Finite Group Databasebeta

← #27301149

Subgroups — #27301149

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
1C22×#13133158C211yesmaximal, characteristic
2C2×#16810424C211yesmaximal, characteristic, complemented
3C2×#16810423C211yesmaximal, characteristic, complemented
4#20275581C211yesmaximal, characteristic, complemented
5#20272806C211yesmaximal, characteristic, complemented
6#20272793C211yesmaximal, characteristic, complemented
7C2×C4×D8×C53C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C2×#13133158—22no—
10C2×#13133158—22no—
11C2×#13133158C414yescomplemented
12C22×#11719908C2211yescharacteristic
13C22×#11719908C2211yescharacteristic
14C2×#13133151C2211yescharacteristic
15C2×#13133139—21no—
16#16810525—21no—
17#16810424—22no—
18#16810423—22no—
19C22×D8×C53C2211yescharacteristic
20C2×C4×D4×C53—21no—
21C4×D8×C53—22no—
22C4×D8×C53—22no—
23C2×#107×C53C2211yescharacteristic
24C2×#102×C53—21no—
25C2×C4×C8×C53C2211yescharacteristic
26C22×D212—21no—
27C2×D424—21no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424C2×C412yescomplemented
33C2×C4⋊Dic53—21no—
34C2×#11719921—41no—
35C2×#11719921—41no—
36#13133158—28no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41C2×#11719908—21no—
42C2×#11719908C2×C412yescomplemented
43#13133151—21no—
44#13133151—21no—
45#13133139—42no—
46C22×D4×C53—21no—
47C23×C4×C53—41no—
48C2×D8×C53—21no—
49C2×D8×C53—21no—
50C2×D8×C53—21no—
51C2×D8×C53C2×C412yes—
52C2×D8×C53D412yes—
53C2×D8×C53D412yes—
54C22×C8×C53D411yescharacteristic
55C22×C8×C53C2311yescharacteristic
56C4×D4×C53—24no—
57C4×D4×C53—42no—
58C2×C4⋊C4×C53—21no—
59C2×C22⋊C4×C53—41no—
60C2×C42×C53D411yescharacteristic
61#107×C53—21no—
62#107×C53D412yes—
63#102×C53—24no—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C24×C53—41no—
67C23×D53—41no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C22×C4×C53—44no—
75C22×C4×C53—41no—
76C22×C4×C53—82no—
77C22×C4×C53—82no—
78C22×C4×C53D811yescharacteristic
79C22×C4×C53D811yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—41no—
82C2×D212—22no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88D8×C53—22no—
89D8×C53—22no—
90D8×C53—22no—
91D8×C53C22⋊C414yescomplemented
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53C22×C411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C4⋊C4×C53—22no—
102C4⋊C4×C53—41no—
103C22⋊C4×C53—44no—
104C42×C53—21no—
105C42×C53—21no—
106#11719921—82no—
107#11719921—82no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110D424—22no—
111D424—22no—
112D424—41no—
113D424—41no—
114#11719908—22no—
115#11719908—41no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53C2×D811yescharacteristic
124C22×D53—42no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131D4×C53—24no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—42no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—42no—
141C2×C4×C53—88no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53D1612yescomplemented
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×Dic53—81no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154D212—42no—
155D212—42no—
156D212—81no—
157D212—81no—
158C8×C53—41no—
159C8×C53C2×C22⋊C411yescharacteristic
160C8×C53C4×D411yescharacteristic
161C8×C53#12111yescharacteristic
162C8×C53#12311yescharacteristic
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×C53C4×D811yescharacteristic
172C22×C53#44711yescharacteristic
173C22×C53#45911yescharacteristic
174C22×C53C2×D1611yescharacteristic
175C22×C53#50811yescharacteristic
176C22×C53#50911yescharacteristic
177C22×C53#50911yescharacteristic
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—82no—
181C2×D53—161no—
182C2×D53—161no—
183C2×D53—161no—
184C2×D53—161no—
185C4×C53—22no—
186C4×C53—41no—
187C4×C53—82no—
188C4×C53#38611yescharacteristic
189C4×C53#44211yescharacteristic
190C4×C53#44911yescharacteristic
191C4×C53#46311yescharacteristic
192Dic53—161no—
193C22×D16—531no—
194C2×C4×D8D5311yescharacteristic, complemented
195C2×#357—531no—
196C2×#357—531no—
197#1970—531no—
198#1380—531no—
199#1374—531no—
200C2×C53—84no—
201C2×C53#187911yescharacteristic
202C2×C53C4×D1611yescharacteristic
203C2×C53#217711yescharacteristic
204C2×C53#218411yescharacteristic
205C2×C53#218611yescharacteristic
206C2×C53#220611yescharacteristic
207C2×C53#221011yescharacteristic
208D53—162no—
209C22×D8—531no—
210C22×D8C2×D5311yescharacteristic
211C2×C4×D4—21no—
212C2×D16—534no—
213C2×D16—1062no—
214C2×D16—1062no—
215C22×C16—531no—
216C22×C16—531no—
217C4×D8—22no—
218C4×D8—22no—
