Finite Group Databasebeta

← #27277523

Subgroups — #27277523

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277120C211yesmaximal, characteristic
2#20274692C211yesmaximal, characteristic
3#20274692C211yesmaximal, characteristic
4#20272599C211yesmaximal, characteristic, complemented
5#20272599C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C2×#13133175—21no—
9C2×C42×D53C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133129—21no—
13C2×#13133129—21no—
14#16810493C2211yescharacteristic
15#16810488—21no—
16#16810480—21no—
17#16810480C412yes—
18C4⋊C4×Dic53—21no—
19C4⋊C4×Dic53C412yes—
20#16810377—21no—
21#16810377—21no—
22C2×C4⋊C8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C22×C4×D53C2×C411yescharacteristic
26C22×C4×D53D411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×#88322—21no—
29C22×#88322—21no—
30#13133175—42no—
31#13133175—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C412yes—
35C2×#11719921—22no—
36C2×C4×Dic53—22no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C42×D53—21no—
41C42×D53—21no—
42C42×D53—21no—
43C42×D53—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133129—42no—
49#13133129—42no—
50#13133124—22no—
51#13133124—22no—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C412yes—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C23×D53C22⋊C411yescharacteristic
63C22×C4×C53—22no—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—22no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53SD1612yes—
86C2×C4×D53Q1612yes—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C42×C53#4111yescharacteristic
99C42×C53#4111yescharacteristic
100C42×C53#4111yescharacteristic
101C42×C53#4111yescharacteristic
102#11719921—44no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107C4⋊Dic53—42no—
108C4×Dic53—21no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53—21no—
114C4×Dic53—44no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C23×C53C2×C22⋊C411yescharacteristic
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53#10312yes—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53#11711yescharacteristic
135C2×C4×C53#11711yescharacteristic
136C2×C4×C53#12311yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53#12311yescharacteristic
139C2×C4×C53#12311yescharacteristic
140C2×C4×C53C2×SD1611yescharacteristic
141C2×C4×C53C2×Q1611yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53#10312yes—
160C4×D53—22no—
161C4×D53—22no—
162C4×D53—22no—
163C4×D53—22no—
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—41no—
167C4×D53—41no—
168C4×D53—41no—
169C4×D53—41no—
170C8×C53—81no—
171C8×C53—81no—
172#88322—81no—
173#88322—81no—
174#8727—531nomaximal, Sylow
175C22×C53#38111yescharacteristic
176C22×C53#41311yescharacteristic
177C22×C53#41311yescharacteristic
178C22×C53C2×#10311yescharacteristic
179C22×C53#41611yescharacteristic
180C22×C53#41711yescharacteristic
181C22×C53#41811yescharacteristic
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—82no—
192C4×C53#48111yescharacteristic
193C4×C53#48111yescharacteristic
194C4×C53#48211yescharacteristic
195C4×C53#48211yescharacteristic
196C4×C53#48411yescharacteristic
197C4×C53#48411yescharacteristic
198C4×C53#48511yescharacteristic
199C4×C53#48511yescharacteristic
200Dic53—22no—
201Dic53—41no—
202Dic53—41no—
203Dic53—82no—
204#2285—531no—
205#1797—531no—
206#1797—531no—
207#1296—531no—
208#1296—531no—
209#1296—531no—
210#1296D5311yescharacteristic, complemented
211C2×C53#154211yescharacteristic
212C2×C53#155011yescharacteristic
213C2×C53#155311yescharacteristic
214C2×C53#155711yescharacteristic
215C2×C53#182211yescharacteristic
216C2×C53#182411yescharacteristic
217C2×C53#182511yescharacteristic
218D53—42no—
219C2×#117—1061no—
220C22×C42—531no—
221C2×C4⋊C8—531no—
222C2×C4⋊C8—531no—
223C2×C4⋊C8C2×D5311yescharacteristic
