Finite Group Databasebeta

← #27280821

Subgroups — #27280821

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277441C211yesmaximal, characteristic, complemented
2#20275548C211yesmaximal, characteristic, complemented
3#20273383C211yesmaximal, characteristic, complemented
4#20273275C211yesmaximal, characteristic, complemented
5#20272574C211yesmaximal, characteristic, complemented
6#20272543C211yesmaximal, characteristic, complemented
7C53×#3205C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9#16810676C2211yescharacteristic
10#16810647C2211yescharacteristic
11C2×#13133144—21no—
12C2×#13133144—21no—
13#16810526—21no—
14#16810503—22no—
15#16810503—22no—
16#16810489C2211yescharacteristic
17#16810469C2211yescharacteristic
18#16810367—22no—
19#16810357—22no—
20C53×#532C2211yescharacteristic
21C2×#127×C53C2211yescharacteristic
22C53×#493C412yescomplemented
23C53×#486C412yescomplemented
24C53×#459—22no—
25C53×#447—22no—
26C53×#431C2211yescharacteristic
27C23×Dic53—22no—
28C2×#11719917—21no—
29D4×Dic53—42no—
30D4×Dic53—42no—
31C2×C4⋊Dic53—21no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40#13133144—42no—
41#13133144—42no—
42C22⋊#88322—21no—
43C4×#88322—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124D412yes—
48C2×#41×C53—21no—
49C22×D4×C53—21no—
50C22×D4×C53—21no—
51C22×D4×C53D412yes—
52C2×D8×C53—24no—
53C4⋊Q8×C53C2×C411yescharacteristic, complemented
54#127×C53—21no—
55#127×C53—21no—
56#127×C53C2×C412yes—
57#127×C53C2×C411yescharacteristic, complemented
58#127×C53D412yes—
59#124×C53—21no—
60#121×C53—21no—
61#121×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66#102×C53—24no—
67C22⋊C8×C53—21no—
68C4×C8×C53C2×C412yescomplemented
69C24×C53—22no—
70#41×C53—41no—
71#41×C53—41no—
72#41×C53—41no—
73#41×C53—41no—
74C2×Q8×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—22no—
77C2×D4×C53—21no—
78C2×D4×C53—22no—
79C2×D4×C53—21no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—82no—
89C2×D4×C53—82no—
90C2×D4×C53C22⋊C414yes—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—22no—
98C22×Dic53—44no—
99C22×Dic53—82no—
100C22×Dic53—82no—
101D8×C53—84no—
102C2×C8×C53—22no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53C22×C411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110C42×C53#4111yescharacteristic
111C22⋊Dic53—44no—
112C22⋊Dic53—82no—
113C22⋊Dic53—82no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4×Dic53—21no—
117C4×Dic53—21no—
118C4×Dic53—21no—
119C4×Dic53D812yescomplemented
120#11719917—41no—
121#11719917—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C23×C53—24no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53#12011yescharacteristic
133Q8×C53—41no—
134Q8×C53—41no—
135D4×C53—44no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—41no—
139D4×C53—42no—
140D4×C53—41no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—82no—
144D4×C53—82no—
145D4×C53—82no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×Dic53—22no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—41no—
173C2×Dic53—88no—
174C8×C53—42no—
175#88322—81no—
176#88322—81no—
177#9299—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—44no—
181C22×C53—44no—
182C22×C53—44no—
183C22×C53—82no—
184C22×C53—82no—
185C22×C53—82no—
186C22×C53—82no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53#38611yescharacteristic
191C22×C53#44711yescharacteristic
192C22×C53#44911yescharacteristic
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53C4×D811yescharacteristic
199C4×C53#44211yescharacteristic
200C4×C53#45911yescharacteristic
201C4×C53#46311yescharacteristic
202Dic53—42no—
203Dic53—81no—
204Dic53—82no—
205#3205D5311yescharacteristic
206#2328—531no—
207#1963—531no—
208#1545—531no—
209#1517—531no—
210#1286—531no—
211#1274—531no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—84no—
215C2×C53—161no—
216C2×C53#187911yescharacteristic
217C2×C53#190411yescharacteristic
218C2×C53#191111yescharacteristic
219#532C2×D5311yescharacteristic
