Finite Group Databasebeta

← #27278806

Subgroups — #27278806

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283847C211yesmaximal, characteristic, complemented
2#20275544C211yesmaximal, characteristic, complemented
3C4×#13133144C211yesmaximal, characteristic, complemented
4#20273132C211yesmaximal, characteristic, complemented
5#20272606C211yesmaximal, characteristic, complemented
6#20272547C211yesmaximal, characteristic, complemented
7C53×#3106C211yesmaximal, characteristic
8Q8×#88322—21no—
9#16810701—22no—
10#16810701—21no—
11#16810676—21no—
12#16810676C2211yescharacteristic
13#16810647—21no—
14#16810647C2211yescharacteristic
15C2×#13133144—22no—
16C2×#13133144—21no—
17C4×#11719917C2211yescharacteristic
18C42×Dic53C2211yescharacteristic
19#16810382—21no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C53×#532C2211yescharacteristic
23C4×#41×C53C2211yescharacteristic
24C53×#482—21no—
25C53×#477C411yescharacteristic, complemented
26C53×#476C411yescharacteristic, complemented
27C53×#460C411yescharacteristic, complemented
28C53×#459C411yescharacteristic, complemented
29C53×#449—21no—
30C4×SD16×C53—21no—
31C4×D8×C53—21no—
32C53×#431C2211yescharacteristic
33C2×#11719917—22no—
34C2×#11719917—21no—
35C2×#11719917D412yes—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C4⋊#88322—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322D411yescharacteristic
43C4⋊#88322D411yescharacteristic
44#13133144—24no—
45#13133144—44no—
46C22⋊#88322—22no—
47C22⋊#88322—21no—
48#13133128—21no—
49C4×#88322—22no—
50C4×#88322—21no—
51C4×#88322—21no—
52C2×#41×C53—21no—
53C2×#41×C53D411yescharacteristic
54C2×SD16×C53—21no—
55C2×D8×C53—21no—
56C4⋊Q8×C53C2×C411yescharacteristic
57#127×C53C2×C411yescharacteristic
58#124×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61C4×Q8×C53C2×C411yescharacteristic
62C4×Q8×C53D411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53C2×C411yescharacteristic
68C4×D4×C53D411yescharacteristic
69#117×C53—21no—
70#117×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#107×C53—21no—
74#106×C53—21no—
75C4⋊C8×C53C2×C411yescharacteristic
76C4⋊C8×C53C2×C411yescharacteristic
77#103×C53—21no—
78#102×C53—21no—
79#102×C53—21no—
80#102×C53—21no—
81C22⋊C8×C53—21no—
82C4×C8×C53—21no—
83#41×C53—22no—
84#41×C53—41no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×Q8×C53C22⋊C411yescharacteristic
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—41no—
94C2×D4×C53C22⋊C411yescharacteristic
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—41no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×Dic53—41no—
103SD16×C53—42no—
104D8×C53—42no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53C22⋊C411yescharacteristic
112C4⋊C4×C53C22⋊C411yescharacteristic
113C22⋊C4×C53—21no—
114C22⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53—21no—
121C42×C53C22×C411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125C4×Dic53—22no—
126C4×Dic53—22no—
127C4×Dic53—22no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130C4×Dic53—41no—
131#11719917—22no—
132#11719917—41no—
133#11719917—42no—
134#11719917—82no—
135C2×#88322—21no—
136C2×#88322—22no—
137C2×#88322—21no—
138C2×#88322—41no—
139C23×C53—21no—
140C23×C53—41no—
141C23×C53#12111yescharacteristic
142Q8×C53—21no—
143Q8×C53—42no—
144Q8×C53#10412yescomplemented
145D4×C53—21no—
146D4×C53—21no—
147D4×C53—42no—
148D4×C53—42no—
149D4×C53—41no—
150D4×C53—41no—
151D4×C53—41no—
152D4×C53—41no—
153D4×C53—82no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—42no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53C4×D411yescharacteristic
173C2×C4×C53C4×D411yescharacteristic
174C2×C4×C53#12011yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12311yescharacteristic
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C8×C53—41no—
181C8×C53—41no—
182#88322—42no—
183#88322—42no—
184#88322—41no—
185#88322—41no—
186#8970—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53#38611yescharacteristic
197C4×C53—21no—
198C4×C53—22no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—81no—
204C4×C53C2×#10411yescharacteristic
205C4×C53#42111yescharacteristic
206C4×C53#44311yescharacteristic
207C4×C53#44411yescharacteristic
208C4×C53#45311yescharacteristic
209C4×C53#45411yescharacteristic
210Dic53—44no—
211#3106D5311yescharacteristic
212#2991—531no—
213#1962—531no—
214C4×#104—531no—
215#1484—531no—
216#1299—531no—
217#1276—531no—
218C2×C53—41no—
219C2×C53—81no—
