Finite Group Databasebeta

← #27538963

Subgroups — #27538963

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283315C211yesmaximal, characteristic, complemented
2#20276920C212yesmaximal, complemented
3#20276910C212yesmaximal, complemented
4#20276903C212yesmaximal, complemented
5#20276898C212yesmaximal, complemented
6#20276501C211yesmaximal, characteristic, complemented
7C2×#16810450C211yesmaximal, characteristic, complemented
8#20276427C211yesmaximal, characteristic, complemented
9#20276422C211yesmaximal, characteristic, complemented
10#20276329C211yesmaximal, characteristic, complemented
11C53×#3061C211yesmaximal, characteristic
12#16811106—21no—
13C2×#13133157C2211yescharacteristic
14C22×#11719908C2211yescharacteristic
15#16810665—21no—
16#16810665—22no—
17#16810661C2211yescharacteristic
18#16810661C2212yes—
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2212yes—
21#16810450C2212yes—
22#16810450C2212yes—
23#16810450C2212yes—
24#16810448C2212yes—
25#16810427C2211yescharacteristic
26#16810427C2212yes—
27#16810427C2211yescharacteristic
28#16810424C2211yescharacteristic
29#16810424C2212yes—
30#16810424C2211yescharacteristic
31C22⋊#11719908C2212yes—
32C22⋊#11719908C2212yes—
33C2×#135×C53C2211yescharacteristic
34C2×#131×C53C2211yescharacteristic
35C53×#537—21no—
36C53×#468—22no—
37C53×#467C2212yes—
38C53×#466C2212yes—
39C2×#107×C53C2211yescharacteristic
40C53×#417—21no—
41Q8×Dic53—41no—
42D4×Dic53—41no—
43D4×Dic53—42no—
44#13133176—22no—
45#13133176—21no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—21no—
48#13133157—22no—
49C2×#11719908—22no—
50C2×#11719908C2311yescharacteristic
51C2×#11719908C2311yescharacteristic
52C2×#11719908C2312yes—
53#13133151—22no—
54#13133151—21no—
55#13133151C2312yes—
56#13133151C2311yescharacteristic
57#13133151C2311yescharacteristic
58#13133149—22no—
59#13133149—21no—
60C2×#41×C53—21no—
61C2×#41×C53D411yescharacteristic
62#135×C53—22no—
63#135×C53—22no—
64C2×Q16×C53C2311yescharacteristic
65C2×SD16×C53—21no—
66C2×D8×C53C2311yescharacteristic
67#131×C53—22no—
68#131×C53—22no—
69C2×M16×C53D411yescharacteristic
70C2×M16×C53D412yes—
71C22×C8×C53C2312yes—
72C22×C8×C53C2311yescharacteristic
73C22⋊Q8×C53—22no—
74C22⋊Q8×C53—41no—
75#121×C53—22no—
76#121×C53—42no—
77#121×C53—41no—
78C2×C4⋊C4×C53—21no—
79#107×C53—21no—
80#107×C53C2312yes—
81#103×C53—22no—
82#102×C53—22no—
83#41×C53—22no—
84#41×C53—22no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×Q8×C53D811yescharacteristic
89C2×D4×C53—21no—
90C2×D4×C53—41no—
91C2×D4×C53—42no—
92C2×D4×C53D812yes—
93C2×D4×C53D811yescharacteristic
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D412yes—
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—42no—
99C22×Dic53—41no—
100Q16×C53—21no—
101Q16×C53—21no—
102SD16×C53—42no—
103D8×C53—21no—
104D8×C53—21no—
105M16×C53—22no—
106M16×C53—24no—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53C2×D412yes—
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53#4112yes—
112C2×C8×C53#4112yes—
113C2×C8×C53C2411yescharacteristic
114C4⋊C4×C53—22no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—42no—
117C22⋊C4×C53—42no—
118C22⋊C4×C53—42no—
119C22⋊Dic53—42no—
120C22⋊Dic53—41no—
121C4⋊Dic53—21no—
122C4⋊Dic53—22no—
123C4⋊Dic53—21no—
124C4⋊Dic53—42no—
125C4⋊Dic53—41no—
126C4×Dic53—42no—
127C4×Dic53—41no—
128#11719908—21no—
129#11719908—21no—
130#11719908—22no—
131C23×C53—41no—
132C23×C53C2×D812yes—
133C23×C53C2×D811yescharacteristic
134Q8×C53—22no—
135Q8×C53—41no—
136Q8×C53—41no—
137D4×C53—24no—
138D4×C53—22no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—82no—
142D4×C53—162no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—82no—
148C2×C4×C53—161no—
149C2×C4×C53#12312yes—
150C2×C4×C53C2×D812yes—
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53#13712yes—
153C2×C4×C53C22×D411yescharacteristic
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—41no—
157C2×Dic53—161no—
158C2×Dic53—162no—
159C8×C53—22no—
160C8×C53#12312yes—
161C8×C53C2×#4111yescharacteristic
162C8×C53C2×#4111yescharacteristic
163#33653—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53#48012yes—
173C22×C53C22×D811yescharacteristic
174C4×C53—22no—
175C4×C53—81no—
176C4×C53—82no—
177C4×C53#48012yes—
178C4×C53C2×#12311yescharacteristic
179C4×C53C2×#13711yescharacteristic
180Dic53—81no—
181Dic53—82no—
