Finite Group Databasebeta

← #27276832

Subgroups — #27276832

Order 13568.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133191C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic, complemented
3C2×#16810356C212yesmaximal
4#20272594C212yesmaximal, complemented
5C53×#1434C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7C2×C4×D212C411yescharacteristic, complemented
8C2×C4×C22⋊D53—21no—
9C2×C4⋊#88322—21no—
10C2×C22⋊#88322C2212yes—
11#16810525C411yescharacteristic, complemented
12#16810521C411yescharacteristic, complemented
13#16810514—21no—
14#16810502—21no—
15#16810483C2211yescharacteristic
16C4×C4⋊Dic53C411yescharacteristic, complemented
17C4×#11719921—21no—
18#16810378—21no—
19#16810356—24no—
20C22×C42×C53C2211yescharacteristic
21C2×#95×C53—21no—
22C2×#95×C53—21no—
23C2×#95×C53C2212yes—
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C22×C4×D53—21no—
27C22×#88322—22no—
28#13133191—21no—
29#13133191—21no—
30#13133191D412yes—
31#13133191Q812yes—
32C4×D212—42no—
33C4×C22⋊D53—44no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×C22⋊Dic53—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—21no—
44C4⋊#88322—42no—
45C22⋊#88322—22no—
46C22⋊#88322—22no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53—22no—
51C23×C4×C53D411yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C2×C42×C53—42no—
55C2×C42×C53C2×C411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#95×C53—24no—
60#95×C53—22no—
61#95×C53—22no—
62#95×C53—42no—
63C24×C53C2×D411yescharacteristic
64C23×D53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—24no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×Q811yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C2×C22⋊D53—22no—
91C2×C22⋊D53—22no—
92C2×C22⋊D53—42no—
93C2×C22⋊D53—42no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C2×D212—21no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212C4⋊C412yes—
101C2×C4×D53—42no—
102C2×C4×D53—42no—
103C2×C4×D53—42no—
104C42×C53—22no—
105C42×C53—22no—
106C42×C53—41no—
107C42×C53—41no—
108C22⋊Dic53—42no—
109#11719921—22no—
110#11719921—22no—
111#11719921—42no—
112#11719921—42no—
113#11719921—42no—
114C4⋊Dic53—21no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53C4⋊C412yes—
118#11719919—42no—
119C4×Dic53—42no—
120C2×#88322—42no—
121C2×#88322—42no—
122C2×#88322—42no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#12111yescharacteristic
129C23×C53C22⋊Q811yescharacteristic
130C23×C53#13512yes—
131C23×C53AGL(1,ℤ/8)12yes—
132C22×D53—22no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—44no—
158C2×C4×C53—44no—
159C2×C4×C53—44no—
160C2×C4×C53—44no—
161C2×C4×C53—44no—
162C2×C4×C53—82no—
163C2×C4×C53—82no—
164C2×C4×C53—82no—
165C2×C4×C53—82no—
166C2×C4×C53#9912yes—
167C2×C4×C53#10412yes—
168C2×C4×C53C2×C4⋊C411yescharacteristic
169C2×C4×C53#11711yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53C22⋊Q811yescharacteristic
172C22⋊D53—44no—
173C22⋊D53—44no—
174C2×Dic53—22no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183D212—42no—
184D212—42no—
185C4×D53—84no—
186#88322—82no—
187#8575—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53#38011yescharacteristic
197C22×C53C2×#9911yescharacteristic
198C22×C53#41011yescharacteristic
199C22×C53#41611yescharacteristic
200C22×C53#41811yescharacteristic
201C22×C53C2×#10411yescharacteristic
202C22×C53#42111yescharacteristic
203C22×C53#46312yes—
204C22×C53#47812yes—
205C2×D53—42no—
206C2×D53—42no—
207C2×D53—42no—
208C2×D53—81no—
209C2×D53—81no—
210C2×D53—81no—
211C2×D53—81no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—22no—
217C4×C53—22no—
218C4×C53—41no—
219C4×C53—41no—
220C4×C53—84no—
221Dic53—42no—
222Dic53—81no—
223Dic53—81no—
224C4×#121—531no—
225#1797—531no—
226C2×#327—532no—
227#1434D5311yescharacteristic, complemented
228#1294—532no—
229C2×C53—22no—
230C2×C53—22no—
231C2×C53#150111yescharacteristic
232C2×C53#150511yescharacteristic
233C2×C53#151011yescharacteristic
234C2×C53#152111yescharacteristic
235C2×C53#179911yescharacteristic
236C2×C53#180411yescharacteristic
237C2×C53#180611yescharacteristic
238D53—82no—
239C2×#121—531no—
240C2×C4×D4—531no—
241C2×C4×D4—1061no—
242C22×C42C2×D5311yescharacteristic
243C2×C4⋊C8—1061no—
