Finite Group Databasebeta

← #27303249

Subgroups — #27303249

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×#16810860C211yesmaximal, characteristic
2C2×#16810425C211yesmaximal, characteristic, complemented
3C2×#16810423C211yesmaximal, characteristic, complemented
4C2×#16810402C211yesmaximal, characteristic, complemented
5#20274741C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C2×C53×#366C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×#13133192—21no—
11C22×#11719908C2211yescharacteristic
12#16810860—22no—
13#16810860—22no—
14C2×#13133151C2211yescharacteristic
15C2×#13133137—21no—
16#16810480—21no—
17#16810425—22no—
18#16810423—22no—
19#16810402—22no—
20#16810385—21no—
21C22×C16×C53C2211yescharacteristic
22C2×#107×C53C2211yescharacteristic
23C2×#107×C53C2211yescharacteristic
24C53×#366—22no—
25C22×D212—21no—
26C22×D212—41no—
27C22×C4×D53D411yescharacteristic
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424D412yes—
34C22×#88322D411yescharacteristic
35C2×C8×D53—21no—
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53—21no—
40C2×C8×D53—21no—
41#13133192—24no—
42#13133192—42no—
43C2×C4⋊Dic53—21no—
44C2×#11719921—41no—
45C2×#11719921—41no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49#13133151—21no—
50#13133151—21no—
51#13133137—24no—
52#13133124—41no—
53C22×C8×C53C2311yescharacteristic
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C16×C53—21no—
57C2×C16×C53—21no—
58C2×C16×C53D412yescomplemented
59#107×C53—21no—
60#107×C53—21no—
61#107×C53—21no—
62#107×C53D412yes—
63C23×D53—41no—
64C23×D53D811yescharacteristic
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—41no—
69C22×Dic53D811yescharacteristic
70C2×D212—22no—
71C2×D212—41no—
72C2×D212—41no—
73C2×D212—41no—
74C2×D212—41no—
75C2×D212—44no—
76C2×D212—41no—
77C2×D212—82no—
78C2×D212—82no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C8×C53C2×D411yescharacteristic
86C2×C8×C53C2×D411yescharacteristic
87C2×C8×C53C2×D411yescharacteristic
88C2×C8×C53#4111yescharacteristic
89C2×C8×C53#4111yescharacteristic
90C2×C8×C53#4111yescharacteristic
91C4⋊C4×C53—22no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95#11719921—44no—
96#11719921—82no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C2×#88322—21no—
100C2×#88322—21no—
101C2×#88322—21no—
102D424—22no—
103D424—22no—
104D424—41no—
105D424—41no—
106C8×D53—41no—
107C8×D53—41no—
108C8×D53—41no—
109C8×D53—41no—
110C16×C53—22no—
111#11719908—41no—
112C23×C53C2×D811yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53D1612yes—
127C2×C4×C53—42no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12311yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145D212—42no—
146D212—42no—
147D212—88no—
148D212—81no—
149D212—81no—
150C4×D53—41no—
151C4×D53—41no—
152C4×D53—41no—
153C4×D53—41no—
154C8×C53#12111yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12311yescharacteristic
157C8×C53#12411yescharacteristic
158#88322—41no—
159#13380—531nomaximal, Sylow
160C22×C53#44711yescharacteristic
161C22×C53#46611yescharacteristic
162C22×C53#48011yescharacteristic
163C22×C53C2×D1611yescharacteristic
164C22×C53#50811yescharacteristic
165C22×C53#50911yescharacteristic
166C22×C53#51011yescharacteristic
167C2×D53—22no—
168C2×D53—22no—
169C2×D53—22no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—82no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—161no—
178C2×D53—161no—
179C2×D53—161no—
180C2×D53—161no—
181C4×C53—82no—
182C4×C53—161no—
183C4×C53#39411yescharacteristic
184C4×C53#45211yescharacteristic
185C4×C53#46811yescharacteristic
186C4×C53#48211yescharacteristic
187Dic53—41no—
188Dic53—161no—
189C2×#466—531no—
190C2×#366D5311yescharacteristic, complemented
191C2×#357—531no—
192C2×#357—531no—
193C2×C22⋊C16—531no—
194#1808—531no—
195#1380—531no—
196C2×C53#203311yescharacteristic
197C2×C53#218411yescharacteristic
198C2×C53#218911yescharacteristic
199C2×C53#221511yescharacteristic
200C2×C53#222011yescharacteristic
201C2×C53#223211yescharacteristic
202C2×C53#223411yescharacteristic
203D53—42no—
204D53—162no—
205C22×D8—531no—
206C23×C8—531no—
207C2×#121—1061no—
208C22×C16—531no—
