Finite Group Databasebeta

← #27301451

Subgroups — #27301451

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C16×D53C211yesmaximal, characteristic, complemented
2C2×#16810856C211yesmaximal, characteristic
3C2×#16810424C211yesmaximal, characteristic, complemented
4#20274741C211yesmaximal, characteristic, complemented
5#20272807C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C2×C53×#357C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×#13133188—21no—
10C22×#11719908C2211yescharacteristic
11C2×C16×D53—22no—
12C2×C16×D53—22no—
13C2×C16×D53—22no—
14#16810856—22no—
15#16810856C414yescomplemented
16C2×#13133151C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133138—21no—
19#16810480—21no—
20#16810424—22no—
21#16810385—21no—
22C22×D8×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#357—22no—
26C22×C22⋊D53—41no—
27C22×C4×D53D411yescharacteristic
28C22×#88322D411yescharacteristic
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53C2×C412yes—
34C2×C8×D53D412yes—
35#13133188—24no—
36#13133188—42no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C22⋊Dic53—41no—
40C2×#11719921—41no—
41C2×#11719908—21no—
42C2×#11719908—21no—
43C2×#11719908D412yescomplemented
44C16×D53—28no—
45#13133151—21no—
46#13133151—21no—
47#13133151—21no—
48#13133151C2×C412yes—
49#13133138—24no—
50#13133124—41no—
51C22×D4×C53—21no—
52C2×D8×C53—21no—
53C2×D8×C53—21no—
54C2×D8×C53—21no—
55C2×D8×C53—21no—
56C2×D8×C53—21no—
57C2×D8×C53C2×C412yes—
58C22×C8×C53C2311yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53D412yescomplemented
63#107×C53—21no—
64#107×C53—21no—
65C24×C53—41no—
66C23×D53D811yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—44no—
76C2×C22⋊D53—82no—
77C2×C22⋊D53—82no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53D811yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53C22⋊C412yes—
86C2×C4×D53SD1612yes—
87D8×C53—22no—
88D8×C53—22no—
89D8×C53—41no—
90D8×C53—41no—
91C2×C8×C53C22×C411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53#4111yescharacteristic
96C2×C8×C53#4111yescharacteristic
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C22⋊Dic53—44no—
100#11719921—82no—
101C4⋊Dic53—22no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322C22⋊C412yes—
108C8×D53—22no—
109C8×D53—22no—
110C8×D53—22no—
111C8×D53—22no—
112C16×C53—22no—
113#11719908—22no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53—81no—
117C23×C53—81no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53C2×D811yescharacteristic
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53#10212yes—
127C22×D53D1612yes—
128C22×D53SD3212yes—
129D4×C53—42no—
130D4×C53—42no—
131D4×C53—81no—
132D4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C22⋊D53—88no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—42no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150C2×Dic53—81no—
151C2×Dic53#10212yes—
152C4×D53—22no—
153C4×D53—22no—
154C4×D53—22no—
155C4×D53—22no—
156C8×C53C4×D411yescharacteristic
157C8×C53C4×D411yescharacteristic
158C8×C53#12011yescharacteristic
159C8×C53#12311yescharacteristic
160#88322—22no—
161#13059—531nomaximal, Sylow
162C22×C53—82no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—161no—
166C22×C53—161no—
167C22×C53—161no—
168C22×C53—161no—
169C22×C53C2×#10211yescharacteristic
170C22×C53#46611yescharacteristic
171C22×C53#46611yescharacteristic
172C22×C53C2×D1611yescharacteristic
173C22×C53C2×SD3211yescharacteristic
174C22×C53#50811yescharacteristic
175C22×C53#50811yescharacteristic
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53#35714yescomplemented
183C4×C53—161no—
184C4×C53#38611yescharacteristic
185C4×C53#41611yescharacteristic
186C4×C53#46511yescharacteristic
187C4×C53#46511yescharacteristic
188Dic53—22no—
189Dic53—82no—
190Dic53—161no—
191C23×C16—531no—
192C2×#466—531no—
193C2×#357—531no—
194C2×#357D5311yescharacteristic, complemented
195#1808—531no—
196#1380—531no—
197#1380—531no—
198C2×C53—162no—
199C2×C53#189311yescharacteristic
200C2×C53C2×#35711yescharacteristic
201C2×C53#213811yescharacteristic
202C2×C53#221511yescharacteristic
203C2×C53#221511yescharacteristic
204C2×C53#221711yescharacteristic
205C2×C53#221711yescharacteristic
206D53—24no—
207C22×D8C2×D5311yescharacteristic
