Finite Group Databasebeta

← #27541019

Subgroups — #27541019

Order 13568.

546 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
1#20287146C211yesmaximal, characteristic, complemented
2#20276942C211yesmaximal, characteristic, complemented
3#20276920C212yesmaximal, complemented
4#20276852C211yesmaximal, characteristic, complemented
5#20276804C212yesmaximal, complemented
6#20276762C211yesmaximal, characteristic, complemented
7#20276655C212yesmaximal, complemented
8C4×#13133161C211yesmaximal, characteristic, complemented
9D4×#11719908C211yesmaximal, characteristic, complemented
10#20276422C212yesmaximal, complemented
11C53×#3283C211yesmaximal, characteristic
12#16811343—21no—
13C2×#13133163C2212yes—
14C2×#13133161C2211yescharacteristic
15C22×#11719908C2212yes—
16#16811028C2211yescharacteristic
17#16810965—22no—
18#16810909—21no—
19#16810864C2212yes—
20C4×Dic212C2211yescharacteristic
21#16810661C2212yes—
22#16810450C2211yescharacteristic
23#16810448C2212yes—
24C4⋊#11719908C2211yescharacteristic
25#16810427C2212yes—
26#16810427C2212yes—
27#16810427C2212yes—
28#16810426C2211yescharacteristic
29#16810424C2211yescharacteristic
30#16810424C2212yes—
31C22⋊#11719908C2212yes—
32C4×#11719908C2211yescharacteristic
33C2×#135×C53C2212yes—
34C53×#547—21no—
35C53×#494C2211yescharacteristic
36C53×#467C2212yes—
37C53×#461—21no—
38C53×#452—22no—
39C4×D8×C53C2211yescharacteristic
40C8×D4×C53C2211yescharacteristic
41C2×Dic212C2311yescharacteristic
42#13133217—21no—
43#13133210—42no—
44#13133204—41no—
45#13133196—22no—
46C4×Dic106—21no—
47#13133182—41no—
48#13133176—22no—
49#13133163—42no—
50#13133161—22no—
51C2×#11719908—42no—
52C2×#11719908C2311yescharacteristic
53C2×#11719908C2312yes—
54C2×#11719908C2311yescharacteristic
55#13133151—21no—
56#13133151C2312yes—
57#13133151C2311yescharacteristic
58#13133149—22no—
59#13133142—22no—
60#13133142—21no—
61C2×#41×C53—22no—
62#135×C53—24no—
63#135×C53—42no—
64C2×Q16×C53—21no—
65C2×Q16×C53C2312yes—
66C2×SD16×C53—22no—
67C2×D8×C53C2311yescharacteristic
68C22×C8×C53C2312yes—
69C4⋊Q8×C53—21no—
70#123×C53—42no—
71C22⋊Q8×C53—22no—
72#121×C53—41no—
73C4×D4×C53—21no—
74C4×D4×C53D411yescharacteristic
75C2×C4⋊C4×C53—41no—
76#107×C53C2311yescharacteristic
77C4⋊C8×C53D411yescharacteristic
78#103×C53—22no—
79#102×C53—21no—
80C22⋊C8×C53D412yes—
81C4×C8×C53C2311yescharacteristic
82#41×C53—44no—
83#41×C53—82no—
84C2×Q8×C53—22no—
85C2×D4×C53—21no—
86C2×D4×C53—42no—
87C2×D4×C53D811yescharacteristic
88C22×C4×C53—41no—
89C22×C4×C53—42no—
90C22×C4×C53C2×D412yes—
91C2×Dic106—21no—
92C2×Dic106—41no—
93Q16×C53—22no—
94Q16×C53—22no—
95Q16×C53—81no—
96SD16×C53—44no—
97D8×C53—21no—
98D8×C53#4112yes—
99C2×C8×C53—22no—
100C2×C8×C53—22no—
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D412yes—
103C2×C8×C53C2411yescharacteristic
104C4⋊C4×C53—21no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—42no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53D811yescharacteristic
109C22⋊C4×C53—42no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53D812yes—
112C42×C53C2×D411yescharacteristic
113C22⋊Dic53—42no—
114C22⋊Dic53—42no—
115C4⋊Dic53—21no—
116C4⋊Dic53—22no—
117C4⋊Dic53—41no—
118#11719919—42no—
119#11719919—41no—
120C4×Dic53—42no—
121C4×Dic53—41no—
122Dic212—21no—
123Dic212—21no—
124#11719908—21no—
125#11719908—21no—
126#11719908—22no—
127C23×C53—41no—
128C23×C53C2×D812yes—
129Q8×C53—42no—
130Q8×C53—42no—
131D4×C53—22no—
132D4×C53—22no—
133D4×C53—41no—
134D4×C53—84no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—84no—
140C2×C4×C53—82no—
141C2×C4×C53C2×D812yes—
142C2×C4×C53C2×D811yescharacteristic
143C2×C4×C53C2×D811yescharacteristic
144C2×C4×C53C22×D411yescharacteristic
145C2×Dic53—41no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148Dic106—41no—
149Dic106—41no—
150Dic106—82no—
151C8×C53—22no—
152C8×C53—21no—
153C8×C53C2×#4111yescharacteristic
154C8×C53H(5,𝔽2)11yescharacteristic
155#33869—531nomaximal, Sylow
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—42no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53C22×D811yescharacteristic
163C4×C53—21no—
164C4×C53—22no—
165C4×C53—81no—
166C4×C53—82no—
167C4×C53#50812yes—
168C4×C53#54611yescharacteristic
