Finite Group Databasebeta

← #27299339

Subgroups — #27299339

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283486C211yesmaximal, characteristic
2C2×#16810434C211yesmaximal, characteristic
3C2×#16810432C211yesmaximal, characteristic
4M32×Dic53C211yesmaximal, characteristic, complemented
5#106×Dic53C211yesmaximal, characteristic, complemented
6#20272820C211yesmaximal, characteristic, complemented
7C53×#1386C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9#16811091—21no—
10C2×#13133157C2211yescharacteristic
11#16810717C411yescharacteristic, complemented
12#16810717C411yescharacteristic, complemented
13#16810710—21no—
14#16810710—21no—
15#16810705C411yescharacteristic, complemented
16#16810705C411yescharacteristic, complemented
17C2×#13133149C2211yescharacteristic
18#16810617—21no—
19C2×C8×Dic53C2211yescharacteristic
20C4⋊C4×Dic53—21no—
21#16810434C414yescomplemented
22#16810432C414yescomplemented
23#16810398—21no—
24C16×Dic53—21no—
25#16810385—21no—
26C2×M32×C53C2211yescharacteristic
27C53×#427C2211yescharacteristic
28C2×#106×C53C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239C2×C412yes—
33#13133239C2×C412yes—
34C2×Dic212C2×C411yescharacteristic
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic
37C22×#88322D411yescharacteristic
38#13133182—21no—
39#13133182—21no—
40#13133179—21no—
41#13133179—21no—
42#13133177—41no—
43#13133177—41no—
44#13133174—41no—
45C2×C4⋊Dic53—21no—
46C2×C4×Dic53—41no—
47C2×C4×Dic53D411yescharacteristic
48C42×D53—41no—
49#13133157—21no—
50#13133157C2×C412yescomplemented
51C2×#11719908—21no—
52#13133149—21no—
53#13133149C2×C412yescomplemented
54#13133140—21no—
55#13133140—21no—
56#13133137—21no—
57#13133137—21no—
58C8×Dic53C2×C411yescharacteristic
59C8×Dic53C2×C411yescharacteristic
60C8×Dic53D411yescharacteristic
61C8×Dic53D411yescharacteristic
62#13133124—41no—
63C22×C8×C53C2311yescharacteristic
64#117×C53—21no—
65C2×C4⋊C4×C53—21no—
66M32×C53—21no—
67M32×C53C2×C412yescomplemented
68C2×C16×C53—21no—
69#107×C53C2×C411yescharacteristic
70#107×C53C2×C411yescharacteristic
71#106×C53—21no—
72#106×C53—21no—
73#106×C53C2×C412yescomplemented
74C22×C4×C53—41no—
75C22×C4×C53C2×D411yescharacteristic
76C2×C22⋊D53—41no—
77C22×Dic53—41no—
78C22×Dic53D811yescharacteristic
79C22×Dic53D811yescharacteristic
80#11719945—22no—
81#11719945—22no—
82#11719945—41no—
83#11719945—41no—
84C2×D212—21no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×Dic106—21no—
89C2×C8×C53C22×C411yescharacteristic
90C2×C8×C53C22×C411yescharacteristic
91C2×C8×C53C22×C411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C2×C8×C53#4111yescharacteristic
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C42×C53—41no—
101#11719921—41no—
102#11719921—41no—
103C4⋊Dic53—22no—
104C4⋊Dic53—41no—
105#11719919—41no—
106#11719919—41no—
107C4×Dic53—41no—
108C4×Dic53—41no—
109C4×Dic53—82no—
110C4×Dic53C22⋊C411yescharacteristic
111C4×Dic53C22⋊C411yescharacteristic
112C4×Dic53SD1611yescharacteristic
113C4×Dic53SD1611yescharacteristic
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322C22⋊C411yescharacteristic
117C2×#88322C22⋊C411yescharacteristic
118Dic212—21no—
119Dic212C4212yescomplemented
120D424—21no—
121D424C4212yescomplemented
122#11719913—22no—
123#11719913—22no—
124C16×C53—22no—
125#11719908—22no—
126C23×C53C2×D811yescharacteristic
127C22×D53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C2×SD1611yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C22⋊D53—42no—
140C22⋊D53—42no—
141C2×Dic53—22no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—42no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53#10211yescharacteristic
148C2×Dic53#10211yescharacteristic
149C2×Dic53#10211yescharacteristic
150C2×Dic53#10211yescharacteristic
151D212—22no—
152D212—41no—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53—82no—
156C4×D53—82no—
157Dic106—22no—
158Dic106—41no—
159C8×C53C2×C4211yescharacteristic
160C8×C53#11711yescharacteristic
161C8×C53C4×D411yescharacteristic
162C8×C53C4×D411yescharacteristic
163#88322—21no—
164#88322—21no—
165C4×#360—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53C2×#10211yescharacteristic
169C22×C53C4×D811yescharacteristic
170C22×C53C4×D811yescharacteristic
171C2×D53—42no—
172C2×D53—81no—
173C2×D53—81no—
174C4×C53—82no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53C4×C22⋊C411yescharacteristic
178C4×C53#41611yescharacteristic
179C4×C53C4×SD1611yescharacteristic
180C4×C53C4×SD1611yescharacteristic
181Dic53—21no—
182Dic53—21no—
183Dic53—21no—
184Dic53—42no—
185Dic53—82no—
186Dic53#36012yescomplemented
187C4×#135—531no—
188C2×#360—531no—
189C2×#360—531no—
190C4×M32—531no—
