Finite Group Databasebeta

← #27443010

Subgroups — #27443010

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285255C211yesmaximal, characteristic, complemented
2#20284568C212yesmaximal
3#20283312C211yesmaximal, characteristic
4#20276220C212yesmaximal, complemented
5#20276059C212yesmaximal, complemented
6#20275501C211yesmaximal, characteristic, complemented
7#20274942C212yesmaximal
8#20274875C211yesmaximal, characteristic, complemented
9#20274776C211yesmaximal, characteristic
10#20274552C211yesmaximal, characteristic
11C53×#1945C211yesmaximal, characteristic, complemented
12C2×#13133223C2211yescharacteristic
13#16811251C2212yes—
14#16811103C2212yes—
15#16811030—22no—
16#16811027—21no—
17#16811024—21no—
18#16810931—22no—
19#16810924—22no—
20#16810919—22no—
21#16810904—22no—
22#16810666C2212yes—
23#16810666C2212yes—
24#16810665C2212yes—
25#16810662—22no—
26#16810660—21no—
27C2×#13133151C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29C2×#13133151C2212yes—
30#107×D53—21no—
31C2×#13133145C2211yescharacteristic
32#16810625C2212yes—
33#16810620C2211yescharacteristic
34#16810578C2211yescharacteristic
35#16810493C2212yes—
36#16810391C2212yes—
37#16810391C2212yes—
38#16810388C2211yescharacteristic
39#16810383C2211yescharacteristic
40C2×#108×C53C2211yescharacteristic
41C53×#427C2211yescharacteristic
42C53×#424C2212yes—
43C2×C4×C8×C53C2211yescharacteristic
44C53×#339C2212yes—
45C2×#11719945C2311yescharacteristic
46#13133223—22no—
47#13133223Q814yescomplemented
48C2×#11719917C2311yescharacteristic
49C2×#11719917C2311yescharacteristic
50C2×#11719912—21no—
51C2×C8×D53—21no—
52#13133217—22no—
53#13133216—22no—
54#13133209—22no—
55#13133208—22no—
56#13133194—42no—
57C4×Dic106—22no—
58C4×D212—22no—
59#13133177—42no—
60#13133176C2311yescharacteristic
61#13133175—22no—
62C2×C4⋊Dic53—42no—
63C2×C4⋊Dic53C2312yes—
64C2×C4⋊Dic53C2312yes—
65C4⋊C4×D53—22no—
66#13133151—21no—
67#13133151—21no—
68#13133151—22no—
69#13133151—21no—
70#13133151—21no—
71#13133151—22no—
72#13133151—22no—
73#13133150—22no—
74#13133150—22no—
75#13133149—22no—
76#13133148—22no—
77#13133148—22no—
78#13133145Q814yescomplemented
79#13133142—22no—
80#13133142D411yescharacteristic
81#13133142D411yescharacteristic
82#13133139—22no—
83#13133139D411yescharacteristic
84#13133139D411yescharacteristic
85#13133127—21no—
86C8×Dic53—21no—
87C2×M16×C53C2312yes—
88C22×C8×C53C2311yescharacteristic
89C22×C8×C53C2311yescharacteristic
90#117×C53C2312yes—
91C2×C42×C53C2311yescharacteristic
92#108×C53—22no—
93#107×C53D411yescharacteristic
94#107×C53D411yescharacteristic
95#106×C53D412yes—
96C4⋊C8×C53—22no—
97C4⋊C8×C53—22no—
98C4×C8×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53C2411yescharacteristic
101C2×C22⋊D53—21no—
102C22×Dic53—22no—
103C22×Dic53—22no—
104#11719945—21no—
105#11719945—21no—
106#11719945C2×Q812yes—
107#11719945C2×Q812yes—
108C2×D212C2×D411yescharacteristic
109C2×C4×D53—21no—
110C2×C4×D53—42no—
111C2×Dic106—42no—
112C2×Dic106C2×D411yescharacteristic
113M16×C53—22no—
114M16×C53—22no—
115C2×C8×C53—21no—
116C2×C8×C53—21no—
117C2×C8×C53—22no—
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×Q812yes—
123C2×C8×C53#4112yes—
124C4⋊C4×C53C2×D412yes—
125C4⋊C4×C53C2×D412yes—
126C22⋊C4×C53—22no—
127C42×C53—22no—
128C42×C53—21no—
129C42×C53C2×Q812yes—
130C22⋊Dic53—21no—
