Finite Group Databasebeta

← #27280793

Subgroups — #27280793

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277441C211yesmaximal, characteristic
2#20275697C211yesmaximal, characteristic, complemented
3#20273383C211yesmaximal, characteristic
4#20273279C211yesmaximal, characteristic, complemented
5#20272574C211yesmaximal, characteristic, complemented
6#20272543C211yesmaximal, characteristic, complemented
7C53×#3208C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9#16810676C2211yescharacteristic
10#16810647C2211yescharacteristic
11#16810526—21no—
12#16810503—22no—
13#16810503—22no—
14#16810489C2211yescharacteristic
15#16810469C2211yescharacteristic
16#16810382—21no—
17#16810382—21no—
18#16810367—22no—
19#16810360—22no—
20C53×#533C2211yescharacteristic
21C2×#127×C53C2211yescharacteristic
22C53×#492C412yescomplemented
23C53×#486C412yescomplemented
24C53×#460—22no—
25C53×#450—22no—
26C53×#431C2211yescharacteristic
27C23×Dic53—22no—
28C2×#11719917—21no—
29D4×Dic53—42no—
30D4×Dic53—42no—
31C2×C4⋊Dic53—21no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C22⋊#88322—21no—
41C4×#88322—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124D412yes—
46C22×D4×C53—21no—
47C22×D4×C53—21no—
48C22×D4×C53D412yes—
49C2×SD16×C53—24no—
50C4⋊Q8×C53C2×C411yescharacteristic, complemented
51C4⋊Q8×C53C2×C411yescharacteristic, complemented
52#127×C53—21no—
53#127×C53—21no—
54#127×C53C2×C412yes—
55#127×C53D412yes—
56#125×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61#117×C53—21no—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64#102×C53—24no—
65C22⋊C8×C53—21no—
66C4×C8×C53C2×C412yescomplemented
67C24×C53—22no—
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—22no—
74C2×D4×C53—22no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—82no—
80C2×D4×C53—82no—
81C2×D4×C53C22⋊C414yes—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—22no—
88C22×Dic53—44no—
89C22×Dic53—82no—
90C22×Dic53—82no—
91SD16×C53—84no—
92C2×C8×C53—22no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C42×C53—41no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104C42×C53#4111yescharacteristic
105C22⋊Dic53—44no—
106C22⋊Dic53—82no—
107C22⋊Dic53—82no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53SD1612yescomplemented
114#11719917—41no—
115#11719917—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C23×C53—24no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53#12011yescharacteristic
126Q8×C53—42no—
127Q8×C53—81no—
128D4×C53—44no—
129D4×C53—42no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—82no—
134D4×C53—82no—
135D4×C53—82no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—22no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159C2×Dic53—88no—
160C8×C53—42no—
161#88322—81no—
162#88322—81no—
163#9295—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—44no—
167C22×C53—44no—
168C22×C53—44no—
169C22×C53—82no—
170C22×C53—82no—
171C22×C53—82no—
172C22×C53—82no—
173C22×C53#38611yescharacteristic
174C22×C53#45011yescharacteristic
175C22×C53#45211yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—81no—
181C4×C53—81no—
182C4×C53C4×SD1611yescharacteristic
183C4×C53#44011yescharacteristic
184C4×C53#46111yescharacteristic
185C4×C53#46311yescharacteristic
186Dic53—42no—
187Dic53—81no—
188Dic53—82no—
189#3208D5311yescharacteristic
190#2328—531no—
191#1995—531no—
192#1545—531no—
193#1518—531no—
194#1286—531no—
195#1274—531no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—84no—
199C2×C53#187711yescharacteristic
200C2×C53#190411yescharacteristic
201C2×C53#191111yescharacteristic
202#533C2×D5311yescharacteristic
203C2×#127C2×D5311yescharacteristic
204C2×C4×D4—1061no—
205#492Dic5312yescomplemented
206#486Dic5312yescomplemented
207#460—22no—
208#450—22no—
