Finite Group Databasebeta

← #27368008

Subgroups — #27368008

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20284198C211yesmaximal, characteristic
3C8×#11719945C211yesmaximal, characteristic
4#20274094C214yesmaximal, complemented
5#20274087C214yesmaximal, complemented
6#20273471C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273387C211yesmaximal, characteristic, complemented
9C2×C53×#498C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11C4×#11719945C2211yescharacteristic
12#16810915C2212yes—
13#16810910C2212yes—
14#16810905—21no—
15#16810900—21no—
16#16810830—21no—
17#16810830C2212yes—
18#16810825—21no—
19#16810825C2212yes—
20C8×Dic106C2212yes—
21C8×D212C2212yes—
22C8×C22⋊D53—22no—
23#16810676C2211yescharacteristic
24#16810674—21no—
25#16810663—21no—
26#16810647C2211yescharacteristic
27#16810647C2211yescharacteristic
28#16810632—21no—
29#16810622—21no—
30C4×C8×D53—21no—
31#16810374C2214yes—
32#16810372C2214yes—
33#16810369C2214yes—
34C2×C4⋊Q8×C53C2211yescharacteristic
35C2×C4⋊Q8×C53C2211yescharacteristic
36C53×#498—21no—
37C53×#498—21no—
38C53×#498C2214yes—
39C2×#106×C53—22no—
40C2×C4×C8×C53C2211yescharacteristic
41C2×#11719945D411yescharacteristic
42#13133248—44no—
43C2×#11719925—22no—
44C2×#11719924—22no—
45C2×#11719917—21no—
46C2×#11719917—21no—
47C2×C8×D53—22no—
48C4×Dic106D411yescharacteristic
49C4×Dic106C2311yescharacteristic
50C4×D212D411yescharacteristic
51C4×D212C2311yescharacteristic
52C4×C22⋊D53—21no—
53C4×C22⋊D53—21no—
54#13133176D411yescharacteristic
55#13133174—21no—
56C42×D53—21no—
57#13133150—22no—
58#13133148—22no—
59C4⋊#88322—21no—
60C4⋊#88322—21no—
61C4⋊#88322C2312yes—
62C4⋊#88322C2312yes—
63C4⋊#88322C2312yes—
64#13133146—22no—
65#13133141—22no—
66#13133141—22no—
67#13133140—22no—
68#13133137—22no—
69C22⋊#88322—21no—
70#13133129—22no—
71C4×#88322—21no—
72C8×Dic53—22no—
73C22×Q8×C53—21no—
74C22×Q8×C53—21no—
75C22×C8×C53D412yes—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53—21no—
79C4⋊Q8×C53—21no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53C2312yes—
83C4⋊Q8×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—21no—
88C2×C42×C53C2311yescharacteristic
89#106×C53—24no—
90#106×C53—24no—
91C4×C8×C53D411yescharacteristic
92C4×C8×C53D411yescharacteristic
93C4×C8×C53C2312yes—
94C2×Q8×C53—22no—
95C2×Q8×C53—22no—
96C2×Q8×C53—41no—
97C2×Q8×C53—41no—
98C2×Q8×C53—42no—
99C2×Q8×C53—42no—
100C2×Q8×C53—82no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C2×C22⋊D53—21no—
107#11719945—22no—
108#11719945SD1614yes—
109C2×D212C2×D411yescharacteristic
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×Dic106C2×D411yescharacteristic
114C2×C8×C53—24no—
115C2×C8×C53C2×D412yes—
116C2×C8×C53C2×D412yes—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C42×C53C2411yescharacteristic
129C22⋊Dic53—21no—
130#11719925—44no—
131#11719924—44no—
132#11719921—21no—
133#11719921—21no—
134C4⋊Dic53C2×D411yescharacteristic
135C4⋊Dic53C2×D411yescharacteristic
136#11719919—21no—
137#11719919—21no—
138C4×Dic53—21no—
139C4×Dic53—21no—
140C4×Dic53—21no—
141#11719917—42no—
142#11719917—42no—
143C2×#88322—22no—
144C2×#88322—22no—
145C2×#88322—22no—
146C8×D53—44no—
147C23×C53#12111yescharacteristic
148C22×D53—21no—
149Q8×C53—42no—
150Q8×C53—42no—
151Q8×C53—82no—
152Q8×C53—82no—
153Q8×C53—82no—
154C2×C4×C53—22no—
155C2×C4×C53—22no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—82no—
159C2×C4×C53—82no—
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53C2×SD1612yes—
164C2×C4×C53#13512yes—
165C2×C4×C53C22×D411yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C2×#4111yescharacteristic
168C22⋊D53—24no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172D212—21no—
173D212C2×SD1612yes—
174C4×D53—24no—
175C4×D53—42no—
176C4×D53—42no—
177Dic106—21no—
178Dic106C2×SD1612yes—
179C8×C53—22no—
180C8×C53#13714yescomplemented
181#88322—42no—
182#88322—42no—
183#88322—42no—
184#19270—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53#46512yes—
188C22×C53C2×#12111yescharacteristic
189C2×D53—22no—
190C2×D53—41no—
191C2×D53—41no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—82no—
195C4×C53—82no—
196C4×C53#46512yes—
197C4×C53#47112yes—
198C4×C53C22×SD1611yescharacteristic
199C4×C53C2×#13511yescharacteristic
200C4×C53C2×#13712yes—
201Dic53—22no—
202Dic53—41no—
203Dic53—41no—
204C2×#498D5311yescharacteristic, complemented
205#3079—531no—
