Finite Group Databasebeta

← #27301536

Subgroups — #27301536

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811040C211yesmaximal, characteristic, complemented
2C2×#16810436C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4#20274779C211yesmaximal, characteristic, complemented
5#20272825C211yesmaximal, characteristic, complemented
6#20272808C211yesmaximal, characteristic, complemented
7C2×C53×#472C211yesmaximal, characteristic
8#16811040—22no—
9#16811040—22no—
10C2×#13133157—21no—
11C2×#13133157C2211yescharacteristic
12C22×#11719908—21no—
13C22×#11719908C2211yescharacteristic
14C2×#13133153C2211yescharacteristic
15C2×#13133151C2211yescharacteristic
16#16810624—21no—
17#16810436—22no—
18#16810424—21no—
19#16810424—21no—
20#16810391—21no—
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—
28C2×#11719917—21no—
29C2×C4⋊Dic53—21no—
30#13133157—24no—
31#13133157—21no—
32#13133157D412yes—
33C2×#11719908—24no—
34C2×#11719908—21no—
35C2×#11719908—21no—
36C2×#11719908—21no—
37C2×#11719908D411yescharacteristic
38C2×#11719908D411yescharacteristic
39#13133153—21no—
40#13133153—21no—
41#13133151—21no—
42#13133151—21no—
43#13133141—41no—
44#13133138—41no—
45C2×#41×C53D411yescharacteristic
46C22×D4×C53—21no—
47AGL(1,ℤ/8)×C53—24no—
48AGL(1,ℤ/8)×C53—42no—
49C2×SD16×C53—41no—
50C2×D8×C53—21no—
51C2×D8×C53—21no—
52C2×D8×C53—21no—
53C2×D8×C53—21no—
54C2×D8×C53—21no—
55C2×D8×C53—41no—
56C2×D8×C53C2×C412yes—
57#131×C53—21no—
58#131×C53—21no—
59#131×C53C2×C412yes—
60#131×C53D412yes—
61C2×M16×C53—21no—
62C2×M16×C53—21no—
63C2×M16×C53D411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53C2311yescharacteristic
66#108×C53—21no—
67#108×C53C2×C412yes—
68#100×C53—24no—
69C24×C53—41no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53C22⋊C412yes—
73#41×C53SD1612yes—
74C2×Q8×C53D811yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53—22no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53D811yescharacteristic
83C22×C4×C53—21no—
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—41no—
86SD16×C53—44no—
87D8×C53—22no—
88D8×C53—22no—
89D8×C53—44no—
90D8×C53—41no—
91D8×C53—41no—
92M16×C53—21no—
93M16×C53—22no—
94M16×C53—22no—
95M16×C53—22no—
96M16×C53—41no—
97M16×C53C22⋊C412yes—
98C2×C8×C53—21no—
99C2×C8×C53—22no—
100C2×C8×C53—22no—
101C2×C8×C53—41no—
102C2×C8×C53C22×C411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110#11719917—41no—
111#11719917—41no—
112C2×#88322—41no—
113#11719908—21no—
114#11719908—22no—
115#11719908—21no—
116C23×C53—21no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53—161no—
121C23×C53—161no—
122C23×C53#13511yescharacteristic
123Q8×C53—21no—
124Q8×C53#10212yes—
125D4×C53—22no—
126D4×C53—22no—
127D4×C53—21no—
128D4×C53—42no—
129D4×C53—42no—
130D4×C53—81no—
131D4×C53—81no—
132D4×C53#10212yes—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53C2×D811yescharacteristic
140C2×C4×C53C2×SD1611yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C2×Dic53—81no—
143C2×Dic53—161no—
144C8×C53—22no—
145C8×C53—42no—
146C8×C53C4×D411yescharacteristic
147C8×C53C4×D411yescharacteristic
148C8×C53#12011yescharacteristic
149C8×C53#12311yescharacteristic
150#88322—81no—
151#13073—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—82no—
158C22×C53—162no—
159C22×C53—162no—
160C22×C53—161no—
161C22×C53—161no—
162C22×C53#41611yescharacteristic
163C22×C53#46511yescharacteristic
164C22×C53#46611yescharacteristic
165C4×C53—22no—
166C4×C53#38611yescharacteristic
167C4×C53C2×#10211yescharacteristic
168C4×C53#46511yescharacteristic
169C4×C53#46611yescharacteristic
170Dic53—161no—
171C2×#504—531no—
172C2×#472D5311yescharacteristic
173C2×#361—531no—
174C2×#357—531no—
175#1816—531no—
176#1387—531no—
177#1381—531no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—42no—
181C2×C53—162no—
182C2×C53#189311yescharacteristic
183C2×C53#222111yescharacteristic
184C2×C53#222311yescharacteristic
