Finite Group Databasebeta

← C2×C53×#1351 #27219225

Subgroups — C2×C53×#1351 #27219225

Order 13568.

776 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#393C211yesmaximal, characteristic
2C2×C53×#342C211yesmaximal, characteristic
3C2×C53×#323C211yesmaximal, characteristic
4C53×#1351C214yesmaximal, direct product factor
5C23×Q8×C53C411yescharacteristic, complemented
6C22×C22⋊C4×C53—21no—
7C22×C22⋊C4×C53—21no—
8C22×C22⋊C4×C53C2211yescharacteristic
9C2×C22⋊C8×C53—21no—
10C53×#393—21no—
11C53×#393—21no—
12C53×#393C412yescomplemented
13C53×#393C2212yes—
14C2×#95×C53—21no—
15C2×#95×C53C411yescharacteristic, complemented
16C53×#342—21no—
17C53×#342C2212yes—
18C53×#323—21no—
19C53×#323C2212yes—
20C25×C53D411yescharacteristic
21C22×Q8×C53—21no—
22C22×Q8×C53—44no—
23C22×Q8×C53—41no—
24C22×Q8×C53—41no—
25C22×Q8×C53C2×C412yescomplemented
26C23×C4×C53—21no—
27C23×C4×C53—21no—
28C23×C4×C53—41no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C22×C8×C53—41no—
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—24no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—41no—
37C2×C22⋊C4×C53—41no—
38C2×C22⋊C4×C53—44no—
39C2×C22⋊C4×C53—41no—
40C2×C22⋊C4×C53—42no—
41C2×C22⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45C22⋊C8×C53—22no—
46C22⋊C8×C53—41no—
47#95×C53—22no—
48#95×C53—21no—
49#95×C53—22no—
50#95×C53C2×C412yescomplemented
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—22no—
54C24×C53—42no—
55C24×C53—42no—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C24×C53SD1612yes—
59C24×C53Q1612yes—
60C24×C53C2×D411yescharacteristic
61C2×Q8×C53—44no—
62C2×Q8×C53—42no—
63C2×Q8×C53—44no—
64C2×Q8×C53—42no—
65C2×Q8×C53—48no—
66C2×Q8×C53—44no—
67C2×Q8×C53—44no—
68C22×C4×C53—24no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53—42no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53—44no—
81C22×C4×C53—42no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—82no—
86C22×C4×C53—82no—
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C2×C8×C53—42no—
92C2×C8×C53—82no—
93C22⋊C4×C53—24no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—44no—
99C22⋊C4×C53—44no—
100C22⋊C4×C53—44no—
101C22⋊C4×C53—42no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—44no—
115C23×C53—44no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—42no—
119C23×C53—42no—
120C23×C53—81no—
121C23×C53—84no—
122C23×C53—81no—
123C23×C53#9911yescharacteristic
124C23×C53#9911yescharacteristic
125C23×C53#10312yes—
126C23×C53#10312yes—
127C23×C53#10412yes—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53C2×SD1611yescharacteristic
130C23×C53C2×Q1611yescharacteristic
131Q8×C53—48no—
132Q8×C53—44no—
133Q8×C53—44no—
134C2×C4×C53—24no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—42no—
140C2×C4×C53—44no—
141C2×C4×C53—42no—
142C2×C4×C53—44no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53—81no—
155C2×C4×C53—84no—
156C2×C4×C53—82no—
157C2×C4×C53—88no—
158C2×C4×C53—82no—
159C8×C53—82no—
160C2×#1351C5311yesmaximal, characteristic, direct product factor, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—24no—
164C22×C53—24no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—22no—
169C22×C53—21no—
170C22×C53—42no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—44no—
176C22×C53—44no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—82no—
182C22×C53—82no—
183C22×C53#32812yes—
184C22×C53#32812yes—
185C22×C53AΣL(1,𝔽16)12yes—
186C22×C53#35412yes—
187C22×C53C2×#9911yescharacteristic
188C22×C53C2×#10311yescharacteristic
189C22×C53C2×#10411yescharacteristic
190C4×C53—44no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—41no—
195C4×C53—82no—
196C4×C53—84no—
197C2×#393C2×C5311yescharacteristic
198C2×#342C2×C5311yescharacteristic
199C2×#323C2×C5311yescharacteristic
200#1351C2×C5314yesdirect product factor