206#3039—531no—
207C8×#41—531no—
208#1672—534no—
209#1671—534no—
210#1566—531no—
211#1548—531no—
212#1546—531no—
213C2×C53—22no—
214C2×C53#303911yescharacteristic
215C2×C53C2×#46511yescharacteristic
216C2×C53C2×#47111yescharacteristic
217D53—42no—
218C2×#137—1061no—
219C2×C4⋊Q8C2×D5311yescharacteristic
220C2×C4⋊Q8C2×D5311yescharacteristic
221C4×#41—531no—
222#498—21no—
223#498—21no—
224#498C2×D5314yes—
225#477—532no—
226#476—532no—
227#475—1061no—
228#474—1061no—
229#462—532no—
230#462—1061no—
231#461—532no—
232#461—1061no—
233C8×Q8—532no—
234C8×D4—532no—
235C8×D4—1062no—
236#431—531no—
237#431—1061no—
238#428—1061no—
239C2×#106—22no—
240#423—531no—
241#423—531no—
242#419—1061no—
243#417—1061no—
244C2×C4×C8—1061no—
245C2×C4×C8C2×D5311yescharacteristic
246#334—534no—
247#333—534no—
248#332—534no—
249C53#1927011yescharacteristic, complemented, Sylow
250C2×#41—531no—
251C22×Q8—21no—
252C22×Q8—21no—
253#137—2124no—
254C2×Q16—1062no—
255C2×SD16—1062no—
256C2×M16—1061no—
257C2×M16—1061no—
258C22×C8—1062no—
259C22×C8C22⋊D5312yes—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8C22×D5312yes—
267C4⋊Q8C22×D5312yes—
268C4×Q8—531no—
269C4×Q8—531no—
270C4×D4—531no—
271C4×D4—531no—
272C4×D4—1061no—
273C4×D4—1061no—
274#117—531no—
275#117—1061no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C42—1061no—
281C2×C42C22×D5311yescharacteristic
282#107—1062no—
283#106—24no—
284#106—24no—
285#106—1062no—
286C4⋊C8—532no—
287C4⋊C8—532no—
288C4⋊C8—532no—
289C4⋊C8—1062no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292#103—1062no—
293#103—1062no—
294#103—1062no—
295#102—1062no—
296C22⋊C8—1061no—
297C22⋊C8—1062no—
298C4×C8—1062no—
299C4×C8—1061no—
300C4×C8C22⋊D5311yescharacteristic
301C4×C8C22⋊D5311yescharacteristic
302C4×C8C22×D5312yes—
303#41—534no—
304#41—1062no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—82no—
312C2×Q8—531no—
313C2×D4—531no—
314C2×D4—1061no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4D4×D5311yescharacteristic
321C22×C4#1171994711yescharacteristic
322C22×C4C2×C22⋊D5311yescharacteristic
323Q16—2124no—
324SD16—2124no—
325M16—2122no—
326M16—2122no—
327C2×C8—24no—
328C2×C8—1062no—
329C2×C8—1062no—
330C2×C8—1062no—
331C2×C8—2124no—
332C2×C8C2×C22⋊D5312yes—
333C2×C8C2×C22⋊D5312yes—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—531no—
343C4⋊C4—531no—
344C4⋊C4—1061no—
345C4⋊C4—1061no—
346C22⋊C4—1061no—
347C22⋊C4—1061no—
348C22⋊C4—1061no—
349C42—1061no—
350C42—1061no—
351C42—1061no—
352C42D4×D5311yescharacteristic
353C42#1171994711yescharacteristic
354C42C2×C22⋊D5311yescharacteristic
355C42C23×D5311yescharacteristic
356C23—1061no—
357C23#1313318811yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—82no—
361Q8—82no—
362Q8—82no—
363Q8—532no—
364Q8—1061no—
365D4—532no—
366D4—1064no—
367D4—1061no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—1061no—
375C2×C4—1064no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4#1313318811yescharacteristic
381C2×C4#1313318811yescharacteristic
382C2×C4#1313318811yescharacteristic
383C2×C4SD16×D5312yes—
384C2×C4#1313323812yes—
385C2×C4C2×D4×D5311yescharacteristic
386C2×C4C22×C22⋊D5311yescharacteristic
387C2×C4C2×#1171994711yescharacteristic
388C8—22no—
389C8—2122no—
390C8—2122no—
391C8—2122no—
392C8#1313324814yescomplemented
393C22—21no—
394C22—21no—
395C22—1062no—
396C22—2121no—
397C22—2121no—
398C22#1681084912yes—
399C22C2×#1313318811yescharacteristic
400C4—21no—
401C4—21no—
402C4—82no—
403C4—82no—
404C4—1062no—
405C4—2121no—
406C4—2121no—
407C4#1681084912yes—
408C4#1681088312yes—
409C4C2×SD16×D5311yescharacteristic
410C4C2×#1313323811yescharacteristic
411C4C2×#1313324812yes—
412C2—22no—
413C2—2122no—
414C2#2028419811yescharacteristic
415C2C2×#1681084911yescharacteristic
416C2C2×#1681088311yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#498C2
socleC22×C53C2×#121
prime radicalC2×#11719945D4
hypercenterC2×#498D53
hypocenterC53#19270
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
4. #20274094C2semidirect
5. #20274087C2semidirect
6. #20273471C2semidirect
7. #20273396C2semidirect
8. #20273387C2semidirect
9. C2×C53×#498C2semidirect
180. C8×C53#137semidirect
204. C2×#498D53semidirect
249. C53#19270semidirect
392. C8#13133248semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202841982
C8×#117199452
#202740942
#202740872
#202734712
#202733962
#202733872
C2×C53×#4982
#1927053