Finite Group Databasebeta

← #27279961

Subgroups — #27279961

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20283383C211yesmaximal, characteristic
3C2×#16810372C211yesmaximal, characteristic, complemented
4#20273103C211yesmaximal, characteristic
5#20272611C211yesmaximal, characteristic, complemented
6#20272589C211yesmaximal, characteristic, complemented
7C53×#1549C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810915—21no—
10#16810910—21no—
11#16810905C412yes—
12#16810900C412yes—
13#16810683—22no—
14#16810679—21no—
15#16810675—21no—
16#16810663—21no—
17#16810647C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20#16810622—21no—
21#16810591—21no—
22#16810581—21no—
23#16810372—21no—
24#16810372—21no—
25#16810355—21no—
26#16810352—21no—
27C2×C4⋊Q8×C53C2211yescharacteristic
28C53×#431C2211yescharacteristic
29C53×#423C2211yescharacteristic
30C53×#332—22no—
31C2×#11719945D411yescharacteristic
32C2×#11719917—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35C4×Dic106C2×C411yescharacteristic
36C4×Dic106D411yescharacteristic
37C4×D212C2×C411yescharacteristic
38C4×D212D411yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41#13133176D411yescharacteristic
42#13133174—21no—
43C42×D53—21no—
44#13133150—22no—
45#13133148—22no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322C2×C412yes—
51C4⋊#88322D411yescharacteristic
52C4⋊#88322D411yescharacteristic
53#13133140—22no—
54#13133137—22no—
55#13133129—22no—
56#13133129—22no—
57#13133127—22no—
58C8×Dic53—22no—
59C22×Q8×C53—21no—
60C2×M16×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53C2×C412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53D412yes—
71C22⋊C8×C53—21no—
72C4×C8×C53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—82no—
76C22×C4×C53—41no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C2×C22⋊D53—21no—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945—21no—
85C2×D212C22⋊C411yescharacteristic
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×Dic106C22⋊C411yescharacteristic
90M16×C53—41no—
91M16×C53—41no—
92C2×C8×C53—22no—
93C2×C8×C53—41no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C42×C53C22×C411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C22⋊Dic53—21no—
103#11719921—21no—
104#11719921—21no—
105C4⋊Dic53C22⋊C411yescharacteristic
106C4⋊Dic53C22⋊C411yescharacteristic
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112#11719917—41no—
113#11719917—41no—
114C2×#88322—22no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—81no—
121C23×C53#12311yescharacteristic
122C22×D53—21no—
123Q8×C53—82no—
124Q8×C53—82no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—82no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C22⋊D53—41no—
141C22⋊D53—41no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145D212—21no—
146D212—21no—
147C4×D53—41no—
148C4×D53—41no—
149C4×D53—41no—
150C4×D53—41no—
151C4×D53—81no—
152Dic106—21no—
153Dic106—21no—
154C8×C53—42no—
155C8×C53—81no—
156#88322—42no—
157#88322—41no—
158#88322—41no—
159#88322—81no—
160#9170—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53#38611yescharacteristic
164C22×C53#48511yescharacteristic
165C22×C53#48511yescharacteristic
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—82no—
173C4×C53#41611yescharacteristic
174C4×C53#41911yescharacteristic
175C4×C53#45211yescharacteristic
176C4×C53#45211yescharacteristic
177C4×C53#47111yescharacteristic
178C4×C53#47311yescharacteristic
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182#3079—531no—
183#2925—531no—
184#1549D5311yescharacteristic, complemented
185C2×#333—531no—
186#1477—531no—
187#1301—531no—
188#1292—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53#187311yescharacteristic
192C2×C53#190311yescharacteristic
193C2×C53#190811yescharacteristic
194D53—81no—
