Finite Group Databasebeta

← #27537140

Subgroups — #27537140

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811041C211yesmaximal, characteristic
2#20283312C211yesmaximal, characteristic
3#20283311C211yesmaximal, characteristic
4#20276503C211yesmaximal, characteristic, complemented
5#20276502C212yesmaximal, complemented
6#20276500C212yesmaximal, complemented
7#20276497C211yesmaximal, characteristic, complemented
8#20276496C212yesmaximal, complemented
9C2×#16810450C211yesmaximal, characteristic, complemented
10#367×D53C212yesmaximal, complemented
11C53×#2159C211yesmaximal, characteristic, complemented
12C2×#13133223C2211yescharacteristic
13#16811104—21no—
14#16811103—21no—
15#16811041C418yescomplemented
16C2×#13133156C2212yes—
17C22×#11719908C2211yescharacteristic
18C2×C16×D53C2212yes—
19#16810666—21no—
20#16810664—22no—
21#16810663—21no—
22#16810662—22no—
23#16810661C2211yescharacteristic
24#16810660C2212yes—
25#16810660C2212yes—
26C2×#13133151C2211yescharacteristic
27#107×D53C2212yes—
28#107×D53C2212yes—
29#16810451C2212yes—
30#16810450C2211yescharacteristic
31#16810450C2212yes—
32#16810450C2212yes—
33#16810450C2211yescharacteristic
34#16810449C2212yes—
35#16810448C2211yescharacteristic
36#16810448C2211yescharacteristic
37C2×M32×C53C2211yescharacteristic
38C53×#427C2211yescharacteristic
39C2×#107×C53C2211yescharacteristic
40C53×#367C2212yes—
41C53×#366C2212yes—
42C2×#11719945D411yescharacteristic
43#13133223C2×C414yes—
44#13133223Q814yes—
45C2×#11719917D411yescharacteristic
46C2×#11719912D412yes—
47C2×C8×D53C2312yes—
48C4×Dic106—41no—
49#13133182—41no—
50C4×D212—41no—
51#13133179—41no—
52C4×C22⋊D53—42no—
53#13133177—42no—
54#13133176—21no—
55#13133175—22no—
56#13133175—22no—
57C2×C4⋊Dic53—21no—
58C4⋊C4×D53—22no—
59C4⋊C4×D53—22no—
60#13133156C2×C418yescomplemented
61C2×#11719908C2×C414yescomplemented
62C2×#11719908C2311yescharacteristic
63C2×#11719908C2311yescharacteristic
64C16×D53C2×C418yescomplemented
65#13133151—21no—
66#13133151C2311yescharacteristic
67#13133151C2312yes—
68#13133151C2311yescharacteristic
69#13133150—22no—
70#13133150—22no—
71#13133149—21no—
72#13133148—22no—
73#13133148—22no—
74C22×C8×C53C2311yescharacteristic
75#117×C53—21no—
76C2×C4⋊C4×C53—21no—
77M32×C53C2×C414yescomplemented
78C2×C16×C53C2312yes—
79#107×C53—21no—
80#107×C53C2311yescharacteristic
81#107×C53C2312yes—
82#107×C53C2311yescharacteristic
83#106×C53—21no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C2×C22⋊D53D812yes—
87C22×Dic53—41no—
88#11719945C4⋊C414yes—
89#11719945Q1614yes—
90C2×D212D811yescharacteristic
91C2×C4×D53—42no—
92C2×C4×D53—42no—
93C2×C4×D53C2×D412yes—
94C2×Dic106D811yescharacteristic
95C2×C8×C53C22×C412yes—
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×Q812yes—
98C2×C8×C53C2411yescharacteristic
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C42×C53—41no—
105C22⋊Dic53—41no—
106#11719921—42no—
107#11719921—42no—
108C4⋊Dic53—21no—
109C4⋊Dic53—22no—
110C4⋊Dic53—21no—
111C4⋊Dic53—41no—
112#11719919—42no—
113#11719919—42no—
114C4×Dic53—41no—
115C4×Dic53—42no—
116C4×Dic53—42no—
117#11719917C4⋊C414yes—
118C2×#88322C2×D412yes—
119#11719912C4⋊C418yes—
120C8×D53C22×C414yes—
121C8×D53C2×Q814yes—
122C16×C53C22×C414yes—
123#11719908C22×C412yes—
124#11719908C22×C412yes—
125C23×C53C2×D811yescharacteristic
126C22×D53C2×D812yes—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—161no—
131C2×C4×C53C2×C4⋊C412yes—
132C2×C4×C53C2×D811yescharacteristic
133C2×C4×C53C2×Q1612yes—
134C2×C4×C53C22×D411yescharacteristic
135C22⋊D53#10718yes—
136C2×Dic53—41no—
137C2×Dic53—42no—
138C2×Dic53—41no—
139C2×Dic53—161no—
140C2×Dic53C2×D812yes—
141D212#10714yes—
142C4×D53—162no—
143C4×D53—162no—
144C4×D53C2×C4⋊C414yes—
145C4×D53C2×Q1614yes—
146Dic106#10714yes—
147C8×C53C2×C4⋊C412yes—
148C8×C53C23×C411yescharacteristic
149C8×C53C22×Q811yescharacteristic
150#88322C2×C4⋊C414yes—
151#33411—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53C2×#10712yes—
155C22×C53C22×D811yescharacteristic
156C2×D53—22no—
157C2×D53—22no—
158C2×D53C2×#10714yes—
159C4×C53—81no—
160C4×C53—82no—
161C4×C53—81no—
162C4×C53C2×#10712yes—
163C4×C53C22×C4⋊C411yescharacteristic
164C4×C53C22×Q1611yescharacteristic
165Dic53—81no—
166Dic53—82no—
167Dic53—81no—
