Finite Group Databasebeta

← #27303255

Subgroups — #27303255

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810859C211yesmaximal, characteristic
2C2×#16810451C211yesmaximal, characteristic, complemented
3C2×#16810425C211yesmaximal, characteristic, complemented
4C2×#16810425C211yesmaximal, characteristic, complemented
5#20274744C211yesmaximal, characteristic, complemented
6#20272807C211yesmaximal, characteristic, complemented
7C2×C53×C22⋊C16C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C2×#13133191—21no—
10#16810859—22no—
11#16810859—22no—
12C2×#13133151C2211yescharacteristic
13C2×#13133151C2211yescharacteristic
14C2×#13133151C2211yescharacteristic
15C2×#13133139—21no—
16#16810483—21no—
17#16810451—22no—
18#16810425—22no—
19#16810425—22no—
20#16810386—21no—
21C23×C8×C53C2211yescharacteristic
22C22×C16×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C53×C22⋊C16—22no—
25C22×D212—21no—
26C22×C22⋊D53—41no—
27C2×D424—21no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424D412yes—
33#13133191—24no—
34#13133191—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C22⋊Dic53—41no—
39C2×#11719921—41no—
40#13133151—21no—
41#13133151—21no—
42#13133151—21no—
43#13133151—21no—
44#13133151—21no—
45#13133151D412yes—
46#13133139—24no—
47#13133124—41no—
48C23×C4×C53D411yescharacteristic
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51C22×C8×C53—21no—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C22×C8×C53D411yescharacteristic
56C22×C8×C53C2311yescharacteristic
57C2×C16×C53—21no—
58C2×C16×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53D412yescomplemented
63C24×C53D811yescharacteristic
64C23×D53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53D811yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C2×C22⋊D53—44no—
74C2×C22⋊D53—82no—
75C2×C22⋊D53—82no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C2×D212—22no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53#4111yescharacteristic
96C2×C8×C53#4111yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C22⋊Dic53—82no—
99#11719921—44no—
100C4⋊Dic53—22no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106D424—22no—
107D424—22no—
108D424—41no—
109D424—41no—
110C16×C53—22no—
111C16×C53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53SD3212yes—
119C23×C53C2×D811yescharacteristic
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53—81no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12311yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C22⋊D53—88no—
141C2×Dic53—42no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150D212—42no—
151D212—42no—
152D212—81no—
153D212—81no—
154C8×C53—41no—
155C8×C53#12111yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12311yescharacteristic
158C8×C53#12411yescharacteristic
159#13381—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53#44711yescharacteristic
168C22×C53#46611yescharacteristic
169C22×C53#48011yescharacteristic
170C22×C53C2×SD3211yescharacteristic
171C22×C53#50811yescharacteristic
172C22×C53#50911yescharacteristic
173C22×C53#50911yescharacteristic
174C2×D53—82no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—161no—
178C2×D53—161no—
179C2×D53—161no—
180C2×D53—161no—
181C4×C53—41no—
182C4×C53#39411yescharacteristic
183C4×C53#45211yescharacteristic
184C4×C53#46811yescharacteristic
185C4×C53#48211yescharacteristic
186Dic53—82no—
187Dic53—161no—
188Dic53—161no—
189C2×#466—531no—
190C2×#367—531no—
191C2×#357—531no—
192C2×#357—531no—
193C2×C22⋊C16D5311yescharacteristic, complemented
194#1808—531no—
195#1380—531no—
196C2×C53—42no—
197C2×C53#203311yescharacteristic
198C2×C53#218611yescharacteristic
199C2×C53#218711yescharacteristic
200C2×C53#221711yescharacteristic
201C2×C53#221811yescharacteristic
202C2×C53#223311yescharacteristic
203C2×C53#223411yescharacteristic
204D53—162no—
205C22×D8—531no—
