Finite Group Databasebeta

← #28319283

Subgroups — #28319283

Order 13568.

1166 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291397C212yesmaximal
2#20290616C212yesmaximal
3#20290582C212yesmaximal, complemented
4#20290581C212yesmaximal, complemented
5#20290550C212yesmaximal
6#20290468C212yesmaximal, complemented
7C4×#13133278C212yesmaximal, complemented
8C2×#16811099C211yesmaximal, characteristic
9#16811648—28no—
10#16811626C2214yes—
11C2×#13133278C2214yes—
12C2×#13133278C2214yes—
13#16811595C2214yes—
14C4×#11719940C2214yes—
15#16811503C2214yes—
16#16811503C2214yes—
17C22×Q8×D53C2211yescharacteristic
18#16811099—28no—
19C2×Q8×Dic53C411yescharacteristic
20C2×C4×Dic106C411yescharacteristic
21C2×C4×Dic106C2211yescharacteristic
22C2×C4×Dic106C2211yescharacteristic
23C2×#13133182C411yescharacteristic
24C2×#13133182C2211yescharacteristic
25C2×#13133182C2211yescharacteristic
26C2×#13133175C411yescharacteristic
27C2×#13133175C411yescharacteristic
28C2×#13133175C2211yescharacteristic
29C2×#13133174C411yescharacteristic
30C2×#13133174C411yescharacteristic
31C2×#13133174C2211yescharacteristic
32C2×C4×Q8×C53C411yescharacteristic
33C2×#11719940C2318yes—
34#13133284—24no—
35C4⋊#88324—24no—
36#13133278—28no—
37#13133278—28no—
38#13133274—28no—
39#13133272—24no—
40C4×#88324—24no—
41C22×Dic106C2×C411yescharacteristic
42C22×Dic106C2311yescharacteristic
43C22×Dic106C2311yescharacteristic
44C2×Q8×D53—24no—
45C2×Q8×D53—24no—
46C2×Q8×D53—24no—
47C22×C4×D53C2×C411yescharacteristic
48C22×C4×D53C2×C411yescharacteristic
49C22×C4×D53C2311yescharacteristic
50Q8×Dic53—24no—
51C4×Dic106—24no—
52C4×Dic106—24no—
53C4×Dic106—24no—
54#13133182—24no—
55#13133182—24no—
56#13133182—24no—
57#13133175—24no—
58#13133175—24no—
59#13133175D418yes—
60#13133174—24no—
61#13133174—24no—
62#13133174—24no—
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×C4⋊Dic53C2×C411yescharacteristic
66C2×#11719919C2×C411yescharacteristic
67C2×#11719919C2×C411yescharacteristic
68C2×#11719919C2×C411yescharacteristic
69C2×#11719919C2×C411yescharacteristic
70C2×#11719919C2311yescharacteristic
71C2×#11719919C2311yescharacteristic
72C2×#11719921C2×C411yescharacteristic
73C2×#11719921C2×C411yescharacteristic
74C2×#11719921C2×C411yescharacteristic
75C2×#11719921C2×C411yescharacteristic
76C2×C4×Dic53C2×C411yescharacteristic
77C2×C4×Dic53C2×C411yescharacteristic
78C2×C4×Dic53C2×C411yescharacteristic
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53C2×C411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C22×Q8×C53C2×C411yescharacteristic
86C4×Q8×C53—24no—
87C2×C4⋊C4×C53C2×C411yescharacteristic
88C2×C4⋊C4×C53C2×C411yescharacteristic
89C2×C4⋊C4×C53C2×C411yescharacteristic
90C2×C42×C53C2×C411yescharacteristic
91C2×C42×C53C2×C411yescharacteristic
92C2×C42×C53C2×C411yescharacteristic
93C23×D53C22×C411yescharacteristic
94C2×Q8×C53—22no—
95C2×Q8×C53—22no—
96C2×Q8×C53—22no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C22×C411yescharacteristic
104C22×Dic53C22×C411yescharacteristic
105C22×Dic53C22×C411yescharacteristic
106C22×Dic53C22×C411yescharacteristic
107C22×Dic53C22×C411yescharacteristic
108C22×Dic53C22×C411yescharacteristic
109C22×Dic53C22×C411yescharacteristic
110C22×Dic53C2411yescharacteristic
111Q8×D53—232no—
112C2×C4×D53—22no—
113C2×C4×D53—22no—
114C2×C4×D53—22no—
115C2×C4×D53—22no—
116C2×C4×D53—22no—
117C2×C4×D53—22no—
118C2×C4×D53—22no—
