Finite Group Databasebeta

← #27288848

Subgroups — #27288848

Order 13568.

402 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275509C211yesmaximal, characteristic
2#20274843C211yesmaximal, characteristic, complemented
3#20274716C211yesmaximal, characteristic
4#20274716C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1277C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×C4⋊#88322C2211yescharacteristic
10C2×#13133139—21no—
11C2×#13133139—21no—
12C2×#13133129—21no—
13C2×#13133129—21no—
14#16810493C2211yescharacteristic
15#16810493C2211yescharacteristic
16#16810386—21no—
17#16810386—21no—
18#16810386—21no—
19#16810386—21no—
20C2×C4⋊C8×C53C2211yescharacteristic
21C2×C4⋊C8×C53C2211yescharacteristic
22C2×C4×C8×C53C2211yescharacteristic
23C22×D212D411yescharacteristic
24C22×C4×D53—21no—
25C22×#88322—21no—
26C4×D212—21no—
27C4×D212—21no—
28C4×D212—21no—
29C4×D212—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719921—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40#13133139—42no—
41#13133139—42no—
42#13133129—42no—
43#13133129—42no—
44C22×C8×C53—21no—
45C22×C8×C53—21no—
46C22×C8×C53D411yescharacteristic
47C22×C8×C53D411yescharacteristic
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—21no—
50C4⋊C8×C53—21no—
51C4⋊C8×C53—21no—
52C4⋊C8×C53—21no—
53C4×C8×C53—21no—
54C4×C8×C53—21no—
55C23×D53—21no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53#4111yescharacteristic
59C22×Dic53—21no—
60C22×Dic53—41no—
61C22×Dic53—41no—
62C2×D212—21no—
63C2×D212—21no—
64C2×D212—21no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212SD1612yes—
68C2×C4×D53—41no—
69C2×C4×D53—41no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C8×C53—21no—
75C2×C8×C53—21no—
76C2×C8×C53—21no—
77C2×C8×C53—21no—
78C2×C8×C53—21no—
79C2×C8×C53—21no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C42×C53#4111yescharacteristic
87C42×C53#4111yescharacteristic
88C42×C53#4111yescharacteristic
89C42×C53#4111yescharacteristic
90#11719921—41no—
91#11719921—41no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53—81no—
103C4⋊Dic53—81no—
104C2×#88322—41no—
105C2×#88322—41no—
106C2×#88322—41no—
107C23×C53#12111yescharacteristic
108C22×D53—22no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—41no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53#12311yescharacteristic
118C2×C4×C53#12311yescharacteristic
119C2×C4×C53#12311yescharacteristic
120C2×C4×C53#12411yescharacteristic
121C2×C4×C53#12611yescharacteristic
122C2×C4×C53#12611yescharacteristic
123C2×C4×C53C2×SD1611yescharacteristic
124C2×C4×C53#13511yescharacteristic
125C2×C4×C53#13511yescharacteristic
126C2×C4×C53#13511yescharacteristic
127C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C53#13711yescharacteristic
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139C2×Dic53—81no—
140D212—22no—
141D212—22no—
142D212—41no—
143D212—41no—
144C4×D53—81no—
145C4×D53—81no—
146C4×D53—81no—
147C4×D53—81no—
148C8×C53—41no—
149C8×C53—41no—
150C8×C53—81no—
151C8×C53—81no—
152#88322—81no—
153#10625—531nomaximal, Sylow
154C22×C53#39711yescharacteristic
155C22×C53#46311yescharacteristic
156C22×C53#46311yescharacteristic
157C22×C53#46511yescharacteristic
158C22×C53#46811yescharacteristic
159C22×C53#47211yescharacteristic
160C22×C53#47311yescharacteristic
161C2×D53—42no—
162C2×D53—42no—
163C2×D53—42no—
164C2×D53—81no—
165C2×D53—81no—
166C2×D53—81no—
167C2×D53—81no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53#48111yescharacteristic
171C4×C53#48211yescharacteristic
172C4×C53#48211yescharacteristic
173C4×C53#48211yescharacteristic
174C4×C53#48311yescharacteristic
175C4×C53#48511yescharacteristic
176C4×C53#48811yescharacteristic
177C4×C53#49011yescharacteristic
178Dic53—81no—
179Dic53—161no—
180Dic53—161no—
181#1954—531no—
182#1834—531no—
183#1802—531no—
184#1802—531no—
185#1296—531no—
186#1296—531no—
187#1277D5311yescharacteristic, complemented
188C2×C53#167711yescharacteristic
189C2×C53#167911yescharacteristic
190C2×C53#169711yescharacteristic
191C2×C53#169911yescharacteristic
192C2×C53#207311yescharacteristic
193C2×C53#207711yescharacteristic
