Finite Group Databasebeta

← #27367927

Subgroups — #27367927

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285275C211yesmaximal, characteristic
2#20284187C211yesmaximal, characteristic
3C2×Q8×#88322C211yesmaximal, characteristic, complemented
4#20274143C212yesmaximal, complemented
5#20274133C212yesmaximal, complemented
6#20274099C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273470C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273393C211yesmaximal, characteristic
11C53×#3078C211yesmaximal, characteristic, complemented
12C2×#13133248—21no—
13#16811173C2211yescharacteristic
14#16811019C2211yescharacteristic
15#16810997C2211yescharacteristic
16#16810973C2211yescharacteristic
17#16810971C2211yescharacteristic
18#16810854—21no—
19#16810854—21no—
20#16810845—22no—
21#16810833C2212yes—
22#16810823C2212yes—
23Q8×#88322—21no—
24Q8×#88322—21no—
25Q8×#88322C2212yes—
26Q8×#88322C2212yes—
27C4×#11719924—21no—
28C4×#11719924—21no—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322—21no—
31C2×C4⋊#88322C2211yescharacteristic
32#16810622—21no—
33C2×C4×#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810373C2212yes—
36#16810373C2212yes—
37#16810372C2212yes—
38#16810370C2212yes—
39#16810369C2212yes—
40#16810366C2212yes—
41C53×#533C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#479—22no—
44C53×#477C2212yes—
45C53×#475C2212yes—
46C53×#428—21no—
47C53×#423C2211yescharacteristic
48C53×#419—21no—
49C2×#11719945—21no—
50#13133248—44no—
51C2×#11719925—22no—
52C2×#11719924—21no—
53C2×#11719924—21no—
54C2×#11719924—22no—
55C2×#11719917—21no—
56C22×#88322—21no—
57C22×#88322D411yescharacteristic
58C22×#88322D411yescharacteristic
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133205—21no—
62#13133191—21no—
63#13133191—21no—
64C4×D212—21no—
65C4×D212—21no—
66#13133148—21no—
67#13133148—21no—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322—22no—
71C4⋊#88322—22no—
72C4⋊#88322C2312yes—
73C4⋊#88322C2312yes—
74#13133141—21no—
75#13133141—21no—
76#13133140—22no—
77#13133137—22no—
78#13133137—21no—
79#13133137—21no—
80C4×#88322—22no—
81C4×#88322—22no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C22×Q8×C53D411yescharacteristic
87C2×M16×C53—21no—
88C4⋊Q8×C53C2311yescharacteristic
89C4⋊Q8×C53C2311yescharacteristic
90#125×C53—21no—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×Q8×C53D412yes—
98C4×Q8×C53C2312yes—
99#117×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105#106×C53—22no—
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#103×C53—22no—
109#103×C53—22no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53SD1614yescomplemented
115C2×Q8×C53C2×D412yes—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×C22⋊D53—41no—
122#11719945—42no—
123#11719945—42no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212—21no—
127C2×C4×D53—41no—
128C2×Dic106—21no—
129M16×C53—42no—
130C2×C8×C53—22no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53C2×D412yes—
141C22⋊C4×C53—41no—
142C42×C53—22no—
143C42×C53—22no—
144C42×C53—41no—
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53#4111yescharacteristic
148C42×C53C2411yescharacteristic
149#11719925—44no—
150#11719924—42no—
151#11719924—44no—
152#11719924—42no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157#11719917—42no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—41no—
162C2×#88322—42no—
163C2×#88322C2×D411yescharacteristic
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×D411yescharacteristic
167C23×C53#12111yescharacteristic
168C22×D53—41no—
169Q8×C53—22no—
170Q8×C53—24no—
171Q8×C53—41no—
172Q8×C53—41no—
173Q8×C53—81no—
174Q8×C53C2×SD1614yes—
175C2×C4×C53—22no—
176C2×C4×C53—24no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—22no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53C2×SD1612yes—
188C2×C4×C53#13512yes—
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C2×#4111yescharacteristic
192C22⋊D53—82no—
193C2×Dic53—41no—
194D212—41no—
195D212—41no—
196D212—41no—
197D212—41no—
198D212—81no—
199C4×D53—82no—
200Dic106—41no—
201Dic106—41no—
202C8×C53—42no—
203#88322—21no—
204#88322—21no—
205#88322—21no—
206#88322—42no—
207#88322—42no—
208#88322#13712yescomplemented
209#19260—531nomaximal, Sylow
210C22×C53—21no—
211C22×C53—21no—
212C22×C53#46512yes—
213C22×C53C2×#12111yescharacteristic
214C2×D53—81no—
215C2×D53—81no—
216C2×D53—81no—
217C4×C53—21no—
218C4×C53—24no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—42no—
222C4×C53—81no—
223C4×C53—81no—
224C4×C53#46512yes—
225C4×C53C22×SD1611yescharacteristic
226C4×C53C2×#13511yescharacteristic
227C4×C53C2×#13711yescharacteristic
228C4×C53#57511yescharacteristic
229Dic53—81no—
230#3167—531no—
231#3078D5311yescharacteristic, complemented
232#3037—531no—
233C2×C8×Q8—531no—
234#1680—532no—
235#1678—532no—
236#1672—532no—
237#1666—532no—
238#1566—531no—
239#1550—531no—
