Finite Group Databasebeta

← #27279960

Subgroups — #27279960

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20283280C211yesmaximal, characteristic
3#20273400C211yesmaximal, characteristic, complemented
4#20273100C211yesmaximal, characteristic
5#20272612C211yesmaximal, characteristic, complemented
6#20272590C211yesmaximal, characteristic, complemented
7C2×C53×#333C211yesmaximal, characteristic, complemented
8C2×#13133223—21no—
9C4×#11719945C2211yescharacteristic
10#16810915—21no—
11#16810910—21no—
12#16810905C412yes—
13#16810900C412yes—
14#16810676C2211yescharacteristic
15#16810663—21no—
16#16810649—21no—
17#16810648—22no—
18#16810647C2211yescharacteristic
19#16810647C2211yescharacteristic
20#16810645—21no—
21C4⋊C8×D53—21no—
22#16810622—21no—
23#16810369—22no—
24#16810353—21no—
25#16810350—21no—
26C2×C4⋊Q8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C53×#333—21no—
30C53×#333—21no—
31C2×#11719945D411yescharacteristic
32#13133223—42no—
33#13133223—42no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36C2×#11719912—22no—
37C2×C8×D53—22no—
38C4×Dic106C2×C411yescharacteristic
39C4×Dic106D411yescharacteristic
40C4×D212C2×C411yescharacteristic
41C4×D212D411yescharacteristic
42C4×C22⋊D53—21no—
43C4×C22⋊D53—21no—
44#13133176D411yescharacteristic
45#13133174—21no—
46C42×D53—21no—
47#13133150—22no—
48#13133148—22no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322C2×C412yes—
53C4⋊#88322D412yes—
54#13133146—22no—
55#13133146—22no—
56#13133140—22no—
57#13133137—22no—
58C22⋊#88322—21no—
59C4×#88322—21no—
60C22×Q8×C53—21no—
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53C2×C412yes—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73C4⋊C8×C53D411yescharacteristic
74C4⋊C8×C53D411yescharacteristic
75C2×Q8×C53—22no—
76C2×Q8×C53—41no—
77C2×Q8×C53—82no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C2×C22⋊D53—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945SD1612yes—
86#11719945Q1612yes—
87C2×D212C22⋊C411yescharacteristic
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×Dic106C22⋊C411yescharacteristic
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—81no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C42×C53C22×C411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C22⋊Dic53—21no—
107#11719921—21no—
108#11719921—21no—
109C4⋊Dic53C22⋊C411yescharacteristic
110C4⋊Dic53C22⋊C411yescharacteristic
111#11719919—21no—
112#11719919—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115C4×Dic53—21no—
116#11719917—41no—
117#11719917—41no—
118#11719917—41no—
119#11719917—41no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—41no—
123#11719912—44no—
124C8×D53—44no—
125C23×C53#12311yescharacteristic
126C22×D53—21no—
127Q8×C53—82no—
128Q8×C53—82no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—82no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53C2×SD1611yescharacteristic
141C2×C4×C53C2×Q1611yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C22⋊D53—22no—
145C22⋊D53—22no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149D212—21no—
150D212#10312yes—
151C4×D53—22no—
152C4×D53—22no—
153C4×D53—41no—
154C4×D53—41no—
155C4×D53—81no—
156Dic106—21no—
157Dic106#10312yes—
158C8×C53—41no—
159C8×C53—41no—
160C8×C53—81no—
161#88322—42no—
162#88322—42no—
163#88322—81no—
164#9169—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53#38611yescharacteristic
168C22×C53#48311yescharacteristic
169C22×C53#48411yescharacteristic
170C2×D53—22no—
171C2×D53—41no—
172C2×D53—41no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—82no—
177C4×C53C2×#10311yescharacteristic
178C4×C53#41711yescharacteristic
179C4×C53#45011yescharacteristic
180C4×C53#45111yescharacteristic
181C4×C53#47111yescharacteristic
182C4×C53#47311yescharacteristic
183Dic53—22no—
184Dic53—41no—
185Dic53—41no—
186#3079—531no—
187#2906—531no—
188#1549—531no—
189C2×#333D5311yescharacteristic, complemented
190#1477—531no—
191#1301—531no—
192#1292—531no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53#187211yescharacteristic
196C2×C53#190311yescharacteristic
