Finite Group Databasebeta

← #27280001

Subgroups — #27280001

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277625C211yesmaximal, characteristic
2#20273279C211yesmaximal, characteristic, complemented
3#20273214C211yesmaximal, characteristic, complemented
4#20273106C211yesmaximal, characteristic
5#20272616C211yesmaximal, characteristic, complemented
6#20272575C211yesmaximal, characteristic, complemented
7C53×#3077C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9#16810676C2211yescharacteristic
10#16810676C2211yescharacteristic
11#16810522—21no—
12#16810498—22no—
13#16810489C2211yescharacteristic
14#16810483—21no—
15C4×C4⋊Dic53C2211yescharacteristic
16C2×#13133124—21no—
17#16810360—21no—
18#16810360—21no—
19#16810360—22no—
20D4⋊#88322—22no—
21C53×#533C2211yescharacteristic
22C2×C4×D4×C53C2211yescharacteristic
23C53×#478—22no—
24C53×#476C412yescomplemented
25C53×#474C412yescomplemented
26C53×#428—21no—
27C53×#423C2211yescharacteristic
28C53×#418—21no—
29C23×Dic53—22no—
30D4×Dic53—42no—
31D4×Dic53—42no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C4×Dic53—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C4×#88322—21no—
42C4×#88322—21no—
43#13133124—22no—
44#13133124—21no—
45#13133124D412yes—
46#13133124D411yescharacteristic
47#13133124D411yescharacteristic
48C22×D4×C53D411yescharacteristic
49C23×C4×C53—21no—
50C2×M16×C53—21no—
51C4⋊Q8×C53C2×C411yescharacteristic, complemented
52C4⋊Q8×C53C2×C411yescharacteristic, complemented
53#125×C53—21no—
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53C2×C412yes—
61C4×D4×C53D412yes—
62#117×C53—21no—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C22⋊C4×C53—21no—
65C2×C42×C53C2311yescharacteristic
66#107×C53—22no—
67#106×C53—22no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53C2×C412yescomplemented
70#102×C53—22no—
71#102×C53—22no—
72C24×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53C22⋊C412yes—
79C2×D4×C53SD1612yes—
80C2×D4×C53Q1612yes—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—82no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×Dic53—21no—
89C22×Dic53—22no—
90C22×Dic53—21no—
91C22×Dic53—44no—
92C22×Dic53—44no—
93C22×Dic53—82no—
94M16×C53—41no—
95M16×C53—41no—
96C2×C8×C53—22no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53C22⋊C412yes—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—41no—
108C42×C53—41no—
109C42×C53C22×C411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112C42×C53#4111yescharacteristic
113C22⋊Dic53—44no—
114C4⋊Dic53—42no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—82no—
118C4×Dic53—41no—
119C4×Dic53—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C23×C53—22no—
123C23×C53—22no—
124C23×C53—22no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53#12011yescharacteristic
128Q8×C53—81no—
129Q8×C53—81no—
130D4×C53—22no—
131D4×C53—22no—
132D4×C53—22no—
133D4×C53#10314yes—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—82no—
141C2×C4×C53—82no—
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53C2×SD1611yescharacteristic
149C2×C4×C53C2×Q1611yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×Dic53—22no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—42no—
160C2×Dic53—41no—
161C2×Dic53—88no—
162C8×C53—42no—
163#88322—81no—
164#88322—81no—
165#9175—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—24no—
169C22×C53—44no—
170C22×C53—44no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53#38611yescharacteristic
174C22×C53#45011yescharacteristic
175C22×C53#45111yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53—81no—
182C4×C53—81no—
183C4×C53C2×#10311yescharacteristic
184C4×C53#41711yescharacteristic
185C4×C53#48311yescharacteristic
186C4×C53#48411yescharacteristic
187Dic53—42no—
188Dic53—81no—
189Dic53—82no—
190#3077D5311yescharacteristic
191#2349—531no—
192#1518—531no—
193#1503—531no—
194#1478—531no—
195#1303—531no—
196#1286—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—44no—
200C2×C53#187211yescharacteristic