220C2×#127C2×D5311yescharacteristic
221C2×C4×D4—1061no—
222#493Dic5312yescomplemented
223#486Dic5312yescomplemented
224#459—22no—
225#447—22no—
226#431—531no—
227#431C2×D5311yescharacteristic
228#423—531no—
229C2×#104—1061no—
230C2×#104—1061no—
231#390—1061no—
232#386—1062no—
233#386—1062no—
234#382—531no—
235#376—531no—
236#331—1062no—
237#327—1062no—
238C53#929911yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C22⋊D5312yes—
243C23×C4—1062no—
244C2×D8—24no—
245C2×M16—1061no—
246C4⋊Q8C2×Dic5311yescharacteristic, complemented
247#127—21no—
248#127—21no—
249#127C2×Dic5312yes—
250#127C2×Dic5311yescharacteristic, complemented
251#127C22⋊D5312yes—
252#124—21no—
253#121—21no—
254#121—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—2122no—
258C4×D4—2122no—
259C2×C4⋊C4—1061no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1062no—
263C2×C42—531no—
264C2×C42—531no—
265C2×C42C22×D5311yescharacteristic
266C4⋊C8—21no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270#104—2122no—
271#104—2122no—
272#102—24no—
273C22⋊C8—21no—
274C22⋊C8—1061no—
275C4×C8—1061no—
276C4×C8C2×Dic5312yescomplemented
277#95—532no—
278#95—1061no—
279#95—1061no—
280#95—1061no—
281C24—22no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4C22⋊Dic5314yes—
303C22×C4—41no—
304C22×C4—1061no—
305C22×C4—1062no—
306C22×C4—1061no—
307C22×C4—4242no—
308C22×C4—4242no—
309C22×C4—2124no—
310C22×C4D4×D5311yescharacteristic
311C22×C4D4×D5311yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313D8—84no—
314M16—2121no—
315M16—2121no—
316C2×C8—22no—
317C2×C8—2121no—
318C2×C8—2121no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—2121no—
322C4⋊C4—2121no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—4242no—
326C22⋊C4—4242no—
327C22⋊C4—2124no—
328C42—532no—
329C42—1061no—
330C42—1061no—
331C42—1061no—
332C42#1171994711yescharacteristic
333C42#1171994711yescharacteristic
334C42C22×Dic5311yescharacteristic
335C42C2×C22⋊D5311yescharacteristic
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23#1313318511yescharacteristic
345Q8—41no—
346Q8—41no—
347D4—44no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—42no—
354D4—41no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—1062no—
367C2×C4—4248no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4C2×C22⋊Dic5311yescharacteristic
377C2×C4D4×Dic5311yescharacteristic
378C2×C4D4×Dic5311yescharacteristic
379C2×C4#1313318911yescharacteristic
380C2×C4#1313318911yescharacteristic
381C2×C4#1313319811yescharacteristic
382C2×C4D8×D5311yescharacteristic
383C2×C4#1313323611yescharacteristic
384C2×C4#1313324111yescharacteristic
385C2×C4#1313324111yescharacteristic
386C8—42no—
387C8—4241no—
388C8—4241no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#1681050311yescharacteristic
402C22#1681075511yescharacteristic
403C22#1681076511yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—4242no—
410C4—4241no—
411C4—2122no—
412C4D8×Dic5311yescharacteristic
413C4#1681073511yescharacteristic
414C4#1681081611yescharacteristic
415C4#1681083611yescharacteristic
416C2—21no—
417C2—21no—
418C2—84no—
419C2—161no—
420C2#2027506511yescharacteristic
421C2#2027520411yescharacteristic
422C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3205C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3205D53
hypocenterC53#9299
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #20277441C2semidirect
2. #20275548C2semidirect
3. #20273383C2semidirect
4. #20273275C2semidirect
5. #20272574C2semidirect
6. #20272543C2semidirect
22. C53×#493C4semidirect
23. C53×#486C4semidirect
53. C4⋊Q8×C53C2×C4semidirect
57. #127×C53C2×C4semidirect
68. C4×C8×C53C2×C4semidirect
119. C4×Dic53D8semidirect
222. #493Dic53semidirect
223. #486Dic53semidirect
238. C53#9299semidirect
246. C4⋊Q8C2×Dic53semidirect
250. #127C2×Dic53semidirect
276. C4×C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202774412
#202755482
#202733832
#202732752
#202725742
#202725432
C53×#32052
#929953