169C4×C53#57711yescharacteristic
170Dic53—82no—
171Dic53—81no—
172Dic53—81no—
173#3342—531no—
174#3283D5311yescharacteristic
175#2241—531no—
176#2237—532no—
177#2224—531no—
178#2216—532no—
179#2209—531no—
180#2189—532no—
181C4×SD32—531no—
182D4×C16—531no—
183#2138—532no—
184C2×C53—42no—
185C2×C53—82no—
186C2×C53#329411yescharacteristic
187C2×C53C2×#50811yescharacteristic
188C2×C53#341711yescharacteristic
189C2×#135C2×D5312yes—
190#556—1061no—
191#547—21no—
192C2×Q32—532no—
193C2×SD32—531no—
194C22×C16—532no—
195#500—531no—
196#494C2×D5311yescharacteristic
197#485—1062no—
198#475—1061no—
199#467—532no—
200#467C2×D5312yes—
201#461—21no—
202#452—22no—
203C4×Q16—531no—
204C4×D8C2×D5311yescharacteristic
205C8×D4C2×D5311yescharacteristic
206#427—532no—
207#367—531no—
208#366—532no—
209C4⋊C16—531no—
210#358—532no—
211#358—532no—
212#358—532no—
213#358—531no—
214#357—531no—
215#357—532no—
216C22⋊C16—532no—
217C4×C16—531no—
218C53#3386911yescharacteristic, complemented, Sylow
219C2×#41—22no—
220#135—24no—
221#135—42no—
222C2×Q16—21no—
223C2×Q16—531no—
224C2×Q16C22×D5312yes—
225C2×SD16—22no—
226C2×D8C22×D5311yescharacteristic
227C22×C8C22×D5312yes—
228C4⋊Q8—21no—
229C4⋊Q8—1061no—
230#126—2122no—
231#124—2121no—
232#123—42no—
233C22⋊Q8—22no—
234C22⋊Q8—1062no—
235#121—41no—
236C4×Q8—1061no—
237C4×Q8—2121no—
238C4×D4—21no—
239C4×D4C22⋊D5311yescharacteristic
240#117—1062no—
241C2×C4⋊C4—41no—
242Q32—2122no—
243SD32—1062no—
244C2×C16—531no—
245C2×C16—531no—
246C2×C16—532no—
247C2×C16—2122no—
248#107—532no—
249#107—531no—
250#107—1061no—
251#107C22×D5311yescharacteristic
252#106—1062no—
253C4⋊C8C22⋊D5311yescharacteristic
254#103—22no—
255#103—1061no—
256#103—1062no—
257#102—21no—
258C22⋊C8C22⋊D5312yes—
259C4×C8C22×D5311yescharacteristic
260#41—44no—
261#41—82no—
262C2×Q8—22no—
263C2×Q8—1061no—
264C2×Q8—2121no—
265C2×D4—21no—
266C2×D4—42no—
267C2×D4#1171992211yescharacteristic
268C22×C4—41no—
269C22×C4—42no—
270C22×C4C2×C22⋊D5312yes—
271Q16—22no—
272Q16—22no—
273Q16—81no—
274Q16—1061no—
275Q16—1061no—
276SD16—44no—
277D8—21no—
278D8#1171994712yes—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8C2×C22⋊D5311yescharacteristic
282C2×C8C2×C22⋊D5312yes—
283C2×C8C23×D5311yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—1061no—
289C4⋊C4—1062no—
290C4⋊C4—2121no—
291C4⋊C4—2122no—
292C4⋊C4—2121no—
293C4⋊C4#1171992211yescharacteristic
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—2122no—
297C22⋊C4—2122no—
298C22⋊C4#1171992212yes—
299C42—2122no—
300C42—2121no—
301C42C2×C22⋊D5311yescharacteristic
302C16—1061no—
303C16—1061no—
304C16—1062no—
305C23—41no—
306C23C2×#1171992212yes—
307Q8—42no—
308Q8—42no—
309Q8—4242no—
310Q8—2121no—
311Q8—2121no—
312D4—22no—
313D4—22no—
314D4—41no—
315D4—84no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—2121no—
323C2×C4—2122no—
324C2×C4—2121no—
325C2×C4C2×#1171992212yes—
326C2×C4C2×#1171992211yescharacteristic
327C2×C4C2×#1171992211yescharacteristic
328C2×C4C22×C22⋊D5311yescharacteristic
329C8—22no—
330C8—21no—
331C8C2×#1171994711yescharacteristic
332C8#1313326211yescharacteristic
333C22—22no—
334C22—22no—
335C22—22no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22C22×#1171992211yescharacteristic
340C4—22no—
341C4—21no—
342C4—82no—
343C4—81no—
344C4—4241no—
345C4—4242no—
346C4—4241no—
347C4#1681105312yes—
348C4#1681126711yescharacteristic
349C4#1681143411yescharacteristic
350C2—42no—
351C2—82no—
352C2#2028652811yescharacteristic
353C2C2×#1681105311yescharacteristic
354C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3283C2
socleC22×C53C22×D8
hypercenter#3283D53
hypocenterC53#33869
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #20287146C2semidirect
2. #20276942C2semidirect
3. #20276920C2semidirect
4. #20276852C2semidirect
5. #20276804C2semidirect
6. #20276762C2semidirect
7. #20276655C2semidirect
8. C4×#13133161C2semidirect
9. D4×#11719908C2semidirect
10. #20276422C2semidirect
218. C53#33869semidirect

Maximal subgroups

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

subgroupindex
#202871462
#202769422
#202769202
#202768522
#202768042
#202767622
#202766552
C4×#131331612
D4×#117199082
#202764222
C53×#32832
#3386953