Finite Group Databasebeta

← C53×#36696 #27244208

Subgroups — C53×#36696 #27244208

Order 13568.

1908 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2843C211yesmaximal, characteristic, complemented
2C53×#2830C213yesmaximal, complemented
3C53×#2829C216yesmaximal, complemented
4C53×#2828C213yesmaximal, complemented
5C53×#2823C211yesmaximal, characteristic, complemented
6C53×#2818C212yesmaximal, complemented
7C53×#2395C213yesmaximal, complemented
8C53×#2392C216yesmaximal, complemented
9C53×#2388C213yesmaximal, complemented
10C4×#127×C53C211yesmaximal, characteristic, complemented
11C4×#126×C53C212yesmaximal, complemented
12C22×#41×C53C2211yescharacteristic
13C2×#127×C53C2211yescharacteristic
14C2×#126×C53C2212yes—
15C2×C4×D4×C53C2213yes—
16C2×C4×D4×C53C2213yes—
17C2×C4×D4×C53C2216yes—
18C2×#117×C53C2213yes—
19C53×#400C2213yes—
20C53×#399C2213yes—
21C53×#398C2213yes—
22C53×#398C2216yes—
23C53×#397C2216yes—
24C53×#397C2213yes—
25C53×#397C2216yes—
26C53×#396C2216yes—
27C53×#396C2216yes—
28C53×#395C2213yes—
29C53×#394C2216yes—
30C53×#394C2216yes—
31C53×#394C2213yes—
32C53×#394C2216yes—
33C53×#394C2216yes—
34C53×#394C2216yes—
35C53×#393C2213yes—
36C53×#392C2216yes—
37C53×#392C2216yes—
38C53×#392C2213yes—
39C53×#390C2213yes—
40C53×#390C2213yes—
41C53×#388C2216yes—
42C53×#387C2216yes—
43C53×#384C2211yescharacteristic
44C53×#383C2212yes—
45C4×C4⋊C4×C53C2216yes—
46C4×C22⋊C4×C53C2216yes—
47C4×C22⋊C4×C53C2213yes—
48C4×C22⋊C4×C53C2213yes—
49C43×C53C2211yescharacteristic
50C2×#41×C53—46no—
51C22×Q8×C53C2311yescharacteristic
52C22×D4×C53C2313yes—
53C22×D4×C53C2313yes—
54C22×D4×C53C2316yes—
55C22×D4×C53C2312yes—
56C22×D4×C53C2311yescharacteristic
57C23×C4×C53C2313yes—
58C23×C4×C53C2311yescharacteristic
59C23×C4×C53C2312yes—
60#127×C53—24no—
61#126×C53—28no—
62C4×D4×C53—46no—
63C4×D4×C53—46no—
64C4×D4×C53—412no—
65#117×C53—46no—
66C2×C4⋊C4×C53C2316yes—
67C2×C4⋊C4×C53C2313yes—
68C2×C4⋊C4×C53C2313yes—
69C2×C4⋊C4×C53C2313yes—
70C2×C4⋊C4×C53C2313yes—
71C2×C4⋊C4×C53C2316yes—
72C2×C4⋊C4×C53C2316yes—
73C2×C22⋊C4×C53C2316yes—
74C2×C22⋊C4×C53C2316yes—
75C2×C22⋊C4×C53C2313yes—
76C2×C22⋊C4×C53C2316yes—
77C2×C22⋊C4×C53C2313yes—
78C2×C22⋊C4×C53C2316yes—
79C2×C22⋊C4×C53C2316yes—
80C2×C22⋊C4×C53C2316yes—
81C2×C22⋊C4×C53C2316yes—
82C2×C22⋊C4×C53C2316yes—
83C2×C42×C53C2313yes—
84C2×C42×C53C2311yescharacteristic
85C2×C42×C53C2316yes—
86C2×C42×C53C2313yes—
87#95×C53C2316yes—
88#95×C53C2316yes—
89#95×C53C2313yes—
90#95×C53C2313yes—
91#95×C53C2313yes—
92#95×C53C2313yes—
93#95×C53C2316yes—
94#95×C53C2316yes—