206C23×C8C2×D5311yescharacteristic
207C2×#121—1061no—
208C22×C16C2×D5311yescharacteristic
209C22×C16C2×D5311yescharacteristic
210#466—1062no—
211#466—1062no—
212C2×#107—531no—
213C2×#107—531no—
214C2×#107—531no—
215C2×#102—1061no—
216#380—1061no—
217#367—1062no—
218#357—1062no—
219#357—1062no—
220C22⋊C16—22no—
221#340—1061no—
222C53#1338111yescharacteristic, complemented, Sylow
223C22×D4—1061no—
224C22×D4—2121no—
225C23×C4D21211yescharacteristic
226C2×D8—532no—
227C2×D8—1061no—
228C2×D8—1061no—
229C2×D8—1061no—
230C2×D8—1061no—
231C2×D8—1061no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8D21211yescharacteristic
239C22×C8C22×D5311yescharacteristic
240#121—1064no—
241#121—2122no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C22⋊C4—2121no—
246C2×C22⋊C4—2121no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16C22⋊D5312yescomplemented
253#107—532no—
254#107—1061no—
255#107—1061no—
256#107—1061no—
257#107—1061no—
258#107—1061no—
259#102—1064no—
260#95—2121no—
261C24—2121no—
262C24D42411yescharacteristic
263C2×D4—1062no—
264C2×D4—4242no—
265C2×D4—4242no—
266C2×D4—2124no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4D42411yescharacteristic
282C22×C4C2×D21211yescharacteristic
283D8—1062no—
284D8—1062no—
285D8—2121no—
286D8—2121no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8#1171994511yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8#1171994711yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C4⋊C4—1062no—
301C4⋊C4—2121no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—4242no—
307C22⋊C4—2124no—
308C16—22no—
309C16—41no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—4241no—
317C23—4241no—
318C23—4241no—
319C23—4241no—
320C23—4241no—
321C23—4241no—
322C23—2122no—
323C23#1313316212yes—
324C23C2×D42411yescharacteristic
325D4—4248no—
326D4—4241no—
327D4—4241no—
328D4—2122no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—4241no—
341C2×C4—4241no—
342C2×C4—4241no—
343C2×C4—4241no—
344C2×C4—4241no—
345C2×C4—2122no—
346C2×C4#1313318711yescharacteristic
347C2×C4#1313319111yescharacteristic
348C2×C4#1313320111yescharacteristic
349C2×C4#1313323911yescharacteristic
350C2×C4#1313324511yescharacteristic
351C2×C4#1313325011yescharacteristic
352C8—41no—
353C8#1313318811yescharacteristic
354C8#1313318911yescharacteristic
355C8#1313319911yescharacteristic
356C8#1313320311yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—8481no—
365C22—8481no—
366C22—8481no—
367C22—8481no—
368C22—4242no—
369C22—4242no—
370C22—4242no—
371C22#1681075911yescharacteristic
372C22#1681085911yescharacteristic
373C22#1681093711yescharacteristic
374C22C2×#1313316211yescharacteristic
375C22#1681105511yescharacteristic
376C22#1681106111yescharacteristic
377C22#1681106111yescharacteristic
378C4—41no—
379C4—8481no—
380C4—8481no—
381C4—4242no—
382C4#1681053911yescharacteristic
383C4#1681078711yescharacteristic
384C4#1681086911yescharacteristic
385C4#1681094811yescharacteristic
386C2—42no—
387C2—8482no—
388C2#2027589811yescharacteristic
389C2#2027664011yescharacteristic
390C2#2027664611yescharacteristic
391C2#2027681111yescharacteristic
392C2#2027681811yescharacteristic
393C2#2027690111yescharacteristic
394C2#2027690711yescharacteristic
subgroupG/H
centerC23C2×D424
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC2×C53×C22⋊C16C2
socleC23×C53C2×D8
prime radicalC2×#16810859C2
hypercenterC2×C22⋊C16D53
hypocenterC53#13381
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#16810451C2semidirect
3. C2×#16810425C2semidirect
4. C2×#16810425C2semidirect
5. #20274744C2semidirect
6. #20272807C2semidirect
7. C2×C53×C22⋊C16C2semidirect
62. C2×C16×C53D4semidirect
193. C2×C22⋊C16D53semidirect
222. C53#13381semidirect
252. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108592
C2×#168104512
C2×#168104252
#202747442
#202728072
C2×C53×C22⋊C162
#1338153