Finite Group Databasebeta

← #27301347

Subgroups — #27301347

Order 13568.

508 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×#16810745C211yesmaximal, characteristic, complemented
2C2×#16810436C211yesmaximal, characteristic, complemented
3C2×#16810433C211yesmaximal, characteristic, complemented
4#20275592C211yesmaximal, characteristic, complemented
5#20272814C211yesmaximal, characteristic, complemented
6#20272794C211yesmaximal, characteristic, complemented
7C2×C53×#509C211yesmaximal, characteristic
8C2×#13133224—21no—
9C2×#13133157C2211yescharacteristic
10#16810745—22no—
11#16810745—22no—
12C2×#13133153C2211yescharacteristic
13C2×#13133149C2211yescharacteristic
14C2×#13133144—21no—
15C2×#13133138—21no—
16C2×#13133127C2211yescharacteristic
17#16810526—21no—
18#16810436—22no—
19#16810433—22no—
20C2×#135×C53C2211yescharacteristic
21C22×D8×C53C2211yescharacteristic
22C53×#509—22no—
23C53×#509—22no—
24C2×SD32×C53C412yescomplemented
25C2×D16×C53C412yescomplemented
26C2×M32×C53C2211yescharacteristic
27#13133224—24no—
28#13133224—42no—
29C2×#11719917—21no—
30C2×#11719917—41no—
31C22×#88322—41no—
32C22×#88322D411yescharacteristic
33C2×C4⋊Dic53—21no—
34C2×C22⋊Dic53—42no—
35C2×C4×Dic53D411yescharacteristic
36#13133157—21no—
37#13133157—21no—
38C2×#11719908—21no—
39#13133153—21no—
40#13133153D412yes—
41#13133149—21no—
42#13133149—21no—
43#13133144—24no—
44#13133138—42no—
45#13133127—21no—
46#13133127—21no—
47C2×#41×C53—21no—
48C22×D4×C53—21no—
49#135×C53—21no—
50#135×C53—21no—
51#135×C53D412yes—
52#135×C53D412yes—
53C2×Q16×C53C2×C411yescharacteristic
54C2×SD16×C53—21no—
55C2×D8×C53—21no—
56C2×D8×C53—21no—
57C2×D8×C53—21no—
58C2×D8×C53—21no—
59C2×D8×C53—21no—
60C2×D8×C53C2×C412yes—
61C2×D8×C53C2×C411yescharacteristic
62C22×C8×C53C2311yescharacteristic
63SD32×C53—24no—
64D16×C53—24no—
65M32×C53—21no—
66M32×C53—21no—
67C2×C16×C53C2×C412yes—
68C24×C53—41no—
69#41×C53—22no—
70#41×C53—22no—
71#41×C53—41no—
72#41×C53—41no—
73C2×Q8×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—22no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—41no—
86Q16×C53—21no—
87Q16×C53C22⋊C412yescomplemented
88SD16×C53—22no—
89SD16×C53—22no—
90D8×C53—22no—
91D8×C53—21no—
92D8×C53—22no—
93D8×C53—41no—
94D8×C53—41no—
95D8×C53C22⋊C412yescomplemented
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—84no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107#11719917—22no—
108#11719917—44no—
109#11719917—41no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—44no—
113C2×#88322—41no—
114C2×#88322D812yes—
115C16×C53—22no—
116#11719908—41no—
117C23×C53—41no—
118C23×C53—42no—
119C23×C53—82no—
120C23×C53—82no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53AGL(1,ℤ/8)11yescharacteristic
124Q8×C53—22no—
125Q8×C53—41no—
126D4×C53—22no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—42no—
130D4×C53—41no—
131D4×C53—42no—
132D4×C53—81no—
133D4×C53—81no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53C2×D811yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—82no—
146C2×Dic53—81no—
147C8×C53C2×C22⋊C411yescharacteristic
148C8×C53C4×D411yescharacteristic
149C8×C53#12111yescharacteristic
150C8×C53#12311yescharacteristic
151#88322—22no—
152#88322—42no—
153#13040—531nomaximal, Sylow
154C22×C53—22no—
155C22×C53—42no—
156C22×C53—82no—
157C22×C53—82no—
158C22×C53—84no—
159C22×C53—162no—
160C22×C53—161no—
161C22×C53—161no—
162C22×C53#44211yescharacteristic
163C22×C53#44911yescharacteristic
164C22×C53#45911yescharacteristic
165C4×C53—42no—
166C4×C53#38611yescharacteristic
167C4×C53C4×D811yescharacteristic
168C4×C53#44711yescharacteristic
169C4×C53#46311yescharacteristic
170Dic53—81no—
171Dic53—161no—
172C2×#509D5311yescharacteristic
173C2×#444—531no—
174C2×#361—531no—
175C2×#360—531no—
176#1972—531no—
177#1383—531no—
178#1375—531no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—82no—
182C2×C53—162no—
183C2×C53#187911yescharacteristic
184C2×C53AGL(1,ℤ/16)12yes—
