Finite Group Databasebeta

← #27442997

Subgroups — #27442997

Order 13568.

674 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#20284573C212yesmaximal
2#20283312C211yesmaximal, characteristic
3#20276228C212yesmaximal, complemented
4#20276062C212yesmaximal, complemented
5#20275500C211yesmaximal, characteristic, complemented
6#20275464C211yesmaximal, characteristic, complemented
7#20274945C212yesmaximal
8#20274878C211yesmaximal, characteristic, complemented
9#20274780C211yesmaximal, characteristic
10#20274554C211yesmaximal, characteristic
11C53×#3164C211yesmaximal, characteristic, complemented
12C2×#13133223C2211yescharacteristic
13#16811103C2212yes—
14#16810927—22no—
15#16810920—22no—
16#16810910—22no—
17#16810905—22no—
18C2×#13133153C2211yescharacteristic
19#16810666C2212yes—
20#16810666C2212yes—
21#16810662—22no—
22#16810660—21no—
23C2×#13133151C2211yescharacteristic
24#107×D53—21no—
25C2×#13133148C2212yes—
26#16810650C2212yes—
27C2×#13133143C2211yescharacteristic
28#16810622C2212yes—
29#16810617C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810488C2212yes—
32#16810392C2212yes—
33#16810391C2212yes—
34#16810387C2211yescharacteristic
35#16810382C2212yes—
36#16810382C2211yescharacteristic
37C53×#544C2212yes—
38C53×#501—22no—
39C53×#500—21no—
40C53×#499—21no—
41C53×#428C2212yes—
42C53×#427C2211yescharacteristic
43C2×#107×C53C2211yescharacteristic
44C53×#405C2211yescharacteristic
45C2×#11719945C2311yescharacteristic
46#13133223—22no—
47#13133223Q814yescomplemented
48C2×#11719917C2312yes—
49C2×#11719917C2311yescharacteristic
50C2×#11719912—21no—
51C22×#88322C2311yescharacteristic
52C2×C8×D53—21no—
53C4×Dic106—22no—
54C4×D212—22no—
55#13133177—42no—
56#13133175—22no—
57C2×C4⋊Dic53C2312yes—
58C4⋊C4×D53—22no—
59C2×C4×Dic53C2311yescharacteristic
60#13133153—22no—
61#13133151—21no—
62#13133151—21no—
63#13133150—22no—
64#13133150—22no—
65#13133148—22no—
66#13133148—22no—
67#13133148—22no—
68#13133148—22no—
69C4⋊#88322—22no—
70C4⋊#88322—22no—
71#13133143Q814yescomplemented
72#13133140—22no—
73#13133140D411yescharacteristic
74#13133140D411yescharacteristic
75#13133137—22no—
76#13133137D411yescharacteristic
77#13133137D411yescharacteristic
78C8×Dic53—22no—
79#13133124—42no—
80C2×M16×C53C2311yescharacteristic
81C22×C8×C53C2311yescharacteristic
82C4⋊Q8×C53—22no—
83C4⋊Q8×C53—22no—
84#125×C53—22no—
85#125×C53—22no—
86C22⋊Q8×C53—42no—
87#117×C53C2311yescharacteristic
88#117×C53C2312yes—
89C2×C4⋊C4×C53C2312yes—
90#107×C53—21no—
91#107×C53—21no—
92#107×C53—22no—
93#107×C53D411yescharacteristic
94#107×C53D411yescharacteristic
95#106×C53—22no—
96#106×C53D412yes—
97#97×C53—21no—
98C4×C8×C53—21no—
99C2×Q8×C53—42no—
100C22×C4×C53—22no—
101C22×C4×C53C2411yescharacteristic
102C2×C22⋊D53—21no—
103C22×Dic53—21no—
104C22×Dic53—22no—
105#11719945—21no—
106#11719945—21no—
107#11719945C2×Q812yes—
108#11719945C2×Q812yes—
109C2×D212C2×D411yescharacteristic
110C2×C4×D53—21no—
111C2×C4×D53—42no—
112C2×Dic106C2×D411yescharacteristic
113M16×C53—21no—
114M16×C53C2×Q812yes—
115C2×C8×C53—21no—
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×Q812yes—
119C2×C8×C53#4112yes—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—42no—
123C4⋊C4×C53—42no—
124C4⋊C4×C53—42no—
125C4⋊C4×C53C2×D411yescharacteristic
126C4⋊C4×C53C2×D411yescharacteristic
127C4⋊C4×C53C2×D412yes—
