Finite Group Databasebeta

← #27300993

Subgroups — #27300993

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D53C211yesmaximal, characteristic, complemented
2C2×#16810692C211yesmaximal, characteristic
3C2×#16810405C211yesmaximal, characteristic, complemented
4#20276325C211yesmaximal, characteristic, complemented
5#20272798C211yesmaximal, characteristic, complemented
6#20272798C211yesmaximal, characteristic, complemented
7C2×C53×#349C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×C4×C22⋊D53C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12C2×#13133156—21no—
13M32×D53—22no—
14M32×D53—22no—
15C2×C16×D53—21no—
16#16810692—21no—
17#16810692—21no—
18#16810692—21no—
19#16810692—21no—
20C2×#13133146C411yescharacteristic
21C2×C22⋊#88322C411yescharacteristic
22C2×#13133129C411yescharacteristic
23C2×#13133127C2211yescharacteristic
24#16810405—22no—
25#16810402—21no—
26#16810402—21no—
27C22×M16×C53C2211yescharacteristic
28C2×M32×C53C2211yescharacteristic
29C2×M32×C53C2211yescharacteristic
30C53×#349—22no—
31C22×C22⋊D53C811yescharacteristic
32C22×C4×D53C2×C411yescharacteristic
33C22×#88322C2×C411yescharacteristic
34C22×#88322C2×C411yescharacteristic
35C2×C8×D53—21no—
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53D411yescharacteristic
40C2×C8×D53D411yescharacteristic
41C2×C8×D53D411yescharacteristic
42C2×C8×D53D411yescharacteristic
43C4×C22⋊D53—21no—
44C4×C22⋊D53—21no—
45C4×C22⋊D53—21no—
46C4×C22⋊D53—21no—
47C2×#11719919C811yescharacteristic
48C2×C22⋊Dic53C811yescharacteristic
49C2×#11719921C811yescharacteristic
50C2×C4×Dic53C2×C411yescharacteristic
51#13133157—21no—
52#13133157—21no—
53#13133157—21no—
54#13133157D412yescomplemented
55#13133156—24no—
56C2×#11719908—21no—
57C2×#11719908—21no—
58C16×D53—24no—
59#13133146—21no—
60#13133146—21no—
61C22⋊#88322—21no—
62C22⋊#88322—21no—
63#13133129—21no—
64#13133129—21no—
65#13133127—21no—
66#13133127—21no—
67C23×C4×C53C2×C411yescharacteristic
68C2×M16×C53—21no—
69C2×M16×C53—21no—
70C2×M16×C53—21no—
71C2×M16×C53—21no—
72C2×M16×C53—21no—
73C2×M16×C53—21no—
74C22×C8×C53C2×C411yescharacteristic
75C22×C8×C53C2311yescharacteristic
76M32×C53—21no—
77M32×C53—21no—
78M32×C53—21no—
79M32×C53D412yescomplemented
80C2×C16×C53—21no—
81C2×C16×C53—21no—
82C24×C53C2×C811yescharacteristic
83C23×D53C2×C811yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53C2×C811yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C2×C22⋊D53—21no—
93C2×C22⋊D53—21no—
94C2×C22⋊D53—42no—
95C22×Dic53C2×C811yescharacteristic
96C22×Dic53C2×C811yescharacteristic
97C22×Dic53C2×C811yescharacteristic
98C2×C4×D53—21no—
99C2×C4×D53—21no—
100C2×C4×D53—21no—
101C2×C4×D53—21no—
102C2×C4×D53C22⋊C411yescharacteristic
103C2×C4×D53C22⋊C411yescharacteristic
104C2×C4×D53M1611yescharacteristic
105C2×C4×D53M1611yescharacteristic
106M16×C53—41no—
107M16×C53—41no—
108M16×C53—41no—
109M16×C53—41no—
110C2×C8×C53—21no—
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53#4111yescharacteristic
118C2×C8×C53#4111yescharacteristic
119C22⋊Dic53—21no—
120C22⋊Dic53—21no—
121#11719921—21no—
122#11719921—21no—
123#11719919—21no—
124#11719919—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322C22⋊C411yescharacteristic
133C2×#88322C22⋊C411yescharacteristic
134C8×D53—21no—
135C8×D53—21no—