191C4×#106—531no—
192#1386—531no—
193#1386D5311yescharacteristic, complemented
194C2×C53—21no—
195C2×C53—21no—
196C2×C53C4×#10211yescharacteristic
197C2×C53C2×#36011yescharacteristic
198C2×C53#214511yescharacteristic
199D53—82no—
200C2×#135—531no—
201C4×#41—1061no—
202C2×M32—531no—
203C2×M32C2×D5311yescharacteristic
204C4×Q16—531no—
205C4×Q16—531no—
206C4×SD16—1061no—
207C4×SD16—1061no—
208C4×D8—531no—
209C4×D8—531no—
210#427C2×D5311yescharacteristic
211C2×#106—531no—
212C2×#106C2×D5311yescharacteristic
213#416—1061no—
214C2×C4×C8—531no—
215C4×C4⋊C4—1061no—
216#360—534no—
217#360—534no—
218#346—1061no—
219C4×C16—1061no—
220#340—1061no—
221C53C4×#36011yescharacteristic, complemented, Sylow
222C2×#41—1061no—
223#135—532no—
224#135—532no—
225#135—1061no—
226#135—1061no—
227C2×Q16—531no—
228C2×SD16—1061no—
229C2×D8—531no—
230C22×C8—531no—
231C22×C8C22×D5311yescharacteristic
232C4×Q8—1061no—
233C4×Q8—1061no—
234C4×D4—1061no—
235C4×D4—1061no—
236C4×D4—2121no—
237C4×D4—2121no—
238#117—21no—
239#117—2121no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—1061no—
242C2×C42—531no—
243C2×C42—2121no—
244C2×C42—2121no—
245M32—21no—
246M32—532no—
247M32—1061no—
248M32C4×D5312yescomplemented
249C2×C16—21no—
250C2×C16—1061no—
251#107C4×D5311yescharacteristic
252#107C4×D5311yescharacteristic
253#106—21no—
254#106—21no—
255#106—532no—
256#106—1061no—
257#106C4×D5312yescomplemented
258#103—1061no—
259#103—1061no—
260#102—1061no—
261#102—1061no—
262C4×C8—531no—
263C4×C8—531no—
264C4×C8—531no—
265C4×C8—531no—
266#95—2121no—
267#41—1062no—
268#41—1062no—
269#41—2121no—
270#41—2121no—
271C2×Q8—1061no—
272C2×D4—1061no—
273C2×D4—2121no—
274C22×C4—41no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4D4×D5311yescharacteristic
282Q16—532no—
283Q16—1061no—
284SD16—1062no—
285SD16—1062no—
286D8—532no—
287D8—1061no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—1061no—
291C2×C8—1061no—
292C2×C8C2×C4×D5311yescharacteristic
293C2×C8C2×C4×D5311yescharacteristic
294C2×C8C2×C4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8#1171994911yescharacteristic
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—1062no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—41no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—41no—
310C42—531no—
311C42—531no—
312C42—531no—
313C42—531no—
314C42—4242no—
315C42—2121no—
316C42—2121no—
317C16—22no—
318C16—1062no—
319C23—2121no—
320C23D8×D5311yescharacteristic
321Q8—1062no—
322Q8—2121no—
323D4—1062no—
324D4—2122no—
325D4—2122no—
326D4—2121no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—1062no—
337C2×C4—4242no—
338C2×C4—4242no—
339C2×C4—4241no—
340C2×C4—4241no—
341C2×C4—2121no—
342C2×C4—2122no—
343C2×C4—2121no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4C22⋊C4×D5311yescharacteristic
347C2×C4#1313317711yescharacteristic
348C2×C4#1313317911yescharacteristic
349C2×C4SD16×D5311yescharacteristic
350C2×C4#1313323811yescharacteristic
351C2×C4#1313324011yescharacteristic
352C8—1061no—
353C8—1061no—
354C8C42×D5311yescharacteristic
355C8#1313317511yescharacteristic
356C8#1313317711yescharacteristic
357C8#1313317911yescharacteristic
358C22—21no—
359C22—21no—
360C22—4241no—
361C22—4241no—
362C22—2122no—
363C22#102×D5311yescharacteristic
364C22#1681070311yescharacteristic
365C22#1681070511yescharacteristic
366C4—82no—
367C4—81no—
368C4—81no—
369C4—532no—
370C4—1061no—
371C4—1061no—
372C4—1061no—
373C4—4242no—
374C4—2122no—
375C4#1681047011yescharacteristic
376C4#1681061811yescharacteristic
377C4#1681070811yescharacteristic
378C4#1681071011yescharacteristic
379C2—21no—
380C2—21no—
381C2—4242no—
382C2#2027452911yescharacteristic
383C2#360×D5311yescharacteristic
384C2#2027644911yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC8×C53C2×C42
FrattiniC22×C8C22×D53
FittingC53×#1386C2
socleC22×C53C2×#102
prime radicalC2×#13133239C22
cosocleC22×C8×C53C23
hypercenter#1386D53
hypocenterC53C4×#360
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
4. M32×Dic53C2semidirect
5. #106×Dic53C2semidirect
6. #20272820C2semidirect
7. C53×#1386C2semidirect
11. #16810717C4semidirect
12. #16810717C4semidirect
15. #16810705C4semidirect
16. #16810705C4semidirect
21. #16810434C4semidirect
22. #16810432C4semidirect
50. #13133157C2×C4semidirect
53. #13133149C2×C4semidirect
67. M32×C53C2×C4semidirect
73. #106×C53C2×C4semidirect
119. Dic212C42semidirect
121. D424C42semidirect
186. Dic53#360semidirect
193. #1386D53semidirect
221. C53C4×#360semidirect
248. M32C4×D53semidirect
257. #106C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202834862
C2×#168104342
C2×#168104322
M32×Dic532
#106×Dic532
#202728202
C53×#13862
C4×#36053