharacteristic
396C2×C4C2×#1171994711yescharacteristic
397C2×C4#1313326211yescharacteristic
398C8—41no—
399C8—531no—
400C8—531no—
401C8—531no—
402C8—531no—
403C8—2121no—
404C8—2122no—
405C8#1313319411yescharacteristic
406C8#1313319411yescharacteristic
407C8#1313319411yescharacteristic
408C8#1313319411yescharacteristic
409C22—41no—
410C22—42no—
411C22—82no—
412C22#1681118611yescharacteristic
413C22#1681133211yescharacteristic
414C22#1681133711yescharacteristic
415C4—42no—
416C4—81no—
417C4—82no—
418C4—1062no—
419C4—4241no—
420C4—4242no—
421C4—2121no—
422C4C2×#1313319411yescharacteristic
423C4#124×D5311yescharacteristic
424C4#1681133211yescharacteristic
425C4#1681133711yescharacteristic
426C2—21no—
427C2—21no—
428C2—21no—
429C2—82no—
430C2#2027988611yescharacteristic
subgroupG/H
centerC4#16811337
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1976C2
socleC2×C53#2563
prime radicalC2×#135×C53C22
hypercenter#1976D53
hypocenterC53#29154
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
1. #20285227C2semidirect
3. #20275889C2semidirect
4. #20275885C2semidirect
6. #20274839C2semidirect
7. #20274838C2semidirect
8. #20274767C2semidirect
9. #20274759C2semidirect
10. #20274595C2semidirect
84. C2×Q16×C53Q8semidirect
85. C2×SD16×C53Q8semidirect
86. C2×D8×C53Q8semidirect
258. C53#29154semidirect
264. C2×Q16Dic106semidirect
265. C2×SD16Dic106semidirect
266. C2×D8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852272
C2×#168107402
#202758892
#202758852
#202756142
#202748392
#202748382
#202747672
#202747592
#202745952
C53×#19762
#2915453