208C23×C8—531no—
209C2×#121—1061no—
210C22×C16—531no—
211C22×C16—1062no—
212C22×C16—1062no—
213C22×C16—1062no—
214C22×C16C2×D5311yescharacteristic
215#466—534no—
216#466—1062no—
217C2×#107—531no—
218C2×#107—531no—
219C2×#107C2×D5311yescharacteristic
220C2×#102—1061no—
221#380—1061no—
222#357—22no—
223#357—1062no—
224#340—1061no—
225C53#1305911yescharacteristic, complemented, Sylow
226C22×D4—21no—
227C22×D4—2121no—
228C23×C4—531no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8C4×D5312yes—
235C22×C8—531no—
236C22×C8—532no—
237C22×C8—532no—
238C22×C8—1061no—
239C22×C8—1061no—
240C22×C8—1061no—
241C22×C8—1061no—
242C22×C8C22×D5311yescharacteristic
243#121—1064no—
244#121—2122no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C22⋊C4—2121no—
249C2×C22⋊C4—2121no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—532no—
253C2×C16—1068no—
254C2×C16—1061no—
255C2×C16—1061no—
256C2×C16C22⋊D5312yescomplemented
257#107—21no—
258#107—21no—
259#107—532no—
260#107—1061no—
261#107—1061no—
262#107—1061no—
263#102—1064no—
264#95—2121no—
265C24—41no—
266C24—531no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—4242no—
274C2×D4—4242no—
275C2×D4—2124no—
276C22×C4—41no—
277C22×C4—531no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4D4×D5311yescharacteristic
287D8—22no—
288D8—22no—
289D8—41no—
290D8—41no—
291C2×C8—532no—
292C2×C8—1061no—
293C2×C8—1061no—
294C2×C8—1062no—
295C2×C8—1062no—
296C2×C8—1062no—
297C2×C8—1062no—
298C2×C8C2×C4×D5311yescharacteristic
299C2×C8#1171994511yescharacteristic
300C2×C8D4×D5311yescharacteristic
301C2×C8D4×D5311yescharacteristic
302C2×C8#1171994711yescharacteristic
303C2×C8C2×C22⋊D5311yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—1062no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C22⋊C4—4242no—
311C22⋊C4—2124no—
312C16—22no—
313C16—1062no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—532no—
322C23—532no—
323C23—532no—
324C23—1061no—
325C23—1061no—
326C23—1061no—
327C23—1061no—
328C23D8×D5311yescharacteristic
329D4—42no—
330D4—42no—
331D4—81no—
332D4—81no—
333D4—4248no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—532no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—4241no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—2122no—
350C2×C4C22⋊C4×D5311yescharacteristic
351C2×C4#1313318811yescharacteristic
352C2×C4#1313319011yescharacteristic
353C2×C4SD16×D5311yescharacteristic
354C2×C4#1313323611yescharacteristic
355C2×C4#1313323811yescharacteristic
356C8—1062no—
357C8#1313317711yescharacteristic
358C8C4×C22⋊D5311yescharacteristic
359C8#1313318511yescharacteristic
360C8#1313319911yescharacteristic
361C22—82no—
362C22—82no—
363C22—82no—
364C22—161no—
365C22—161no—
366C22—161no—
367C22—161no—
368C22—534no—
369C22—1062no—
370C22—1062no—
371C22—1062no—
372C22—1062no—
373C22—1062no—
374C22—1062no—
375C22#102×D5311yescharacteristic
376C22#1681085611yescharacteristic
377C22#1681085811yescharacteristic
378C22D16×D5311yescharacteristic
379C22SD32×D5311yescharacteristic
380C22#1681105211yescharacteristic
381C22#1681105411yescharacteristic
382C4—161no—
383C4—8481no—
384C4—1062no—
385C4—4242no—
386C4#1681049911yescharacteristic
387C4#1681061611yescharacteristic
388C4#1681084911yescharacteristic
389C4#1681085111yescharacteristic
390C2—162no—
391C2—1064no—
392C2#2027514011yescharacteristic
393C2#357×D5311yescharacteristic
394C2#2027641911yescharacteristic
395C2#2027679811yescharacteristic
396C2#2027680011yescharacteristic
397C2#2027680811yescharacteristic
398C2#2027681011yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#357C2
socleC23×C53C2×D8
prime radicalC2×#16810856C2
cosocleC22×C8×C53C23
hypercenterC2×#357D53
hypocenterC53#13059
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. C22×C16×D53C2semidirect
3. C2×#16810424C2semidirect
4. #20274741C2semidirect
5. #20272807C2semidirect
6. #20272806C2semidirect
7. C2×C53×#357C2semidirect
15. #16810856C4semidirect
43. C2×#11719908D4semidirect
62. C2×C16×C53D4semidirect
182. C2×D53#357semidirect
194. C2×#357D53semidirect
225. C53#13059semidirect
256. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C22×C16×D532
C2×#168108562
C2×#168104242
#202747412
#202728072
#202728062
C2×C53×#3572
#1305953