131#11719921—42no—
132C4⋊Dic53—22no—
133C4⋊Dic53—22no—
134C4⋊Dic53—22no—
135C4⋊Dic53—22no—
136C4⋊Dic53—42no—
137C4⋊Dic53—84no—
138C4⋊Dic53C2×D411yescharacteristic
139C4⋊Dic53C2×D411yescharacteristic
140#11719919—42no—
141#11719919—42no—
142#11719919—42no—
143C4×Dic53—21no—
144C4×Dic53—42no—
145#11719917—21no—
146#11719917—21no—
147#11719917C2×Q812yes—
148#11719917C2×Q812yes—
149C2×#88322—21no—
150C2×#88322—21no—
151#11719912—24no—
152C8×D53—24no—
153C23×C53H(5,𝔽2)11yescharacteristic
154C22×D53—21no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C22×Q811yescharacteristic
163C2×C4×C53C22×Q811yescharacteristic
164C2×C4×C53C2×#4111yescharacteristic
165C2×C4×C532−1+411yescharacteristic
166C22⋊D53—22no—
167C22⋊D53—22no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—82no—
174C2×Dic53—82no—
175D212—21no—
176D212C4⋊Q812yes—
177C4×D53—22no—
178C4×D53—22no—
179C4×D53—162no—
180Dic106—21no—
181Dic106—162no—
182Dic106C4⋊Q812yes—
183C8×C53—21no—
184C8×C53—21no—
185C8×C53—42no—
186C8×C53C22⋊Q812yes—
187C8×C53C22⋊Q812yes—
188#88322—22no—
189#88322—22no—
190#26083—531nomaximal, Sylow
191C22×C53—21no—
192C22×C53—21no—
193C22×C53#54411yescharacteristic
194C22×C53D4211yescharacteristic
195C22×C53#55411yescharacteristic
196C2×D53—22no—
197C2×D53—41no—
198C2×D53—41no—
199C4×C53—24no—
200C4×C53—42no—
201C4×C53—42no—
202C4×C53C2×C22⋊Q811yescharacteristic
203C4×C53C2×C4⋊Q811yescharacteristic
204C4×C53#54711yescharacteristic
205C4×C53D4×Q811yescharacteristic
206Dic53—22no—
207Dic53—41no—
208Dic53—41no—
209Dic53—82no—
210Dic53—82no—
211#3164—531no—
212#3083—532no—
213#2913—531no—
214#2089—532no—
215#2060—532no—
216#1952—531no—
217#1945D5311yescharacteristic, complemented
218#1859—532no—
219#1845—531no—
220#1816—531no—
221#1764—531no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53#244011yescharacteristic
225C2×C53#329211yescharacteristic
226C2×C53#329311yescharacteristic
227D53—42no—
228C2×#131—531no—
229#544—532no—
230#520—532no—
231#501—1062no—
232#500—1061no—
233#499—1061no—
234#479—1062no—
235#478—1062no—
236#477—1062no—
237#474—1062no—
238C2×#108C2×D5311yescharacteristic
239#428—532no—
240#428—532no—
241#428—532no—
242#428—1062no—
243#427—1061no—
244#427C2×D5311yescharacteristic
245C2×#107—532no—
246C2×#107—531no—
247C2×#107—531no—
248C2×#107—1061no—
249#424C2×D5312yes—
250C2×#104—531no—
251#417—532no—
252#416—531no—
253#405—531no—
254C2×C4×C8C2×D5311yescharacteristic
255#384—532no—
256#343—532no—
257#343—532no—
258#341—531no—
259#339—531no—
260#339C2×D5312yes—
261C53#2608311yescharacteristic, complemented, Sylow
262C2×#41—531no—
263#131—534no—
264#131—1062no—
265C2×M16—531no—
266C2×M16—531no—
267C2×M16—1061no—
268C2×M16C22×D5312yes—
269C22×C8—1061no—
270C22×C8C22×D5311yescharacteristic
271C22×C8C22×D5311yescharacteristic
272C4⋊Q8—1062no—
273C4⋊Q8—1062no—
274#125—1062no—
275#125—1062no—
276C22⋊Q8—2122no—
277C4×Q8—1062no—
278C4×D4—1062no—
279C4×D4—2122no—
280#117—531no—
281#117—1062no—
282#117C22×D5312yes—
283C2×C4⋊C4—532no—
284C2×C4⋊C4—532no—
285C2×C4⋊C4—1062no—
286C2×C4⋊C4—2122no—
287C2×C42C22×D5311yescharacteristic
288#108—22no—
289#107—1062no—
290#107—1061no—
291#107—1061no—
292#107—1061no—
293#107—1061no—
294#107—1062no—
295#107—1062no—
296#107—1062no—
297#107—1062no—
298#107D21211yescharacteristic
299#107D21211yescharacteristic