197C2×C53#190811yescharacteristic
198D53—42no—
199C2×#131—1061no—
200C2×C4⋊Q8C2×D5311yescharacteristic
201C4×#41—531no—
202#477—1061no—
203#476—1061no—
204#475—532no—
205#474—532no—
206#431—531no—
207#428—1061no—
208#424—1062no—
209#424—1061no—
210#423—531no—
211#423—531no—
212#423—1061no—
213C2×C4⋊C8—1061no—
214C2×C4⋊C8C2×D5311yescharacteristic
215C2×C4⋊C8C2×D5311yescharacteristic
216#417—1061no—
217#333—21no—
218#333—21no—
219#332—1062no—
220Q8⋊C8—1061no—
221D4⋊C8—1061no—
222C53#916911yescharacteristic, complemented, Sylow
223C2×#41—531no—
224C22×Q8—21no—
225#131—2122no—
226#131—2122no—
227C2×M16—1062no—
228C2×M16—1061no—
229C2×M16—1061no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—1062no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8C4×D5312yes—
237C4×Q8—531no—
238C4×Q8—531no—
239C4×D4—531no—
240C4×D4—531no—
241C4×D4—1061no—
242C4×D4—1061no—
243#117—531no—
244#117—1061no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C42—1061no—
248C2×C42C22×D5311yescharacteristic
249#107—1062no—
250#106—1062no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—532no—
255C4⋊C8—532no—
256C4⋊C8—1062no—
257C4⋊C8—1062no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260C4⋊C8—1061no—
261C4⋊C8C22⋊D5311yescharacteristic
262C4⋊C8C22⋊D5311yescharacteristic
263#103—1062no—
264#102—1062no—
265C22⋊C8—1061no—
266C4×C8—1061no—
267#41—532no—
268#41—532no—
269#41—1061no—
270#41—1061no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—82no—
274C2×Q8—531no—
275C2×D4—531no—
276C2×D4—1061no—
277C22×C4—41no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4#1171994511yescharacteristic
282C22×C4D4×D5311yescharacteristic
283C22×C4#1171994711yescharacteristic
284M16—2124no—
285M16—2121no—
286M16—2121no—
287M16—2121no—
288M16—2121no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—81no—
295C2×C8—1062no—
296C2×C8—1062no—
297C2×C8—2124no—
298C2×C8—2121no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—531no—
304C4⋊C4—531no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C42—1061no—
311C42—1061no—
312C42—1061no—
313C42C2×C4×D5311yescharacteristic
314C42D4×D5311yescharacteristic
315C42D4×D5311yescharacteristic
316C42C2×C22⋊D5311yescharacteristic
317C23—1061no—
318C23#1313319911yescharacteristic
319Q8—82no—
320Q8—82no—
321Q8—532no—
322Q8—1061no—
323D4—532no—
324D4—1062no—
325D4—1062no—
326D4—1061no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—1061no—
333C2×C4—1062no—
334C2×C4—1062no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—4241no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4C22⋊C4×D5311yescharacteristic
341C2×C4#1313317711yescharacteristic
342C2×C4C4×C22⋊D5311yescharacteristic
343C2×C4#1313318511yescharacteristic
344C2×C4#1313318811yescharacteristic
345C2×C4#1313319011yescharacteristic
346C2×C4SD16×D5311yescharacteristic
347C2×C4Q16×D5311yescharacteristic
348C2×C4#1313324411yescharacteristic
349C2×C4#1313324911yescharacteristic
350C8—41no—
351C8—41no—
352C8—81no—
353C8—4241no—
354C8—2122no—
355C8—2122no—
356C22—21no—
357C22—21no—
358C22—1062no—
359C22—2121no—
360C22—2121no—
361C22#1681049911yescharacteristic
362C22#1681095211yescharacteristic
363C22#1681095711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—82no—
368C4—1062no—
369C4—2121no—
370C4—2121no—
371C4#103×D5311yescharacteristic
372C4#1681062311yescharacteristic
373C4#1681077211yescharacteristic
374C4#1681077711yescharacteristic
375C4#1681088311yescharacteristic
376C4#1681089511yescharacteristic
377C2—21no—
378C2—21no—
379C2—2122no—
380C2#2027501911yescharacteristic
381C2#2027519811yescharacteristic
382C2#2027523311yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#333C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenterC2×#333D53
hypocenterC53#9169
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
3. #20273400C2semidirect
5. #20272612C2semidirect
6. #20272590C2semidirect
7. C2×C53×#333C2semidirect
189. C2×#333D53semidirect
222. C53#9169semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202832802
#202734002
#202731002
#202726122
#202725902
C2×C53×#3332
#916953