194C2×C53#207811yescharacteristic
195D53—82no—
196C2×C4×D4—531no—
197C2×C4⋊C8—531no—
198C2×C4⋊C8C2×D5311yescharacteristic
199C2×C4⋊C8C2×D5311yescharacteristic
200C2×#102—1061no—
201C2×#102—1061no—
202C2×C22⋊C8—1061no—
203C2×C22⋊C8—1061no—
204C2×C4×C8C2×D5311yescharacteristic
205#384—531no—
206#384—531no—
207#340—1061no—
208#340—1061no—
209#340—1061no—
210#340—1061no—
211C53#1062511yescharacteristic, complemented, Sylow
212C22×D4—531no—
213C23×C4—1061no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—1061no—
217C22×C8D21211yescharacteristic
218C22×C8D21211yescharacteristic
219C4×D4—1061no—
220C4×D4—1061no—
221C4×D4—1061no—
222C4×D4—1061no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C42C22×D5311yescharacteristic
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—1061no—
237C4⋊C8—1061no—
238#102—2122no—
239#102—2122no—
240C22⋊C8—2122no—
241C22⋊C8—2122no—
242C4×C8—21no—
243C4×C8—21no—
244C24—1061no—
245C2×D4—532no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C22×C4—1061no—
252C22×C4—2121no—
253C22×C4—2121no—
254C22×C4—2121no—
255C22×C4—2121no—
256C22×C4—2121no—
257C22×C4—2121no—
258C22×C4—2121no—
259C22×C4—2121no—
260C22×C4C2×D21211yescharacteristic
261C22×C4D4×D5311yescharacteristic
262C22×C4#1171994911yescharacteristic
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—2121no—
276C2×C8—2121no—
277C2×C8—2121no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—4241no—
281C4⋊C4—4241no—
282C4⋊C4—2121no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C22⋊C4—2121no—
291C22⋊C4—2121no—
292C42#1171994511yescharacteristic
293C42#1171994511yescharacteristic
294C42#1171994711yescharacteristic
295C42#1171994711yescharacteristic
296C23—1062no—
297C23—2121no—
298C23—2121no—
299C23—2121no—
300C23—2121no—
301C23—2121no—
302C23—2121no—
303C23#1313319211yescharacteristic
304D4—1062no—
305D4—1062no—
306D4—2121no—
307D4—2121no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—4241no—
311C2×C4—4241no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4—4241no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4#1313319911yescharacteristic
324C2×C4#1313319911yescharacteristic
325C2×C4#1313320111yescharacteristic
326C2×C4#1313320511yescharacteristic
327C2×C4#1313321211yescharacteristic
328C2×C4#1313321211yescharacteristic
329C2×C4SD16×D5311yescharacteristic
330C2×C4#1313323811yescharacteristic
331C2×C4#1313323911yescharacteristic
332C2×C4#1313323911yescharacteristic
333C2×C4#1313324111yescharacteristic
334C2×C4#1313324511yescharacteristic
335C2×C4#1313324511yescharacteristic
336C2×C4#1313324911yescharacteristic
337C8—41no—
338C8—41no—
339C8—81no—
340C8—81no—
341C8—4241no—
342C22—4241no—
343C22—4241no—
344C22—4241no—
345C22—4241no—
346C22—2122no—
347C22—2122no—
348C22—2122no—
349C22#1681055011yescharacteristic
350C22#1681083911yescharacteristic
351C22#1681083911yescharacteristic
352C22#1681085311yescharacteristic
353C22#1681087011yescharacteristic
354C22#1681089411yescharacteristic
355C22#1681089911yescharacteristic
356C4—21no—
357C4—21no—
358C4—8481no—
359C4—8481no—
360C4—4241no—
361C4#1681094011yescharacteristic
362C4#1681094611yescharacteristic
363C4#1681094811yescharacteristic
364C4#1681094811yescharacteristic
365C4#1681095211yescharacteristic
366C4#1681096211yescharacteristic
367C4#1681097811yescharacteristic
368C4#1681098811yescharacteristic
369C2—4242no—
370C2#2027412511yescharacteristic
371C2#2027413911yescharacteristic
372C2#2027424911yescharacteristic
373C2#2027426311yescharacteristic
374C2#2027613011yescharacteristic
375C2#2027615011yescharacteristic
376C2#2027615711yescharacteristic
subgroupG/H
centerC23#13133192
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1277C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenter#1277D53
hypocenterC53#10625
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #20274843C2semidirect
5. #20272599C2semidirect
6. #20272599C2semidirect
7. C53×#1277C2semidirect
187. #1277D53semidirect
211. C53#10625semidirect

Maximal subgroups

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

subgroupindex
#202755092
#202748432
#202747162
#202725992
C53×#12772
#1062553