136C8×D53—21no—
137C8×D53—21no—
138C8×D53—21no—
139C8×D53—21no—
140C8×D53—21no—
141C8×D53—21no—
142C16×C53—22no—
143C16×C53—41no—
144#11719908—22no—
145#11719908—41no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53C22×C811yescharacteristic
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—22no—
155C22×D53—41no—
156C22×D53C22⋊C811yescharacteristic
157C22×D53C22⋊C811yescharacteristic
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53C2×M1611yescharacteristic
169C2×C4×C53#13111yescharacteristic
170C2×C4×C53#13111yescharacteristic
171C22⋊D53—42no—
172C22⋊D53—42no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53C22⋊C811yescharacteristic
180C2×Dic53C22⋊C811yescharacteristic
181C4×D53—21no—
182C4×D53—21no—
183C4×D53—21no—
184C4×D53—21no—
185C4×D53—21no—
186C4×D53—21no—
187C4×D53—21no—
188C4×D53—21no—
189C8×C53—41no—
190C8×C53#12011yescharacteristic
191C8×C53#12111yescharacteristic
192C8×C53#12111yescharacteristic
193C8×C53#12311yescharacteristic
194#88322—21no—
195#88322—21no—
196#88322—41no—
197#12985—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53C2×C22⋊C811yescharacteristic
206C22×C53C8×D411yescharacteristic
207C22×C53C8×D411yescharacteristic
208C2×D53—21no—
209C2×D53—21no—
210C2×D53—21no—
211C2×D53—21no—
212C2×D53—21no—
213C2×D53—42no—
214C2×D53—42no—
215C2×D53#34912yescomplemented
216C4×C53—41no—
217C4×C53#38611yescharacteristic
218C4×C53#40811yescharacteristic
219C4×C53#43511yescharacteristic
220C4×C53#43511yescharacteristic
221Dic53—21no—
222Dic53—21no—
223Dic53—41no—
224Dic53—41no—
225C22×M32—531no—
226C2×#435—531no—
227C2×#349—531no—
228C2×#349D5311yescharacteristic, complemented
229#2112—531no—
230#1377—531no—
231#1377—531no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—81no—
235C2×C53#186911yescharacteristic
236C2×C53C2×#34911yescharacteristic
237C2×C53#211511yescharacteristic
238D53—22no—
239D53—41no—
240C22×M16C2×D5311yescharacteristic
241C23×C8—531no—
242C2×C4×D4—531no—
243C2×M32—531no—
244C2×M32—531no—
245C2×M32—1061no—
246C2×M32—1062no—
247C2×M32—1062no—
248C2×M32C2×D5311yescharacteristic
249C2×M32C2×D5311yescharacteristic
250C22×C16—1061no—
251#435—1061no—
252#435—1061no—
253#435—1061no—
254#435—1061no—
255C2×C4⋊C8—531no—
256C2×C22⋊C8—531no—
257C2×C22⋊C8—531no—
258C2×#97—531no—
259#349—22no—
260#349—1062no—
261C22⋊C16—1061no—
262C22⋊C16—1061no—
263C53#1298511yescharacteristic, complemented, Sylow
264C22×D4—531no—
265C23×C4—531no—
266C23×C4C4×D5311yescharacteristic
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—21no—
270C2×M16—21no—
271C2×M16—21no—
272C2×M16—21no—
273C22×C8—531no—
274C22×C8—531no—
275C22×C8—531no—
276C22×C8—531no—
277C22×C8—531no—
278C22×C8—531no—
279C22×C8—1061no—
280C22×C8—1061no—
281C22×C8—1061no—
282C22×C8—1061no—
283C22×C8C4×D5311yescharacteristic
284C22×C8C22×D5311yescharacteristic
285C4×D4—1061no—
286C4×D4—1061no—
287C4×D4—1061no—
288C4×D4—1061no—
289C2×C4⋊C4—531no—
290C2×C22⋊C4—531no—
291C2×C22⋊C4—531no—
292C2×C42—531no—
293M32—21no—
294M32—21no—
295M32—21no—
296M32—532no—
297M32—1064no—
298M32—1061no—
299M32—1061no—
300M32—1061no—
301M32C22⋊D5312yescomplemented
302C2×C16—21no—
303C2×C16—21no—
304C2×C16—1064no—
305C2×C16—1061no—
306C2×C16—1061no—
307C4⋊C8—1061no—
308C4⋊C8—1061no—
309C22⋊C8—1061no—
310C22⋊C8—1061no—
