Finite Group Databasebeta

← C53×#21794 #27229306

Subgroups — C53×#21794 #27229306

Order 13568.

1394 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#532C211yesmaximal, characteristic
2C53×#2283C212yesmaximal, complemented
3C53×#1964C218yesmaximal, complemented
4C53×#1782C214yesmaximal, complemented
5C22×#41×C53C2212yes—
6C53×#532—22no—
7C53×#532—22no—
8C53×#532—22no—
9C53×#532—22no—
10C2×C4⋊Q8×C53C411yescharacteristic, complemented
11C2×#127×C53C411yescharacteristic, complemented
12C2×#124×C53C412yescomplemented
13C2×#121×C53C2214yes—
14C2×C4×D4×C53C414yescomplemented
15C22×C42×C53C2211yescharacteristic
16C2×#99×C53C22116yes—
17C53×#383—22no—
18C53×#382—22no—
19C53×#381C2214yes—
20C53×#380C2214yes—
21C4×C4⋊C4×C53—22no—
22C4×C22⋊C4×C53C2214yes—
23C53×#376—22no—
24C2×#41×C53—24no—
25C2×#41×C53—24no—
26C2×#41×C53—48no—
27C22×Q8×C53C2×C412yes—
28C22×D4×C53C2×C414yes—
29C22×D4×C53C2×C412yes—
30C22×D4×C53C2314yes—
31C23×C4×C53C2×C414yes—
32C23×C4×C53C2311yescharacteristic
33C23×C4×C53C2312yes—
34C4⋊Q8×C53—21no—
35C4⋊Q8×C53—22no—
36C4⋊Q8×C53—21no—
37#127×C53—21no—
38#127×C53—22no—
39#127×C53—21no—
40#124×C53—22no—
41#124×C53—24no—
42#124×C53—22no—
43#121×C53—24no—
44#121×C53—24no—
45#121×C53—24no—
46#121×C53—24no—
47C4×D4×C53—24no—
48C4×D4×C53—24no—
49C4×D4×C53—24no—
50C4×D4×C53—24no—
51C2×C4⋊C4×C53—24no—
52C2×C4⋊C4×C53C2×C414yes—
53C2×C22⋊C4×C53C2×C418yes—
54C2×C22⋊C4×C53C2318yes—
55C2×C42×C53—24no—
56C2×C42×C53C2×C412yes—
57C2×C42×C53C2×C411yescharacteristic
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53D412yes—
60C2×C42×C53D412yes—
61C2×C42×C53D412yes—
62C2×C42×C53D412yes—
63#99×C53—416no—
64#95×C53—24no—
65#95×C53—24no—
66#95×C53—24no—
67#95×C53—24no—
68C24×C53C22×C414yes—
69C24×C53C2411yescharacteristic
70#41×C53—416no—
71#41×C53—416no—
72C2×Q8×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—44no—
75C2×D4×C53—28no—
76C2×D4×C53—216no—
77C2×D4×C53—24no—
78C2×D4×C53—24no—
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—44no—
82C2×D4×C53—48no—
83C22×C4×C53—24no—
84C22×C4×C53—28no—
85C22×C4×C53—28no—
86C22×C4×C53—44no—
87C22×C4×C53—48no—
88C22×C4×C53C22×C414yes—
89C22×C4×C53C22×C412yes—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C22×C412yes—
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D412yes—
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D412yes—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—24no—
101C4⋊C4×C53—48no—
102C22⋊C4×C53—28no—
103C22⋊C4×C53—28no—
104C22⋊C4×C53—216no—
105C42×C53—48no—
106C42×C53C22⋊C412yes—
107C42×C53C22⋊C412yes—
108C42×C53C22⋊C412yes—
109C42×C53C22⋊C412yes—
110C42×C53C22⋊C411yescharacteristic
111C42×C53C22⋊C412yes—
112C42×C53C22⋊C411yescharacteristic
113C42×C53C22⋊C411yescharacteristic
114C42×C53C22⋊C412yes—
115C42×C53C22⋊C411yescharacteristic
116C23×C53—24no—
117C23×C53—28no—
118C23×C53—42no—
119C23×C53—48no—
120C23×C53C23×C411yescharacteristic
121C23×C53C22×D412yes—
122C23×C53H(5,𝔽2)14yes—
123Q8×C53—44no—
124Q8×C53—44no—
125D4×C53—48no—
126D4×C53—44no—
127D4×C53—44no—
128D4×C53—416no—
129D4×C53—48no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—24no—
133C2×C4×C53—24no—
134C2×C4×C53—24no—
135C2×C4×C53—24no—
136C2×C4×C53—24no—
137C2×C4×C53—24no—
138C2×C4×C53—28no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—416no—
143C2×C4×C53—48no—
144C2×C4×C53—48no—
145C2×C4×C53—48no—
146C2×C4×C53C2×C22⋊C412yes—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C2×C22⋊C412yes—
150C2×C4×C53C2×C22⋊C412yes—
151C2×C4×C53C2×C22⋊C412yes—
152C2×C4×C53C2×C22⋊C412yes—
153#21794C5311yesmaximal, characteristic, direct product factor, Sylow
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—44no—
158C22×C53—44no—
159C22×C53—48no—
160C22×C53—88no—
161C22×C53C22×C22⋊C411yescharacteristic
162C22×C53#51812yes—
163C22×C53#53514yes—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—22no—
170C4×C53—44no—
171C4×C53—88no—
172C4×C53#41014yes—
173C2×#532C2×C5311yescharacteristic
174#2283C2×C5312yescomplemented
175#1964C2×C5318yescomplemented
176#1782C2×C5314yescomplemented
177C2×C53—24no—
178C2×C53—22no—
179C2×C53—84no—