209#431—531no—
210#431C2×D5311yescharacteristic
211#423—531no—
212#390—1061no—
213#386—1062no—
214#386—1062no—
215#382—531no—
216#376—531no—
217#339—1061no—
218#339—1061no—
219#331—1062no—
220#328—1062no—
221C53#929511yescharacteristic, complemented, Sylow
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C22⋊D5312yes—
225C23×C4—1062no—
226C2×SD16—24no—
227C2×M16—1061no—
228C4⋊Q8C2×Dic5311yescharacteristic, complemented
229C4⋊Q8C2×Dic5311yescharacteristic, complemented
230#127—21no—
231#127—21no—
232#127C2×Dic5312yes—
233#127C22⋊D5312yes—
234#125—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—2122no—
240C4×D4—2122no—
241#117—21no—
242C2×C4⋊C4—1061no—
243C2×C22⋊C4—1062no—
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1062no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42C22×D5311yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—1061no—
251C4⋊C8—1061no—
252C4⋊C8—1061no—
253#102—24no—
254C22⋊C8—21no—
255C22⋊C8—1061no—
256C4×C8—1061no—
257C4×C8C2×Dic5312yescomplemented
258#95—532no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262C24—22no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4C22⋊Dic5314yes—
277C22×C4—1061no—
278C22×C4—1062no—
279C22×C4—1061no—
280C22×C4—4242no—
281C22×C4—4242no—
282C22×C4—2124no—
283C22×C4D4×D5311yescharacteristic
284C22×C4D4×D5311yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286SD16—84no—
287M16—2121no—
288M16—2121no—
289C2×C8—22no—
290C2×C8—2121no—
291C2×C8—2121no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—2121no—
299C4⋊C4—2121no—
300C22⋊C4—41no—
301C22⋊C4—4242no—
302C22⋊C4—4242no—
303C22⋊C4—2124no—
304C42—41no—
305C42—532no—
306C42—1061no—
307C42—1061no—
308C42—1061no—
309C42#1171994711yescharacteristic
310C42#1171994711yescharacteristic
311C42C22×Dic5311yescharacteristic
312C42C2×C22⋊D5311yescharacteristic
313C23—24no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#1313318511yescharacteristic
321Q8—42no—
322Q8—81no—
323D4—44no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—1062no—
336C2×C4—4248no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4C2×C22⋊Dic5311yescharacteristic
346C2×C4D4×Dic5311yescharacteristic
347C2×C4D4×Dic5311yescharacteristic
348C2×C4#1313318911yescharacteristic
349C2×C4#1313318911yescharacteristic
350C2×C4#1313319811yescharacteristic
351C2×C4SD16×D5311yescharacteristic
352C2×C4#1313323811yescharacteristic
353C2×C4#1313324411yescharacteristic
354C2×C4#1313324611yescharacteristic
355C8—42no—
356C8—4241no—
357C8—4241no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#1681050311yescharacteristic
368C22#1681077211yescharacteristic
369C22#1681078211yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—81no—
375C4—81no—
376C4—4242no—
377C4—4241no—
378C4—2122no—
379C4SD16×Dic5311yescharacteristic
380C4#1681072411yescharacteristic
381C4#1681082611yescharacteristic
382C4#1681084011yescharacteristic
383C2—21no—
384C2—21no—
385C2—84no—
386C2#2027505111yescharacteristic
387C2#2027520411yescharacteristic
388C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3208C2
socleC22×C53#386
prime radicalC2×#127×C53C22
cosocleC2×C42×C53C23
hypercenter#3208D53
hypocenterC53#9295
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
2. #20275697C2semidirect
4. #20273279C2semidirect
5. #20272574C2semidirect
6. #20272543C2semidirect
22. C53×#492C4semidirect
23. C53×#486C4semidirect
50. C4⋊Q8×C53C2×C4semidirect
51. C4⋊Q8×C53C2×C4semidirect
66. C4×C8×C53C2×C4semidirect
113. C4×Dic53SD16semidirect
205. #492Dic53semidirect
206. #486Dic53semidirect
221. C53#9295semidirect
228. C4⋊Q8C2×Dic53semidirect
229. C4⋊Q8C2×Dic53semidirect
257. C4×C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202774412
#202756972
#202733832
#202732792
#202725742
#202725432
C53×#32082
#929553