224C2×C4⋊C8C2×D5311yescharacteristic
225C2×C22⋊C8—1061no—
226C2×C22⋊C8—1061no—
227#384—531no—
228#384C2×D5311yescharacteristic
229#382—1061no—
230#380—532no—
231#380—1061no—
232C4×C4⋊C4—532no—
233C4×C4⋊C4—1061no—
234#336—1061no—
235#336—1061no—
236C53#872711yescharacteristic, complemented, Sylow
237C23×C4—531no—
238C23×C4—531no—
239C23×C4—531no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—1061no—
243C22×C8—1061no—
244#117—2122no—
245#117—2122no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—532no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4C4×D5312yes—
252C2×C22⋊C4—1062no—
253C2×C42—531no—
254C2×C42—531no—
255C2×C42—531no—
256C2×C42—1061no—
257C2×C42—1061no—
258C2×C42—1061no—
259C2×C42—1061no—
260C2×C42—1062no—
261C2×C42C22×D5311yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270C22⋊C8—2122no—
271C22⋊C8—2122no—
272#95—1062no—
273#95—1062no—
274C24—531no—
275C22×C4—22no—
276C22×C4—531no—
277C22×C4—532no—
278C22×C4—532no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1062no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4C2×C4×D5311yescharacteristic
297C22×C4D4×D5311yescharacteristic
298C22×C4D4×D5311yescharacteristic
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—2122no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C22⋊C4—2124no—
322C42—1061no—
323C42—1061no—
324C42—1061no—
325C42—1061no—
326C42—1061no—
327C42—1061no—
328C42—2124no—
329C42#1171994511yescharacteristic
330C42#1171994511yescharacteristic
331C42#1171994711yescharacteristic
332C42#1171994911yescharacteristic
333C23—532no—
334C23—1061no—
335C23—1061no—
336C23—1061no—
337C23—1061no—
338C23—1061no—
339C23—1061no—
340C23C22⋊C4×D5311yescharacteristic
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—532no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1062no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4#1313317411yescharacteristic
369C2×C4#1313317511yescharacteristic
370C2×C4#1313319911yescharacteristic
371C2×C4#1313319911yescharacteristic
372C2×C4#1313320211yescharacteristic
373C2×C4#1313320211yescharacteristic
374C2×C4SD16×D5311yescharacteristic
375C2×C4Q16×D5311yescharacteristic
376C2×C4#1313323611yescharacteristic
377C2×C4#1313323811yescharacteristic
378C2×C4#1313324111yescharacteristic
379C2×C4#1313324411yescharacteristic
380C2×C4#1313324411yescharacteristic
381C2×C4#1313324911yescharacteristic
382C8—81no—
383C8—81no—
384C8—4241no—
385C8—4241no—
386C22—1062no—
387C22—1062no—
388C22—1062no—
389C22—2121no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22#1681048511yescharacteristic
394C22#1681060411yescharacteristic
395C22#1681060611yescharacteristic
396C22#103×D5311yescharacteristic
397C22#1681061611yescharacteristic
398C22#1681062311yescharacteristic
399C22#1681062611yescharacteristic
400C4—21no—
401C4—21no—
402C4—82no—
403C4—1062no—
404C4—4242no—
405C4—2121no—
406C4—2121no—
407C4#1681094011yescharacteristic
408C4#1681094311yescharacteristic
409C4#1681094511yescharacteristic
410C4#1681094911yescharacteristic
411C4#1681095711yescharacteristic
412C4#1681096011yescharacteristic
413C4#1681096211yescharacteristic
414C4#1681096611yescharacteristic
415C2—2122no—
416C2#2027337111yescharacteristic
417C2#2027340211yescharacteristic
418C2#2027341711yescharacteristic
419C2#2027343611yescharacteristic
420C2#2027479711yescharacteristic
421C2#2027480411yescharacteristic
422C2#2027480811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#8727
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
4. #20272599C2semidirect
5. #20272599C2semidirect
6. #20272598C2semidirect
7. C53×#1296C2semidirect
210. #1296D53semidirect
236. C53#8727semidirect

Maximal subgroups

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

subgroupindex
#202771202
#202746922
#202725992
#202725982
C53×#12962
#872753