Finite Group Databasebeta

← #27414028

Subgroups — #27414028

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#389×D53C211yesmaximal, characteristic
2#20277120C211yesmaximal, characteristic
3#20276286C212yesmaximal, complemented
4#20276285C212yesmaximal, complemented
5#20276277C211yesmaximal, characteristic, complemented
6#20276274C211yesmaximal, characteristic, complemented
7#20276273C211yesmaximal, characteristic, complemented
8#20274433C212yesmaximal
9#20274433C212yesmaximal
10#20274407C211yesmaximal, characteristic
11C53×#2100C211yesmaximal, characteristic, complemented
12C22×#11719912C2212yes—
13C2×#13133175C2212yes—
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53C2211yescharacteristic
16C2×C4⋊C4×D53C2211yescharacteristic
17C2×C42×D53C2211yescharacteristic
18#16810522C2211yescharacteristic
19#16810521C2211yescharacteristic
20#16810520—21no—
21#16810519C2211yescharacteristic
22#16810518—21no—
23#16810493C2211yescharacteristic
24#16810488C2212yes—
25#16810480—21no—
26#16810480—21no—
27C4⋊C4×Dic53—21no—
28C4⋊C4×Dic53—21no—
29#95×D53—21no—
30#16810386C2212yes—
31#16810386C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810378C2211yescharacteristic
37#16810377C2212yes—
38C53×#389C2211yescharacteristic
39C53×#384C2211yescharacteristic
40C53×#340C2212yes—
41C53×#340C2212yes—
42C53×#336C2211yescharacteristic
43C22×C4×D53—21no—
44C22×C4×D53—21no—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C2×#11719912—24no—
49C2×#11719912—24no—
50C2×#11719912—24no—
51C22×#88322C2312yes—
52#13133175—28no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C4⋊Dic53C2312yes—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×#11719919—21no—
60C2×#11719919—21no—
61C2×#11719919—21no—
62C2×#11719919—21no—
63C4⋊C4×D53—22no—
64C4⋊C4×D53—24no—
65C4⋊C4×D53—44no—
66C4⋊C4×D53Q814yes—
67C2×#11719921—22no—
68C2×C4×Dic53—22no—
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C42×D53—24no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—22no—
77#13133124—22no—
78C22×C8×C53C2312yes—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53C2311yescharacteristic
83C2×C4⋊C4×C53C2312yes—
84C2×C4⋊C4×C53C2311yescharacteristic
85C2×C42×C53C2311yescharacteristic
86#95×C53—21no—
87C23×D53C2×D411yescharacteristic
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2411yescharacteristic
94C22×Dic53—21no—
95C22×Dic53—22no—
96C22×Dic53—21no—
97C22×Dic53C2×D411yescharacteristic
98C22×Dic53C2×D411yescharacteristic
99C22×Dic53C2×D411yescharacteristic
100C2×C4×D53—24no—
101C2×C4×D53—22no—
102C2×C4×D53—22no—
103C2×C4×D53—22no—
104C2×C4×D53—22no—
105C2×C4×D53—41no—
106C2×C4×D53—41no—
107C2×C4×D53—42no—
108C2×C4×D53—42no—
109C2×C4×D53—42no—
110C2×C4×D53—42no—
111C2×C4×D53—42no—
112C2×C4×D53C2×Q812yes—
113C2×C4×D53#4114yes—
114C2×C4×D53#4114yes—
115C2×C4×D53#4112yes—
116C2×C8×C53—22no—
117C2×C8×C53—22no—
118C2×C8×C53—22no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—24no—
122C4⋊C4×C53—42no—
123C4⋊C4×C53—42no—
124C4⋊C4×C53—42no—
125C4⋊C4×C53C2×Q812yes—
126C42×C53—22no—
127#11719921—44no—
128C4⋊Dic53—21no—
129C4⋊Dic53—22no—
130C4⋊Dic53—24no—
131C4⋊Dic53—42no—
132C4⋊Dic53—42no—
133C4⋊Dic53—42no—
134C4⋊Dic53C2×Q812yes—
135#11719919—24no—
136#11719919—42no—
137#11719919—42no—
138#11719919—42no—
139C4×Dic53—22no—