95C24×C53C2411yescharacteristic
96C24×C53C2413yes—
97C24×C53C2412yes—
98#41×C53—416no—
99C2×Q8×C53—43no—
100C2×D4×C53—26no—
101C2×D4×C53—26no—
102C2×D4×C53—43no—
103C2×D4×C53—43no—
104C2×D4×C53—46no—
105C2×D4×C53—43no—
106C2×D4×C53—43no—
107C2×D4×C53—46no—
108C2×D4×C53—46no—
109C2×D4×C53—46no—
110C2×D4×C53—43no—
111C22×C4×C53—26no—
112C22×C4×C53—212no—
113C22×C4×C53—26no—
114C22×C4×C53—212no—
115C22×C4×C53C2416yes—
116C22×C4×C53C2412yes—
117C22×C4×C53C2411yescharacteristic
118C22×C4×C53C2413yes—
119C22×C4×C53C2416yes—
120C22×C4×C53C2413yes—
121C22×C4×C53C2413yes—
122C22×C4×C53C2411yescharacteristic
123C4⋊C4×C53—212no—
124C4⋊C4×C53—46no—
125C4⋊C4×C53—43no—
126C4⋊C4×C53—43no—
127C4⋊C4×C53—43no—
128C4⋊C4×C53—43no—
129C4⋊C4×C53—46no—
130C22⋊C4×C53—212no—
131C22⋊C4×C53—46no—
132C22⋊C4×C53—46no—
133C22⋊C4×C53—46no—
134C22⋊C4×C53—46no—
135C22⋊C4×C53—43no—
136C22⋊C4×C53—46no—
137C22⋊C4×C53—46no—
138C22⋊C4×C53—46no—
139C22⋊C4×C53—43no—
140C42×C53—212no—
141C42×C53—26no—
142C42×C53—43no—
143C42×C53#4114yescomplemented
144C23×C53—21no—
145C23×C53—23no—
146C23×C53—22no—
147C23×C53—23no—
148C23×C53—26no—
149C23×C53—26no—
150C23×C53—23no—
151C23×C53—23no—
152C23×C53—26no—
153C23×C53—26no—
154C23×C53—23no—
155C23×C53C2511yescharacteristic
156Q8×C53—44no—
157D4×C53—412no—
158D4×C53—412no—
159D4×C53—424no—
160D4×C53—48no—
161D4×C53—44no—
162C2×C4×C53—23no—
163C2×C4×C53—23no—
164C2×C4×C53—26no—
165C2×C4×C53—26no—
166C2×C4×C53—23no—
167C2×C4×C53—26no—
168C2×C4×C53—23no—
169C2×C4×C53—26no—
170C2×C4×C53—23no—
171C2×C4×C53—26no—
172C2×C4×C53—26no—
173C2×C4×C53—26no—
174C2×C4×C53—26no—
175C2×C4×C53—26no—
176C2×C4×C53—412no—
177C2×C4×C53—44no—
178C2×C4×C53—48no—
179C2×C4×C53C2×#4116yes—
180C2×C4×C53H(5,𝔽2)16yes—
181#36696C5311yesmaximal, characteristic, direct product factor, Sylow
182C22×C53—43no—
183C22×C53—46no—
184C22×C53—43no—
185C22×C53—46no—
186C22×C53—41no—
187C22×C53—42no—
188C22×C53—43no—
189C22×C53—46no—
190C22×C53—46no—
191C22×C53—43no—
192C22×C53—43no—
193C22×C53C22×#4111yescharacteristic
194C22×C53C2×H(5,𝔽2)13yes—
195C22×C53#58513yes—
196C4×C53—26no—
197C4×C53—46no—
198C4×C53—43no—
199C4×C53—42no—
200C4×C53—41no—
201C4×C53—43no—
202C4×C53—46no—
203C4×C53#56014yes—
204#2843C2×C5311yescharacteristic, complemented
205#2830C2×C5313yescomplemented
206#2829C2×C5316yescomplemented
207#2828C2×C5313yescomplemented
208#2823C2×C5311yescharacteristic, complemented
209#2818C2×C5312yescomplemented
210#2395C2×C5313yescomplemented
211#2392C2×C5316yescomplemented
212#2388C2×C5313yescomplemented
213C4×#127C2×C5311yescharacteristic, complemented