219C2×#107—531no—
220C2×#107C2×D5311yescharacteristic
221C2×#102—21no—
222C2×#102—1061no—
223C2×C4×C8C2×D5311yescharacteristic
224#390—1061no—
225#357—1062no—
226#357—1062no—
227C53#1301211yescharacteristic, complemented, Sylow
228C22×D4—21no—
229C22×D4—1061no—
230C23×C4—41no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—532no—
235C2×D8—1061no—
236C2×D8—1061no—
237C2×D8—1061no—
238C2×D8—1061no—
239C2×D8—1061no—
240C2×D8C4×D5312yes—
241C2×D8D21212yes—
242C2×D8C22⋊D5312yes—
243C22×C8C22⋊D5311yescharacteristic
244C22×C8C22×D5311yescharacteristic
245C4×D4—24no—
246C4×D4—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—1061no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—2121no—
251C2×C22⋊C4—2121no—
252C2×C42C22⋊D5311yescharacteristic
253D16—1068no—
254C2×C16—532no—
255C2×C16—1061no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259C2×C16—1061no—
260#107—21no—
261#107—1061no—
262#107—1061no—
263#107C22⋊D5312yes—
264#102—24no—
265#102—2122no—
266C4×C8—21no—
267C4×C8—21no—
268C24—41no—
269C24—2121no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—1062no—
277C2×D4—2121no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4—2121no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—2121no—
287C22×C4#1171992211yescharacteristic
288C22×C4#1171992211yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290D8—22no—
291D8—22no—
292D8—22no—
293D8—1062no—
294D8—1062no—
295D8—2121no—
296D8—2121no—
297D8#1171992114yescomplemented
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8C2×C4×D5311yescharacteristic
302C2×C8C2×D21211yescharacteristic
303C2×C8#1171994511yescharacteristic
304C2×C8#1171994511yescharacteristic
305C2×C8C2×C22⋊D5311yescharacteristic
306C2×C8C2×C22⋊D5311yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C22⋊C4—44no—
312C22⋊C4—4242no—
313C22⋊C4—4242no—
314C42—21no—
315C42—21no—
316C16—1062no—
317C16—2121no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—4241no—
326C23—4241no—
327C23—4241no—
328C23—4241no—
329C23—4241no—
330C23—4241no—
331C23—2122no—
332C23C2×#1171992211yescharacteristic
333D4—24no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—4241no—
338D4—4241no—
339D4—2122no—
340D4—2122no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—88no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—4241no—
351C2×C4—4241no—
352C2×C4—4241no—
353C2×C4#1313315812yescomplemented
354C2×C4C4×C22⋊D5311yescharacteristic
355C2×C4#1313318611yescharacteristic
356C2×C4#1313319111yescharacteristic
357C2×C4#1313323711yescharacteristic
358C2×C4#1313324211yescharacteristic
359C2×C4#1313324311yescharacteristic
360C8—41no—
361C8C2×#1171992111yescharacteristic
362C8C4×C22⋊D5311yescharacteristic
363C8#1313319111yescharacteristic
364C8#1313320011yescharacteristic
365C22—44no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—8481no—
373C22—8481no—
374C22—8481no—
375C22—8481no—
376C22—4242no—
377C22—4242no—
378C22—4242no—
379C22C4×#1171992211yescharacteristic
380C22#1681075711yescharacteristic
381C22#1681081811yescharacteristic
382C22C2×#1313315811yescharacteristic
383C22#1681105311yescharacteristic
384C22#1681105811yescharacteristic
385C22#1681105911yescharacteristic
386C4—22no—
387C4—41no—
388C4—82no—
389C4—8481no—
390C4#1681050211yescharacteristic
391C4#1681073311yescharacteristic
392C4#1681076811yescharacteristic
393C4#1681083811yescharacteristic
394C2—84no—
395C2—8482no—
396C2#2027506311yescharacteristic
397C2C4×#1313315811yescharacteristic
398C2#2027658911yescharacteristic
399C2#2027662711yescharacteristic
400C2#2027663811yescharacteristic
401C2#2027675011yescharacteristic
402C2#2027677011yescharacteristic
subgroupG/H
centerC23C2×#11719922
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C4×D8×C53C2
socleC23×C53C2×D8
prime radicalC22×#13133158C2
cosocleC22×C8×C53C23
hypercenterC2×C4×D8D53
hypocenterC53#13012
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. C2×#16810424C2semidirect
3. C2×#16810423C2semidirect
4. #20275581C2semidirect
5. #20272806C2semidirect
6. #20272793C2semidirect
7. C2×C4×D8×C53C2semidirect
11. C2×#13133158C4semidirect
32. C2×D424C2×C4semidirect
42. C2×#11719908C2×C4semidirect
91. D8×C53C22⋊C4semidirect
144. C2×C4×C53D16semidirect
194. C2×C4×D8D53semidirect
227. C53#13012semidirect
297. D8#11719921semidirect
353. C2×C4#13133158semidirect

Maximal subgroups

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

subgroupindex
C22×#131331582
C2×#168104242
C2×#168104232
#202755812
#202728062
#202727932
C2×C4×D8×C532
#1301253