Finite Group Databasebeta

← #27301059

Subgroups — #27301059

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
1#20283315C211yesmaximal, characteristic, complemented
2#20276431C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4C2×#16810407C211yesmaximal, characteristic, complemented
5#20272825C211yesmaximal, characteristic, complemented
6#20272812C211yesmaximal, characteristic, complemented
7C2×C53×#472C211yesmaximal, characteristic
8#16811106—21no—
9C2×#13133157C2211yescharacteristic
10C22×#11719908C2211yescharacteristic
11#16810665—22no—
12#16810665—21no—
13#16810661—21no—
14#16810661C2211yescharacteristic
15C2×#13133151—21no—
16C2×#13133151C2211yescharacteristic
17#16810424—22no—
18#16810424—21no—
19#16810424—21no—
20#16810407—22no—
21C2×AGL(1,ℤ/8)×C53—21no—
22C22×D8×C53C2211yescharacteristic
23C2×#131×C53C2211yescharacteristic
24C53×#472—22no—
25C53×#472C414yescomplemented
26C2×#108×C53C2211yescharacteristic
27C2×#100×C53—21no—
28Q8×Dic53—41no—
29D4×Dic53—42no—
30D4×Dic53—41no—
31#13133176—22no—
32#13133176—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—21no—
35#13133157—21no—
36#13133157—21no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41#13133151—22no—
42#13133151—21no—
43#13133151D411yescharacteristic
44#13133151D412yes—
45#13133151D411yescharacteristic
46#13133149—22no—
47#13133149—21no—
48C2×#41×C53D411yescharacteristic
49C22×D4×C53—21no—
50AGL(1,ℤ/8)×C53—24no—
51AGL(1,ℤ/8)×C53—42no—
52C2×SD16×C53—41no—
53C2×D8×C53—21no—
54C2×D8×C53—21no—
55C2×D8×C53—21no—
56C2×D8×C53—21no—
57C2×D8×C53—21no—
58C2×D8×C53—41no—
59C2×D8×C53C2×C412yescomplemented
60#131×C53—21no—
61#131×C53—21no—
62#131×C53C2×C412yes—
63#131×C53D412yes—
64C2×M16×C53—21no—
65C2×M16×C53—21no—
66C2×M16×C53D411yescharacteristic
67C22×C8×C53—21no—
68C22×C8×C53C2311yescharacteristic
69#108×C53—21no—
70#108×C53C2×C412yescomplemented
71#100×C53—24no—
72C24×C53—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×C53—22no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53D811yescharacteristic
86C22×C4×C53—21no—
87C22×C4×C53C2×D411yescharacteristic
88C22×Dic53—42no—
89C22×Dic53—41no—
90SD16×C53—44no—
91D8×C53—22no—
92D8×C53—22no—
93D8×C53—44no—
94D8×C53—41no—
95D8×C53—41no—
96M16×C53—21no—
97M16×C53—22no—
98M16×C53—22no—
99M16×C53—22no—
100M16×C53—41no—
101M16×C53C22⋊C412yes—
102C2×C8×C53—21no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C2×C8×C53C22×C411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C22⋊Dic53—42no—
113C22⋊Dic53—41no—
114C4⋊Dic53—21no—
115C4⋊Dic53—22no—
116C4⋊Dic53—21no—
117C4⋊Dic53—42no—
118C4⋊Dic53—41no—
119C4×Dic53—42no—
120C4×Dic53—41no—
121#11719908—41no—
122#11719908—41no—
123C23×C53—21no—
124C23×C53—42no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53—161no—
128C23×C53—161no—
129C23×C53AGL(1,ℤ/8)11yescharacteristic
130Q8×C53—21no—
131Q8×C53#10212yes—
132D4×C53—22no—
133D4×C53—22no—
134D4×C53—21no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—81no—
138D4×C53—81no—
139D4×C53#10212yes—
140C2×C4×C53—21no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—41no—
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—161no—
153C2×Dic53—162no—
154C8×C53—22no—
155C8×C53—42no—
156C8×C53C4×D411yescharacteristic
157C8×C53C4×D411yescharacteristic
158C8×C53#12111yescharacteristic
159C8×C53#12111yescharacteristic
160#12996—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—82no—
167C22×C53—162no—
168C22×C53—162no—
169C22×C53—161no—
170C22×C53—161no—
171C22×C53#41711yescharacteristic
172C22×C53#44711yescharacteristic
173C22×C53#48111yescharacteristic
174C4×C53—22no—
175C4×C53#38611yescharacteristic
176C4×C53C2×#10211yescharacteristic
177C4×C53#44811yescharacteristic
178C4×C53#48011yescharacteristic
179Dic53—82no—
180Dic53—81no—
181Dic53—81no—
182C2×#472D5311yescharacteristic
183#2913—531no—
184#2140—531no—
185C2×#357—531no—
186C2×#350—531no—
187#1387—531no—
188#1382—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—42no—