220C2×C53—161no—
221C2×C53#187111yescharacteristic
222C2×C53#187611yescharacteristic
223C2×C53#188311yescharacteristic
224#532C2×D5311yescharacteristic
225C4×#41C2×D5311yescharacteristic
226#482—21no—
227#477Dic5311yescharacteristic, complemented
228#476Dic5311yescharacteristic, complemented
229#460Dic5311yescharacteristic, complemented
230#459Dic5311yescharacteristic, complemented
231#449—21no—
232C8×Q8—1061no—
233C4×SD16—21no—
234C4×D8—21no—
235#436—1062no—
236#436—1061no—
237#431—531no—
238#431—1061no—
239#431C2×D5311yescharacteristic
240#423—531no—
241#423—1061no—
242C2×#104—1061no—
243C2×#104—1062no—
244C4×M16—531no—
245C43—531no—
246#339—1061no—
247D4⋊C8—1061no—
248D4⋊C8—1061no—
249C53#897011yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C2×#41C22⋊D5311yescharacteristic
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—532no—
255C2×M16—1062no—
256C2×M16—1061no—
257C4⋊Q8C2×Dic5311yescharacteristic
258#127C2×Dic5311yescharacteristic
259#124—21no—
260#121—21no—
261#121—21no—
262C4×Q8C2×Dic5311yescharacteristic
263C4×Q8C22⋊D5311yescharacteristic
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4C2×Dic5311yescharacteristic
269C4×D4C22⋊D5311yescharacteristic
270#117—21no—
271#117C22⋊D5311yescharacteristic
272C2×C42—21no—
273C2×C42—1061no—
274C2×C42—1061no—
275C2×C42—1061no—
276C2×C42C22×D5311yescharacteristic
277#107—21no—
278#106—21no—
279C4⋊C8—531no—
280C4⋊C8—531no—
281C4⋊C8—1062no—
282C4⋊C8—1061no—
283C4⋊C8—1061no—
284C4⋊C8C2×Dic5311yescharacteristic
285C4⋊C8C2×Dic5311yescharacteristic
286#104—1064no—
287#104—2124no—
288#103—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292C22⋊C8—21no—
293C22⋊C8—1062no—
294C22⋊C8—1061no—
295#97—1061no—
296C4×C8—21no—
297C4×C8—1062no—
298C4×C8—1061no—
299C4×C8—1061no—
300#41—22no—
301#41—41no—
302#41—42no—
303#41—42no—
304C2×Q8—21no—
305C2×Q8C22⋊Dic5311yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4C22⋊Dic5311yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—2121no—
317C22×C4D4×D5311yescharacteristic
318C22×C4#1171994711yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320SD16—42no—
321D8—42no—
322M16—1062no—
323M16—4242no—
324M16—2121no—
325M16—2122no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—1061no—
329C2×C8—1061no—
330C2×C8—1062no—
331C2×C8—2121no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4C22⋊Dic5311yescharacteristic
337C4⋊C4C22⋊Dic5311yescharacteristic
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—1062no—
347C42—1062no—
348C42—1062no—
349C42—2121no—
350C42—2121no—
351C42—2121no—
352C42D4×D5311yescharacteristic
353C42#1171994711yescharacteristic
354C42C22×Dic5311yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C23—21no—
357C23—41no—
358C23#1313318911yescharacteristic
359Q8—21no—
360Q8—42no—
361Q8#1313314412yescomplemented
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4C2×C22⋊Dic5311yescharacteristic
392C2×C4D4×Dic5311yescharacteristic
393C2×C4D4×Dic5311yescharacteristic
394C2×C4#1313318511yescharacteristic
395C2×C4#1313318911yescharacteristic
396C2×C4#1313319811yescharacteristic
397C8—41no—
398C8—41no—
399C8—2122no—
400C8—2121no—
401C8—2122no—
402C8—2121no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#1681050311yescharacteristic
413C4—21no—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—2124no—
421C4C2×#1313314411yescharacteristic
422C4#1681064011yescharacteristic
423C4#1681074011yescharacteristic
424C4#1681074511yescharacteristic
425C4#1681078911yescharacteristic
426C4#1681079411yescharacteristic
427C2—41no—
428C2—81no—
429C2—161no—
430C2#2027501811yescharacteristic
431C2#2027504411yescharacteristic
432C2#2027509311yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3106C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3106D53
hypocenterC53#8970
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #20283847C2semidirect
2. #20275544C2semidirect
3. C4×#13133144C2semidirect
4. #20273132C2semidirect
5. #20272606C2semidirect
6. #20272547C2semidirect
25. C53×#477C4semidirect
26. C53×#476C4semidirect
27. C53×#460C4semidirect
28. C53×#459C4semidirect
144. Q8×C53#104semidirect
227. #477Dic53semidirect
228. #476Dic53semidirect
229. #460Dic53semidirect
230. #459Dic53semidirect
249. C53#8970semidirect
361. Q8#13133144semidirect

Maximal subgroups

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

subgroupindex
#202838472
#202755442
C4×#131331442
#202731322
#202726062
#202725472
C53×#31062
#897053