119C2×C4×D53—22no—
120C2×C4×D53C2×D414yes—
121C2×Dic106—22no—
122C2×Dic106—22no—
123C2×Dic106—22no—
124C2×Dic106—22no—
125C2×Dic106—22no—
126C2×Dic106—22no—
127C2×Dic106—22no—
128C2×Dic106#4114yes—
129C2×Dic106#4114yes—
130#11719940—28no—
131#11719940—28no—
132C2×#88324—28no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53C22⋊C414yes—
136C42×C53—22no—
137C42×C53—22no—
138C42×C53—22no—
139#11719921—22no—
140#11719921—22no—
141#11719921C22⋊C414yes—
142#11719921C22⋊C414yes—
143C4⋊Dic53—22no—
144C4⋊Dic53—22no—
145C4⋊Dic53C22⋊C414yes—
146#11719919—22no—
147#11719919—22no—
148#11719919—22no—
149#11719919—22no—
150#11719919—22no—
151#11719919—22no—
152C4×Dic53—22no—
153C4×Dic53—22no—
154C4×Dic53—22no—
155C4×Dic53—22no—
156C4×Dic53—22no—
157C4×Dic53—22no—
158C4×Dic53—22no—
159C4×Dic53C2×D414yes—
160C4×Dic53C2×D414yes—
161C23×C53C23×C411yescharacteristic
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—41no—
165C22×D53—41no—
166C22×D53C2×C22⋊C412yes—
167Q8×C53—28no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53C2×C22⋊C412yes—
181C2×C4×C53C2×C22⋊C412yes—
182C2×C4×C53C2×C22⋊C412yes—
183C2×C4×C53#11712yes—
184C2×C4×C53#11712yes—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53C2×C22⋊C412yes—
196C2×Dic53C2×C22⋊C412yes—
197C2×Dic53#11712yes—
198C2×Dic53#11712yes—
199C2×Dic53C22×D412yes—
200C2×Dic53C2×#4112yes—
201C2×Dic53C2×#4112yes—
202C4×D53—28no—
203C4×D53—28no—
204C4×D53—28no—
205Dic106—28no—
206Dic106—28no—
207Dic106—28no—
208#88324—48no—
209#20730—531nomaximal, Sylow
210C22×C53—21no—
211C22×C53—21no—
212C22×C53C22×C22⋊C411yescharacteristic
213C22×C53C2×#11711yescharacteristic
214C22×C53C2×#11711yescharacteristic
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—42no—
219C2×D53—42no—
220C4×C53—22no—
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—41no—
226C4×C53—41no—
227Dic53—22no—
228Dic53—22no—
229Dic53—22no—
230Dic53—22no—
231Dic53—41no—
232Dic53—41no—
233Dic53#52514yes—
234C2×#519—531no—
235#2921—532no—
236#1822—532no—
237#1801—532no—
238#1801—532no—
239#1784—532no—
240C4×#101—532no—
241C4×#101—532no—
242C2×C53—21no—
243C2×C53—21no—
244C2×C53#228011yescharacteristic
245C2×C53#288911yescharacteristic
246C2×C53#288911yescharacteristic
247D53—42no—
248C23×Q8—531no—
249#519—1068no—
250C2×C4×Q8—531no—
251C2×C4×Q8—531no—
252C2×C4×Q8—531no—
253C2×C4×Q8—531no—
254C2×C4×Q8—531no—
255C2×C4×Q8—531no—
256C2×C4×Q8—531no—
257C2×C4×Q8#615711yescharacteristic
258C2×#117—531no—
259C2×#117—531no—
260C2×#117—531no—
261C2×#117—531no—
262C2×#117—531no—
263C2×#117—531no—
264#433—1068no—
265#423—534no—
266C2×#101—534no—
267C2×#101—534no—
268#407—534no—
269C4×M16—534no—
270#343—534no—
271#343—534no—
272C53#2073011yescharacteristic, complemented, Sylow
273C22×Q8—531no—
274C22×Q8—531no—
275C22×Q8—531no—
276C22×Q8—1064no—
277C22×Q8—1064no—
278C22×Q8—1064no—
279C22×Q8C2×#615711yescharacteristic
280C23×C4—531no—
281C23×C4—531no—
282C23×C4—531no—
283C2×M16—538no—
284C4×Q8—24no—
285C4×Q8—1064no—
286C4×Q8—1064no—
287C4×Q8—1064no—
288C4×Q8—1064no—
289C4×Q8—1064no—
290C4×Q8—1064no—
291C4×Q8—1064no—
292#117—538no—
293#117—1064no—
294#117—1064no—
295#117—1064no—
296#117—1064no—
297#117—1064no—
298C2×C4⋊C4—531no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—531no—
304C2×C4⋊C4—531no—
305C2×C4⋊C4—531no—
306C2×C4⋊C4—531no—
307C2×C4⋊C4C2×#615711yescharacteristic
308C2×C4⋊C4C2×#615711yescharacteristic
