Finite Group Databasebeta

← #27538987

Subgroups — #27538987

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284320C211yesmaximal, characteristic
2C2×#16810670C211yesmaximal, characteristic, complemented
3#20276933C214yesmaximal, complemented
4#20276928C214yesmaximal, complemented
5#20276489C211yesmaximal, characteristic, complemented
6#20276488C211yesmaximal, characteristic, complemented
7#20276464C211yesmaximal, characteristic, complemented
8C2×#16810404C211yesmaximal, characteristic, complemented
9C53×#2148C211yesmaximal, characteristic, complemented
10C22×#11719912C2211yescharacteristic
11C2×M16×D53—21no—
12#16811130—21no—
13C2×#13133157C2211yescharacteristic
14#16810878C2212yes—
15#16810873C2212yes—
16#16810866—22no—
17#16810670—22no—
18#16810670C2214yes—
19C2×#13133153C2211yescharacteristic
20C2×#13133152—21no—
21#16810661C2211yescharacteristic
22#16810619—21no—
23#16810447C2214yescomplemented
24#16810446C2214yescomplemented
25#16810439C2212yescomplemented
26#16810436C2212yescomplemented
27#16810404C2214yes—
28C2×#135×C53C2211yescharacteristic
29C2×M32×C53C2211yescharacteristic
30C2×#108×C53C2211yescharacteristic
31C53×#362C2212yescomplemented
32C53×#361C2212yescomplemented
33C22×C4×D53D411yescharacteristic
34C2×#11719917—21no—
35C2×#11719912—21no—
36C2×#11719912—21no—
37C2×#11719912—21no—
38C2×#11719912—21no—
39C2×#11719912—41no—
40C2×#11719912D412yes—
41C2×#11719912C2312yes—
42M16×D53—24no—
43M16×D53—42no—
44C22×#88322D411yescharacteristic
45C2×C8×D53—41no—
46#13133198—41no—
47#13133193—22no—
48#13133188—22no—
49#13133185—41no—
50C4×C22⋊D53—42no—
51#13133176—21no—
52#13133157—21no—
53#13133157C2312yes—
54C2×#11719908—21no—
55#13133153—21no—
56#13133153C2312yes—
57#13133152—24no—
58#13133151—21no—
59#13133149C2312yes—
60#13133141—22no—
61#13133138—22no—
62C2×#41×C53—21no—
63#135×C53—21no—
64#135×C53—21no—
65#135×C53—21no—
66#135×C53—21no—
67C2×Q16×C53C2311yescharacteristic
68C2×SD16×C53—21no—
69C2×D8×C53C2311yescharacteristic
70C2×M16×C53—21no—
71C22×C8×C53C2311yescharacteristic
72M32×C53—21no—
73M32×C53C2312yes—
74C2×C16×C53—21no—
75#108×C53—21no—
76#108×C53C2312yes—
77C23×D53SD1611yescharacteristic
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82C2×Q8×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C2×C22⋊D53—42no—
88C22×Dic53SD1611yescharacteristic
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×C4×D53—21no—
92C2×C4×D53—21no—
93C2×C4×D53SD1612yes—
94C2×C4×D53C2×D412yes—
95Q16×C53—21no—
96Q16×C53—21no—
97SD16×C53—41no—
98SD16×C53—41no—
99D8×C53—21no—
100D8×C53—21no—
101M16×C53—22no—
102M16×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53#4111yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C2×C8×C53#4111yescharacteristic
109C2×C8×C53C2411yescharacteristic
110C22⋊Dic53—41no—
111C22⋊Dic53—42no—
112#11719921—42no—
113C4⋊Dic53—22no—
114#11719919—42no—
115C4×Dic53—41no—
116#11719917—22no—
117#11719917—41no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—41no—
121C2×#88322C2×D412yes—
122#11719912—22no—
123#11719912—22no—
124#11719912—22no—
125#11719912—22no—
126#11719912—44no—
127C8×D53—44no—
128C16×C53—22no—
129#11719908—22no—
130C23×C53—41no—
131C23×C53C2×SD1611yescharacteristic
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—81no—
135C22×D53C2×SD1612yes—
136Q8×C53—41no—
137Q8×C53—41no—
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—81no—
141D4×C53—81no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53C22×D411yescharacteristic
151C22⋊D53—84no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—42no—
155C2×Dic53C2×SD1612yes—
156C4×D53—22no—
157C4×D53—22no—
158C4×D53—22no—
159C4×D53—22no—
160C8×C53—42no—
161C8×C53#12311yescharacteristic
162C8×C53#12311yescharacteristic
163C8×C53C2×#4111yescharacteristic
164C8×C53C2×#4111yescharacteristic
165#88322—22no—
166#88322—42no—
167#33656—531nomaximal, Sylow
168C22×C53—41no—
169C22×C53—82no—
170C22×C53—81no—
171C22×C53#48111yescharacteristic
172C22×C53#48111yescharacteristic
173C22×C53C22×SD1611yescharacteristic
174C2×D53—22no—
175C2×D53—42no—
176C2×D53—42no—
177C2×D53—81no—
178C4×C53—81no—
179C4×C53#48111yescharacteristic
180C4×C53#48111yescharacteristic
181C4×C53C2×#12311yescharacteristic
182C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
183Dic53—22no—
184Dic53—82no—
185#3054—531no—
186C2×#430—531no—
187#2239—534no—
188#2238—534no—
189#2155—531no—
190#2155—531no—
191#2148—531no—
192#2148D5311yescharacteristic, complemented
