Finite Group Databasebeta

← #27302447

Subgroups — #27302447

Order 13568.

448 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276410C211yesmaximal, characteristic, complemented
2#20276407C211yesmaximal, characteristic
3#20276369C211yesmaximal, characteristic, complemented
4#20276365C211yesmaximal, characteristic
5#20274765C211yesmaximal, characteristic, complemented
6#20274515C211yesmaximal, characteristic, complemented
7C53×#3057C211yesmaximal, characteristic
8C2×#13133153—21no—
9#16810607C2211yescharacteristic
10#16810607C2211yescharacteristic
11#16810607C2211yescharacteristic
12#16810597C2211yescharacteristic
13#16810578—21no—
14#16810422—22no—
15#16810420—22no—
16#16810413—22no—
17#16810410—22no—
18#16810394—21no—
19#16810381—21no—
20C22×SD16×C53C411yescharacteristic, complemented
21C2×#131×C53C2211yescharacteristic
22C53×#536C2211yescharacteristic
23C53×#535C2211yescharacteristic
24C53×#468—21no—
25C53×#468—21no—
26C53×#465—21no—
27C53×#465—21no—
28C53×#427C411yescharacteristic, complemented
29C53×#419C411yescharacteristic, complemented
30C53×#418C411yescharacteristic, complemented
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34#13133176—21no—
35#13133153—41no—
36#13133153—41no—
37#13133136—21no—
38#13133136—21no—
39#13133136D412yes—
40#13133134—21no—
41#13133134—21no—
42#13133134—21no—
43#13133132—21no—
44#13133132D412yes—
45#13133127—41no—
46C8×Dic53—41no—
47C2×#41×C53C2311yescharacteristic
48C22×Q8×C53C2×C411yescharacteristic
49C22×D4×C53C2×C411yescharacteristic
50C2×SD16×C53—21no—
51C2×SD16×C53—21no—
52C2×SD16×C53—41no—
53C2×SD16×C53—41no—
54#131×C53—41no—
55#131×C53—41no—
56C2×M16×C53C2×C411yescharacteristic
57C2×M16×C53D411yescharacteristic
58C2×M16×C53D411yescharacteristic
59C22×C8×C53C2×C411yescharacteristic
60C22×C8×C53D411yescharacteristic
61C22×C8×C53D411yescharacteristic
62C4⋊Q8×C53—21no—
63#127×C53—21no—
64#124×C53—21no—
65#124×C53—21no—
66#123×C53—21no—
67#123×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70#121×C53—21no—
71#121×C53—21no—
72#120×C53—21no—
73#120×C53—21no—
74#117×C53C2×C411yescharacteristic
75#117×C53C2×C411yescharacteristic
76#107×C53—21no—
77#106×C53—21no—
78#103×C53—21no—
79#103×C53—21no—
80#102×C53—21no—
81#102×C53—21no—
82C24×C53—21no—
83#41×C53—41no—
84#41×C53—41no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—41no—
88C2×Q8×C53—41no—
89C2×Q8×C53C2×D411yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53C2×D411yescharacteristic
97C2×D4×C53#4111yescharacteristic
98C2×D4×C53#4111yescharacteristic
99C22×C4×C53—21no—
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103SD16×C53—42no—
104SD16×C53—41no—
105SD16×C53—41no—
106M16×C53—41no—
107M16×C53—41no—
108M16×C53—41no—
109C2×C8×C53—21no—
110C2×C8×C53—21no—
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C2×C8×C53—21no—
114C2×C8×C53—21no—
115C2×C8×C53C22⋊C411yescharacteristic
116C2×C8×C53C22⋊C411yescharacteristic
117C2×C8×C53C22⋊C411yescharacteristic
118C2×C8×C53C22⋊C411yescharacteristic
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C42×C53—21no—
130C42×C53—21no—
131C22⋊Dic53—22no—
132C4⋊Dic53—41no—
133C4×Dic53—41no—
134#11719917—22no—
135#11719917—41no—
136#11719917—41no—
137#11719917—41no—
138#11719917—41no—
139#11719917—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C23×C53—41no—
144C23×C53—81no—
145C23×C53—161no—
146C23×C53C4×D411yescharacteristic
147C23×C53#12111yescharacteristic
148C23×C53#12411yescharacteristic
149Q8×C53—41no—
150Q8×C53—41no—
151Q8×C53—41no—
152Q8×C53—81no—
153D4×C53—41no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—81no—
159D4×C53—161no—
160D4×C53—161no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—81no—
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53C4×D411yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12711yescharacteristic
175C2×Dic53—42no—
176C8×C53—21no—
177C8×C53—21no—
178C8×C53—21no—
179C8×C53#9911yescharacteristic
180C8×C53#9911yescharacteristic
181#88322—42no—
182#88322—81no—
183#88322—81no—
184#13232—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—162no—
195C22×C53#39011yescharacteristic
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—81no—
200C4×C53—81no—
201C4×C53—81no—
202C4×C53C2×#9911yescharacteristic
203C4×C53#41011yescharacteristic
204Dic53—82no—
205#3057D5311yescharacteristic
206#2135—531no—
207#2134—531no—
208#2123—531no—
209#2122—531no—
210#1813—531no—
211#1757—531no—
212C2×C53—21no—
