Finite Group Databasebeta

← #27280002

Subgroups — #27280002

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#20284508C211yesmaximal, characteristic
2#20277626C211yesmaximal, characteristic
3#20273279C211yesmaximal, characteristic, complemented
4#20273105C211yesmaximal, characteristic
5#20272618C211yesmaximal, characteristic, complemented
6#20272574C211yesmaximal, characteristic, complemented
7C53×#1503C211yesmaximal, characteristic, complemented
8C2×C4×D212—21no—
9C2×C4×D212C2211yescharacteristic
10#16810920—22no—
11#16810910C412yescomplemented
12#16810900C412yescomplemented
13#16810676C2211yescharacteristic
14#16810666—21no—
15#16810647C2211yescharacteristic
16#16810627—21no—
17#16810521—21no—
18#16810494—22no—
19#16810489C2211yescharacteristic
20#16810482—21no—
21#95×D53—21no—
22#16810360—22no—
23#16810350—22no—
24C53×#533C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C4×C4⋊C4×C53C2211yescharacteristic
27C53×#328—21no—
28C53×#328—21no—
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C22×C4×D53—22no—
32C2×#11719917—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—42no—
36C4×D212—42no—
37C4×D212C2×C412yes—
38C4×D212D412yes—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719919—22no—
42C2×#11719921—22no—
43C2×#11719921—21no—
44#13133150—22no—
45#13133148—22no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322C2×C412yescomplemented
49#13133137—22no—
50#13133137—22no—
51C22⋊#88322—21no—
52C4×#88322—21no—
53#13133124—22no—
54#13133124—21no—
55#13133124D412yes—
56C4⋊Q8×C53C2×C411yescharacteristic, complemented
57C4⋊Q8×C53C2×C411yescharacteristic, complemented
58#125×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63#117×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C22⋊C8×C53—21no—
69C4×C8×C53—21no—
70#95×C53D411yescharacteristic
71#95×C53D411yescharacteristic
72C23×D53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—22no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212C22⋊C412yes—
86C2×D212SD1612yes—
87C2×D212Q1612yes—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—44no—
91C2×C4×D53—44no—
92C2×C4×D53—82no—
93C2×C4×D53—82no—
94C2×C8×C53—41no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C42×C53—41no—
104C42×C53—41no—
105C42×C53—41no—
106C42×C53C22×C411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110#11719921—22no—
111#11719921—22no—
112#11719921—44no—
113C4⋊Dic53—21no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53C22⋊C412yes—
117#11719919—82no—
118#11719917—41no—
119#11719917—41no—
120C2×#88322—22no—
121C2×#88322—41no—
122C23×C53#12011yescharacteristic
123C22×D53—22no—
124C22×D53—22no—
125C22×D53—22no—
126C22×D53—42no—
127C22×D53—42no—
128Q8×C53—81no—
129Q8×C53—81no—
130C2×C4×C53—22no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
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×C53C2×Q1611yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—22no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155D212—22no—
156D212—22no—
157D212—22no—
158D212#10314yes—
159C4×D53—82no—
160C4×D53—82no—
161C4×D53—88no—
162C8×C53—81no—
163#88322—42no—
164#88322—81no—
165#9175—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53#38611yescharacteristic
169C22×C53#45011yescharacteristic
170C22×C53#45111yescharacteristic
171C2×D53—24no—
172C2×D53—44no—
173C2×D53—44no—
174C2×D53—42no—
175C2×D53—42no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53—81no—
182C4×C53—81no—
183C4×C53—81no—
184C4×C53C2×#10311yescharacteristic
185C4×C53#41711yescharacteristic
186C4×C53#48311yescharacteristic
187C4×C53#48411yescharacteristic
188Dic53—42no—
189Dic53—82no—
190#3077—531no—
191#2349—531no—
192#1518—531no—
193#1503D5311yescharacteristic, complemented
194#1478—531no—
195#1303—531no—
196#1286—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53#187211yescharacteristic
200C2×C53#190411yescharacteristic
201C2×C53#190611yescharacteristic
202D53—44no—
203#533C2×D5311yescharacteristic
204C2×C4×D4—531no—