195C2×C4⋊Q8C2×D5311yescharacteristic
196C4×#41—531no—
197#477—1061no—
198#476—1061no—
199#475—532no—
200#474—532no—
201#433—1062no—
202#432—1061no—
203#431—1061no—
204#431C2×D5311yescharacteristic
205#428—1061no—
206#423—531no—
207#423C2×D5311yescharacteristic
208C2×C4⋊C8—531no—
209C2×C4⋊C8—531no—
210#417—1061no—
211#408—1061no—
212#405—1061no—
213#333—1061no—
214#333—1061no—
215#332—22no—
216Q8⋊C8—1061no—
217D4⋊C8—1061no—
218C53#917011yescharacteristic, complemented, Sylow
219C2×#41—531no—
220C22×Q8—21no—
221C2×M16—21no—
222C2×M16—1061no—
223C22×C8—1061no—
224C22×C8—1061no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8C4×D5312yes—
229C4×Q8—531no—
230C4×Q8—531no—
231C4×D4—531no—
232C4×D4—531no—
233C4×D4—1061no—
234C4×D4—1061no—
235#117—531no—
236#117—1061no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C42—1061no—
240C2×C42C22×D5311yescharacteristic
241#107—1062no—
242#106—1062no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—532no—
246C4⋊C8—531no—
247C4⋊C8—531no—
248C4⋊C8—1061no—
249C4⋊C8—1061no—
250C4⋊C8—1061no—
251C4⋊C8—1061no—
252C4⋊C8C22⋊D5312yes—
253#103—1062no—
254#102—1062no—
255C22⋊C8—21no—
256C22⋊C8—1062no—
257C22⋊C8—1062no—
258#97—1062no—
259C4×C8—21no—
260C4×C8—1062no—
261#41—1061no—
262#41—1061no—
263#41—1061no—
264#41—1061no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—82no—
268C2×Q8—531no—
269C2×D4—531no—
270C2×D4—1061no—
271C22×C4—41no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4#1171994511yescharacteristic
276C22×C4D4×D5311yescharacteristic
277C22×C4#1171994711yescharacteristic
278M16—41no—
279M16—41no—
280M16—2121no—
281M16—2121no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—1062no—
285C2×C8—1061no—
286C2×C8—1061no—
287C2×C8—4241no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8—2121no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C42—1061no—
303C42—1061no—
304C42—1061no—
305C42C2×C4×D5311yescharacteristic
306C42D4×D5311yescharacteristic
307C42D4×D5311yescharacteristic
308C42C2×C22⋊D5311yescharacteristic
309C23—1061no—
310C23#1313319911yescharacteristic
311Q8—82no—
312Q8—82no—
313Q8—1061no—
314Q8—1061no—
315D4—1061no—
316D4—1061no—
317D4—2121no—
318D4—2121no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—4241no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4C22⋊C4×D5311yescharacteristic
333C2×C4#1313317711yescharacteristic
334C2×C4C4×C22⋊D5311yescharacteristic
335C2×C4#1313318511yescharacteristic
336C2×C4#1313318811yescharacteristic
337C2×C4#1313319011yescharacteristic
338C2×C4#1313323811yescharacteristic
339C2×C4#1313324011yescharacteristic
340C2×C4#1313324611yescharacteristic
341C2×C4#1313324911yescharacteristic
342C8—42no—
343C8—81no—
344C8—4241no—
345C8—2121no—
346C8—2121no—
347C8—2122no—
348C22—21no—
349C22—21no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22#1681049911yescharacteristic
354C22#1681096211yescharacteristic
355C22#1681096311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—82no—
360C4—2121no—
361C4—2121no—
362C4—2121no—
363C4#1681061811yescharacteristic
364C4#1681063011yescharacteristic
365C4#1681078211yescharacteristic
366C4#1681078411yescharacteristic
367C4#1681088911yescharacteristic
368C4#1681089711yescharacteristic
369C2—21no—
370C2—21no—
371C2—4241no—
372C2#2027502411yescharacteristic
373C2#2027519211yescharacteristic
374C2#2027522711yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1549C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1549D53
hypocenterC53#9170
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C2×#16810372C2semidirect
5. #20272611C2semidirect
6. #20272589C2semidirect
7. C53×#1549C2semidirect
184. #1549D53semidirect
218. C53#9170semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202833832
C2×#168103722
#202731032
#202726112
#202725892
C53×#15492
#917053