Finite Group Databasebeta

← #27539615

Subgroups — #27539615

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285228C211yesmaximal, characteristic
2#500×D53C211yesmaximal, characteristic
3#20277002C211yesmaximal, characteristic, complemented
4#20277001C211yesmaximal, characteristic, complemented
5#20276998C211yesmaximal, characteristic, complemented
6#20276996C212yesmaximal, complemented
7#20276989C212yesmaximal, complemented
8#20276502C212yesmaximal
9#20276500C212yesmaximal
10#20276308C211yesmaximal, characteristic
11C53×#2255C211yesmaximal, characteristic, complemented
12#16811179—21no—
13C4⋊Q8×D53—21no—
14C2×#13133156C2212yes—
15#16811028C2211yescharacteristic
16#16811027—21no—
17#16811027C2211yescharacteristic
18#16811027C2212yes—
19#16811027C2211yescharacteristic
20#16811026—21no—
21#16811025C2211yescharacteristic
22#16811022—21no—
23#16811020—21no—
24#16810660C2211yescharacteristic
25#16810660C2211yescharacteristic
26#16810658—21no—
27#107×D53—21no—
28#107×D53C2212yes—
29C4×C8×D53C2211yescharacteristic
30#16810451C2212yes—
31#16810450C2212yes—
32#16810450C2212yes—
33#16810449C2212yes—
34#16810448C2212yes—
35#16810448C2212yes—
36#16810399C2211yescharacteristic
37C16×Dic53C2212yes—
38C53×#500C2211yescharacteristic
39C53×#499C2211yescharacteristic
40C53×#367C2212yes—
41C53×#366C2212yes—
42C53×#346C2211yescharacteristic
43C2×Q8×D53—41no—
44C2×C8×D53D411yescharacteristic
45C2×C8×D53C2311yescharacteristic
46#13133217—21no—
47#13133216—21no—
48#13133216—21no—
49#13133216—22no—
50#13133216—41no—
51#13133215—41no—
52#13133209—21no—
53#13133195—41no—
54#13133195—41no—
55#13133182—41no—
56#13133182—41no—
57#13133175—21no—
58#13133175—21no—
59C4⋊C4×D53—22no—
60C4⋊C4×D53—41no—
61C42×D53D411yescharacteristic
62#13133156Q818yescomplemented
63C2×#11719908C2312yes—
64#13133151—21no—
65#13133151C2311yescharacteristic
66#13133151C2311yescharacteristic
67#13133151C2312yes—
68#13133150—21no—
69#13133150—21no—
70#13133150—22no—
71#13133150—22no—
72#13133149—21no—
73#13133148—21no—
74#13133148—21no—
75C4×#88322D411yescharacteristic
76C8×Dic53D411yescharacteristic
77C8×Dic53C2311yescharacteristic
78C4⋊Q8×C53—21no—
79#125×C53—21no—
80C2×C16×C53C2312yes—
81#107×C53—21no—
82#107×C53C2311yescharacteristic
83#107×C53C2311yescharacteristic
84#107×C53C2312yes—
85#106×C53—21no—
86C4×C8×C53C2311yescharacteristic
87C2×Q8×C53—41no—
88Q8×D53—84no—
89C2×C4×D53—42no—
90C2×C4×D53D811yescharacteristic
91C2×C4×D53D811yescharacteristic
92C2×C4×D53C2×D411yescharacteristic
93C2×Dic106—41no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×Dic106—42no—
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2411yescharacteristic
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C42×C53C2×D411yescharacteristic
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—22no—
111C4⋊Dic53—41no—
112#11719919—41no—
113#11719919—41no—
114#11719919—42no—
115#11719919—42no—
116#11719919—41no—
117#11719919—41no—
118C4×Dic53—41no—
119C4×Dic53—41no—
120C4×Dic53D811yescharacteristic
121C4×Dic53D811yescharacteristic
122C4×Dic53C2×D411yescharacteristic
123C2×#88322C2×D411yescharacteristic
124C2×#88322C2×D411yescharacteristic
125C8×D53Q1614yes—
126C8×D53C2×Q814yes—
127C16×C53C2×Q814yescomplemented
128#11719908C2×Q814yescomplemented
129C22×D53C2×D811yescharacteristic
130Q8×C53—82no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53C2×D811yescharacteristic
135C2×C4×C53C2×D811yescharacteristic
136C2×C4×C53C22×D411yescharacteristic
137C2×Dic53—42no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53C2×D811yescharacteristic
141C2×Dic53C2×D811yescharacteristic
142C2×Dic53C2×D811yescharacteristic
143C4×D53—22no—
144C4×D53—22no—
145C4×D53—84no—
146C4×D53C4⋊Q812yes—
147C4×D53C2×Q1612yes—
148Dic106—82no—
149Dic106—82no—
150Dic106—84no—
151Dic106—82no—
152C8×C53C2×Q1612yes—
153C8×C53C22×Q812yes—
154#88322C4⋊Q812yes—
155#88322C2×Q1612yes—
156#33731—531nomaximal, Sylow
157C22×C53C22×D811yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53#50012yes—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—82no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53C2×C4⋊Q811yescharacteristic
