Finite Group Databasebeta

← #27301151

Subgroups — #27301151

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic53C211yesmaximal, characteristic, complemented
2C2×#16810424C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4#20275582C211yesmaximal, characteristic, complemented
5#20272807C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C22×D16×C53C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9C22×#11719908C2211yescharacteristic
10D8×Dic53—22no—
11D8×Dic53—22no—
12C2×#13133151C2211yescharacteristic
13C2×#13133151C2211yescharacteristic
14C2×#13133138—21no—
15C2×#13133138—21no—
16C2×C8×Dic53C2211yescharacteristic
17#16810526—21no—
18#16810424—22no—
19#16810424—22no—
20C22×D8×C53C2211yescharacteristic
21C22×D8×C53C2211yescharacteristic
22C2×D16×C53—22no—
23C2×D16×C53—22no—
24C2×D16×C53C414yescomplemented
25C22×C16×C53C2211yescharacteristic
26C23×Dic53—41no—
27C22×#88322D411yescharacteristic
28D4×Dic53—24no—
29D4×Dic53—42no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C22⋊Dic53—41no—
33C2×C22⋊Dic53—41no—
34C2×C22⋊Dic53—41no—
35C2×C4×Dic53D411yescharacteristic
36C2×#11719908—21no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39#13133151—21no—
40#13133151—21no—
41#13133151—21no—
42#13133151D412yes—
43#13133138—24no—
44#13133138—42no—
45C8×Dic53—21no—
46C8×Dic53—21no—
47C22×D4×C53—21no—
48C22×D4×C53—21no—
49C2×D8×C53—21no—
50C2×D8×C53—21no—
51C2×D8×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—
58C2×D8×C53C2×C412yescomplemented
59C2×D8×C53D412yes—
60C2×D8×C53D412yes—
61C22×C8×C53C2311yescharacteristic
62D16×C53—28no—
63C2×C16×C53—21no—
64C2×C16×C53—21no—
65C2×C16×C53C2×C412yescomplemented
66C24×C53—41no—
67C24×C53—41no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—44no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—82no—
85C22×Dic53—82no—
86C22×Dic53D811yescharacteristic
87C22×Dic53D811yescharacteristic
88D8×C53—22no—
89D8×C53—22no—
90D8×C53—22no—
91D8×C53—22no—
92D8×C53—22no—
93D8×C53—41no—
94D8×C53—41no—
95D8×C53C22⋊C414yescomplemented
96C2×C8×C53C22×C411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C22⋊Dic53—44no—
103C22⋊Dic53—82no—
104C22⋊Dic53—82no—
105C4⋊Dic53—22no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—21no—
114C16×C53—22no—
115#11719908—41no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—42no—
120C23×C53—81no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53—81no—
124C23×C53—81no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53—81no—
128C23×C53—81no—
129C23×C53—81no—
130C23×C53C2×D811yescharacteristic
131D4×C53—24no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—81no—
138D4×C53—81no—
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151C2×Dic53—88no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154C2×Dic53—81no—
155C2×Dic53—81no—
156C2×Dic53—81no—
157C2×Dic53D1612yescomplemented
158C8×C53C2×C22⋊C411yescharacteristic
159C8×C53C4×D411yescharacteristic
160C8×C53#12111yescharacteristic
161C8×C53#12311yescharacteristic
162#88322—41no—
163#13012—531nomaximal, Sylow
164C22×C53—44no—
165C22×C53—82no—
166C22×C53—82no—
167C22×C53—82no—
168C22×C53—82no—
169C22×C53—82no—
170C22×C53—82no—
171C22×C53—82no—
172C22×C53—82no—
173C22×C53—82no—
174C22×C53—161no—
175C22×C53—161no—
176C22×C53—161no—
177C22×C53—161no—
178C22×C53C4×D811yescharacteristic
179C22×C53#44711yescharacteristic
180C22×C53#45911yescharacteristic
181C22×C53C2×D1611yescharacteristic
182C22×C53#50811yescharacteristic
183C22×C53#50911yescharacteristic
184C22×C53#50911yescharacteristic
185C4×C53#38611yescharacteristic
186C4×C53#44211yescharacteristic
187C4×C53#44911yescharacteristic
188C4×C53#46311yescharacteristic
189Dic53—22no—
190Dic53—41no—
191Dic53—82no—
192Dic53—161no—
193C22×D16D5311yescharacteristic
194C2×C4×D8—531no—
195C2×#357—531no—
196C2×#357—531no—
197#1970—531no—
198#1380—531no—
199#1374—531no—
200C2×C53—84no—
201C2×C53—162no—
202C2×C53#187911yescharacteristic
203C2×C53C4×D1611yescharacteristic
204C2×C53#217711yescharacteristic
205C2×C53#218411yescharacteristic
206C2×C53#218611yescharacteristic
207C2×C53#220611yescharacteristic
208C2×C53#221011yescharacteristic