214C4×#126C2×C5312yescomplemented
215C2×C53—81no—
216C2×C53—83no—
217C2×C53—82no—
218C2×C53#346013yes—
219C2×C53C2×#56011yescharacteristic
220C2×C53#352913yes—
221C22×#41C22×C5311yescharacteristic
222C2×#127C22×C5311yescharacteristic
223C2×#126C22×C5312yes—
224C2×C4×D4C22×C5313yes—
225C2×C4×D4C22×C5313yes—
226C2×C4×D4C22×C5316yes—
227C2×#117C22×C5313yes—
228#400C22×C5313yes—
229#399C22×C5313yes—
230#398C22×C5313yes—
231#398C22×C5316yes—
232#397C22×C5316yes—
233#397C22×C5313yes—
234#397C22×C5316yes—
235#396C22×C5316yes—
236#396C22×C5316yes—
237#395C22×C5313yes—
238#394C22×C5316yes—
239#394C22×C5316yes—
240#394C22×C5313yes—
241#394C22×C5316yes—
242#394C22×C5316yes—
243#394C22×C5316yes—
244#393C22×C5313yes—
245#392C22×C5316yes—
246#392C22×C5316yes—
247#392C22×C5313yes—
248#390C22×C5313yes—
249#390C22×C5313yes—
250#388C22×C5316yes—
251#387C22×C5316yes—
252#384C22×C5311yescharacteristic
253#383C22×C5312yes—
254C4×C4⋊C4C22×C5316yes—
255C4×C22⋊C4C22×C5316yes—
256C4×C22⋊C4C22×C5313yes—
257C4×C22⋊C4C22×C5313yes—
258C43C22×C5311yescharacteristic
259C53#3669611yescharacteristic, direct product factor, Sylow
260C2×#41—46no—
261C22×Q8C23×C5311yescharacteristic
262C22×D4C23×C5313yes—
263C22×D4C23×C5313yes—
264C22×D4C23×C5316yes—
265C22×D4C23×C5312yes—
266C22×D4C23×C5311yescharacteristic
267C23×C4C23×C5313yes—
268C23×C4C23×C5311yescharacteristic
269C23×C4C23×C5312yes—
270#127—24no—
271#126—28no—
272C4×D4—412no—
273C4×D4—46no—
274C4×D4—46no—
275#117—46no—
276C2×C4⋊C4C23×C5316yes—
277C2×C4⋊C4C23×C5313yes—
278C2×C4⋊C4C23×C5313yes—
279C2×C4⋊C4C23×C5313yes—
280C2×C4⋊C4C23×C5313yes—
281C2×C4⋊C4C23×C5316yes—
282C2×C4⋊C4C23×C5316yes—
283C2×C22⋊C4C23×C5316yes—
284C2×C22⋊C4C23×C5316yes—
285C2×C22⋊C4C23×C5313yes—
286C2×C22⋊C4C23×C5316yes—
287C2×C22⋊C4C23×C5313yes—
288C2×C22⋊C4C23×C5316yes—
289C2×C22⋊C4C23×C5316yes—
290C2×C22⋊C4C23×C5316yes—
291C2×C22⋊C4C23×C5316yes—
292C2×C22⋊C4C23×C5316yes—
293C2×C42C23×C5316yes—
294C2×C42C23×C5313yes—
295C2×C42C23×C5313yes—
296C2×C42C23×C5311yescharacteristic
297#95C23×C5316yes—
298#95C23×C5316yes—
299#95C23×C5313yes—
300#95C23×C5313yes—
301#95C23×C5313yes—
302#95C23×C5313yes—
303#95C23×C5316yes—
304#95C23×C5316yes—
305C24C24×C5312yes—
306C24C24×C5313yes—
307C24C24×C5311yescharacteristic
308#41—416no—
309C2×Q8—43no—
310C2×D4—26no—
311C2×D4—26no—
312C2×D4—43no—
313C2×D4—43no—
314C2×D4—46no—
315C2×D4—43no—
316C2×D4—43no—
317C2×D4—46no—
318C2×D4—46no—
319C2×D4—46no—
320C2×D4—43no—
321C22×C4—26no—
322C22×C4—212no—
323C22×C4—26no—
324C22×C4—212no—
325C22×C4C24×C5316yes—