300#106—1062no—
301#106—1062no—
302#106—1062no—
303#106D21212yes—
304C4⋊C8—22no—
305C4⋊C8—22no—
306#104—534no—
307#103—531no—
308#103—531no—
309#103—1062no—
310#102—531no—
311#102—531no—
312#102—1062no—
313#97—1061no—
314C4×C8—22no—
315C4×C8—1061no—
316#41—532no—
317#41—532no—
318#41—1061no—
319#41—1061no—
320C2×Q8—531no—
321C2×Q8—2122no—
322C2×D4—531no—
323C2×D4—1061no—
324C22×C4—21no—
325C22×C4—1061no—
326C22×C4—1062no—
327C22×C4—1062no—
328C22×C4—2122no—
329C22×C4C23×D5311yescharacteristic
330M16—22no—
331M16—22no—
332M16—532no—
333M16—532no—
334M16—1061no—
335M16—1061no—
336M16—1064no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—1061no—
341C2×C8—1061no—
342C2×C8—1064no—
343C2×C8C2×D21211yescharacteristic
344C2×C8C2×D21211yescharacteristic
345C2×C8D4×D5311yescharacteristic
346C2×C8D4×D5311yescharacteristic
347C2×C8#1171994712yes—
348C2×C8Q8×D5312yes—
349C4⋊C4—531no—
350C4⋊C4—531no—
351C4⋊C4—1062no—
352C4⋊C4—1062no—
353C4⋊C4—1062no—
354C4⋊C4—1062no—
355C4⋊C4—4244no—
356C4⋊C4—2122no—
357C4⋊C4—2122no—
358C4⋊C4—2122no—
359C4⋊C4—2122no—
360C4⋊C4C2×D21212yes—
361C4⋊C4C2×D21212yes—
362C22⋊C4—22no—
363C22⋊C4—1061no—
364C22⋊C4—2122no—
365C42—22no—
366C42—21no—
367C42—1061no—
368C42—2122no—
369C42Q8×D5312yes—
370C23—1061no—
371C23#1313326311yescharacteristic
372Q8—532no—
373Q8—8482no—
374Q8—1061no—
375D4—532no—
376D4—1062no—
377D4—1062no—
378D4—1061no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—8482no—
385C2×C4—1061no—
386C2×C4—1062no—
387C2×C4—1062no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—4242no—
391C2×C4—4242no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4C2×D4×D5311yescharacteristic
395C2×C4C22×D21211yescharacteristic
396C2×C4C2×Q8×D5311yescharacteristic
397C2×C4C2×Q8×D5311yescharacteristic
398C2×C4C2×#1171994711yescharacteristic
399C2×C4#1313326411yescharacteristic
400C8—21no—
401C8—21no—
402C8—42no—
403C8—1062no—
404C8—1062no—
405C8#1313319712yes—
406C8#1313319712yes—
407C22—21no—
408C22—21no—
409C22—1062no—
410C22—2121no—
411C22—2121no—
412C22#1681125311yescharacteristic
413C22D4×D21211yescharacteristic
414C22#1681133111yescharacteristic
415C4—24no—
416C4—42no—
417C4—42no—
418C4—1062no—
419C4—4242no—
420C4—4242no—
421C4—2121no—
422C4—2121no—
423C4C2×#1313319711yescharacteristic
424C4C4⋊Q8×D5311yescharacteristic
425C4#1681127311yescharacteristic
426C4Q8×D21211yescharacteristic
427C2—21no—
428C2—21no—
429C2—2122no—
430C2#2027841511yescharacteristic
431C2#2028651511yescharacteristic
432C2#2028652011yescharacteristic
subgroupG/H
centerC22D4×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1945C2
socleC22×C53D42
prime radicalC2×#11719945C23
hypercenter#1945D53
hypocenterC53#26083
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1945

Complements

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

NH ≅ G/Ntype
1. #20285255C2semidirect
4. #20276220C2semidirect
5. #20276059C2semidirect
6. #20275501C2semidirect
8. #20274875C2semidirect
11. C53×#1945C2semidirect
47. #13133223Q8semidirect
78. #13133145Q8semidirect
217. #1945D53semidirect
261. C53#26083semidirect

Maximal subgroups

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

subgroupindex
#202852552
#202845682
#202833122
#202762202
#202760592
#202755012
#202749422
#202748752
#202747762
#202745522
C53×#19452
#2608353