180C2×C53#230911yescharacteristic
181C2×C53C2×#41012yes—
182C22×#41C22×C5312yes—
183#532—22no—
184#532—22no—
185#532—22no—
186#532—22no—
187C2×C4⋊Q8C4×C5311yescharacteristic, complemented
188C2×#127C4×C5311yescharacteristic, complemented
189C2×#124C4×C5312yescomplemented
190C2×#121C22×C5314yes—
191C2×C4×D4C4×C5314yescomplemented
192C22×C42C22×C5311yescharacteristic
193C2×#99C22×C53116yes—
194#383—22no—
195#382—22no—
196#381C22×C5314yes—
197#380C22×C5314yes—
198C4×C4⋊C4—22no—
199C4×C22⋊C4C22×C5314yes—
200#376—22no—
201C53#2179411yescharacteristic, direct product factor, Sylow
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—48no—
205C22×Q8C2×C4×C5312yes—
206C22×D4C2×C4×C5314yes—
207C22×D4C2×C4×C5312yes—
208C22×D4C23×C5314yes—
209C23×C4C2×C4×C5314yes—
210C23×C4C23×C5311yescharacteristic
211C23×C4C23×C5312yes—
212C4⋊Q8—21no—
213C4⋊Q8—22no—
214C4⋊Q8—21no—
215#127—21no—
216#127—22no—
217#127—21no—
218#124—22no—
219#124—24no—
220#124—22no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—24no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4C2×C4×C5314yes—
231C2×C22⋊C4C2×C4×C5318yes—
232C2×C22⋊C4C23×C5318yes—
233C2×C42—24no—
234C2×C42C2×C4×C5312yes—
235C2×C42C2×C4×C5311yescharacteristic
236C2×C42C2×C4×C5311yescharacteristic
237C2×C42D4×C5312yes—
238C2×C42D4×C5312yes—
239C2×C42D4×C5312yes—
240C2×C42D4×C5312yes—
241#99—416no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246C24C22×C4×C5314yes—
247C24C24×C5311yescharacteristic
248#41—416no—
249#41—416no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—44no—
253C2×D4—28no—
254C2×D4—216no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—44no—
260C2×D4—48no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—44no—
265C22×C4—48no—
266C22×C4C22×C4×C5314yes—
267C22×C4C22×C4×C5312yes—
268C22×C4C22×C4×C5311yescharacteristic
269C22×C4C22×C4×C5311yescharacteristic
270C22×C4C22×C4×C5312yes—
271C22×C4C2×D4×C5312yes—
272C22×C4C2×D4×C5312yes—
273C22×C4C2×D4×C5312yes—
274C22×C4C2×D4×C5312yes—
275C22×C4C2×D4×C5312yes—
276C22×C4C2×D4×C5312yes—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—48no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C22⋊C4—216no—
283C42—48no—
284C42C22⋊C4×C5312yes—
285C42C22⋊C4×C5312yes—
286C42C22⋊C4×C5312yes—
287C42C22⋊C4×C5312yes—
288C42C22⋊C4×C5311yescharacteristic
289C42C22⋊C4×C5312yes—
290C42C22⋊C4×C5311yescharacteristic
291C42C22⋊C4×C5311yescharacteristic
292C42C22⋊C4×C5312yes—
293C42C22⋊C4×C5311yescharacteristic
294C23—24no—
295C23—28no—
296C23—42no—
297C23—48no—
298C23C23×C4×C5311yescharacteristic
299C23C22×D4×C5312yes—
300C23H(5,𝔽2)×C5314yes—
301Q8—44no—
302Q8—44no—
303D4—48no—
304D4—44no—
305D4—44no—
306D4—416no—
307D4—48no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—416no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4C2×C22⋊C4×C5312yes—
325C2×C4C2×C22⋊C4×C5312yes—
326C2×C4C2×C22⋊C4×C5312yes—
327C2×C4C2×C22⋊C4×C5312yes—
328C2×C4C2×C22⋊C4×C5311yescharacteristic
329C2×C4C2×C22⋊C4×C5311yescharacteristic
330C2×C4C2×C22⋊C4×C5312yes—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—44no—
335C22—44no—
336C22—48no—
337C22—88no—
338C22C22×C22⋊C4×C5311yescharacteristic
339C22C53×#51812yes—
340C22C53×#53514yes—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—44no—
348C4—88no—
349C4C53×#41014yes—
350C2—24no—
351C2—22no—
352C2—84no—
353C2C53×#230911yescharacteristic
354C2C2×C53×#41012yes—
subgroupG/H
center, socleC22×C53C22×C22⋊C4
commutatorC23C23×C4×C53
Frattini, cosocleC24C24×C53
prime radicalC2×C53×#532C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2283C2semidirect
3. C53×#1964C2semidirect
4. C53×#1782C2semidirect
10. C2×C4⋊Q8×C53C4semidirect
11. C2×#127×C53C4semidirect
12. C2×#124×C53C4semidirect
14. C2×C4×D4×C53C4semidirect
153. #21794C53direct (N × H)
174. #2283C2×C53semidirect
175. #1964C2×C53semidirect
176. #1782C2×C53semidirect
187. C2×C4⋊Q8C4×C53semidirect
188. C2×#127C4×C53semidirect
189. C2×#124C4×C53semidirect
191. C2×C4×D4C4×C53semidirect
201. C53#21794direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5322
C53×#22832
C53×#19642
C53×#17822
#2179453