Finite Group Databasebeta

← #27276100

Subgroups — #27276100

Order 13568.

760 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274725C211yesmaximal, characteristic, complemented
2#20273372C211yesmaximal, characteristic, complemented
3#20273279C211yesmaximal, characteristic, complemented
4#20273275C211yesmaximal, characteristic, complemented
5#20273114C211yesmaximal, characteristic, complemented
6#20273106C211yesmaximal, characteristic, complemented
7C53×#3506C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9#16810676C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133138—21no—
12#16810489C2211yescharacteristic
13#16810385—21no—
14#16810372—21no—
15#16810369—21no—
16#16810360—22no—
17#16810357—22no—
18D4⋊#88322—22no—
19D4⋊#88322—22no—
20C53×#559—22no—
21C53×#556C412yescomplemented
22C53×#555—21no—
23C53×#553—22no—
24C53×#550C412yescomplemented
25C53×#547—21no—
26C53×#538—22no—
27C53×#535—21no—
28C53×#533C2211yescharacteristic
29C53×#532C2211yescharacteristic
30C53×#525—22no—
31C53×#520—21no—
32C2×C4×D4×C53C2211yescharacteristic
33C22×#88322—21no—
34C2×C4⋊Dic53—21no—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39#13133138—42no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C4×#88322—21no—
43C4×#88322—21no—
44#13133124—21no—
45#13133124—21no—
46C2×#41×C53—22no—
47C2×#41×C53—21no—
48C22×D4×C53D411yescharacteristic
49C23×C4×C53—21no—
50C4⋊Q8×C53C2×C411yescharacteristic
51C4⋊Q8×C53D411yescharacteristic
52C4⋊Q8×C53D411yescharacteristic
53#127×C53—22no—
54#127×C53C2×C411yescharacteristic
55#126×C53—24no—
56#125×C53—21no—
57#124×C53—24no—
58#124×C53—21no—
59#124×C53—22no—
60#123×C53—24no—
61#123×C53—24no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53—21no—
64C22⋊Q8×C53D412yes—
65#121×C53—22no—
66#121×C53—22no—
67#121×C53—22no—
68#121×C53—24no—
69#121×C53—21no—
70#121×C53D412yes—
71#120×C53—24no—
72C4×Q8×C53—22no—
73C4×Q8×C53—21no—
74C4×Q8×C53C2×C412yes—
75C4×D4×C53—24no—
76C4×D4×C53—24no—
77C4×D4×C53—22no—
78C4×D4×C53—21no—
79C4×D4×C53—21no—
80C4×D4×C53—21no—
81C4×D4×C53C2×C412yes—
82C4×D4×C53D412yes—
83C4×D4×C53D412yes—
84#117×C53—22no—
85#117×C53—22no—
86#117×C53—21no—
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53D411yescharacteristic
89C2×C22⋊C4×C53—21no—
90C2×C42×C53C2311yescharacteristic
91C24×C53—21no—
92#41×C53—41no—
93#41×C53—41no—
94#41×C53—41no—
95#41×C53—41no—
96#41×C53—84no—
97C2×Q8×C53—21no—
98C2×Q8×C53—21no—
99C2×Q8×C53C22⋊C412yes—
100C2×D4×C53—22no—
101C2×D4×C53—24no—
102C2×D4×C53—24no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53C22⋊C412yes—
111C2×D4×C53C22⋊C412yes—
112C22×C4×C53—24no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—82no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×Dic53—41no—
122C4⋊C4×C53—24no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53C22⋊C412yes—
131C4⋊C4×C53C22⋊C412yes—
132C4⋊C4×C53C22⋊C412yes—
133C22⋊C4×C53—24no—
134C22⋊C4×C53—24no—
135C22⋊C4×C53—24no—
136C22⋊C4×C53—22no—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—22no—
141C42×C53—24no—
142C42×C53—22no—
143C42×C53C22×C411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C4⋊Dic53—41no—
148C4⋊Dic53—41no—
149C2×#88322—41no—
