Finite Group Databasebeta

← #27286639

Subgroups — #27286639

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810969C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3#20275510C211yesmaximal, characteristic, complemented
4C2×#16810371C211yesmaximal, characteristic, complemented
5C2×#16810367C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#459C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9#16810969—22no—
10#16810969—22no—
11#16810913—22no—
12#16810913—22no—
13C2×#13133148—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133138—22no—
17C2×#13133138—21no—
18C2×C22⋊#88322—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810371—22no—
21#16810367—22no—
22C22×D8×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C2×C4×D4×C53C2211yescharacteristic
25C53×#459—22no—
26C53×#459—22no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×Dic106—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32C22×#88322D412yes—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217D412yes—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39#13133148—24no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yes—
44#13133138—24no—
45#13133138—28no—
46C22⋊#88322—44no—
47C4×#88322—21no—
48C4×#88322—21no—
49C22×D4×C53—21no—
50C22×D4×C53—21no—
51C22×D4×C53—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C2×D8×C53—24no—
55C2×D8×C53—42no—
56C2×D8×C53—42no—
57C22×C8×C53—21no—
58#127×C53—21no—
59#127×C53—21no—
60#127×C53—21no—
61#127×C53D412yes—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53D412yes—
71#102×C53—24no—
72C24×C53—21no—
73C24×C53—41no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—42no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—42no—
90C2×D4×C53—41no—
91C2×D4×C53—82no—
92C2×D4×C53SD1612yes—
93C22×C4×C53—21no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53#4111yescharacteristic
101C22×Dic53—41no—
102C2×Dic106—22no—
103C2×Dic106—41no—
104C2×Dic106—82no—
105D8×C53—48no—
106C2×C8×C53—22no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C4⋊Dic53—22no—
118C4⋊Dic53—22no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—42no—
129C23×C53—22no—
130C23×C53—41no—
131C23×C53—42no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—82no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53—82no—
139C23×C53#12111yescharacteristic
140D4×C53—22no—
141D4×C53—22no—
142D4×C53—41no—
143D4×C53—44no—
144D4×C53—42no—
145D4×C53—41no—
146D4×C53—42no—
147D4×C53—82no—
148D4×C53—82no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152D4×C53—81no—
153C2×C4×C53—22no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—82no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12711yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53#13512yes—
167C2×C4×C53#13511yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)12yes—
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C53#13711yescharacteristic
171C2×Dic53—42no—
172C2×Dic53—82no—
173Dic106—82no—
174Dic106—82no—
175C8×C53—42no—
176#88322—42no—
177#88322—42no—
178#88322—81no—
179#10267—531nomaximal, Sylow
180C22×C53—44no—
181C22×C53—42no—
182C22×C53—82no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—82no—
186C22×C53—82no—
187C22×C53—82no—
188C22×C53—162no—
189C22×C53—162no—
190C22×C53#39211yescharacteristic
191C22×C53#46312yes—
192C22×C53#46511yescharacteristic
193C22×C53#46811yescharacteristic
194C22×C53#47212yes—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53#44811yescharacteristic
199C4×C53#44912yes—
200C4×C53#44911yescharacteristic
201C4×C53#45011yescharacteristic
202C4×C53#45211yescharacteristic
203C4×C53#49511yescharacteristic
204C4×C53#49611yescharacteristic
205Dic53—82no—
206C2×#486—531no—
207C2×#476—531no—
208C2×#459D5311yescharacteristic
209#1954—531no—
210C2×#333—531no—
211C2×#331—531no—
212#1277—531no—
213C2×C53—82no—
214C2×C53—162no—
215C2×C53#167712yes—
216C2×C53#169712yes—
217C2×C53#200011yescharacteristic
218C2×C53#201211yescharacteristic
219C2×C53#201811yescharacteristic
220C22×D8—21no—
221C2×C4⋊Q8—531no—
222C2×#127C2×D5311yescharacteristic
223C2×C4×D4C2×D5311yescharacteristic
