Finite Group Databasebeta

← #27538965

Subgroups — #27538965

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#20284381C211yesmaximal, characteristic
2#20283312C211yesmaximal, characteristic, complemented
3#20276921C212yesmaximal, complemented
4#20276911C212yesmaximal, complemented
5#20276905C212yesmaximal, complemented
6#20276899C212yesmaximal, complemented
7#20276504C211yesmaximal, characteristic, complemented
8C2×#16810451C211yesmaximal, characteristic, complemented
9#20276427C211yesmaximal, characteristic, complemented
10#20276328C211yesmaximal, characteristic, complemented
11C53×#2138C211yesmaximal, characteristic, complemented
12C2×#13133223C2211yescharacteristic
13#16811201—21no—
14#16811103—21no—
15C2×#13133157C2211yescharacteristic
16#16810866—22no—
17#16810861C2212yes—
18#16810856C2212yes—
19#16810666—21no—
20#16810662—22no—
21#16810660C2212yes—
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24#107×D53C2212yes—
25#16810624—21no—
26#16810451C2212yes—
27#16810451C2212yes—
28#16810450C2212yes—
29#16810448C2212yes—
30#16810427C2212yes—
31#16810424C2212yes—
32#16810402C2212yes—
33#16810402C2212yes—
34C2×#135×C53C2211yescharacteristic
35C22×C16×C53C2211yescharacteristic
36C53×#427C2211yescharacteristic
37C53×#358C2211yescharacteristic
38C53×#358C2211yescharacteristic
39C53×#357C2211yescharacteristic
40C53×#357C2211yescharacteristic
41C2×#11719945D411yescharacteristic
42#13133223—22no—
43#13133223—22no—
44C2×#11719917D411yescharacteristic
45C2×#11719912D412yes—
46C2×C8×D53C2312yes—
47#13133196—41no—
48#13133193—22no—
49#13133191—41no—
50#13133189—42no—
51#13133188—22no—
52C4×Dic106—41no—
53C4×D212—41no—
54#13133177—42no—
55#13133175—22no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C4⋊C4×D53—22no—
59#13133157—22no—
60C2×#11719908C2312yes—
61#13133151—21no—
62#13133151—21no—
63#13133151C2312yes—
64#13133151C2312yes—
65#13133150—22no—
66#13133148—22no—
67#13133141—22no—
68#13133138—22no—
69C2×#41×C53—21no—
70#135×C53—22no—
71#135×C53—22no—
72C2×Q16×C53C2311yescharacteristic
73C2×SD16×C53—21no—
74C2×D8×C53C2311yescharacteristic
75C22×C8×C53C2311yescharacteristic
76#117×C53—21no—
77C2×C16×C53—22no—
78C2×C16×C53C2311yescharacteristic
79C2×C16×C53C2311yescharacteristic
80#107×C53C2311yescharacteristic
81#107×C53C2311yescharacteristic
82#106×C53—21no—
83#41×C53—42no—
84#41×C53—42no—
85C2×Q8×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C2×C22⋊D53—42no—
91C2×C22⋊D53D812yes—
92C22×Dic53—41no—
93C22×Dic53—41no—
94#11719945—22no—
95#11719945—22no—
96C2×D212D811yescharacteristic
97C2×C4×D53—42no—
98C2×C4×D53C2×D412yes—
99C2×Dic106D811yescharacteristic
100Q16×C53—21no—
101Q16×C53—21no—
102SD16×C53—42no—
103D8×C53—21no—
104D8×C53—21no—
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4112yes—
107C2×C8×C53#4112yes—
108C2×C8×C53C2411yescharacteristic
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C22⋊C4×C53—41no—
112C42×C53—41no—
113C22⋊Dic53—42no—
114#11719921—42no—
115#11719921—42no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120#11719919—42no—
121#11719919—42no—
122C4×Dic53—42no—
123#11719917—22no—
124C2×#88322C2×D412yes—
125#11719912—24no—
126C8×D53—22no—
127C8×D53—22no—
128C16×C53—21no—
129C16×C53—21no—
130#11719908—22no—
131C23×C53—41no—
132C23×C53C2×D811yescharacteristic
133C22×D53C2×D812yes—
134Q8×C53—41no—
135Q8×C53—41no—
136D4×C53—41no—
137D4×C53—41no—
138D4×C53—82no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—82no—
143C2×C4×C53#12312yes—
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53#13712yes—
146C2×C4×C53C22×D411yescharacteristic
147C22⋊D53—24no—
148C22⋊D53—162no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—161no—
152C2×Dic53—161no—
153C2×Dic53C2×D812yes—
154D212—22no—
155C4×D53—22no—
156C4×D53—22no—
157C4×D53—162no—
158Dic106—22no—
159C8×C53#12312yes—
160C8×C53C2×#4111yescharacteristic
161C8×C53C2×#4111yescharacteristic
162#88322—22no—
163#33653—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53#48012yes—
170C22×C53C22×D811yescharacteristic
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53#48012yes—
178C4×C53C2×#12311yescharacteristic
179C4×C53C2×#13711yescharacteristic
180Dic53—22no—
181Dic53—82no—
182Dic53—82no—
183#3061—531no—