201C2×C53#190411yescharacteristic
202C2×C53#190611yescharacteristic
203#533C2×D5311yescharacteristic
204C2×C4×D4—1061no—
205C2×C4×D4C2×D5311yescharacteristic
206#478—22no—
207#476Dic5312yescomplemented
208#474Dic5312yescomplemented
209#431—531no—
210#431—531no—
211#428—21no—
212#423C2×D5311yescharacteristic
213#418—21no—
214#389—1061no—
215#385—1062no—
216#382—531no—
217#380—1061no—
218C4×C4⋊C4—531no—
219C2×#95—1061no—
220#328—1061no—
221#328—1061no—
222#328—1062no—
223D4⋊C8—1062no—
224C53#917511yescharacteristic, complemented, Sylow
225C22×D4C22⋊D5311yescharacteristic
226C23×C4—21no—
227C23×C4—1062no—
228C2×M16—21no—
229C4⋊Q8C2×Dic5311yescharacteristic, complemented
230C4⋊Q8C2×Dic5311yescharacteristic, complemented
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—2122no—
239C4×D4—2122no—
240C4×D4C2×Dic5312yes—
241C4×D4C22⋊D5312yes—
242#117—21no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4C22⋊D5311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1062no—
249C2×C42—1061no—
250C2×C42C22×D5311yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—1061no—
255C4⋊C8—1061no—
256C4⋊C8C2×Dic5312yescomplemented
257#102—22no—
258#102—22no—
259C22⋊C8—1061no—
260C22⋊C8—1061no—
261C4×C8—1061no—
262C4×C8—1061no—
263#95—531no—
264#95—531no—
265#95—532no—
266#95—1062no—
267#95—1061no—
268C24—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#1171992412yes—
275C2×D4#1171992512yes—
276C2×D4C22⋊Dic5312yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1061no—
284C22×C4—4242no—
285C22×C4—2124no—
286C22×C4—2124no—
287C22×C4D4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—2121no—
294C2×C8—2121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—4242no—
303C4⋊C4—2122no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C4⋊C4C22⋊Dic5312yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—2124no—
311C42—41no—
312C42—2121no—
313C42—2121no—
314C42#1171994711yescharacteristic
315C42#1171994711yescharacteristic
316C42C22×Dic5311yescharacteristic
317C42C2×C22⋊D5311yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1313318511yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1313314114yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—1062no—
339C2×C4—4248no—
340C2×C4—2121no—
341C2×C4—2122no—
342C2×C4—2121no—
343C2×C4—2122no—
344C2×C4—2121no—
345C2×C4—2122no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4C2×C22⋊Dic5311yescharacteristic
349C2×C4D4×Dic5311yescharacteristic
350C2×C4D4×Dic5311yescharacteristic
351C2×C4#1313318911yescharacteristic
352C2×C4#1313318911yescharacteristic
353C2×C4#1313319811yescharacteristic
354C2×C4C2×#1171992411yescharacteristic
355C2×C4C2×#1171992511yescharacteristic
356C2×C4#1313324211yescharacteristic
357C2×C4#1313324711yescharacteristic
358C8—42no—
359C8—4241no—
360C8—4241no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1681050311yescharacteristic
369C22#1681077311yescharacteristic
370C22#1681077811yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—4242no—
379C4—4241no—
380C4—2122no—
381C4C2×#1313314111yescharacteristic
382C4#1681062411yescharacteristic
383C4#1681095111yescharacteristic
384C4#1681095611yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2027502311yescharacteristic
389C2#2027520411yescharacteristic
390C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3077C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#3077D53
hypocenterC53#9175
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #20273279C2semidirect
3. #20273214C2semidirect
5. #20272616C2semidirect
6. #20272575C2semidirect
24. C53×#476C4semidirect
25. C53×#474C4semidirect
51. C4⋊Q8×C53C2×C4semidirect
52. C4⋊Q8×C53C2×C4semidirect
69. C4⋊C8×C53C2×C4semidirect
207. #476Dic53semidirect
208. #474Dic53semidirect
224. C53#9175semidirect
229. C4⋊Q8C2×Dic53semidirect
230. C4⋊Q8C2×Dic53semidirect
256. C4⋊C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202776252
#202732792
#202732142
#202731062
#202726162
#202725752
C53×#30772
#917553