Finite Group Databasebeta

← #27280359

Subgroups — #27280359

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Dic53C211yesmaximal, characteristic, complemented
2#20275544C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272547C211yesmaximal, characteristic, complemented
7C53×#3100C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9C22×C4×Dic53—21no—
10#16810676C2211yescharacteristic
11#16810647C2211yescharacteristic
12C2×#13133144—21no—
13C2×#13133144—21no—
14C4⋊C4×Dic53—21no—
15C4×C4⋊Dic53C2211yescharacteristic
16C22⋊C4×Dic53—22no—
17C4×C22⋊Dic53—21no—
18C42×Dic53C2211yescharacteristic
19#16810357—22no—
20D4⋊#88322—22no—
21C53×#532C2211yescharacteristic
22C2×C4×D4×C53C2211yescharacteristic
23C53×#480—21no—
24C53×#474C412yescomplemented
25C53×#459C412yescomplemented
26C53×#447—21no—
27C4×D8×C53—22no—
28C53×#431C2211yescharacteristic
29C23×Dic53—22no—
30C2×#11719917—21no—
31D4×Dic53—28no—
32C2×C4⋊Dic53—21no—
33C2×C22⋊Dic53—22no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53—42no—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133144—42no—
46#13133144—42no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49#13133124—22no—
50#13133124D412yes—
51C2×#41×C53—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C2×D8×C53—22no—
55C4⋊Q8×C53C2×C411yescharacteristic, complemented
56#127×C53C2×C411yescharacteristic, complemented
57#124×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53C2×C412yes—
65C4×D4×C53D412yes—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53—21no—
68C2×C42×C53C2311yescharacteristic
69#107×C53—22no—
70C4⋊C8×C53C2×C412yescomplemented
71#102×C53—22no—
72#102×C53—22no—
73C22⋊C8×C53—21no—
74C4×C8×C53—21no—
75C24×C53—21no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×D4×C53—24no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C2×D4×C53C22⋊C412yes—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×Dic53—21no—
100C22×Dic53—28no—
101C22×Dic53—21no—
102C22×Dic53—22no—
103C22×Dic53—44no—
104C22×Dic53—44no—
105D8×C53—44no—
106C2×C8×C53—22no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53C22⋊C412yes—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53#4111yescharacteristic
119C22⋊Dic53—28no—
120C4⋊Dic53—24no—
121C4×Dic53—24no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125C4×Dic53—42no—
126C4×Dic53—44no—
127C4×Dic53—41no—
128C4×Dic53—41no—
129C4×Dic53D812yescomplemented
130#11719917—41no—
131#11719917—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C23×C53—24no—
135C23×C53—22no—
136C23×C53—42no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53#12011yescharacteristic
141Q8×C53—41no—
142Q8×C53—41no—
143D4×C53—24no—
144D4×C53—22no—
145D4×C53—42no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—81no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53#10414yescomplemented
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53C2×C22⋊C411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12311yescharacteristic
170C2×C4×C53C2×D811yescharacteristic
171C2×C4×C53#13511yescharacteristic
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×Dic53—22no—
175C2×Dic53—22no—
176C2×Dic53—22no—
177C2×Dic53—24no—
178C2×Dic53—22no—
179C2×Dic53—416no—
180C2×Dic53—42no—
181C2×Dic53—41no—
182C2×Dic53—42no—
183C2×Dic53—41no—
184C8×C53—42no—
185#88322—81no—
186#88322—81no—
187#9233—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—24no—
191C22×C53—44no—
192C22×C53—44no—
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53#38611yescharacteristic
199C22×C53#44711yescharacteristic
200C22×C53#44911yescharacteristic
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—42no—
205C4×C53—81no—
206C4×C53C2×#10412yes—
207C4×C53C4×D811yescharacteristic
208C4×C53#44211yescharacteristic
209C4×C53#45911yescharacteristic
210C4×C53#46311yescharacteristic
211Dic53—24no—
212Dic53—42no—
213Dic53—44no—
214#3100D5311yescharacteristic
215C42×D4—531no—
216#1962—531no—
217#1502—531no—
218#1486—531no—
219#1303—531no—
220#1276—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—44no—
224C2×C53—161no—
225C2×C53#187911yescharacteristic
226C2×C53#189611yescharacteristic
227C2×C53#190611yescharacteristic
228#532C2×D5311yescharacteristic
229C2×C4×D4—1061no—
230C2×C4×D4C2×D5311yescharacteristic
231C22×C42—1061no—
232#480—21no—
233#474Dic5312yescomplemented
234#459Dic5312yescomplemented
235#447—21no—
236C4×D8—22no—
237#431—531no—
238#431C2×D5311yescharacteristic