240#1547—531no—
241C2×C53—22no—
242C2×C53#303711yescharacteristic
243C2×C53C2×#46511yescharacteristic
244C2×C53#306711yescharacteristic
245D53—161no—
246C2×#137—1061no—
247#533C2×D5311yescharacteristic
248#532—531no—
249C2×C4×Q8C2×D5311yescharacteristic
250#498—531no—
251#492—531no—
252#487—531no—
253#486—531no—
254#479—22no—
255#477C2×D5312yes—
256#475C2×D5312yes—
257#465—1061no—
258#465—1061no—
259#464—1062no—
260#462—532no—
261#460—532no—
262C8×Q8—532no—
263C8×Q8—532no—
264C8×Q8—1061no—
265C8×Q8—1061no—
266C4×SD16—1061no—
267C4×SD16—1061no—
268#428—21no—
269#423—531no—
270#423C2×D5311yescharacteristic
271C2×C4⋊C8—531no—
272C2×C4⋊C8—1061no—
273#419—21no—
274#417—1061no—
275C2×C4×C8—531no—
276C2×C4×C8—1061no—
277#334—532no—
278#334—532no—
279#333—532no—
280#332—532no—
281#332—532no—
282#331—532no—
283C53#1926011yescharacteristic, complemented, Sylow
284C2×#41—1061no—
285C22×Q8C22⋊D5311yescharacteristic
286#137—2124no—
287C2×Q16—1062no—
288C2×SD16—1061no—
289C2×SD16—1061no—
290C2×SD16—1062no—
291C2×M16—21no—
292C2×M16—1061no—
293C22×C8—531no—
294C22×C8—531no—
295C22×C8—1061no—
296C4⋊Q8—531no—
297C4⋊Q8C22×D5311yescharacteristic
298C4⋊Q8C22×D5311yescharacteristic
299#127—531no—
300#125—21no—
301#124—1061no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—1061no—
305#121—1061no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8C22⋊D5312yes—
311C4×Q8C22×D5312yes—
312C4×D4—1061no—
313C4×D4—1061no—
314#117—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C22⋊D5311yescharacteristic
317C2×C42—21no—
318C2×C42C22×D5311yescharacteristic
319#107—22no—
320#106—22no—
321#106—1061no—
322#106—1061no—
323C4⋊C8—21no—
324C4⋊C8—532no—
325C4⋊C8—532no—
326C4⋊C8—1062no—
327C4⋊C8—1062no—
328C4⋊C8—1061no—
329C4⋊C8—1061no—
330C4⋊C8C22×D5312yes—
331#103—22no—
332#103—22no—
333#103—1061no—
334#103—1061no—
335#103—1062no—
336#102—1061no—
337#102—1061no—
338#102—1062no—
339C4×C8—531no—
340C4×C8—531no—
341C4×C8—531no—
342C4×C8—531no—
343C4×C8—1062no—
344C4×C8—1062no—
345#41—2122no—
346#41—2122no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—1061no—
352C2×Q8#1171992414yescomplemented
353C2×Q8C2×C22⋊D5312yes—
354C2×D4—1061no—
355C2×D4—1061no—
356C2×D4—1061no—
357C2×D4—2121no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—2121no—
361C22×C4D4×D5311yescharacteristic
362C22×C4#1171994711yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364Q16—2124no—
365SD16—2122no—
366SD16—2122no—
367SD16—2124no—
368M16—42no—
369M16—2122no—
370C2×C8—22no—
371C2×C8—531no—
372C2×C8—531no—
373C2×C8—531no—
374C2×C8—531no—
375C2×C8—1062no—
376C2×C8—1062no—
377C2×C8—1062no—
378C2×C8—2121no—
379C2×C8—2122no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—1061no—
390C4⋊C4—1061no—
391C4⋊C4C2×C22⋊D5312yes—
392C22⋊C4—41no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42—22no—
396C42—22no—
397C42—41no—
398C42D4×D5311yescharacteristic
399C42#1171994711yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—2121no—
403C23#1313318911yescharacteristic
404Q8—24no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408Q8—81no—
409Q8—2121no—
410Q8—2121no—
411Q8C2×#1171992414yes—
412D4—4241no—
413D4—4242no—
414D4—2121no—
415D4—2121no—
416D4—2121no—
417D4—2121no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—4242no—
428C2×C4—2121no—
429C2×C4#1313318911yescharacteristic
430C2×C4#1313318911yescharacteristic
431C2×C4#1313318911yescharacteristic
432C2×C4C2×#1171992412yes—
433C2×C4#1313323712yes—
434C2×C4C2×D4×D5311yescharacteristic
435C2×C4C22×C22⋊D5311yescharacteristic
436C2×C4C2×#1171994711yescharacteristic
437C8—42no—
438C8—532no—
439C8—1061no—
440C8—1061no—
441C8—1061no—
442C8—2122no—
443C8—2122no—
444C22—21no—
445C22—21no—
446C22—4241no—
447C22—4241no—
448C22—4241no—
449C22#1681085412yes—
450C22C2×#1313318911yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—24no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4—4241no—
459C4#1681085412yes—
460C4C22×#1171992411yescharacteristic
461C4C2×#1313323711yescharacteristic
462C4#137×D5311yescharacteristic
463C4#1681142211yescharacteristic
464C2—22no—
465C2—8481no—
466C2#2028418411yescharacteristic
467C2C2×#1681085411yescharacteristic
468C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3078C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3078D53
hypocenterC53#19260
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. C2×Q8×#88322C2semidirect
4. #20274143C2semidirect
5. #20274133C2semidirect
6. #20274099C2semidirect
7. #20274065C2semidirect
8. #20273470C2semidirect
9. #20273405C2semidirect
11. C53×#3078C2semidirect
114. C2×Q8×C53SD16semidirect
208. #88322#137semidirect
231. #3078D53semidirect
283. C53#19260semidirect
352. C2×Q8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202852752
#202841872
C2×Q8×#883222
#202741432
#202741332
#202740992
#202740652
#202734702
#202734052
#202733932
C53×#30782
#1926053