309C2×C4⋊C4C2×#615711yescharacteristic
310C2×C22⋊C4—531no—
311C2×C22⋊C4—531no—
312C2×C22⋊C4—531no—
313C2×C22⋊C4—531no—
314C2×C42—531no—
315C2×C42—531no—
316C2×C42—531no—
317C2×C42—531no—
318C2×C42—531no—
319C2×C42—531no—
320C2×C42—531no—
321C2×C42—531no—
322C2×C42—531no—
323C2×C42C2×#615711yescharacteristic
324C2×C42C2×#615711yescharacteristic
325C2×C42C2×#615711yescharacteristic
326C4⋊C8—1064no—
327C4⋊C8—1064no—
328#101—1068no—
329#101—1068no—
330C22⋊C8—1068no—
331#97—1064no—
332C4×C8—1064no—
333C24—531no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—534no—
338C2×Q8—534no—
339C2×Q8—1062no—
340C2×Q8—1062no—
341C2×Q8—1062no—
342C2×Q8—1062no—
343C2×Q8—1062no—
344C2×Q8—1062no—
345C2×Q8—10632no—
346C2×Q8—1062no—
347C22×C4—531no—
348C22×C4—531no—
349C22×C4—531no—
350C22×C4—531no—
351C22×C4—531no—
352C22×C4—534no—
353C22×C4—531no—
354C22×C4—531no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—1062no—
358C22×C4—1062no—
359C22×C4—1062no—
360C22×C4—1062no—
361C22×C4—1062no—
362C22×C4—1062no—
363C22×C4C22×#615711yescharacteristic
364C22×C4C22×#615711yescharacteristic
365C22×C4C22×#615711yescharacteristic
366C22×C4C22×#615711yescharacteristic
367C22×C4C22×#615711yescharacteristic
368C22×C4C22×#615711yescharacteristic
369C22×C4C22×#615711yescharacteristic
370M16—1068no—
371M16—1068no—
372C2×C8—1068no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—534no—
376C4⋊C4—1062no—
377C4⋊C4—1062no—
378C4⋊C4—1062no—
379C4⋊C4—1062no—
380C4⋊C4—1062no—
381C4⋊C4—1062no—
382C4⋊C4—1062no—
383C4⋊C4—1062no—
384C4⋊C4C22⋊#615714yes—
385C22⋊C4—534no—
386C22⋊C4—534no—
387C22⋊C4—1062no—
388C22⋊C4—1062no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—534no—
393C42—534no—
394C42—1062no—
395C42—1062no—
396C42—1062no—
397C42—1062no—
398C42—1062no—
399C42—1062no—
400C42—1062no—
401C23—532no—
402C23—1061no—
403C23—1061no—
404C23—2121no—
405C23—2121no—
406C23C23×#615711yescharacteristic
407Q8—28no—
408Q8—1068no—
409Q8—1068no—
410Q8—1068no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—532no—
424C2×C4—532no—
425C2×C4—532no—
426C2×C4—532no—
427C2×C4—532no—
428C2×C4—532no—
429C2×C4—532no—
430C2×C4—1068no—
431C2×C4—1068no—
432C2×C4—1068no—
433C2×C4—1061no—
434C2×C4—1061no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4C2×C22⋊#615712yes—
444C2×C4C2×C22⋊#615712yes—
445C2×C4C2×C22⋊#615712yes—
446C2×C4#1313329612yes—
447C2×C4#1313329612yes—
448C8—2128no—
449C22—21no—
450C22—21no—
451C22—1062no—
452C22—1062no—
453C22—1062no—
454C22—2122no—
455C22—2122no—
456C22C22×C22⋊#615711yescharacteristic
457C22C2×#1313329611yescharacteristic
458C22C2×#1313329611yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—41no—
465C4—41no—
466C4—534no—
467C4—1062no—
468C4—1062no—
469C4—1062no—
470C4—1062no—
471C4—2121no—
472C4—2121no—
473C2—21no—
474C2—21no—
475C2—2122no—
476C2#2029115911yescharacteristic
477C2#2029132711yescharacteristic
478C2#2029132811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22×C4
cosocleC22×Dic53C24
hypercenterC2×C4×Q8#6157
hypocenterC53#20730
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20290582C2semidirect
4. #20290581C2semidirect
6. #20290468C2semidirect
7. C4×#13133278C2semidirect
272. C53#20730semidirect

Maximal subgroups

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

subgroupindex
#202913972
#202906162
#202905822
#202905812
#202905502
#202904682
C4×#131332782
C2×#168110992
#2073053