185C2×#135C2×D5311yescharacteristic
186C22×D8C2×D5311yescharacteristic
187C2×#131—1061no—
188#509—22no—
189#509—22no—
190C2×SD32Dic5312yescomplemented
191C2×D16Dic5312yescomplemented
192C2×M32—531no—
193C2×M32C2×D5311yescharacteristic
194#444—1062no—
195#444—1062no—
196C2×#108—531no—
197C2×#106—531no—
198C2×#104—1061no—
199C2×#102—1061no—
200C2×#97—531no—
201#390—1061no—
202#361—1062no—
203#360—1062no—
204C53#1304011yescharacteristic, complemented, Sylow
205C2×#41—21no—
206C22×D4—21no—
207#135—21no—
208#135—21no—
209#135C22⋊D5312yes—
210#135C22⋊D5312yes—
211C2×Q16C2×Dic5311yescharacteristic
212C2×SD16—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8C2×Dic5312yes—
219C2×D8C2×Dic5311yescharacteristic
220#131—1064no—
221#131—2122no—
222C2×M16—1061no—
223C2×M16—2121no—
224C22×C8—531no—
225C22×C8—2121no—
226C22×C8C22×D5311yescharacteristic
227C2×C4⋊C4—1061no—
228C2×C22⋊C4—2122no—
229C2×C42—531no—
230SD32—24no—
231D16—24no—
232M32—21no—
233M32—21no—
234M32—1061no—
235M32—1061no—
236C2×C16—1061no—
237C2×C16C2×Dic5312yes—
238#108—532no—
239#108—1061no—
240#106—1061no—
241#106—1061no—
242#104—1064no—
243#102—2122no—
244#97—1061no—
245#97—1061no—
246C24—41no—
247#41—22no—
248#41—22no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—1061no—
262C22×C4—2121no—
263C22×C4D4×D5311yescharacteristic
264Q16—21no—
265Q16C22⋊Dic5312yescomplemented
266SD16—22no—
267SD16—22no—
268D8—22no—
269D8—21no—
270D8—22no—
271D8—41no—
272D8—41no—
273D8C22⋊Dic5312yescomplemented
274M16—1062no—
275M16—2124no—
276M16—2121no—
277C2×C8—532no—
278C2×C8—1061no—
279C2×C8—1061no—
280C2×C8—2124no—
281C2×C8—2121no—
282C2×C8D4×D5311yescharacteristic
283C2×C8#1171994711yescharacteristic
284C2×C8#1171994711yescharacteristic
285C2×C8C22×Dic5311yescharacteristic
286C2×C8C2×C22⋊D5311yescharacteristic
287C2×C8C2×C22⋊D5311yescharacteristic
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C22⋊C4—4244no—
291C42—1061no—
292C42—1061no—
293C16—22no—
294C16—2121no—
295C23—41no—
296C23—42no—
297C23—82no—
298C23—82no—
299C23—81no—
300C23—81no—
301C23#1313324111yescharacteristic
302Q8—22no—
303Q8—41no—
304D4—22no—
305D4—42no—
306D4—42no—
307D4—42no—
308D4—41no—
309D4—42no—
310D4—81no—
311D4—81no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—4242no—
316C2×C4—4241no—
317C2×C4—2122no—
318C2×C4—2121no—
319C2×C4D4×Dic5311yescharacteristic
320C2×C4#1313318511yescharacteristic
321C2×C4#1313318911yescharacteristic
322C2×C4D8×D5311yescharacteristic
323C2×C4#1313323611yescharacteristic
324C2×C4#1313324111yescharacteristic
325C8—1062no—
326C8—2122no—
327C8C2×C22⋊Dic5311yescharacteristic
328C8D4×Dic5311yescharacteristic
329C8#1313318911yescharacteristic
330C8#1313319811yescharacteristic
331C22—22no—
332C22—42no—
333C22—82no—
334C22—82no—
335C22—84no—
336C22—161no—
337C22—161no—
338C22—162no—
339C22#1681073511yescharacteristic
340C22#1681076511yescharacteristic
341C22#1681081611yescharacteristic
342C4—42no—
343C4—8481no—
344C4—4241no—
345C4#1681050311yescharacteristic
346C4D8×Dic5311yescharacteristic
347C4#1681075511yescharacteristic
348C4#1681083611yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2—162no—
353C2#2027506511yescharacteristic
354C2#2027661412yes—
subgroupG/H
centerC22#16810735
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#509C2
socleC22×C53#442
cosocleC22×C8×C53C23
hypercenterC2×#509D53
hypocenterC53#13040
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16810745C2semidirect
2. C2×#16810436C2semidirect
3. C2×#16810433C2semidirect
4. #20275592C2semidirect
5. #20272814C2semidirect
6. #20272794C2semidirect
24. C2×SD32×C53C4semidirect
25. C2×D16×C53C4semidirect
87. Q16×C53C22⋊C4semidirect
95. D8×C53C22⋊C4semidirect
190. C2×SD32Dic53semidirect
191. C2×D16Dic53semidirect
204. C53#13040semidirect
265. Q16C22⋊Dic53semidirect
273. D8C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107452
C2×#168104362
C2×#168104332
#202755922
#202728142
#202727942
C2×C53×#5092
#1304053