192C2×C53—162no—
193C2×C53#187411yescharacteristic
194C2×C53#219211yescharacteristic
195C2×C53#219311yescharacteristic
196C2×AGL(1,ℤ/8)—21no—
197C22×D8C2×D5311yescharacteristic
198C2×#131C2×D5311yescharacteristic
199#520—1061no—
200C2×M32—531no—
201C22×C16—531no—
202#472—22no—
203#472Dic5314yescomplemented
204C2×#108C2×D5311yescharacteristic
205#428—1062no—
206#428—1061no—
207#427—531no—
208#427—1061no—
209C2×#107—531no—
210C2×#107—1061no—
211C2×#100—21no—
212#357—1062no—
213#357—1061no—
214#357—1061no—
215#350—1062no—
216C53#1299611yescharacteristic, complemented, Sylow
217C2×#41C22⋊D5311yescharacteristic
218C22×D4—21no—
219AGL(1,ℤ/8)—24no—
220AGL(1,ℤ/8)—42no—
221C2×SD16—41no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—41no—
228C2×D8C2×Dic5312yescomplemented
229#131—21no—
230#131—21no—
231#131C2×Dic5312yes—
232#131C22⋊D5312yes—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16C22⋊D5311yescharacteristic
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—1061no—
246M32—1061no—
247C2×C16—1061no—
248C2×C16—1061no—
249C2×C16—1061no—
250C2×C16—1061no—
251#108—21no—
252#108C2×Dic5312yescomplemented
253#107—532no—
254#107—531no—
255#107—531no—
256#107—1062no—
257#107—1061no—
258#106—1062no—
259#106—1061no—
260#100—24no—
261C24—41no—
262#41—21no—
263#41—21no—
264#41#1171992312yes—
265#41C22⋊Dic5312yes—
266C2×Q8#1171992211yescharacteristic
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4#1171992211yescharacteristic
275C22×C4—21no—
276C22×C4—2122no—
277C22×C4—2121no—
278C22×C4C2×C22⋊D5311yescharacteristic
279SD16—44no—
280D8—22no—
281D8—22no—
282D8—44no—
283D8—41no—
284D8—41no—
285M16—21no—
286M16—22no—
287M16—22no—
288M16—22no—
289M16—41no—
290M16C22⋊Dic5312yes—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8D4×D5311yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8#1171994711yescharacteristic
299C2×C8C22×Dic5311yescharacteristic
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—2121no—
311C16—2121no—
312C23—21no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—161no—
317C23—161no—
318C23#1313324211yescharacteristic
319Q8—21no—
320Q8#1313313812yes—
321D4—22no—
322D4—22no—
323D4—21no—
324D4—42no—
325D4—42no—
326D4—81no—
327D4—81no—
328D4#1313313812yes—
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×C22⋊Dic5311yescharacteristic
338C2×C4#1313318511yescharacteristic
339C2×C4#1313319811yescharacteristic
340C2×C4C2×#1171992211yescharacteristic
341C2×C4C2×#1171992311yescharacteristic
342C2×C4#1313324711yescharacteristic
343C8—22no—
344C8—42no—
345C8D4×Dic5311yescharacteristic
346C8D4×Dic5311yescharacteristic
347C8#1313318911yescharacteristic
348C8#1313318911yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—82no—
355C22—162no—
356C22—162no—
357C22—161no—
358C22—161no—
359C22#1681062411yescharacteristic
360C22#1681075611yescharacteristic
361C22#1681093911yescharacteristic
362C4—22no—
363C4—4242no—
364C4—4241no—
365C4—4241no—
366C4#1681050311yescharacteristic
367C4C2×#1313313811yescharacteristic
368C4#1681076111yescharacteristic
369C4#1681093411yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—162no—
374C2#2027503311yescharacteristic
375C2#2027667011yescharacteristic
376C2#2027667711yescharacteristic
subgroupG/H
centerC22#16810756
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#472C2
socleC22×C53#447
cosocleC22×C8×C53C23
hypercenterC2×#472D53
hypocenterC53#12996
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. #20283315C2semidirect
2. #20276431C2semidirect
3. C2×#16810424C2semidirect
4. C2×#16810407C2semidirect
5. #20272825C2semidirect
6. #20272812C2semidirect
25. C53×#472C4semidirect
59. C2×D8×C53C2×C4semidirect
70. #108×C53C2×C4semidirect
203. #472Dic53semidirect
216. C53#12996semidirect
228. C2×D8C2×Dic53semidirect
252. #108C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202833152
#202764312
C2×#168104242
C2×#168104072
#202728252
#202728122
C2×C53×#4722
#1299653