Finite Group Databasebeta

← #27301055

Subgroups — #27301055

Order 13568.

512 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
1C2×#16810893C211yesmaximal, characteristic
2#20283315C211yesmaximal, characteristic, complemented
3#20276432C211yesmaximal, characteristic, complemented
4C2×#16810425C211yesmaximal, characteristic, complemented
5#20272824C211yesmaximal, characteristic, complemented
6#20272811C211yesmaximal, characteristic, complemented
7C2×C53×#350C211yesmaximal, characteristic, complemented
8C2×#13133242—21no—
9C22×D424C2211yescharacteristic
10#16811106—21no—
11#16810893—22no—
12#16810893C414yescomplemented
13C2×#13133153C2211yescharacteristic
14#16810665—22no—
15#16810665—21no—
16#16810661—21no—
17#16810661C2211yescharacteristic
18C2×#13133151—21no—
19C2×#13133151C2211yescharacteristic
20C2×#13133133—21no—
21#16810425—22no—
22#16810425—21no—
23#16810425—21no—
24C2×#131×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C22×C16×C53C2211yescharacteristic
27C53×#350—22no—
28C22×D212—21no—
29#13133242—24no—
30#13133242—42no—
31C2×#11719924—41no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424—21no—
37C2×D424C2×C412yescomplemented
38C2×#11719922—41no—
39C2×#11719917—21no—
40Q8×Dic53—41no—
41D4×Dic53—42no—
42D4×Dic53—41no—
43#13133176—22no—
44#13133176—21no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—21no—
47#13133153—21no—
48#13133153C2×C412yescomplemented
49#13133151—22no—
50#13133151—21no—
51#13133151D411yescharacteristic
52#13133151D412yes—
53#13133151D411yescharacteristic
54#13133149—22no—
55#13133149—21no—
56#13133133—24no—
57C2×#41×C53D411yescharacteristic
58#131×C53—21no—
59#131×C53—21no—
60#131×C53C2×C412yes—
61#131×C53D412yes—
62C2×M16×C53—21no—
63C2×M16×C53D411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53C2311yescharacteristic
66M32×C53—21no—
67M32×C53—21no—
68C2×C16×C53—21no—
69C2×C16×C53—21no—
70C2×C16×C53—21no—
71C2×C16×C53—21no—
72C23×D53—41no—
73#41×C53—21no—
74#41×C53—21no—
75#41×C53C22⋊C412yes—
76#41×C53SD1612yes—
77C2×Q8×C53D811yescharacteristic
78C2×D4×C53—21no—
79C2×D4×C53D811yescharacteristic
80C22×C4×C53—21no—
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—42no—
83C22×Dic53—41no—
84C2×D212—22no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90M16×C53—21no—
91M16×C53—22no—
92M16×C53—22no—
93M16×C53C22⋊C412yes—
94C2×C8×C53—21no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53C22×C411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C2×C8×C53#4111yescharacteristic
103C22⋊Dic53—42no—
104C22⋊Dic53—41no—
105#11719924—44no—
106#11719922—44no—
107C4⋊Dic53—21no—
108C4⋊Dic53—22no—
109C4⋊Dic53—21no—
110C4⋊Dic53—42no—
111C4⋊Dic53—41no—
112C4×Dic53—42no—
113C4×Dic53—41no—
114#11719917—22no—
115#11719917—41no—
116C2×#88322—41no—
117D424—22no—
118D424—22no—
119D424—41no—
120D424—41no—
121C16×C53—41no—
122C16×C53—41no—
123C23×C53—21no—
124C23×C53AGL(1,ℤ/8)11yescharacteristic
125C22×D53—42no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—161no—
129C22×D53—161no—
130Q8×C53—21no—
131Q8×C53#10212yes—
132D4×C53—22no—
133D4×C53—22no—
134D4×C53—21no—
135D4×C53#10212yes—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12311yescharacteristic
142C2×C4×C53C2×D811yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53#13711yescharacteristic
145C2×Dic53—41no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—161no—
149C2×Dic53—162no—
150D212—42no—
151D212—42no—
152D212—81no—
153D212—81no—
154C8×C53—22no—
155C8×C53C4×D411yescharacteristic
156C8×C53C4×D411yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12111yescharacteristic
159#88322—42no—
160#12996—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53#41711yescharacteristic
167C22×C53#44711yescharacteristic
168C22×C53#48111yescharacteristic
169C2×D53—82no—
170C2×D53—162no—
171C2×D53—161no—
172C2×D53—161no—
173C2×D53—162no—
174C4×C53—22no—
175C4×C53#38611yescharacteristic
176C4×C53C2×#10211yescharacteristic
177C4×C53#44811yescharacteristic
178C4×C53#48011yescharacteristic
179Dic53—82no—
180Dic53—81no—
181Dic53—81no—
182C2×#472—531no—
183#2913—531no—
184#2140—531no—
185C2×#357—531no—
186C2×#350D5311yescharacteristic, complemented
187#1387—531no—
188#1382—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—42no—
192C2×C53#187411yescharacteristic
193C2×C53#219211yescharacteristic