205C2×C4×D4—1061no—
206#478—1062no—
207#476—532no—
208#474—532no—
209#431—531no—
210#431C2×D5311yescharacteristic
211#428—1061no—
212#423—531no—
213#418—1061no—
214#389—1061no—
215#385—1062no—
216#382—531no—
217#380—1061no—
218C4×C4⋊C4C2×D5311yescharacteristic
219C2×#95—1061no—
220#328—21no—
221#328—21no—
222#328—1062no—
223D4⋊C8—1062no—
224C53#917511yescharacteristic, complemented, Sylow
225C22×D4—531no—
226C23×C4—1062no—
227C23×C4—1061no—
228C2×M16—1061no—
229C4⋊Q8C4×D5311yescharacteristic, complemented
230C4⋊Q8C4×D5311yescharacteristic, complemented
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—532no—
237C4×D4—532no—
238C4×D4—1061no—
239C4×D4—1061no—
240C4×D4—2122no—
241C4×D4—2122no—
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1061no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1062no—
249C2×C42—21no—
250C2×C42C22×D5311yescharacteristic
251#107—1062no—
252#106—1062no—
253C4⋊C8—21no—
254C4⋊C8—532no—
255C4⋊C8—1061no—
256C4⋊C8—1061no—
257#102—1062no—
258#102—1062no—
259C22⋊C8—21no—
260C22⋊C8—1061no—
261C4×C8—21no—
262C4×C8—1061no—
263#95—532no—
264#95—1062no—
265#95—1061no—
266#95C22⋊D5311yescharacteristic
267#95C22⋊D5311yescharacteristic
268C24—1061no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—532no—
272C2×D4—532no—
273C2×D4—532no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—1061no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—4242no—
284C22×C4—4242no—
285C22×C4—2124no—
286C22×C4—2124no—
287C22×C4D4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290M16—2121no—
291M16—2121no—
292C2×C8—41no—
293C2×C8—1062no—
294C2×C8—2121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—532no—
303C4⋊C4—1061no—
304C4⋊C4—4242no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C22⋊C4—41no—
308C22⋊C4—1062no—
309C22⋊C4—1062no—
310C22⋊C4—2124no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42C2×C4×D5311yescharacteristic
315C42#1171994511yescharacteristic
316C42D4×D5311yescharacteristic
317C42#1171994711yescharacteristic
318C23—1062no—
319C23—1062no—
320C23—1062no—
321C23—2122no—
322C23—2122no—
323C23#1313318511yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—534no—
327D4—1062no—
328D4—1062no—
329D4—1062no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—1062no—
341C2×C4—4242no—
342C2×C4—4242no—
343C2×C4—4248no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4C22⋊C4×D5311yescharacteristic
349C2×C4#1313317711yescharacteristic
350C2×C4C4×C22⋊D5311yescharacteristic
351C2×C4#1313318811yescharacteristic
352C2×C4#1313319011yescharacteristic
353C2×C4#1313319911yescharacteristic
354C2×C4SD16×D5311yescharacteristic
355C2×C4Q16×D5311yescharacteristic
356C2×C4#1313324411yescharacteristic
357C2×C4#1313324911yescharacteristic
358C8—81no—
359C8—4241no—
360C8—2122no—
361C22—21no—
362C22—21no—
363C22—1064no—
364C22—2124no—
365C22—2124no—
366C22—2122no—
367C22—2122no—
368C22#1681049911yescharacteristic
369C22#1681077211yescharacteristic
370C22#1681077711yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—4242no—
380C4—2122no—
381C4#103×D5311yescharacteristic
382C4#1681062311yescharacteristic
383C4#1681095211yescharacteristic
384C4#1681095711yescharacteristic
385C2—21no—
386C2—21no—
387C2—2124no—
388C2#2027501911yescharacteristic
389C2#2027520511yescharacteristic
390C2#2027521911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1503C2
socleC22×C53#386
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1503D53
hypocenterC53#9175
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
3. #20273279C2semidirect
5. #20272618C2semidirect
6. #20272574C2semidirect
7. C53×#1503C2semidirect
11. #16810910C4semidirect
12. #16810900C4semidirect
48. C4⋊#88322C2×C4semidirect
56. C4⋊Q8×C53C2×C4semidirect
57. C4⋊Q8×C53C2×C4semidirect
193. #1503D53semidirect
224. C53#9175semidirect
229. C4⋊Q8C4×D53semidirect
230. C4⋊Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202845082
#202776262
#202732792
#202731052
#202726182
#202725742
C53×#15032
#917553