167C4×C53C22×Q1611yescharacteristic
168Dic53—82no—
169Dic53—81no—
170Dic53—81no—
171Dic53#50012yes—
172Dic53#50912yes—
173Dic53#51012yes—
174#3160—531no—
175C2×#500—531no—
176#2255—531no—
177#2255—531no—
178#2255—531no—
179#2255D5311yescharacteristic, complemented
180#2254—532no—
181#2252—532no—
182#2159—532no—
183#2159—532no—
184C4×M32—531no—
185C2×C53C2×#50011yescharacteristic
186C2×C53C2×#50911yescharacteristic
187C2×C53C2×#51011yescharacteristic
188D53—22no—
189#533—1061no—
190C2×C4⋊Q8—1061no—
191C2×M32—532no—
192#500—532no—
193#500—531no—
194#500—531no—
195#500—531no—
196#500—1061no—
197#500—1061no—
198#500C2×D5311yescharacteristic
199#499—531no—
200#499—1061no—
201#499C2×D5311yescharacteristic
202#498—1061no—
203#427—531no—
204#427—531no—
205#427—1061no—
206C2×#107—532no—
207C2×#107—1061no—
208C2×C4×C8—531no—
209#367—532no—
210#367—532no—
211#367—532no—
212#367C2×D5312yes—
213#366—532no—
214#366—532no—
215#366—532no—
216#366C2×D5312yes—
217#346—531no—
218#346C2×D5311yescharacteristic
219C4×C16—532no—
220C53#3373111yescharacteristic, complemented, Sylow
221C22×Q8—2121no—
222C22×C8—531no—
223C22×C8—531no—
224C4⋊Q8—21no—
225C4⋊Q8—1062no—
226C4⋊Q8—1061no—
227C4⋊Q8—1061no—
228C4⋊Q8—1061no—
229C4⋊Q8—2121no—
230C4⋊Q8—2121no—
231#125—21no—
232#125—1061no—
233C22⋊Q8—2121no—
234C22⋊Q8—2121no—
235C4×Q8—2121no—
236C4×Q8—2121no—
237#117—1061no—
238#117—1061no—
239C2×C4⋊C4—1062no—
240C2×C4⋊C4—2121no—
241C2×C42—531no—
242M32—538no—
243C2×C16—532no—
244C2×C16C22×D5312yes—
245#107—21no—
246#107—531no—
247#107—532no—
248#107—531no—
249#107—1062no—
250#107—1062no—
251#107—1061no—
252#107—1061no—
253#107—1061no—
254#107C22×D5311yescharacteristic
255#107C22×D5312yes—
256#107C22×D5311yescharacteristic
257#106—21no—
258#106—1061no—
259#106—1061no—
260#106—1061no—
261C4×C8—531no—
262C4×C8—531no—
263C4×C8—531no—
264C4×C8C22×D5311yescharacteristic
265C2×Q8—41no—
266C2×Q8—4244no—
267C2×Q8—2121no—
268C2×Q8—2121no—
269C2×Q8—2122no—
270C2×Q8—2121no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—2122no—
275C2×C8—531no—
276C2×C8—531no—
277C2×C8—534no—
278C2×C8—534no—
279C2×C8D4×D5311yescharacteristic
280C2×C8C23×D5311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—1062no—
290C4⋊C4—2122no—
291C4⋊C4—2122no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C4⋊C4—2121no—
297C22⋊C4—2121no—
298C22⋊C4—2121no—
299C42—531no—
300C42—531no—
301C42—531no—
302C42—2121no—
303C42—2121no—
304C42D4×D5311yescharacteristic
305C16—534no—
306C16Q8×D5314yescomplemented
307C23—531no—
308Q8—82no—
309Q8—4242no—
310Q8—4244no—
311Q8—4242no—
312Q8—4242no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—4244no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2122no—
327C2×C4D8×D5311yescharacteristic
328C2×C4D8×D5311yescharacteristic
329C2×C4C2×D4×D5311yescharacteristic
330C8—532no—
331C8—532no—
332C8Q16×D5312yes—
333C8C2×Q8×D5312yes—
334C22—532no—
335C22—1061no—
336C22—1061no—
337C22C2×D8×D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—82no—
341C4—81no—
342C4—81no—
343C4—532no—
344C4—532no—
345C4—532no—
346C4—4242no—
347C4—4241no—
348C4—4241no—
349C4C4⋊Q8×D5311yescharacteristic
350C4C2×Q16×D5311yescharacteristic
351C2—1062no—
352C2#500×D5311yescharacteristic
353C2#509×D5311yescharacteristic
354C2#510×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2255C2
socleC22×C53C22×D8
prime radicalC22×D53C2×D8
hypercenter#2255D53
hypocenterC53#33731
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #20277002C2semidirect
4. #20277001C2semidirect
5. #20276998C2semidirect
6. #20276996C2semidirect
7. #20276989C2semidirect
11. C53×#2255C2semidirect
62. #13133156Q8semidirect
127. C16×C53C2×Q8semidirect
128. #11719908C2×Q8semidirect
179. #2255D53semidirect
220. C53#33731semidirect
306. C16Q8×D53semidirect

Maximal subgroups

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

subgroupindex
#202852282
#500×D532
#202770022
#202770012
#202769982
#202769962
#202769892
#202765022
#202765002
#202763082
C53×#22552
#3373153