244C2×C22⋊C8—532no—
245#390—531no—
246#389—531no—
247#388—1061no—
248#386—1061no—
249#380—531no—
250C4×C4⋊C4—531no—
251C4×C22⋊C4—1061no—
252C2×#95—21no—
253C2×#95—21no—
254C2×#95C2×D5312yes—
255#336—1061no—
256#327—1064no—
257C53#857511yescharacteristic, complemented, Sylow
258C22×D4—531no—
259C22×D4—1061no—
260C23×C4—22no—
261C23×C4—1061no—
262C23×C4C22⋊D5311yescharacteristic
263C23×C4C22⋊D5311yescharacteristic
264C23×C4C22×D5311yescharacteristic
265C22×C8—1062no—
266#121—532no—
267#121—532no—
268#121—1061no—
269#121—1061no—
270C4×D4—2122no—
271C4×D4—2124no—
272C2×C4⋊C4—531no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—1061no—
276C2×C22⋊C4—1061no—
277C2×C22⋊C4—1061no—
278C2×C22⋊C4—1061no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1061no—
281C2×C42—42no—
282C2×C42—1061no—
283C2×C42C4×D5311yescharacteristic
284C2×C42C4×D5311yescharacteristic
285C2×C42D21211yescharacteristic
286C2×C42D21211yescharacteristic
287C4⋊C8—2122no—
288C22⋊C8—1062no—
289C22⋊C8—1062no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—42no—
294#95—1061no—
295#95—1061no—
296#95—1061no—
297C24—1061no—
298C24C2×C22⋊D5311yescharacteristic
299C2×D4—532no—
300C2×D4—1062no—
301C2×D4—1062no—
302C2×D4—1061no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2122no—
306C2×D4—2122no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—1061no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—2122no—
326C22×C4—2122no—
327C22×C4—2122no—
328C22×C4#1171992111yescharacteristic
329C22×C4#1171992111yescharacteristic
330C22×C4#1171992111yescharacteristic
331C22×C4#1171992111yescharacteristic
332C22×C4C2×C4×D5311yescharacteristic
333C22×C4C2×D21211yescharacteristic
334C22×C4D4×D5311yescharacteristic
335C22×C4#1171994711yescharacteristic
336C22×C4Q8×D5311yescharacteristic
337C22×C4#1171994911yescharacteristic
338C2×C8—2122no—
339C2×C8—2122no—
340C2×C8—2122no—
341C4⋊C4—532no—
342C4⋊C4—1061no—
343C4⋊C4—2121no—
344C4⋊C4—2121no—
345C4⋊C4—2122no—
346C22⋊C4—1062no—
347C22⋊C4—1062no—
348C22⋊C4—2122no—
349C22⋊C4—2122no—
350C22⋊C4—2122no—
351C22⋊C4—2122no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—2122no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—1062no—
362C23—2121no—
363C23—2121no—
364C23—2121no—
365C23—2121no—
366C23—2121no—
367C23—2121no—
368C23C2×#1171992111yescharacteristic
369C23#1313318811yescharacteristic
370C23#1313319311yescharacteristic
371C23#1313323712yes—
372C23#1313324212yes—
373D4—2122no—
374D4—2122no—
375D4—2124no—
376D4—2124no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—1062no—
405C2×C4—4244no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4#1313313112yes—
415C2×C4#1313314512yes—
416C2×C4C4⋊C4×D5311yescharacteristic
417C2×C4#1313317511yescharacteristic
418C2×C4#1313319211yescharacteristic
419C2×C4#1313319711yescharacteristic
420C8—4242no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—4241no—
430C22—4241no—
431C22—4241no—
432C22—4241no—
433C22—2122no—
434C22—2122no—
435C22—2122no—
436C22#1681048011yescharacteristic
437C22C2×#1313313111yescharacteristic
438C22#1681059611yescharacteristic
439C22#1681061711yescharacteristic
440C22#1681062711yescharacteristic
441C22C2×#1313314511yescharacteristic
442C22#1681064111yescharacteristic
443C22#1681083512yes—
444C22#1681092012yes—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—22no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—84no—
454C4—4241no—
455C4—4241no—
456C4—2122no—
457C2—22no—
458C2—22no—
459C2—4242no—
460C2#2027320411yescharacteristic
461C2#2027322211yescharacteristic
462C2#2027324311yescharacteristic
463C2#2027329011yescharacteristic
464C2#2027470011yescharacteristic
465C2#2027472211yescharacteristic
466C2#2027473611yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1434C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1434D53
hypocenterC53#8575
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #20274695C2semidirect
4. #20272594C2semidirect
5. C53×#1434C2semidirect
7. C2×C4×D212C4semidirect
11. #16810525C4semidirect
12. #16810521C4semidirect
16. C4×C4⋊Dic53C4semidirect
227. #1434D53semidirect
257. C53#8575semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202746952
C2×#168103562
#202725942
C53×#14342
#857553