209C22×C16C2×D5311yescharacteristic
210#466—1062no—
211#466—1062no—
212C2×#107—531no—
213C2×#107C2×D5311yescharacteristic
214C2×#107C2×D5311yescharacteristic
215C2×#102—1061no—
216#380—1061no—
217#366—22no—
218#357—1062no—
219#357—1062no—
220C22⋊C16—1062no—
221#340—1061no—
222C53#1338011yescharacteristic, complemented, Sylow
223C22×D4—1061no—
224C22×D4—2121no—
225C23×C4—531no—
226C2×D8—532no—
227C2×D8—1061no—
228C2×D8—1061no—
229C2×D8—1061no—
230C2×D8—1061no—
231C2×D8—1061no—
232C22×C8—531no—
233C22×C8—1061no—
234C22×C8—1061no—
235C22×C8—1061no—
236C22×C8—1061no—
237C22×C8—1061no—
238C22×C8—1061no—
239C22×C8C22×D5311yescharacteristic
240#121—1064no—
241#121—2122no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1061no—
245C2×C22⋊C4—2121no—
246C2×C22⋊C4—2121no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—1061no—
250C2×C16—1061no—
251C2×C16—1061no—
252C2×C16D21212yescomplemented
253#107—21no—
254#107—21no—
255#107—21no—
256#107—1061no—
257#107—1061no—
258#107C22⋊D5312yes—
259#102—1064no—
260#95—2121no—
261C24—531no—
262C24—2121no—
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—41no—
273C22×C4—41no—
274C22×C4—531no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—2121no—
282C22×C4D4×D5311yescharacteristic
283D8—1062no—
284D8—1062no—
285D8—2121no—
286D8—2121no—
287C2×C8—1061no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—2121no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8C2×D21211yescharacteristic
295C2×C8#1171994511yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8#1171994911yescharacteristic
298C2×C8#1171994911yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—4242no—
307C22⋊C4—2124no—
308C16—22no—
309C16—2121no—
310C23—532no—
311C23—1061no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—1061no—
317C23—4241no—
318C23—4241no—
319C23—4241no—
320C23—4241no—
321C23—4241no—
322C23—4241no—
323C23—2122no—
324C23D8×D5311yescharacteristic
325D4—4248no—
326D4—4241no—
327D4—4241no—
328D4—2122no—
329D4—2122no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—4241no—
340C2×C4—4241no—
341C2×C4—4241no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4#1313318511yescharacteristic
347C2×C4#1313319211yescharacteristic
348C2×C4#1313320211yescharacteristic
349C2×C4#1313324011yescharacteristic
350C2×C4#1313324111yescharacteristic
351C2×C4#1313324911yescharacteristic
352C8—2121no—
353C8#1313319111yescharacteristic
354C8#1313319211yescharacteristic
355C8#1313320211yescharacteristic
356C8#1313320611yescharacteristic
357C22—8481no—
358C22—8481no—
359C22—8481no—
360C22—8481no—
361C22—1062no—
362C22—1062no—
363C22—1062no—
364C22—4242no—
365C22—4242no—
366C22—4242no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22#1681075511yescharacteristic
372C22#1681086011yescharacteristic
373C22#1681093811yescharacteristic
374C22D16×D5311yescharacteristic
375C22#1681105611yescharacteristic
376C22#1681105711yescharacteristic
377C22#1681106511yescharacteristic
378C4—82no—
379C4—161no—
380C4—8481no—
381C4—2121no—
382C4#1681054011yescharacteristic
383C4#1681078411yescharacteristic
384C4#1681087011yescharacteristic
385C4#1681095011yescharacteristic
386C2—8482no—
387C2—2122no—
388C2#2027589911yescharacteristic
389C2#2027662511yescharacteristic
390C2#2027665411yescharacteristic
391C2#2027680211yescharacteristic
392C2#2027682911yescharacteristic
393C2#2027689711yescharacteristic
394C2#2027690911yescharacteristic
subgroupG/H
centerC23D8×D53
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC2×C53×#366C2
socleC23×C53C2×D8
prime radicalC2×#16810860C2
hypercenterC2×#366D53
hypocenterC53#13380
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#366

Complements

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

NH ≅ G/Ntype
2. C2×#16810425C2semidirect
3. C2×#16810423C2semidirect
4. C2×#16810402C2semidirect
5. #20274741C2semidirect
6. #20272806C2semidirect
7. C2×C53×#366C2semidirect
58. C2×C16×C53D4semidirect
190. C2×#366D53semidirect
222. C53#13380semidirect
252. C2×C16D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168108602
C2×#168104252
C2×#168104232
C2×#168104022
#202747412
#202728062
C2×C53×#3662
#1338053