224#486—1062no—
225#486—1062no—
226#476—1062no—
227#476—1062no—
228#459—22no—
229#459—22no—
230C2×#106—1061no—
231C2×C4⋊C8—531no—
232C2×C4⋊C8—531no—
233C2×C4⋊C8C2×D5311yescharacteristic
234C2×#102—21no—
235C2×#102—1062no—
236C2×#102—1061no—
237C2×C22⋊C8—1062no—
238C2×C4×C8—531no—
239#333—1062no—
240#331—1062no—
241C53#1026711yescharacteristic, complemented, Sylow
242C22×Q8—1061no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C22⋊D5311yescharacteristic
247C23×C4—21no—
248C2×D8—24no—
249C2×D8—42no—
250C2×D8—42no—
251C22×C8—21no—
252C22×C8—532no—
253C22×C8—1061no—
254C22×C8—1061no—
255C4⋊Q8—532no—
256C4⋊Q8—1061no—
257C4⋊Q8—1061no—
258C4⋊Q8—1061no—
259#127—21no—
260#127—21no—
261#127—21no—
262#127C22⋊D5312yes—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4C22⋊D5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C42C22×D5311yescharacteristic
272#106—1064no—
273C4⋊C8—21no—
274C4⋊C8—532no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278C4⋊C8C22⋊D5312yes—
279#102—24no—
280#102—1064no—
281#102—1068no—
282C22⋊C8—2124no—
283C4×C8—1061no—
284C4×C8—1061no—
285C24—21no—
286C24—41no—
287C2×Q8—1062no—
288C2×Q8—4242no—
289C2×Q8—2121no—
290C2×D4—21no—
291C2×D4—22no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—82no—
308C2×D4#1171992312yes—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—2121no—
315C22×C4D4×D5311yescharacteristic
316C22×C4#1171994711yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318D8—48no—
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—1062no—
323C2×C8—1062no—
324C2×C8—1062no—
325C2×C8—1062no—
326C2×C8—2121no—
327C2×C8—2122no—
328C2×C8—2121no—
329C2×C8—2121no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—2121no—
335C4⋊C4—2121no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42D4×D5311yescharacteristic
339C42D4×D5311yescharacteristic
340C42C2×C22⋊D5311yescharacteristic
341C42C2×C22⋊D5311yescharacteristic
342C23—22no—
343C23—41no—
344C23—42no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—82no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23#1313318911yescharacteristic
353Q8—4242no—
354Q8—4242no—
355D4—22no—
356D4—22no—
357D4—44no—
358D4—41no—
359D4—42no—
360D4—41no—
361D4—42no—
362D4—82no—
363D4—82no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—4242no—
375C2×C4—2122no—
376C2×C4#1313318511yescharacteristic
377C2×C4#1313318511yescharacteristic
378C2×C4#1313318611yescharacteristic
379C2×C4#1313318811yescharacteristic
380C2×C4#1313318911yescharacteristic
381C2×C4#1313321311yescharacteristic
382C2×C4C2×#1171992311yescharacteristic
383C2×C4#1313323612yes—
384C2×C4#1313323711yescharacteristic
385C2×C4#1313324112yes—
386C2×C4#1313324311yescharacteristic
387C2×C4#1313324711yescharacteristic
388C8—42no—
389C8—4241no—
390C8—2122no—
391C8—2122no—
392C22—42no—
393C22—44no—
394C22—82no—
395C22—81no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—162no—
401C22—162no—
402C22#1681053111yescharacteristic
403C22#1681083612yes—
404C22#1681085011yescharacteristic
405C22#1681086711yescharacteristic
406C22#1681089112yes—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—4242no—
411C4#1681076111yescharacteristic
412C4#1681076512yes—
413C4#1681076611yescharacteristic
414C4#1681077411yescharacteristic
415C4#1681078511yescharacteristic
416C4#1681100611yescharacteristic
417C4#1681101111yescharacteristic
418C2—82no—
419C2—162no—
420C2#2027412212yes—
421C2#2027424612yes—
422C2#2027571811yescharacteristic
423C2#2027579011yescharacteristic
424C2#2027582111yescharacteristic
subgroupG/H
centerC23#13133189
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#459C2
socleC23×C53#121
hypercenterC2×#459D53
hypocenterC53#10267
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16810969C2semidirect
2. C2×#16810913C2semidirect
3. #20275510C2semidirect
4. C2×#16810371C2semidirect
5. C2×#16810367C2semidirect
6. #20272549C2semidirect
241. C53#10267semidirect

Maximal subgroups

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

subgroupindex
C2×#168109692
C2×#168109132
#202755102
C2×#168103712
C2×#168103672
#202725492
C2×C53×#4592
#1026753