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#2138D5311yescharacteristic, complemented
193#2113—531no—
194C2×C53—22no—
195C2×C53—161no—
196C2×C53C2×#48011yescharacteristic
197C2×C53#341611yescharacteristic
198C2×C53#341611yescharacteristic
199D53—42no—
200C2×#135C2×D5311yescharacteristic
201C2×#131—531no—
202#537—1061no—
203#520—1061no—
204C2×M32—531no—
205C22×C16C2×D5311yescharacteristic
206#468—1062no—
207#467—532no—
208#466—532no—
209#428—1062no—
210#428—1061no—
211#427—532no—
212#427C2×D5311yescharacteristic
213C2×#107—531no—
214C2×#107—531no—
215C2×#107—532no—
216#417—1061no—
217#367—532no—
218#367—532no—
219#367—532no—
220#366—532no—
221#358—532no—
222#358C2×D5311yescharacteristic
223#358C2×D5311yescharacteristic
224#357—532no—
225#357C2×D5311yescharacteristic
226#357C2×D5311yescharacteristic
227C22⋊C16—532no—
228C22⋊C16—532no—
229C53#3365311yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×#41—531no—
232#135—22no—
233#135—22no—
234C2×Q16C22×D5311yescharacteristic
235C2×SD16—21no—
236C2×D8C22×D5311yescharacteristic
237#131—1062no—
238#131—1062no—
239C2×M16—531no—
240C2×M16—532no—
241C22×C8—532no—
242C22×C8C22×D5311yescharacteristic
243C22⋊Q8—1062no—
244C22⋊Q8—2121no—
245#121—1062no—
246#121—2121no—
247#121—2122no—
248C4×Q8—2121no—
249C4×D4—2121no—
250C4×D4—2122no—
251#117—21no—
252#117—1062no—
253C2×C4⋊C4—1062no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256M32—1062no—
257C2×C16—22no—
258C2×C16—532no—
259C2×C16C22×D5311yescharacteristic
260C2×C16C22×D5311yescharacteristic
261#107—532no—
262#107—532no—
263#107—1062no—
264#107—1061no—
265#107—1061no—
266#107C22×D5311yescharacteristic
267#107C22×D5311yescharacteristic
268#106—21no—
269#106—1062no—
270#103—1062no—
271#102—1062no—
272#41—42no—
273#41—42no—
274#41—1062no—
275#41—1062no—
276C2×Q8—21no—
277C2×Q8—531no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—532no—
281C2×D4—531no—
282C2×D4—2122no—
283C22×C4—41no—
284C22×C4—532no—
285C22×C4—2122no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4D4×D5311yescharacteristic
289Q16—21no—
290Q16—21no—
291SD16—42no—
292D8—21no—
293D8—21no—
294M16—1062no—
295M16—1064no—
296C2×C8—532no—
297C2×C8—1062no—
298C2×C8—1062no—
299C2×C8#1171994512yes—
300C2×C8D4×D5311yescharacteristic
301C2×C8#1171994712yes—
302C2×C8C23×D5311yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—2122no—
308C4⋊C4—2122no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C22⋊C4—41no—
312C22⋊C4—2122no—
313C22⋊C4—2122no—
314C22⋊C4—2122no—
315C42—41no—
316C42—2122no—
317C16—21no—
318C16—21no—
319C16—1062no—
320C23—41no—
321C23—532no—
322C23D8×D5311yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—1062no—
326D4—41no—
327D4—41no—
328D4—82no—
329D4—8482no—
330D4—1064no—
331D4—1062no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—532no—
337C2×C4—8482no—
338C2×C4—8481no—
339C2×C4—8481no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4#1313319912yes—
345C2×C4D8×D5311yescharacteristic
346C2×C4#1313324612yes—
347C2×C4C2×D4×D5311yescharacteristic
348C8—1062no—
349C8#1313319912yes—
350C8C2×#1171994711yescharacteristic
351C8C2×#1171994511yescharacteristic
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—1062no—
358C22—1062no—
359C22—1062no—
360C22#1681093512yes—
361C22C2×D8×D5311yescharacteristic
362C4—81no—
363C4—81no—
364C4—81no—
365C4—1062no—
366C4—4242no—
367C4—4242no—
368C4#1681093512yes—
369C4C2×#1313319911yescharacteristic
370C4C2×#1313324611yescharacteristic
371C2—22no—
372C2—161no—
373C2—2122no—
374C2C2×#1681093511yescharacteristic
375C2#2028785811yescharacteristic
376C2#2028785811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2138C2
socleC22×C53C22×D8
prime radical#20284381C2
hypercenter#2138D53
hypocenterC53#33653
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #20283312C2semidirect
3. #20276921C2semidirect
4. #20276911C2semidirect
5. #20276905C2semidirect
6. #20276899C2semidirect
7. #20276504C2semidirect
8. C2×#16810451C2semidirect
9. #20276427C2semidirect
10. #20276328C2semidirect
11. C53×#2138C2semidirect
192. #2138D53semidirect
229. C53#33653semidirect

Maximal subgroups

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

subgroupindex
#202843812
#202833122
#202769212
#202769112
#202769052
#202768992
#202765042
C2×#168104512
#202764272
#202763282
C53×#21382
#3365353