Finite Group Databasebeta

← #27483821

Subgroups — #27483821

Order 13568.

878 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283078C211yesmaximal, characteristic, complemented
2C4×#13133214C211yesmaximal, characteristic
3#20276279C212yesmaximal, complemented
4#20275950C212yesmaximal
5#20275412C214yesmaximal, complemented
6#20274716C212yesmaximal, complemented
7#20274716C212yesmaximal, complemented
8C53×#1767C211yesmaximal, characteristic, complemented
9C2×#13133214C2211yescharacteristic
10C2×#13133209—21no—
11C2×C4×D212—21no—
12C2×C4×D212—21no—
13C2×C4×D212C2212yes—
14C2×#13133139C2214yes—
15C2×#13133139C2214yes—
16#16810525—21no—
17#16810525—21no—
18#16810525C2212yes—
19#16810521—21no—
20#16810521—21no—
21#16810521C2212yes—
22#16810521C2212yes—
23#16810493C2211yescharacteristic
24#16810493C2211yescharacteristic
25#16810493C2211yescharacteristic
26#16810489—21no—
27#16810386C2214yes—
28#16810386C2214yes—
29C2×C4⋊C8×C53C2212yes—
30C2×#97×C53C2212yes—
31C43×C53C2211yescharacteristic
32C53×#336C2212yes—
33C22×D212—21no—
34C22×D212—21no—
35C22×D212C2312yes—
36C22×C4×D53—22no—
37#13133214—24no—
38#13133209—44no—
39C4×D212—24no—
40C4×D212—24no—
41C4×D212—44no—
42C4×D212—44no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53C2312yes—
50C2×C4⋊Dic53C2312yes—
51C2×C4⋊Dic53C2312yes—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×#11719921—22no—
55C2×#11719921—22no—
56C2×#11719921—22no—
57#13133139—28no—
58#13133139—44no—
59#13133124—22no—
60#13133124—22no—
61C22×C8×C53C2314yes—
62C2×C42×C53D411yescharacteristic
63C2×C42×C53D411yescharacteristic
64C2×C42×C53D411yescharacteristic
65C2×C42×C53D411yescharacteristic
66C2×C42×C53C2311yescharacteristic
67C2×C42×C53C2311yescharacteristic
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—24no—
70#97×C53—42no—
71C23×D53—22no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2411yescharacteristic
79C22×Dic53—22no—
80C22×Dic53—22no—
81C22×Dic53—22no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—24no—
87C2×D212—24no—
88C2×D212—22no—
89C2×D212—42no—
90C2×D212—42no—
91C2×D212—42no—
92C2×D212—42no—
93C2×D212#4114yes—
94C2×C4×D53—44no—
95C2×C4×D53—44no—
96C2×C4×D53—44no—
97C2×C8×C53—24no—
98C2×C8×C53—24no—
99C2×C8×C53—24no—
100C42×C53—22no—
101C42×C53—22no—
102C42×C53—22no—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—21no—
106C42×C53—21no—
107C42×C53#4114yes—
108#11719921—44no—
109#11719921—44no—
110#11719921—44no—
111C4⋊Dic53—24no—
112C4⋊Dic53—22no—
113C4⋊Dic53—22no—
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4⋊Dic53—42no—
117C4⋊Dic53—42no—
118C4⋊Dic53—42no—
119C4⋊Dic53—42no—
120C4⋊Dic53—42no—
121C4⋊Dic53—42no—
122#11719919—44no—
123#11719919—44no—
124C23×C53C22×D411yescharacteristic
125C22×D53—24no—
126C22×D53—42no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53#12412yes—
145C2×C4×C53#12412yes—
146C2×C4×C53AGL(1,ℤ/8)12yes—
147C2×C4×C53AGL(1,ℤ/8)12yes—
148C2×C4×C53AGL(1,ℤ/8)12yes—
149C2×C4×C53#13712yes—
150C2×C4×C53C2×#4112yes—
151C2×C4×C53C2×#4112yes—
152C2×C4×C53H(5,𝔽2)11yescharacteristic
153C2×C4×C532−1+411yescharacteristic
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163D212—24no—
164D212—24no—
165D212—48no—
166D212—48no—
167D212—42no—
168D212—42no—
169C4×D53—88no—
170C8×C53—44no—
171#29130—531nomaximal, Sylow
172C22×C53#52511yescharacteristic
173C22×C53C2×#12411yescharacteristic
174C22×C53#53711yescharacteristic
175C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
176C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
177C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
178C22×C53C2×#13711yescharacteristic
179C2×D53—44no—
180C2×D53—44no—
181C2×D53—44no—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—82no—
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—22no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53#49014yes—
192C4×C53#55212yes—
193C4×C53#55312yes—
194Dic53—82no—
195Dic53—82no—
196Dic53—82no—
197#2859—531no—
198C4×#127—531no—
199#2100—532no—
200#2041—532no—
201#1936—534no—
202#1802—532no—
203#1802—532no—
204#1767D5311yescharacteristic, complemented
205C2×C53#256211yescharacteristic
206C2×C53#310911yescharacteristic
207C2×C53#311011yescharacteristic
208C2×C53C2×#49011yescharacteristic
209C2×C53#314111yescharacteristic