182Dic53—81no—
183#3061D5311yescharacteristic
184#2913—531no—
185#2237—532no—
186#2235—532no—
187#2234—532no—
188#2233—532no—
189#2159—531no—
190C2×#367—531no—
191#2139—531no—
192#2138—531no—
193#2113—531no—
194C2×C53—22no—
195C2×C53—42no—
196C2×C53—161no—
197C2×C53C2×#48011yescharacteristic
198C2×C53#341611yescharacteristic
199C2×C53#341611yescharacteristic
200C2×#135C2×D5311yescharacteristic
201C2×#131C2×D5311yescharacteristic
202#537—21no—
203#520—1061no—
204C2×M32—531no—
205C22×C16—531no—
206#468—22no—
207#467C2×D5312yes—
208#466C2×D5312yes—
209#428—1061no—
210#428—1062no—
211#427—531no—
212#427—532no—
213C2×#107—531no—
214C2×#107—532no—
215C2×#107C2×D5311yescharacteristic
216#417—21no—
217#367—532no—
218#367—532no—
219#367—532no—
220#366—532no—
221#358—531no—
222#358—531no—
223#358—532no—
224#357—531no—
225#357—531no—
226#357—532no—
227C22⋊C16—532no—
228C22⋊C16—532no—
229C53#3365311yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×#41C22⋊D5311yescharacteristic
232#135—22no—
233#135—22no—
234C2×Q16C22×D5311yescharacteristic
235C2×SD16—21no—
236C2×D8C22×D5311yescharacteristic
237#131—22no—
238#131—22no—
239C2×M16C22⋊D5311yescharacteristic
240C2×M16C22⋊D5312yes—
241C22×C8C22×D5312yes—
242C22×C8C22×D5311yescharacteristic
243C22⋊Q8—22no—
244C22⋊Q8—41no—
245#121—22no—
246#121—41no—
247#121—42no—
248C4×Q8—2121no—
249C4×D4—2122no—
250C4×D4—2121no—
251#117—1062no—
252#117—1061no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1062no—
255C2×C4⋊C4—1061no—
256M32—1062no—
257C2×C16—532no—
258C2×C16—531no—
259C2×C16—531no—
260C2×C16—1062no—
261#107—21no—
262#107—531no—
263#107—531no—
264#107—532no—
265#107—1062no—
266#107—1061no—
267#107C22×D5312yes—
268#106—1062no—
269#106—1061no—
270#103—22no—
271#102—22no—
272#41—22no—
273#41—22no—
274#41—42no—
275#41—42no—
276C2×Q8—21no—
277C2×Q8#1171992211yescharacteristic
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4#1171992212yes—
282C2×D4#1171992211yescharacteristic
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—2122no—
286C22×C4—2121no—
287C22×C4C2×C22⋊D5312yes—
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—21no—
290Q16—21no—
291SD16—42no—
292D8—21no—
293D8—21no—
294M16—22no—
295M16—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8#1171994712yes—
299C2×C8#1171994712yes—
300C2×C8C2×C22⋊D5312yes—
301C2×C8C2×C22⋊D5311yescharacteristic
302C2×C8C23×D5311yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—1061no—
307C4⋊C4—1062no—
308C4⋊C4—1061no—
309C4⋊C4—2122no—
310C4⋊C4—2121no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—2122no—
314C22⋊C4—2121no—
315C42—2122no—
316C42—2121no—
317C16—1061no—
318C16—1061no—
319C16—1062no—
320C23—41no—
321C23C2×#1171992212yes—
322C23C2×#1171992211yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—41no—
326D4—24no—
327D4—22no—
328D4—41no—
329D4—41no—
330D4—82no—
331D4—162no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—161no—
338C2×C4—8481no—
339C2×C4—8482no—
340C2×C4—2121no—
341C2×C4—2122no—
342C2×C4—2121no—
343C2×C4#1313319812yes—
344C2×C4C2×#1171992212yes—
345C2×C4C2×#1171992211yescharacteristic
346C2×C4#1313324712yes—
347C2×C4C22×C22⋊D5311yescharacteristic
348C8—22no—
349C8#1313319812yes—
350C8C2×#1171994711yescharacteristic
351C8C2×#1171994711yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1681093412yes—
361C22C22×#1171992211yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—4242no—
366C4—4241no—
367C4—4241no—
368C4#1681093412yes—
369C4C2×#1313319811yescharacteristic
370C4C2×#1313324711yescharacteristic
371C2—22no—
372C2—42no—
373C2—161no—
374C2C2×#1681093411yescharacteristic
375C2#2028785911yescharacteristic
376C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3061C2
socleC22×C53C22×D8
hypercenter#3061D53
hypocenterC53#33653
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. #20283315C2semidirect
2. #20276920C2semidirect
3. #20276910C2semidirect
4. #20276903C2semidirect
5. #20276898C2semidirect
6. #20276501C2semidirect
7. C2×#16810450C2semidirect
8. #20276427C2semidirect
9. #20276422C2semidirect
10. #20276329C2semidirect
229. C53#33653semidirect

Maximal subgroups

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

subgroupindex
#202833152
#202769202
#202769102
#202769032
#202768982
#202765012
C2×#168104502
#202764272
#202764222
#202763292
C53×#30612
#3365353