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—22no—
204C2×C53—41no—
205C2×C53—42no—
206C2×C53—41no—
207C2×C53#135114yesdirect product factor
208C2×C53C2×#32811yescharacteristic
209C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
210C2×C53C2×#35411yescharacteristic
211C23×Q8C4×C5311yescharacteristic, complemented
212C22×C22⋊C4—21no—
213C22×C22⋊C4—21no—
214C22×C22⋊C4C22×C5311yescharacteristic
215C2×C22⋊C8—21no—
216#393—21no—
217#393—21no—
218#393C4×C5312yescomplemented
219#393C22×C5312yes—
220C2×#95—21no—
221C2×#95C4×C5311yescharacteristic, complemented
222#342—21no—
223#342C22×C5312yes—
224#323—21no—
225#323C22×C5312yes—
226C53C2×#135111yescharacteristic, direct product factor, Sylow
227C25D4×C5311yescharacteristic
228C22×Q8—21no—
229C22×Q8—44no—
230C22×Q8—41no—
231C22×Q8—41no—
232C22×Q8C2×C4×C5312yescomplemented
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—41no—
236C23×C4C2×C4×C5311yescharacteristic
237C23×C4D4×C5311yescharacteristic
238C22×C8—41no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4C2×C4×C5311yescharacteristic
249C2×C22⋊C4D4×C5311yescharacteristic
250C2×C22⋊C4D4×C5311yescharacteristic
251C2×C22⋊C4C23×C5311yescharacteristic
252C22⋊C8—22no—
253C22⋊C8—41no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95C2×C4×C5312yescomplemented
258C24—21no—
259C24—21no—
260C24—22no—
261C24—42no—
262C24—42no—
263C24C22⋊C4×C5311yescharacteristic
264C24C22⋊C4×C5311yescharacteristic
265C24SD16×C5312yes—
266C24Q16×C5312yes—
267C24C2×D4×C5311yescharacteristic
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—48no—
273C2×Q8—44no—
274C2×Q8—44no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4C22⋊C4×C5311yescharacteristic
295C22×C4C22⋊C4×C5311yescharacteristic
296C22×C4C22×C4×C5311yescharacteristic
297C22×C4C2×D4×C5311yescharacteristic
298C2×C8—42no—
299C2×C8—82no—
300C22⋊C4—24no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—44no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23—81no—
328C23—84no—
329C23—81no—
330C23#99×C5311yescharacteristic
331C23#99×C5311yescharacteristic
332C23#103×C5312yes—
333C23#103×C5312yes—
334C23#104×C5312yes—
335C23C2×C22⋊C4×C5311yescharacteristic
336C23C2×SD16×C5311yescharacteristic
337C23C2×Q16×C5311yescharacteristic
338Q8—44no—
339Q8—44no—
340Q8—48no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—88no—
365C2×C4—82no—
366C8—82no—
367C22—21no—
368C22—21no—
369C22—24no—
370C22—24no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—44no—
382C22—44no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—81no—
387C22—82no—
388C22—82no—
389C22C53×#32812yes—
390C22C53×#32812yes—
391C22C53×AΣL(1,𝔽16)12yes—
392C22C53×#35412yes—
393C22C2×#99×C5311yescharacteristic
394C22C2×#103×C5311yescharacteristic
395C22C2×#104×C5311yescharacteristic
396C4—44no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—84no—
403C2—21no—
404C2—21no—
405C2—22no—
406C2—41no—
407C2—42no—
408C2—41no—
409C2C53×#135114yesdirect product factor
410C2C2×C53×#32811yescharacteristic
411C2C2×C53×AΣL(1,𝔽16)11yescharacteristic
412C2C2×C53×#35411yescharacteristic
subgroupG/H
center, socleC23×C53#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC25×C53D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1351C2direct (N × H)
5. C23×Q8×C53C4semidirect
12. C53×#393C4semidirect
15. C2×#95×C53C4semidirect
25. C22×Q8×C53C2×C4semidirect
50. #95×C53C2×C4semidirect
160. C2×#1351C53direct (N × H)
200. #1351C2×C53direct (N × H)
207. C2×C53#1351direct (N × H)
211. C23×Q8C4×C53semidirect
218. #393C4×C53semidirect
221. C2×#95C4×C53semidirect
226. C53C2×#1351direct (N × H)
232. C22×Q8C2×C4×C53semidirect
257. #95C2×C4×C53semidirect
409. C2C53×#1351direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#3932
C2×C53×#3422
C2×C53×#3232
C53×#13512
C2×#135153