128C4⋊C4×C53C2×D412yes—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—22no—
131C42×C53—21no—
132C42×C53—22no—
133#11719921—42no—
134C4⋊Dic53—22no—
135C4⋊Dic53—22no—
136#11719919—42no—
137C4×Dic53—21no—
138C4×Dic53—42no—
139C4×Dic53C2×Q812yes—
140#11719917—21no—
141#11719917—22no—
142#11719917—22no—
143#11719917C2×Q812yes—
144C2×#88322—21no—
145C2×#88322—21no—
146C2×#88322—21no—
147C2×#88322—22no—
148C2×#88322C2×D411yescharacteristic
149C2×#88322C2×D411yescharacteristic
150#11719912—24no—
151C8×D53—24no—
152C23×C53H(5,𝔽2)11yescharacteristic
153C22×D53—21no—
154Q8×C53—162no—
155C2×C4×C53—22no—
156C2×C4×C53—21no—
157C2×C4×C53—22no—
158C2×C4×C53—21no—
159C2×C4×C53—42no—
160C2×C4×C53—82no—
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C22×D411yescharacteristic
163C2×C4×C53C22×Q811yescharacteristic
164C2×C4×C53C22×Q811yescharacteristic
165C2×C4×C53C2×#4111yescharacteristic
166C2×C4×C532−1+411yescharacteristic
167C22⋊D53—22no—
168C22⋊D53—22no—
169C2×Dic53—21no—
170C2×Dic53—22no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—42no—
174C2×Dic53—82no—
175D212—21no—
176D212C4⋊Q812yes—
177C4×D53—22no—
178C4×D53—22no—
179C4×D53—162no—
180Dic106—21no—
181Dic106C4⋊Q812yes—
182C8×C53—22no—
183C8×C53C22⋊Q812yes—
184C8×C53C22⋊Q812yes—
185#88322—22no—
186#88322—21no—
187#88322—21no—
188#88322—42no—
189#26082—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53#54411yescharacteristic
193C22×C53D4211yescharacteristic
194C22×C53#55411yescharacteristic
195C2×D53—22no—
196C2×D53—41no—
197C2×D53—41no—
198C4×C53—42no—
199C4×C53—41no—
200C4×C53—42no—
201C4×C53—41no—
202C4×C53—82no—
203C4×C53C2×C22⋊Q811yescharacteristic
204C4×C53C2×C4⋊Q811yescharacteristic
205C4×C53#54711yescharacteristic
206C4×C53D4×Q811yescharacteristic
207Dic53—22no—
208Dic53—42no—
209Dic53—82no—
210#3164D5311yescharacteristic, complemented
211#3084—532no—
212#2913—531no—
213#2090—532no—
214#2061—532no—
215#1952—531no—
216#1945—531no—
217#1860—532no—
218#1846—531no—
219#1816—531no—
220#1764—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53#244011yescharacteristic
224C2×C53#329011yescharacteristic
225C2×C53#329011yescharacteristic
226D53—42no—
227C2×#131—531no—
228#544C2×D5312yes—
229#520—532no—
230#501—22no—
231#500—21no—
232#499—21no—
233#479—1062no—
234#478—1062no—
235#476—1062no—
236#475—1062no—
237C2×#108—531no—
238#428—532no—
239#428—532no—
240#428—1062no—
241#428C2×D5312yes—
242#427—1061no—
243#427C2×D5311yescharacteristic
244C2×#107—531no—
245C2×#107—1061no—
246C2×#107C2×D5311yescharacteristic
247C2×#106—532no—
248#424—532no—
249C2×#104—531no—
250#417—532no—
251#416—531no—
252#405C2×D5311yescharacteristic
253C2×C4×C8—531no—
254#382—532no—
255#343—532no—
256#343—532no—
257#341—531no—
258#339—531no—
259#339—532no—
260C53#2608211yescharacteristic, complemented, Sylow
261C2×#41—531no—
262#131—534no—
263#131—1062no—
264C2×M16—531no—
265C2×M16—532no—
266C2×M16—1061no—
267C2×M16C22×D5311yescharacteristic
268C22×C8—531no—
269C22×C8—1061no—
270C22×C8C22×D5311yescharacteristic
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273#125—22no—
274#125—22no—
275C22⋊Q8—42no—
276C4×Q8—1062no—
277C4×D4—1062no—
278C4×D4—2122no—
279#117—1062no—
280#117C22×D5311yescharacteristic
281#117C22×D5312yes—
282C2×C4⋊C4—532no—
283C2×C4⋊C4—1062no—
284C2×C4⋊C4C22×D5312yes—
285C2×C42—531no—
286#108—1062no—
287#107—21no—
288#107—21no—
289#107—22no—
290#107—1061no—