311C22⋊C8—1061no—
312C22⋊C8—1061no—
313#97—1061no—
314#97—1061no—
315C24—531no—
316C24C8×D5311yescharacteristic
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—2122no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—1061no—
334C22×C4—1061no—
335C22×C4—1061no—
336C22×C4—1061no—
337C22×C4C8×D5311yescharacteristic
338C22×C4C2×C4×D5311yescharacteristic
339M16—41no—
340M16—41no—
341M16—41no—
342M16—41no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—531no—
347C2×C8—531no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—1061no—
352C2×C8—1061no—
353C2×C8—1061no—
354C2×C8—1061no—
355C2×C8—1061no—
356C2×C8—1061no—
357C2×C8—1061no—
358C2×C8—1061no—
359C2×C8—1061no—
360C2×C8—1061no—
361C2×C8#1171994511yescharacteristic
362C2×C8D4×D5311yescharacteristic
363C2×C8D4×D5311yescharacteristic
364C2×C8D4×D5311yescharacteristic
365C2×C8#1171994711yescharacteristic
366C2×C8C2×C22⋊D5311yescharacteristic
367C4⋊C4—1061no—
368C4⋊C4—1061no—
369C22⋊C4—1061no—
370C22⋊C4—1061no—
371C22⋊C4—1061no—
372C22⋊C4—1061no—
373C42—1061no—
374C42—1061no—
375C16—22no—
376C16—41no—
377C16—1062no—
378C16—2121no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—41no—
383C23—41no—
384C23—531no—
385C23—531no—
386C23—1062no—
387C23—1061no—
388C23—1061no—
389C23—2121no—
390C23C2×C8×D5311yescharacteristic
391D4—2122no—
392D4—2122no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—531no—
401C2×C4—531no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—1061no—
408C2×C4—1061no—
409C2×C4—1061no—
410C2×C4—1061no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4C22⋊C4×D5311yescharacteristic
417C2×C4#1313317711yescharacteristic
418C2×C4C4×C22⋊D5311yescharacteristic
419C2×C4M16×D5311yescharacteristic
420C2×C4#1313322311yescharacteristic
421C2×C4#1313322511yescharacteristic
422C8—41no—
423C8—1061no—
424C8—1061no—
425C8—2121no—
426C8#1313318511yescharacteristic
427C8#1313318811yescharacteristic
428C8#1313319011yescharacteristic
429C8#1313319911yescharacteristic
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—532no—
438C22—1061no—
439C22—1061no—
440C22—1061no—
441C22—1061no—
442C22—1061no—
443C22—2122no—
444C22—2122no—
445C22C22⋊C8×D5311yescharacteristic
446C22#1681068611yescharacteristic
447C22C8×C22⋊D5311yescharacteristic
448C4—41no—
449C4—1061no—
450C4—1061no—
451C4—2121no—
452C4—2121no—
453C4#1681049911yescharacteristic
454C4#1681058811yescharacteristic
455C4#1681069211yescharacteristic
456C4#1681069711yescharacteristic
457C2—21no—
458C2—21no—
459C2—81no—
460C2—1062no—
461C2—2121no—
462C2#2027500411yescharacteristic
463C2#349×D5311yescharacteristic
464C2#2027633411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#349C2
socleC22×C53C2×C22⋊C8
prime radicalC22×C22⋊D53C8
cosocleC22×C8×C53C23
hypercenterC2×#349D53
hypocenterC53#12985
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
1. C2×M32×D53C2semidirect
3. C2×#16810405C2semidirect
4. #20276325C2semidirect
5. #20272798C2semidirect
6. #20272798C2semidirect
7. C2×C53×#349C2semidirect
54. #13133157D4semidirect
79. M32×C53D4semidirect
215. C2×D53#349semidirect
228. C2×#349D53semidirect
263. C53#12985semidirect
301. M32C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D532
C2×#168106922
C2×#168104052
#202763252
#202727982
C2×C53×#3492
#1298553