185C2×AGL(1,ℤ/8)—21no—
186C22×D8C2×D5311yescharacteristic
187C2×#131C2×D5311yescharacteristic
188#504—1062no—
189#504—1062no—
190C2×M32—531no—
191C2×M32—1061no—
192C22×C16—531no—
193C22×C16—1061no—
194#472—22no—
195#472Dic5314yescomplemented
196C2×#108—531no—
197C2×#108C2×D5311yescharacteristic
198C2×#107—531no—
199#417—1061no—
200C2×#100—21no—
201#361—1062no—
202#357—1061no—
203#357—1061no—
204#343—1061no—
205C53#1307311yescharacteristic, complemented, Sylow
206C2×#41C22⋊D5311yescharacteristic
207C22×D4—21no—
208AGL(1,ℤ/8)—24no—
209AGL(1,ℤ/8)—42no—
210C2×SD16—41no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—41no—
217C2×D8C2×Dic5312yes—
218#131—21no—
219#131—21no—
220#131C2×Dic5312yes—
221#131C22⋊D5312yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—1061no—
225C2×M16C22⋊D5311yescharacteristic
226C22×C8—21no—
227C22×C8C22×D5311yescharacteristic
228C2×C4⋊C4—1061no—
229M32—532no—
230M32—1064no—
231M32—1061no—
232C2×C16—531no—
233C2×C16—531no—
234C2×C16—1064no—
235C2×C16—1061no—
236C2×C16—1061no—
237C2×C16—1061no—
238#108—21no—
239#108—1061no—
240#108—1061no—
241#108C2×Dic5312yes—
242#107—1061no—
243#107—1061no—
244#103—2121no—
245#102—2121no—
246#100—24no—
247C24—41no—
248#41—21no—
249#41—21no—
250#41#1171992312yes—
251#41C22⋊Dic5312yes—
252C2×Q8#1171992211yescharacteristic
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4#1171992211yescharacteristic
261C22×C4—21no—
262C22×C4—2121no—
263C22×C4C2×C22⋊D5311yescharacteristic
264SD16—44no—
265D8—22no—
266D8—22no—
267D8—44no—
268D8—41no—
269D8—41no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—41no—
275M16—2121no—
276M16—2121no—
277M16C22⋊Dic5312yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—2121no—
283C2×C8D4×D5311yescharacteristic
284C2×C8D4×D5311yescharacteristic
285C2×C8#1171994711yescharacteristic
286C2×C8#1171994711yescharacteristic
287C2×C8C22×Dic5311yescharacteristic
288C2×C8C2×C22⋊D5311yescharacteristic
289C4⋊C4—2121no—
290C4⋊C4—2121no—
291C16—1061no—
292C16—1061no—
293C16—1062no—
294C23—21no—
295C23—42no—
296C23—81no—
297C23—81no—
298C23—161no—
299C23—161no—
300C23#1313323711yescharacteristic
301Q8—21no—
302Q8#1313313812yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—42no—
307D4—42no—
308D4—81no—
309D4—81no—
310D4#1313313812yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—8481no—
315C2×C4—4241no—
316C2×C4C2×C22⋊Dic5311yescharacteristic
317C2×C4#1313318911yescharacteristic
318C2×C4#1313318911yescharacteristic
319C2×C4C2×#1171992211yescharacteristic
320C2×C4C2×#1171992311yescharacteristic
321C2×C4#1313323711yescharacteristic
322C8—22no—
323C8—42no—
324C8—4241no—
325C8D4×Dic5311yescharacteristic
326C8D4×Dic5311yescharacteristic
327C8#1313318511yescharacteristic
328C8#1313319811yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—82no—
335C22—162no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#1681061911yescharacteristic
340C22#1681085011yescharacteristic
341C22#1681085711yescharacteristic
342C4—22no—
343C4—8481no—
344C4#1681050311yescharacteristic
345C4C2×#1313313811yescharacteristic
346C4#1681085011yescharacteristic
347C4#1681085711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—162no—
352C2#2027514411yescharacteristic
353C2#2027683311yescharacteristic
354C2#2027684311yescharacteristic
subgroupG/H
centerC22#16810857
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#472C2
socleC22×C53#466
cosocleC22×C8×C53C23
hypercenterC2×#472D53
hypocenterC53#13073
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. C2×#16811040C2semidirect
2. C2×#16810436C2semidirect
3. C2×#16810424C2semidirect
4. #20274779C2semidirect
5. #20272825C2semidirect
6. #20272808C2semidirect
25. C53×#472C4semidirect
195. #472Dic53semidirect
205. C53#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#168110402
C2×#168104362
C2×#168104242
#202747792
#202728252
#202728082
C2×C53×#4722
#1307353