291#107—1061no—
292#107—1062no—
293#107—1062no—
294#107C22⋊D5311yescharacteristic
295#107C22⋊D5311yescharacteristic
296#106—22no—
297#106—1062no—
298#106—1062no—
299#106—1062no—
300#106—1062no—
301#106C22⋊D5312yes—
302C4⋊C8—1062no—
303C4⋊C8—1062no—
304#104—534no—
305#103—531no—
306#103—531no—
307#103—1062no—
308#102—531no—
309#102—531no—
310#102—1062no—
311#97—21no—
312C4×C8—21no—
313C4×C8—1062no—
314#95—2122no—
315#41—532no—
316#41—532no—
317#41—1061no—
318#41—1061no—
319C2×Q8—42no—
320C2×Q8—531no—
321C2×D4—531no—
322C2×D4—1061no—
323C22×C4—22no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4—1062no—
327C22×C4—2122no—
328C22×C4C23×D5311yescharacteristic
329M16—21no—
330M16—532no—
331M16—1061no—
332M16—1064no—
333M16—1062no—
334M16—1062no—
335M16Q8×D5312yes—
336C2×C8—21no—
337C2×C8—531no—
338C2×C8—531no—
339C2×C8—1061no—
340C2×C8—1061no—
341C2×C8—1061no—
342C2×C8—1064no—
343C2×C8—1062no—
344C2×C8Q8×D5312yes—
345C2×C8#1171994912yes—
346C2×C8C2×C22⋊D5311yescharacteristic
347C2×C8C2×C22⋊D5311yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—1062no—
354C4⋊C4—1062no—
355C4⋊C4—2122no—
356C4⋊C4D4×D5311yescharacteristic
357C4⋊C4D4×D5311yescharacteristic
358C4⋊C4C2×C22⋊D5312yes—
359C4⋊C4C2×C22⋊D5312yes—
360C22⋊C4—22no—
361C22⋊C4—21no—
362C22⋊C4—2122no—
363C42—22no—
364C42—21no—
365C42—532no—
366C42—1061no—
367C42—2122no—
368C23—1061no—
369C23#1313326211yescharacteristic
370Q8—162no—
371Q8—532no—
372Q8—1061no—
373D4—532no—
374D4—1062no—
375D4—1062no—
376D4—1061no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—8482no—
384C2×C4—1061no—
385C2×C4—1062no—
386C2×C4—1062no—
387C2×C4—1062no—
388C2×C4—4242no—
389C2×C4—2122no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4C2×D4×D5311yescharacteristic
393C2×C4C22×C22⋊D5311yescharacteristic
394C2×C4C2×Q8×D5311yescharacteristic
395C2×C4C2×Q8×D5311yescharacteristic
396C2×C4C2×#1171994911yescharacteristic
397C2×C4#1313326511yescharacteristic
398C8—22no—
399C8—1061no—
400C8—1061no—
401C8—1062no—
402C8—2122no—
403C8#1313319312yes—
404C8#1313319312yes—
405C22—21no—
406C22—21no—
407C22—1062no—
408C22—2121no—
409C22—2121no—
410C22#1681125011yescharacteristic
411C22D4×C22⋊D5311yescharacteristic
412C22#1681133411yescharacteristic
413C4—42no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—82no—
418C4—1062no—
419C4—4242no—
420C4—2122no—
421C4C2×#1313319311yescharacteristic
422C4C4⋊Q8×D5311yescharacteristic
423C4#1681127611yescharacteristic
424C4Q8×C22⋊D5311yescharacteristic
425C2—21no—
426C2—21no—
427C2—2122no—
428C2#2027841311yescharacteristic
429C2#2028649711yescharacteristic
430C2#2028650111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3164C2
socleC22×C53D42
prime radicalC2×#11719945C23
hypercenter#3164D53
hypocenterC53#26082
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3164

Complements

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

NH ≅ G/Ntype
3. #20276228C2semidirect
4. #20276062C2semidirect
5. #20275500C2semidirect
6. #20275464C2semidirect
8. #20274878C2semidirect
11. C53×#3164C2semidirect
47. #13133223Q8semidirect
71. #13133143Q8semidirect
210. #3164D53semidirect
260. C53#26082semidirect

Maximal subgroups

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

subgroupindex
#202845732
#202833122
#202762282
#202760622
#202755002
#202754642
#202749452
#202748782
#202747802
#202745542
C53×#31642
#2608253