194C2×C53#219311yescharacteristic
195D53—162no—
196C2×AGL(1,ℤ/8)—1061no—
197C22×D8—531no—
198C2×#131C2×D5311yescharacteristic
199#520—1061no—
200C2×M32C2×D5311yescharacteristic
201C22×C16C2×D5311yescharacteristic
202#472—534no—
203#472—1062no—
204C2×#108—531no—
205#428—1062no—
206#428—1061no—
207#427—531no—
208#427—1061no—
209C2×#107—531no—
210C2×#107—1061no—
211C2×#100—1061no—
212#357—1062no—
213#357—1061no—
214#357—1061no—
215#350—22no—
216C53#1299611yescharacteristic, complemented, Sylow
217C2×#41D21211yescharacteristic
218C22×D4—1061no—
219AGL(1,ℤ/8)—1064no—
220AGL(1,ℤ/8)—2122no—
221C2×SD16—2121no—
222C2×D8—532no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C2×D8—2121no—
229#131—21no—
230#131—21no—
231#131C4×D5312yes—
232#131C22⋊D5312yes—
233C2×M16—21no—
234C2×M16—1061no—
235C2×M16D21211yescharacteristic
236C22×C8—21no—
237C22×C8C22×D5311yescharacteristic
238C4×Q8—2121no—
239C4×D4—2122no—
240C4×D4—2121no—
241#117—1062no—
242#117—1061no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4—1061no—
245M32—21no—
246M32—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251#108—532no—
252#108—1061no—
253#107—532no—
254#107—531no—
255#107—531no—
256#107—1062no—
257#107—1061no—
258#106—1062no—
259#106—1061no—
260#100—1064no—
261C24—2121no—
262#41—21no—
263#41—21no—
264#41#1171991312yes—
265#41#1171992112yes—
266C2×Q8D42411yescharacteristic
267C2×D4—21no—
268C2×D4—1062no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C2×D4—2121no—
273C2×D4—2121no—
274C2×D4D42411yescharacteristic
275C22×C4—21no—
276C22×C4—2122no—
277C22×C4—2121no—
278C22×C4C2×D21211yescharacteristic
279SD16—2124no—
280D8—1062no—
281D8—1062no—
282D8—2124no—
283D8—2121no—
284D8—2121no—
285M16—21no—
286M16—22no—
287M16—22no—
288M16—1062no—
289M16—2121no—
290M16#1171992112yes—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—2121no—
295C2×C8C2×C4×D5311yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8D4×D5311yescharacteristic
298C2×C8#1171994711yescharacteristic
299C2×C8#1171994711yescharacteristic
300C2×C8C2×C22⋊D5311yescharacteristic
301C4⋊C4—1061no—
302C4⋊C4—1062no—
303C4⋊C4—1061no—
304C4⋊C4—2122no—
305C4⋊C4—2121no—
306C22⋊C4—2122no—
307C22⋊C4—2121no—
308C42—2122no—
309C42—2121no—
310C16—41no—
311C16—41no—
312C23—21no—
313C23—8481no—
314C23—8481no—
315C23—4241no—
316C23—4241no—
317C23—2122no—
318C23#1313324511yescharacteristic
319Q8—21no—
320Q8#1313313912yes—
321D4—22no—
322D4—22no—
323D4—21no—
324D4—4241no—
325D4—4241no—
326D4—2122no—
327D4—2122no—
328D4#1313313912yes—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—8481no—
333C2×C4—8482no—
334C2×C4—2121no—
335C2×C4—2122no—
336C2×C4—2121no—
337C2×C4C2×#1171992111yescharacteristic
338C2×C4#1313318711yescharacteristic
339C2×C4#1313320111yescharacteristic
340C2×C4C2×D42411yescharacteristic
341C2×C4C2×#1171991311yescharacteristic
342C2×C4#1313325011yescharacteristic
343C8—22no—
344C8—2122no—
345C8#1313317711yescharacteristic
346C8#1313317711yescharacteristic
347C8#1313318911yescharacteristic
348C8#1313318911yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—8482no—
355C22—8482no—
356C22—8481no—
357C22—8481no—
358C22—4242no—
359C22#1681062511yescharacteristic
360C22#1681075911yescharacteristic
361C22#1681094211yescharacteristic
362C4—22no—
363C4—4242no—
364C4—4241no—
365C4—4241no—
366C4#1681050011yescharacteristic
367C4C2×#1313313911yescharacteristic
368C4#1681076411yescharacteristic
369C4#1681093711yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—8482no—
374C2#2027503011yescharacteristic
375C2#2027667411yescharacteristic
376C2#2027668111yescharacteristic
subgroupG/H
centerC22#16810759
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#350C2
socleC22×C53#447
prime radicalC2×#16810893C2
cosocleC22×C8×C53C23
hypercenterC2×#350D53
hypocenterC53#12996
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. #20283315C2semidirect
3. #20276432C2semidirect
4. C2×#16810425C2semidirect
5. #20272824C2semidirect
6. #20272811C2semidirect
7. C2×C53×#350C2semidirect
12. #16810893C4semidirect
37. C2×D424C2×C4semidirect
48. #13133153C2×C4semidirect
186. C2×#350D53semidirect
216. C53#12996semidirect

Maximal subgroups

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

subgroupindex
C2×#168108932
#202833152
#202764322
C2×#168104252
#202728242
#202728112
C2×C53×#3502
#1299653