209C22×D8C2×D5311yescharacteristic
210C22×D8C2×D5311yescharacteristic
211C2×C4×D4—1061no—
212C2×D16—22no—
213C2×D16—22no—
214C2×D16Dic5314yescomplemented
215C22×C16—531no—
216C22×C16C2×D5311yescharacteristic
217C4×D8—1062no—
218C4×D8—1062no—
219C2×#107—531no—
220C2×#107—531no—
221C2×#102—1061no—
222C2×#102—1061no—
223C2×C4×C8—531no—
224#390—1061no—
225#357—1062no—
226#357—1062no—
227C53#1301211yescharacteristic, complemented, Sylow
228C22×D4—21no—
229C22×D4—21no—
230C23×C4—2121no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—21no—
239C2×D8C2×Dic5312yes—
240C2×D8C2×Dic5312yescomplemented
241C2×D8C22⋊D5312yes—
242C2×D8C22⋊D5312yes—
243C22×C8—531no—
244C22×C8C22×D5311yescharacteristic
245C4×D4—1064no—
246C4×D4—2122no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C22⋊C4—2121no—
250C2×C22⋊C4—2121no—
251C2×C22⋊C4—2121no—
252C2×C42—531no—
253D16—28no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259C2×C16C2×Dic5312yescomplemented
260#107—532no—
261#107—1061no—
262#107—1061no—
263#107—1061no—
264#102—1064no—
265#102—2122no—
266C4×C8—1061no—
267C4×C8—1061no—
268C24—41no—
269C24—41no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—4242no—
285C22×C4—4242no—
286C22×C4—2124no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4D4×D5311yescharacteristic
290D8—22no—
291D8—22no—
292D8—22no—
293D8—22no—
294D8—22no—
295D8—41no—
296D8—41no—
297D8C22⋊Dic5314yescomplemented
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8D4×D5311yescharacteristic
302C2×C8#1171994711yescharacteristic
303C2×C8#1171994711yescharacteristic
304C2×C8C22×Dic5311yescharacteristic
305C2×C8C2×C22⋊D5311yescharacteristic
306C2×C8C2×C22⋊D5311yescharacteristic
307C4⋊C4—1062no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C22⋊C4—4242no—
312C22⋊C4—4242no—
313C22⋊C4—2124no—
314C42—1061no—
315C42—1061no—
316C16—22no—
317C16—2121no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23D8×D5311yescharacteristic
333D4—24no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—81no—
340D4—81no—
341C2×C4—532no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—4248no—
351C2×C4—4241no—
352C2×C4—4241no—
353C2×C4—2122no—
354C2×C4D4×Dic5311yescharacteristic
355C2×C4#1313318511yescharacteristic
356C2×C4#1313318911yescharacteristic
357C2×C4#1313323611yescharacteristic
358C2×C4#1313324111yescharacteristic
359C2×C4#1313324111yescharacteristic
360C8—2121no—
361C8C2×C22⋊Dic5311yescharacteristic
362C8D4×Dic5311yescharacteristic
363C8#1313318911yescharacteristic
364C8#1313319811yescharacteristic
365C22—44no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22D8×Dic5311yescharacteristic
380C22#1681075511yescharacteristic
381C22#1681081611yescharacteristic
382C22D16×D5311yescharacteristic
383C22#1681105211yescharacteristic
384C22#1681105711yescharacteristic
385C22#1681105711yescharacteristic
386C4—8481no—
387C4—1062no—
388C4—4242no—
389C4—2121no—
390C4#1681050311yescharacteristic
391C4#1681073511yescharacteristic
392C4#1681076511yescharacteristic
393C4#1681083611yescharacteristic
394C2—84no—
395C2—162no—
396C2#2027506511yescharacteristic
397C2D16×Dic5311yescharacteristic
398C2#2027659111yescharacteristic
399C2#2027662511yescharacteristic
400C2#2027663511yescharacteristic
401C2#2027674811yescharacteristic
402C2#2027676811yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC22×D16×C53C2
socleC23×C53C2×D8
cosocleC22×C8×C53C23
hypercenterC22×D16D53
hypocenterC53#13012
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×D16

Complements

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

NH ≅ G/Ntype
1. C2×D8×Dic53C2semidirect
2. C2×#16810424C2semidirect
3. C2×#16810424C2semidirect
4. #20275582C2semidirect
5. #20272807C2semidirect
6. #20272792C2semidirect
24. C2×D16×C53C4semidirect
58. C2×D8×C53C2×C4semidirect
65. C2×C16×C53C2×C4semidirect
95. D8×C53C22⋊C4semidirect
157. C2×Dic53D16semidirect
214. C2×D16Dic53semidirect
227. C53#13012semidirect
240. C2×D8C2×Dic53semidirect
259. C2×C16C2×Dic53semidirect
297. D8C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic532
C2×#168104242
#202755822
#202728072
#202727922
C22×D16×C532
#1301253