193C2×#349—531no—
194C2×C53—21no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—161no—
198C2×C53C2×#48111yescharacteristic
199D53—42no—
200C2×#135C2×D5311yescharacteristic
201C22×M16—531no—
202C22×M16—1061no—
203#525—1061no—
204C2×M32—531no—
205C2×M32C2×D5311yescharacteristic
206#470—532no—
207#469—532no—
208#468—1062no—
209#430—534no—
210#430—1062no—
211C2×#108—531no—
212C2×#108—1061no—
213C2×#108C2×D5311yescharacteristic
214#427—531no—
215#416—1061no—
216#365—534no—
217#365—534no—
218#362—532no—
219#362C2×D5312yescomplemented
220#361—532no—
221#361C2×D5312yescomplemented
222#349—534no—
223C53#3365611yescharacteristic, complemented, Sylow
224C2×#41—21no—
225C23×C4—531no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135—21no—
230C2×Q16C22×D5311yescharacteristic
231C2×SD16—21no—
232C2×D8C22×D5311yescharacteristic
233C2×M16—21no—
234C2×M16—532no—
235C2×M16—532no—
236C2×M16—1061no—
237C2×M16—1061no—
238C2×M16—1061no—
239C2×M16—1061no—
240C2×M16—1064no—
241C2×M16—1061no—
242C2×M16—2121no—
243C2×M16—2122no—
244C22×C8—531no—
245C22×C8—2121no—
246C22×C8C22×D5311yescharacteristic
247#123—2121no—
248C22⋊Q8—1062no—
249#121—1062no—
250#120—2121no—
251C4×D4—2122no—
252#117—1061no—
253M32—21no—
254M32—532no—
255M32—1061no—
256M32C22×D5312yes—
257C2×C16—21no—
258C2×C16—1061no—
259#108—21no—
260#108—532no—
261#108—1064no—
262#108—1061no—
263#108C22×D5312yes—
264#107—1061no—
265#106—532no—
266#103—1062no—
267#102—1062no—
268C24—531no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—2122no—
277C22×C4—41no—
278C22×C4—531no—
279C22×C4—532no—
280C22×C4—532no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4D4×D5311yescharacteristic
286Q16—21no—
287Q16—21no—
288SD16—41no—
289SD16—41no—
290D8—21no—
291D8—21no—
292M16—22no—
293M16—41no—
294M16—1062no—
295M16—1062no—
296M16—1062no—
297M16—1062no—
298M16—1062no—
299M16—2124no—
300M16—2121no—
301C2×C8—41no—
302C2×C8—532no—
303C2×C8—1061no—
304C2×C8—1061no—
305C2×C8—2124no—
306C2×C8—2121no—
307C2×C8#1171994511yescharacteristic
308C2×C8#1171994511yescharacteristic
309C2×C8D4×D5311yescharacteristic
310C2×C8#1171994711yescharacteristic
311C2×C8#1171994711yescharacteristic
312C2×C8C23×D5311yescharacteristic
313C4⋊C4—1062no—
314C4⋊C4—2122no—
315C22⋊C4—2122no—
316C22⋊C4—2122no—
317C22⋊C4—2121no—
318C42—2121no—
319C16—22no—
320C16—1062no—
321C23—41no—
322C23—532no—
323C23—1061no—
324C23—1061no—
325C23—4241no—
326C23SD16×D5311yescharacteristic
327Q8—41no—
328Q8—41no—
329D4—41no—
330D4—41no—
331D4—81no—
332D4—81no—
333D4—4244no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—532no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—2122no—
345C2×C4#1313319911yescharacteristic
346C2×C4#1313319911yescharacteristic
347C2×C4SD16×D5311yescharacteristic
348C2×C4#1313324111yescharacteristic
349C2×C4#1313324111yescharacteristic
350C2×C4C2×D4×D5311yescharacteristic
351C8—42no—
352C8—1062no—
353C8—2122no—
354C8#1313319911yescharacteristic
355C8#1313319911yescharacteristic
356C8C2×#1171994711yescharacteristic
357C8C2×#1171994511yescharacteristic
358C22—41no—
359C22—82no—
360C22—81no—
361C22—1062no—
362C22—4241no—
363C22—2122no—
364C22—2122no—
365C22#1681094011yescharacteristic
366C22#1681094011yescharacteristic
367C22C2×SD16×D5311yescharacteristic
368C4—81no—
369C4—1062no—
370C4—4242no—
371C4#1681094011yescharacteristic
372C4#1681094011yescharacteristic
373C4C2×#1313319911yescharacteristic
374C4C2×#1313324111yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—161no—
379C2—2122no—
380C2C2×#1681094011yescharacteristic
subgroupG/H
centerC4#16810940
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2148C2
socleC2×C53C2×#481
prime radical#20284320C2
hypercenter#2148D53
hypocenterC53#33656
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. C2×#16810670C2semidirect
3. #20276933C2semidirect
4. #20276928C2semidirect
5. #20276489C2semidirect
6. #20276488C2semidirect
7. #20276464C2semidirect
8. C2×#16810404C2semidirect
9. C53×#2148C2semidirect
23. #16810447C22semidirect
24. #16810446C22semidirect
25. #16810439C22semidirect
26. #16810436C22semidirect
31. C53×#362C22semidirect
32. C53×#361C22semidirect
192. #2148D53semidirect
219. #362C2×D53semidirect
221. #361C2×D53semidirect
223. C53#33656semidirect

Maximal subgroups

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

subgroupindex
#202843202
C2×#168106702
#202769332
#202769282
#202764892
#202764882
#202764642
C2×#168104042
C53×#21482
#3365653