168Dic53C2×#10714yes—
169C2×#504—531no—
170#2913—531no—
171#2913—531no—
172#2159—531no—
173#2159—532no—
174#2159—532no—
175#2159D5311yescharacteristic, complemented
176#2158—531no—
177#2158—532no—
178C2×#367—531no—
179C2×#367—532no—
180C2×C53—22no—
181C2×C53C22×#10711yescharacteristic
182C2×C53#341611yescharacteristic
183C2×C53#341611yescharacteristic
184D53—24no—
185C2×#131—531no—
186#520—1061no—
187#520—1061no—
188#504—538no—
189C2×M32—532no—
190C2×M32C2×D5311yescharacteristic
191C22×C16—531no—
192C22×C16—532no—
193#428—1062no—
194#428—1062no—
195#428—1061no—
196#428—1061no—
197#427—531no—
198#427—532no—
199#427—532no—
200#427C2×D5311yescharacteristic
201C2×#107—531no—
202C2×#107—532no—
203C2×#107—532no—
204C2×#107C2×D5311yescharacteristic
205#367—532no—
206#367—531no—
207#367—532no—
208#367—532no—
209#367—531no—
210#367C2×D5312yes—
211#366—532no—
212#366—531no—
213#366—531no—
214#366C2×D5312yes—
215C53#3341111yescharacteristic, complemented, Sylow
216C2×#41—531no—
217#131—534no—
218#131—534no—
219C2×M16—531no—
220C2×M16—532no—
221C22×C8—532no—
222C22×C8C22×D5311yescharacteristic
223C4×Q8—2121no—
224C4×Q8—2121no—
225C4×D4—2121no—
226C4×D4—2122no—
227C4×D4—2121no—
228C4×D4—2122no—
229#117—21no—
230#117—1062no—
231#117—1062no—
232#117—1061no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1062no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1061no—
237M32—538no—
238M32C4×D5314yescomplemented
239C2×C16—534no—
240C2×C16—531no—
241C2×C16—531no—
242C2×C16—538no—
243C2×C16C22×D5312yes—
244#107—21no—
245#107—531no—
246#107—531no—
247#107—532no—
248#107—1062no—
249#107—1062no—
250#107—1061no—
251#107C22×D5311yescharacteristic
252#107C22×D5311yescharacteristic
253#107C22×D5312yes—
254#106—21no—
255#106—1062no—
256#106—1062no—
257#106—1061no—
258#41—534no—
259#41—534no—
260C2×Q8—531no—
261C2×D4—532no—
262C2×D4—531no—
263C22×C4—41no—
264C22×C4—532no—
265C22×C4—2122no—
266C22×C4—2122no—
267C22×C4—2121no—
268C22×C4D4×D5311yescharacteristic
269M16—534no—
270M16—538no—
271C2×C8—532no—
272C2×C8—534no—
273C2×C8—534no—
274C2×C8C2×C4×D5312yes—
275C2×C8D4×D5311yescharacteristic
276C2×C8Q8×D5312yes—
277C2×C8C23×D5311yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—1061no—
283C4⋊C4—1062no—
284C4⋊C4—1061no—
285C4⋊C4—2122no—
286C4⋊C4—2122no—
287C4⋊C4—2121no—
288C22⋊C4—41no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C22⋊C4—2121no—
292C42—41no—
293C42—2122no—
294C42—2122no—
295C42—2121no—
296C16—532no—
297C16—532no—
298C16C2×C4×D5314yes—
299C23—532no—
300C23D8×D5311yescharacteristic
301Q8—534no—
302D4—538no—
303D4—534no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—161no—
308C2×C4—532no—
309C2×C4—534no—
310C2×C4—534no—
311C2×C4—8482no—
312C2×C4—8482no—
313C2×C4—8481no—
314C2×C4—2121no—
315C2×C4—2122no—
316C2×C4—2121no—
317C2×C4C4⋊C4×D5312yes—
318C2×C4D8×D5311yescharacteristic
319C2×C4Q16×D5312yes—
320C2×C4C2×D4×D5311yescharacteristic
321C8—534no—
322C8C4⋊C4×D5312yes—
323C8C22×C4×D5311yescharacteristic
324C8C2×Q8×D5311yescharacteristic
325C22—21no—
326C22—21no—
327C22—534no—
328C22—1062no—
329C22—1062no—
330C22#107×D5312yes—
331C22C2×D8×D5311yescharacteristic
332C4—81no—
333C4—82no—
334C4—81no—
335C4—534no—
336C4—4242no—
337C4—4241no—
338C4—4241no—
339C4#107×D5312yes—
340C4C2×C4⋊C4×D5311yescharacteristic
341C4C2×Q16×D5311yescharacteristic
342C2—22no—
343C2—1064no—
344C2C2×#107×D5311yescharacteristic
345C2#2028786011yescharacteristic
346C2#2028786111yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2159C2
socleC22×C53C22×D8
prime radicalC2×#11719945D4
cosocleC2×C8×C53C24
hypercenter#2159D53
hypocenterC53#33411
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #20276503C2semidirect
5. #20276502C2semidirect
6. #20276500C2semidirect
7. #20276497C2semidirect
8. #20276496C2semidirect
9. C2×#16810450C2semidirect
10. #367×D53C2semidirect
11. C53×#2159C2semidirect
15. #16811041C4semidirect
60. #13133156C2×C4semidirect
61. C2×#11719908C2×C4semidirect
64. C16×D53C2×C4semidirect
77. M32×C53C2×C4semidirect
175. #2159D53semidirect
215. C53#33411semidirect
238. M32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110412
#202833122
#202833112
#202765032
#202765022
#202765002
#202764972
#202764962
C2×#168104502
#367×D532
C53×#21592
#3341153