140C4×Dic53—22no—
141C4×Dic53—22no—
142C4×Dic53—44no—
143C2×#88322—22no—
144C2×#88322—22no—
145C2×#88322—22no—
146#11719912—48no—
147C23×C53C22×D411yescharacteristic
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—21no—
152C22×D53C22⋊Q812yes—
153C22×D53#12312yes—
154C22×D53#12412yes—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53AGL(1,ℤ/8)12yes—
169C2×C4×C53#13712yes—
170C2×C4×C53C22×Q811yescharacteristic
171C2×C4×C53C2×#4112yes—
172C2×C4×C53C2×#4112yes—
173C2×C4×C53C2×#4111yescharacteristic
174C2×Dic53—22no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—42no—
185C2×Dic53—41no—
186C2×Dic53—42no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53C22⋊Q812yes—
190C2×Dic53#12312yes—
191C2×Dic53#12412yes—
192C4×D53—44no—
193C4×D53—44no—
194C4×D53—48no—
195C4×D53—84no—
196C4×D53#12518yes—
197C4×D53#12618yes—
198C8×C53—42no—
199#88322—42no—
200#23747—531nomaximal, Sylow
201C22×C53C2×C22⋊Q811yescharacteristic
202C22×C53C2×#12311yescharacteristic
203C22×C53C2×#12411yescharacteristic
204C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
205C22×C53C2×#13711yescharacteristic
206C22×C53#57511yescharacteristic
207C22×C53#57511yescharacteristic
208C2×D53—22no—
209C2×D53—22no—
210C2×D53—22no—
211C2×D53—22no—
212C2×D53—22no—
213C2×D53—22no—
214C2×D53#40014yes—
215C4×C53—22no—
216C4×C53—22no—
217C4×C53—42no—
218C4×C53—81no—
219C4×C53—82no—
220C4×C53C2×#12512yes—
221C4×C53C2×#12612yes—
222Dic53—42no—
223Dic53—41no—
224Dic53—41no—
225Dic53—81no—
226Dic53—82no—
227Dic53#40014yes—
228C2×#389—531no—
229#2285—531no—
230#2101—532no—
231#2101—532no—
232#2100—531no—
233#2100D5311yescharacteristic, complemented
234#2099—531no—
235#2099—531no—
236#1736—532no—
237#1736—532no—
238#1728—531no—
239C2×C53C2×#40011yescharacteristic
240C2×C53#307711yescharacteristic
241C2×C53#307811yescharacteristic
242C2×C53#309411yescharacteristic
243C2×C53#309511yescharacteristic
244C2×C53#314111yescharacteristic
245C2×C53#314211yescharacteristic
246D53—24no—
247C22×M16—532no—
248C2×#117—532no—
249C22×C4⋊C4—531no—
250C22×C4⋊C4—531no—
251C22×C4⋊C4—1061no—
252C22×C42—531no—
253#389—531no—
254#389—531no—
255#389—531no—
256#389—1061no—
257#389—1061no—
258#389C2×D5311yescharacteristic
259#384—531no—
260#384C2×D5311yescharacteristic
261#382—532no—
262#380—1061no—
263#380—1061no—
264C4×C4⋊C4—1061no—
265C4×C4⋊C4—1061no—
266C2×#95—1061no—
267#340—532no—
268#340—532no—
269#340—532no—
270#340—532no—
271#340—532no—
272#340—532no—
273#340C2×D5312yes—
274#340C2×D5312yes—
275#336—532no—
276#336—531no—
277#336C2×D5311yescharacteristic
278C53#2374711yescharacteristic, complemented, Sylow
279C23×C4—531no—
280C23×C4—531no—
281C23×C4—531no—
282C23×C4—1061no—
283C23×C4—1061no—
284C2×M16—1064no—
285C2×M16—1064no—
286C2×M16—1064no—
287C22×C8—532no—
288C22×C8C22×D5312yes—
289#117—1068no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—531no—
294C2×C4⋊C4—534no—
295C2×C4⋊C4—531no—
296C2×C4⋊C4—532no—
297C2×C4⋊C4—1062no—
298C2×C4⋊C4—1064no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—2124no—
307C2×C4⋊C4C22×D5311yescharacteristic
308C2×C4⋊C4C22×D5312yes—
309C2×C4⋊C4C22×D5311yescharacteristic
310C2×C22⋊C4—1062no—
311C2×C42—531no—
312C2×C42—531no—
313C2×C42—531no—
314C2×C42—1064no—
315C2×C42—1062no—
316C2×C42C22×D5311yescharacteristic