150C2×#88322—41no—
151C2×#88322—41no—
152C2×#88322—41no—
153C2×#88322—41no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—81no—
159C23×C53—81no—
160C23×C53#12011yescharacteristic
161Q8×C53—42no—
162Q8×C53—41no—
163Q8×C53—41no—
164Q8×C53—81no—
165D4×C53—22no—
166D4×C53—22no—
167D4×C53—42no—
168D4×C53—42no—
169D4×C53—42no—
170D4×C53—42no—
171D4×C53—41no—
172D4×C53—41no—
173D4×C53—41no—
174D4×C53—41no—
175D4×C53—84no—
176D4×C53—84no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—44no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—82no—
187C2×C4×C53—81no—
188C2×C4×C53—84no—
189C2×C4×C53C2×C22⋊C411yescharacteristic
190C2×C4×C53C2×C22⋊C411yescharacteristic
191C2×C4×C53C2×C22⋊C411yescharacteristic
192C2×C4×C53#12011yescharacteristic
193C2×C4×C53#12011yescharacteristic
194C2×C4×C53#12011yescharacteristic
195C2×C4×C53#13511yescharacteristic
196C2×C4×C53#13511yescharacteristic
197C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C53#13711yescharacteristic
199C2×Dic53—81no—
200C2×Dic53—81no—
201C2×Dic53—81no—
202#88322—81no—
203#88322—81no—
204#88322—81no—
205#8434—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—42no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—82no—
215C22×C53—82no—
216C22×C53#37911yescharacteristic
217C22×C53#44911yescharacteristic
218C22×C53#45211yescharacteristic
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—42no—
224C4×C53—42no—
225C4×C53—42no—
226C4×C53—42no—
227C4×C53#41611yescharacteristic
228C4×C53#41711yescharacteristic
229C4×C53#44911yescharacteristic
230C4×C53#45211yescharacteristic
231Dic53—161no—
232#3506D5311yescharacteristic
233#1804—531no—
234#1542—531no—
235#1518—531no—
236#1517—531no—
237#1480—531no—
238#1478—531no—
239C2×C53—21no—
240C2×C53—21no—
241C2×C53—82no—
242C2×C53—82no—
243C2×C53#178811yescharacteristic
244C2×C53#179411yescharacteristic
245C2×C53#179511yescharacteristic
246#559—22no—
247#556Dic5312yescomplemented
248#555—21no—
249#553—22no—
250#550Dic5312yescomplemented
251#547—21no—
252#538—22no—
253#535—21no—
254#533C2×D5311yescharacteristic
255#532C2×D5311yescharacteristic
256#525—22no—
257#520—21no—
258C2×C4×D4C2×D5311yescharacteristic
259#431—531no—
260#431—531no—
261C2×C4⋊C8—531no—
262C2×#102—1061no—
263#382—531no—
264#340—1061no—
265#333—1061no—
266#332—1061no—
267#328—1062no—
268#327—1062no—
269D4⋊C8—1062no—
270D4⋊C8—1062no—
271C53#843411yescharacteristic, complemented, Sylow
272C2×#41—22no—
273C2×#41—21no—
274C22×D4C22⋊D5311yescharacteristic
275C23×C4—21no—
276C22×C8—1061no—
277C4⋊Q8C2×Dic5311yescharacteristic
278C4⋊Q8C22⋊D5311yescharacteristic
279C4⋊Q8C22⋊D5311yescharacteristic
280#127—22no—
281#127C2×Dic5311yescharacteristic
282#126—24no—
283#125—21no—
284#124—24no—
285#124—22no—
286#124—21no—
287#123—24no—
288#123—24no—
289C22⋊Q8—22no—
290C22⋊Q8—21no—
291C22⋊Q8C22⋊D5312yes—
292#121—22no—
293#121—22no—
294#121—24no—
295#121—21no—
296#121—22no—
297#121C22⋊D5312yes—
298#120—24no—
299C4×Q8—22no—
300C4×Q8—21no—
301C4×Q8C2×Dic5312yes—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4C2×Dic5312yes—
309C4×D4C22⋊D5312yes—
310C4×D4C22⋊D5312yes—
311#117—22no—
312#117—21no—