239#423—531no—
240C2×#104—1061no—
241C2×#104—1061no—
242C4×C4⋊C4—531no—
243C4×C4⋊C4—1061no—
244C4×C22⋊C4—1062no—
245C4×C22⋊C4—1061no—
246C43—531no—
247#327—1062no—
248D4⋊C8—1062no—
249C53#923311yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C22×D4C22⋊D5311yescharacteristic
252C23×C4—21no—
253C23×C4—1062no—
254C2×D8—22no—
255C2×M16—1061no—
256C4⋊Q8C2×Dic5311yescharacteristic, complemented
257#127C2×Dic5311yescharacteristic, complemented
258#124—21no—
259#121—21no—
260#121—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—1068no—
266C4×D4C2×Dic5312yes—
267C4×D4C22⋊D5312yes—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4C22⋊D5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1062no—
272C2×C42—531no—
273C2×C42—531no—
274C2×C42—1061no—
275C2×C42—1062no—
276C2×C42—1061no—
277C2×C42—1061no—
278C2×C42—2122no—
279C2×C42—2122no—
280C2×C42C22×D5311yescharacteristic
281#107—22no—
282C4⋊C8—1061no—
283C4⋊C8—1061no—
284C4⋊C8—1061no—
285C4⋊C8C2×Dic5312yescomplemented
286#104—2122no—
287#104—2122no—
288#102—22no—
289#102—22no—
290C22⋊C8—21no—
291C22⋊C8—1061no—
292C4×C8—21no—
293C4×C8—1061no—
294#95—532no—
295#95—1062no—
296C24—21no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301C2×Q8—21no—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4C22⋊Dic5312yes—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—1061no—
318C22×C4—1068no—
319C22×C4—1062no—
320C22×C4—1061no—
321C22×C4—2124no—
322C22×C4—2124no—
323C22×C4D4×D5311yescharacteristic
324C22×C4D4×D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326D8—44no—
327M16—2121no—
328M16—2121no—
329C2×C8—22no—
330C2×C8—2121no—
331C2×C8—2121no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—1064no—
336C4⋊C4C22⋊Dic5312yes—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—1068no—
342C42—532no—
343C42—1064no—
344C42—1061no—
345C42—1061no—
346C42—1061no—
347C42—2124no—
348C42—2122no—
349C42—2121no—
350C42—2121no—
351C42#1171994711yescharacteristic
352C42#1171994711yescharacteristic
353C42C22×Dic5311yescharacteristic
354C42C2×C22⋊D5311yescharacteristic
355C23—24no—
356C23—22no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23#1313318511yescharacteristic
362Q8—41no—
363Q8—41no—
364D4—24no—
365D4—22no—
366D4—42no—
367D4—41no—
368D4—41no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4#1313314414yescomplemented
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—1062no—
386C2×C4—1062no—
387C2×C4—1062no—
388C2×C4—1062no—
389C2×C4—1064no—
390C2×C4—21216no—
391C2×C4—2121no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2121no—
395C2×C4C2×C22⋊Dic5311yescharacteristic
396C2×C4D4×Dic5311yescharacteristic
397C2×C4D4×Dic5311yescharacteristic
398C2×C4#1313318911yescharacteristic
399C2×C4#1313318911yescharacteristic
400C2×C4#1313319811yescharacteristic
401C2×C4D8×D5311yescharacteristic
402C2×C4#1313323611yescharacteristic
403C2×C4#1313324111yescharacteristic
404C2×C4#1313324111yescharacteristic
405C8—42no—
406C8—4241no—
407C8—4241no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—44no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#1681050311yescharacteristic
419C22#1681075511yescharacteristic
420C22#1681076511yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—1064no—
427C4—2124no—
428C4—2122no—
429C4C2×#1313314412yes—
430C4D8×Dic5311yescharacteristic
431C4#1681073511yescharacteristic
432C4#1681081611yescharacteristic
433C4#1681083611yescharacteristic
434C2—21no—
435C2—21no—
436C2—44no—
437C2—161no—
438C2#2027506511yescharacteristic
439C2#2027515911yescharacteristic
440C2#2027521811yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3100C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3100D53
hypocenterC53#9233
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. C4×D4×Dic53C2semidirect
2. #20275544C2semidirect
3. #20273210C2semidirect
4. #20273140C2semidirect
5. #20272617C2semidirect
6. #20272547C2semidirect
24. C53×#474C4semidirect
25. C53×#459C4semidirect
55. C4⋊Q8×C53C2×C4semidirect
56. #127×C53C2×C4semidirect
70. C4⋊C8×C53C2×C4semidirect
129. C4×Dic53D8semidirect
151. D4×C53#104semidirect
233. #474Dic53semidirect
234. #459Dic53semidirect
249. C53#9233semidirect
256. C4⋊Q8C2×Dic53semidirect
257. #127C2×Dic53semidirect
285. C4⋊C8C2×Dic53semidirect
372. D4#13133144semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Dic532
#202755442
#202732102
#202731402
#202726172
#202725472
C53×#31002
#923353