317#95—21no—
318#95—1061no—
319#95—1061no—
320#95—1062no—
321#95—1062no—
322#95—1061no—
323C24—531no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—531no—
328C22×C4—532no—
329C22×C4—534no—
330C22×C4—532no—
331C22×C4—534no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—1062no—
335C22×C4—1062no—
336C22×C4—1064no—
337C22×C4—1061no—
338C22×C4—1062no—
339C22×C4—1062no—
340C22×C4—1062no—
341C22×C4—1061no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2122no—
345C22×C4—2122no—
346C22×C4—2122no—
347C22×C4—2122no—
348C22×C4—2122no—
349C22×C4D4×D5311yescharacteristic
350C22×C4D4×D5311yescharacteristic
351C22×C4C23×D5311yescharacteristic
352M16—2128no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—1062no—
357C2×C8—1062no—
358C2×C8—1062no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—532no—
366C4⋊C4—1064no—
367C4⋊C4—1061no—
368C4⋊C4—1062no—
369C4⋊C4—1064no—
370C4⋊C4—2122no—
371C4⋊C4—2122no—
372C4⋊C4—2122no—
373C4⋊C4—2122no—
374C4⋊C4—2122no—
375C4⋊C4—2122no—
376C4⋊C4Q8×D5312yes—
377C22⋊C4—2124no—
378C42—22no—
379C42—1062no—
380C42—1062no—
381C42—1062no—
382C42—2124no—
383C23—532no—
384C23—532no—
385C23—532no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23C2×D4×D5311yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—532no—
405C2×C4—532no—
406C2×C4—532no—
407C2×C4—538no—
408C2×C4—538no—
409C2×C4—1062no—
410C2×C4—1061no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—4244no—
417C2×C4—2124no—
418C2×C4—2124no—
419C2×C4—2128no—
420C2×C4—2121no—
421C2×C4—2122no—
422C2×C4—2121no—
423C2×C4—2122no—
424C2×C4—2122no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4#1313324412yes—
429C2×C4#1313324912yes—
430C2×C4C2×Q8×D5311yescharacteristic
431C2×C4#41×D5312yes—
432C2×C4#41×D5312yes—
433C2×C4#41×D5311yescharacteristic
434C8—42no—
435C8—2122no—
436C22—534no—
437C22—1062no—
438C22—1062no—
439C22—1062no—
440C22—1062no—
441C22—1062no—
442C22—1062no—
443C22C22⋊Q8×D5311yescharacteristic
444C22#123×D5311yescharacteristic
445C22#124×D5311yescharacteristic
446C22C2×#1313324411yescharacteristic
447C22C2×#1313324911yescharacteristic
448C22#1681141911yescharacteristic
449C22#1681141911yescharacteristic
450C4—22no—
451C4—22no—
452C4—42no—
453C4—82no—
454C4—81no—
455C4—534no—
456C4—4242no—
457C4—4241no—
458C4—2122no—
459C4—2121no—
460C4—2121no—
461C4#125×D5312yes—
462C4#126×D5312yes—
463C2—1064no—
464C2#400×D5311yescharacteristic
465C2#2028451311yescharacteristic
466C2#2028452111yescharacteristic
467C2#2028465211yescharacteristic
468C2#2028466011yescharacteristic
469C2#2028511011yescharacteristic
470C2#2028511411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2100C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2100D53
hypocenterC53#23747
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2100

Complements

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

NH ≅ G/Ntype
3. #20276286C2semidirect
4. #20276285C2semidirect
5. #20276277C2semidirect
6. #20276274C2semidirect
7. #20276273C2semidirect
11. C53×#2100C2semidirect
233. #2100D53semidirect
278. C53#23747semidirect

Maximal subgroups

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

subgroupindex
#389×D532
#202771202
#202762862
#202762852
#202762772
#202762742
#202762732
#202744332
#202744072
C53×#21002
#2374753