313#117—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4C22⋊D5311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C42C22×D5311yescharacteristic
319C4⋊C8—1061no—
320C4⋊C8—1061no—
321C4⋊C8—1061no—
322C4⋊C8—1061no—
323#102—2122no—
324C22⋊C8—1061no—
325C22⋊C8—1061no—
326C4×C8—1061no—
327C4×C8—1061no—
328#95—1061no—
329#95—1061no—
330C24—21no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—84no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8C22⋊Dic5312yes—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4C22⋊Dic5312yes—
350C2×D4C22⋊Dic5312yes—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—82no—
357C22×C4—2121no—
358C22×C4C2×C22⋊D5311yescharacteristic
359C22×C4C2×C22⋊D5311yescharacteristic
360C22×C4C2×C22⋊D5311yescharacteristic
361C2×C8—2121no—
362C2×C8—2121no—
363C2×C8—2121no—
364C2×C8—2121no—
365C2×C8—2121no—
366C4⋊C4—24no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—2121no—
375C4⋊C4—2121no—
376C4⋊C4C22⋊Dic5312yes—
377C4⋊C4C22⋊Dic5312yes—
378C4⋊C4C22⋊Dic5312yes—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C42—24no—
388C42—22no—
389C42C22×Dic5311yescharacteristic
390C42C2×C22⋊D5311yescharacteristic
391C42C2×C22⋊D5311yescharacteristic
392C42C2×C22⋊D5311yescharacteristic
393C23—22no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—81no—
398C23—81no—
399C23#1313318611yescharacteristic
400Q8—42no—
401Q8—41no—
402Q8—41no—
403Q8—81no—
404D4—22no—
405D4—22no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—84no—
415D4—84no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—84no—
428C2×C4—4241no—
429C2×C4—4241no—
430C2×C4—4241no—
431C2×C4C2×C22⋊Dic5311yescharacteristic
432C2×C4C2×C22⋊Dic5311yescharacteristic
433C2×C4C2×C22⋊Dic5311yescharacteristic
434C2×C4#1313318611yescharacteristic
435C2×C4#1313318611yescharacteristic
436C2×C4#1313318611yescharacteristic
437C2×C4#1313323711yescharacteristic
438C2×C4#1313323711yescharacteristic
439C2×C4#1313324211yescharacteristic
440C2×C4#1313324711yescharacteristic
441C8—4241no—
442C8—4241no—
443C8—4241no—
444C22—21no—
445C22—21no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—82no—
453C22—82no—
454C22#1681047911yescharacteristic
455C22#1681076811yescharacteristic
456C22#1681078511yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—8481no—
466C4#1681061911yescharacteristic
467C4#1681062411yescharacteristic
468C4#1681076811yescharacteristic
469C4#1681078511yescharacteristic
470C2—21no—
471C2—21no—
472C2—82no—
473C2—82no—
474C2#2027465511yescharacteristic
475C2#2027467911yescharacteristic
476C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3506C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3506D53
hypocenterC53#8434
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #20274725C2semidirect
2. #20273372C2semidirect
3. #20273279C2semidirect
4. #20273275C2semidirect
5. #20273114C2semidirect
6. #20273106C2semidirect
21. C53×#556C4semidirect
24. C53×#550C4semidirect
247. #556Dic53semidirect
250. #550Dic53semidirect
271. C53#8434semidirect

Maximal subgroups

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

subgroupindex
#202747252
#202733722
#202732792
#202732752
#202731142
#202731062
C53×#35062
#843453