210C2×C53#325311yescharacteristic
211C2×C53#325411yescharacteristic
212D53—84no—
213C2×#127—531no—
214C2×#125—1061no—
215C2×C4×D4—532no—
216C2×C4×D4—1061no—
217C2×C4×D4—1061no—
218C2×C4⋊C8C2×D5312yes—
219C2×#102—534no—
220C2×#102—534no—
221C2×#97C2×D5312yes—
222#390—532no—
223#390—1061no—
224#390—1061no—
225#389—532no—
226#389—532no—
227#389—1061no—
228#389—1061no—
229#384—531no—
230#384—531no—
231#384—531no—
232#382—1061no—
233C43C2×D5311yescharacteristic
234#340—534no—
235#340—534no—
236#336C2×D5312yes—
237C53#2913011yescharacteristic, complemented, Sylow
238C22×D4—532no—
239C22×D4—1061no—
240C22×D4—1061no—
241C23×C4—1062no—
242C22×C8C22×D5314yes—
243#127—1064no—
244#125—2124no—
245C4×D4—1064no—
246C4×D4—1064no—
247C4×D4—2124no—
248C4×D4—2124no—
249C2×C4⋊C4—532no—
250C2×C4⋊C4—532no—
251C2×C4⋊C4—532no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1062no—
255C2×C4⋊C4—1062no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1062no—
263C2×C42D21211yescharacteristic
264C2×C42D21211yescharacteristic
265C2×C42D21211yescharacteristic
266C2×C42D21211yescharacteristic
267C2×C42C22×D5311yescharacteristic
268C2×C42C22×D5311yescharacteristic
269C2×C42C22×D5311yescharacteristic
270C4⋊C8—24no—
271#102—1068no—
272#102—2124no—
273#97—42no—
274#95—1062no—
275#95—1062no—
276C24—1062no—
277C2×D4—534no—
278C2×D4—1064no—
279C2×D4—1064no—
280C2×D4—1062no—
281C2×D4—1062no—
282C2×D4—1062no—
283C2×D4—1062no—
284C2×D4—1062no—
285C2×D4—2122no—
286C2×D4—2122no—
287C2×D4—2122no—
288C2×D4—2122no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—1062no—
292C22×C4—2124no—
293C22×C4—2124no—
294C22×C4—2124no—
295C22×C4C2×D21211yescharacteristic
296C22×C4C2×D21211yescharacteristic
297C22×C4C2×D21211yescharacteristic
298C22×C4C2×D21211yescharacteristic
299C22×C4C2×D21211yescharacteristic
300C22×C4C2×D21211yescharacteristic
301C22×C4C23×D5311yescharacteristic
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C4⋊C4—1064no—
306C4⋊C4—1062no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—1062no—
310C4⋊C4—2122no—
311C4⋊C4—2122no—
312C4⋊C4—2124no—
313C4⋊C4—2122no—
314C4⋊C4—2122no—
315C4⋊C4—2124no—
316C4⋊C4—2122no—
317C4⋊C4—2122no—
318C22⋊C4—2124no—
319C22⋊C4—2124no—
320C22⋊C4—2124no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—21no—
327C42—21no—
328C42#1171994514yes—
329C23—1064no—
330C23—2122no—
331C23—2122no—
332C23—2122no—
333C23—2122no—
334C23—2122no—
335C23—2122no—
336C23C22×D21211yescharacteristic
337D4—1064no—
338D4—1064no—
339D4—2128no—
340D4—2128no—
341D4—2122no—
342D4—2122no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—4248no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4#1313320512yes—
366C2×C4#1313320512yes—
367C2×C4#1313324512yes—
368C2×C4#1313324512yes—
369C2×C4#1313324512yes—
370C2×C4#1313325012yes—
371C2×C4#41×D5312yes—
372C2×C4C2×#1171994512yes—
373C2×C4#1313326211yescharacteristic
374C2×C4#1313326511yescharacteristic
375C8—44no—
376C22—4242no—
377C22—4242no—
378C22—4242no—
379C22—4242no—
380C22—2124no—
381C22—2124no—
382C22—2124no—
383C22#1681113511yescharacteristic
384C22C2×#1313320511yescharacteristic
385C22#1681120411yescharacteristic
386C22C2×#1313324511yescharacteristic
387C22C2×#1313324511yescharacteristic
388C22C2×#1313324511yescharacteristic
389C22C2×#1313325011yescharacteristic
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—4242no—
396C4—4242no—
397C4—4242no—
398C4#1681098814yes—
399C4#1681131412yes—
400C4#1681132312yes—
401C2—4244no—
402C2#2027987611yescharacteristic
403C2#2028480711yescharacteristic
404C2#2028481511yescharacteristic
405C2C2×#1681098811yescharacteristic
406C2#2028510911yescharacteristic
407C2#2028609611yescharacteristic
408C2#2028610311yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1767C2
socleC23×C53C22×D4
prime radicalC2×#13133214C22
hypercenter#1767D53
hypocenterC53#29130
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
1. #20283078C2semidirect
3. #20276279C2semidirect
5. #20275412C2semidirect
6. #20274716C2semidirect
7. #20274716C2semidirect
8. C53×#1767C2semidirect
204. #1767D53semidirect
237. C53#29130semidirect

Maximal subgroups

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

subgroupindex
#202830782
C4×#131332142
#202762792
#202759502
#202754122
#202747162
C53×#17672
#2913053