326C22×C4C24×C5312yes—
327C22×C4C24×C5311yescharacteristic
328C22×C4C24×C5313yes—
329C22×C4C24×C5316yes—
330C22×C4C24×C5313yes—
331C22×C4C24×C5313yes—
332C22×C4C24×C5311yescharacteristic
333C4⋊C4—212no—
334C4⋊C4—43no—
335C4⋊C4—43no—
336C4⋊C4—43no—
337C4⋊C4—43no—
338C4⋊C4—46no—
339C4⋊C4—46no—
340C22⋊C4—212no—
341C22⋊C4—46no—
342C22⋊C4—46no—
343C22⋊C4—43no—
344C22⋊C4—43no—
345C22⋊C4—46no—
346C22⋊C4—46no—
347C22⋊C4—46no—
348C22⋊C4—46no—
349C22⋊C4—46no—
350C42—212no—
351C42—26no—
352C42—43no—
353C42#41×C5314yescomplemented
354C23—22no—
355C23—23no—
356C23—21no—
357C23—23no—
358C23—26no—
359C23—23no—
360C23—26no—
361C23—23no—
362C23—26no—
363C23—26no—
364C23—23no—
365C23C25×C5311yescharacteristic
366Q8—44no—
367D4—412no—
368D4—412no—
369D4—424no—
370D4—48no—
371D4—44no—
372C2×C4—23no—
373C2×C4—23no—
374C2×C4—26no—
375C2×C4—23no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—23no—
380C2×C4—23no—
381C2×C4—26no—
382C2×C4—26no—
383C2×C4—26no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—412no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4C2×#41×C5316yes—
390C2×C4H(5,𝔽2)×C5316yes—
391C22—43no—
392C22—46no—
393C22—43no—
394C22—46no—
395C22—41no—
396C22—42no—
397C22—43no—
398C22—46no—
399C22—46no—
400C22—43no—
401C22—43no—
402C22C22×#41×C5311yescharacteristic
403C22C2×H(5,𝔽2)×C5313yes—
404C22C53×#58513yes—
405C4—26no—
406C4—46no—
407C4—41no—
408C4—42no—
409C4—43no—
410C4—46no—
411C4—43no—
412C4C53×#56014yes—
413C2—81no—
414C2—83no—
415C2—82no—
416C2C53×#346013yes—
417C2C2×C53×#56011yescharacteristic
418C2C53×#352913yes—
subgroupG/H
centerC22×C4×C53C24
commutator, Frattini, cosocleC23C25×C53
socleC23×C53C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2843C2semidirect
2. C53×#2830C2semidirect
3. C53×#2829C2semidirect
4. C53×#2828C2semidirect
5. C53×#2823C2semidirect
6. C53×#2818C2semidirect
7. C53×#2395C2semidirect
8. C53×#2392C2semidirect
9. C53×#2388C2semidirect
10. C4×#127×C53C2semidirect
11. C4×#126×C53C2semidirect
143. C42×C53#41semidirect
181. #36696C53direct (N × H)
204. #2843C2×C53semidirect
205. #2830C2×C53semidirect
206. #2829C2×C53semidirect
207. #2828C2×C53semidirect
208. #2823C2×C53semidirect
209. #2818C2×C53semidirect
210. #2395C2×C53semidirect
211. #2392C2×C53semidirect
212. #2388C2×C53semidirect
213. C4×#127C2×C53semidirect
214. C4×#126C2×C53semidirect
259. C53#36696direct (N × H)
353. C42#41×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#28432
C53×#28302
C53×#28292
C53×#28282
C53×#28232
C53×#28182
C53×#23952
C53×#23922
C53×#23882
C4×#127×C532
C4×#126×C532
#3669653