213C2×C53—41no—
214C2×C53—41no—
215C2×C53—41no—
216C2×C53—161no—
217C2×C53#196411yescharacteristic
218C22×SD16Dic5311yescharacteristic, complemented
219C2×#131C2×D5311yescharacteristic
220#536C2×D5311yescharacteristic
221#535C2×D5311yescharacteristic
222#468—21no—
223#468—21no—
224#465—21no—
225#465—21no—
226C2×#108—1061no—
227#427Dic5311yescharacteristic, complemented
228#419Dic5311yescharacteristic, complemented
229#418Dic5311yescharacteristic, complemented
230#413—531no—
231#413—531no—
232#413—531no—
233#410—531no—
234#405—1061no—
235#356—1062no—
236#355—1062no—
237#352—1062no—
238AΣL(1,𝔽16)—1062no—
239#344—1061no—
240#338—1061no—
241C53#1323211yescharacteristic, complemented, Sylow
242C2×#41C22×D5311yescharacteristic
243C22×Q8C2×Dic5311yescharacteristic
244C22×D4C2×Dic5311yescharacteristic
245C2×SD16—21no—
246C2×SD16—21no—
247C2×SD16—41no—
248C2×SD16—41no—
249#131—41no—
250#131—41no—
251C2×M16—1061no—
252C2×M16—1061no—
253C2×M16—1061no—
254C2×M16C2×Dic5311yescharacteristic
255C2×M16C22⋊D5311yescharacteristic
256C2×M16C22⋊D5311yescharacteristic
257C22×C8C2×Dic5311yescharacteristic
258C22×C8C22⋊D5311yescharacteristic
259C22×C8C22⋊D5311yescharacteristic
260C4⋊Q8—21no—
261#127—21no—
262#124—21no—
263#124—21no—
264#123—21no—
265#123—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—21no—
269#121—21no—
270#120—21no—
271#120—21no—
272#117—1061no—
273#117C2×Dic5311yescharacteristic
274#117C2×Dic5311yescharacteristic
275#108—2121no—
276#108—2121no—
277#107—21no—
278#106—21no—
279#103—21no—
280#103—21no—
281#102—21no—
282#102—21no—
283#101—532no—
284#101—1061no—
285#101—1061no—
286#100—1061no—
287#100—1061no—
288#100—1061no—
289#99—532no—
290#99—1061no—
291#97—2121no—
292C4×C8—2121no—
293C24—21no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8D4×D5311yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4D4×D5311yescharacteristic
308C2×D4#1171994711yescharacteristic
309C2×D4#1171994711yescharacteristic
310C22×C4—21no—
311C22×C4C22×Dic5311yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313C22×C4C2×C22⋊D5311yescharacteristic
314SD16—42no—
315SD16—41no—
316SD16—41no—
317M16—41no—
318M16—41no—
319M16—41no—
320M16—1062no—
321M16—2121no—
322M16—2121no—
323M16—2121no—
324M16—2121no—
325M16—2121no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—2121no—
333C2×C8—2121no—
334C2×C8—2121no—
335C2×C8C22⋊Dic5311yescharacteristic
336C2×C8C22⋊Dic5311yescharacteristic
337C2×C8C22⋊Dic5311yescharacteristic
338C2×C8C22⋊Dic5311yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—2121no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—1062no—
351C42—21no—
352C42—21no—
353C42—2121no—
354C23—41no—
355C23—81no—
356C23—161no—
357C23D4×Dic5311yescharacteristic
358C23#1313318811yescharacteristic
359C23#1313320411yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—81no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370D4—161no—
371D4—161no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—2122no—
383C2×C4C2×C22⋊Dic5311yescharacteristic
384C2×C4D4×Dic5311yescharacteristic
385C2×C4#1313318811yescharacteristic
386C2×C4#1313321311yescharacteristic
387C8—21no—
388C8—21no—
389C8—21no—
390C8—4241no—
391C8—4241no—
392C8—2122no—
393C8#1313313211yescharacteristic
394C8#1313313211yescharacteristic
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—162no—
405C22#1681052611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—4242no—
413C4C2×#1313313211yescharacteristic
414C4#1681059711yescharacteristic
415C2—21no—
416C2—41no—
417C2—41no—
418C2—41no—
419C2—161no—
420C2#2027555211yescharacteristic
subgroupG/H
centerC2#20275552
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#3057C2
socleC2×C53#1964
prime radicalC53×#535C22
cosocleC2×#41×C53C23
hypercenter#3057D53
hypocenterC53#13232
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #3057

Complements

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

NH ≅ G/Ntype
1. #20276410C2semidirect
3. #20276369C2semidirect
5. #20274765C2semidirect
6. #20274515C2semidirect
20. C22×SD16×C53C4semidirect
28. C53×#427C4semidirect
29. C53×#419C4semidirect
30. C53×#418C4semidirect
218. C22×SD16Dic53semidirect
227. #427Dic53semidirect
228. #419Dic53semidirect
229. #418Dic53semidirect
241. C53#13232semidirect

Maximal subgroups

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

subgroupindex
#202764102
#202764072
#202763692
#202763652
#202747652
#202745152
C53×#30572
#1323253