Finite Group Databasebeta

← C53×#12727 #27220239

Subgroups — C53×#12727 #27220239

Order 13568.

810 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3508C211yesmaximal, characteristic
2C53×#2127C212yesmaximal, complemented
3C2×C53×#353C212yesmaximal, complemented
4C53×#1828C212yesmaximal, complemented
5C53×#559—22no—
6C53×#550—22no—
7C53×#546—24no—
8C53×#540—22no—
9C53×#535C411yescharacteristic, complemented
10C53×#535C2211yescharacteristic
11C53×#533C411yescharacteristic, complemented
12C53×#532—22no—
13C2×#127×C53C2211yescharacteristic
14C53×#518C411yescharacteristic, complemented
15C53×#518C2211yescharacteristic
16C4×#41×C53C411yescharacteristic, complemented
17C2×#99×C53C2214yes—
18C53×#354—22no—
19C53×#353—22no—
20C53×#353—22no—
21C53×#353—22no—
22C53×#342—24no—
23C2×#41×C53—24no—
24C2×#41×C53C2×C411yescharacteristic
25C22×D4×C53—22no—
26C22×D4×C53C2×C411yescharacteristic
27C22×D4×C53C2311yescharacteristic
28C4⋊Q8×C53—21no—
29#127×C53—21no—
30#127×C53—22no—
31#127×C53D412yes—
32#127×C53D412yes—
33#127×C53D412yes—
34#125×C53—21no—
35#124×C53—24no—
36#124×C53—21no—
37#124×C53D412yes—
38#123×C53—24no—
39#123×C53—24no—
40C22⋊Q8×C53—22no—
41#121×C53—24no—
42#121×C53—24no—
43#121×C53—24no—
44#121×C53—22no—
45#121×C53—22no—
46#120×C53—22no—
47#120×C53—22no—
48C4×Q8×C53—22no—
49C4×Q8×C53—21no—
50C4×D4×C53—24no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C4×D4×C53—21no—
54C4×D4×C53—22no—
55C4×D4×C53—22no—
56C4×D4×C53—22no—
57#117×C53—21no—
58#117×C53C2×C411yescharacteristic
59#117×C53C2×C411yescharacteristic
60#117×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53—22no—
62C2×C22⋊C4×C53—24no—
63C2×C22⋊C4×C53D414yes—
64C2×C42×C53—21no—
65C2×C42×C53C2×C411yescharacteristic
66#99×C53—24no—
67#99×C53—24no—
68C24×C53—21no—
69C24×C53C2×D412yes—
70#41×C53—42no—
71#41×C53—88no—
72C2×Q8×C53—22no—
73C2×Q8×C53C22⋊C411yescharacteristic
74C2×D4×C53—21no—
75C2×D4×C53—24no—
76C2×D4×C53—24no—
77C2×D4×C53—24no—
78C2×D4×C53—24no—
79C2×D4×C53—21no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53—84no—
83C2×D4×C53C22⋊C412yes—
84C2×D4×C53C22⋊C411yescharacteristic
85C2×D4×C53C2×D412yes—
86C2×D4×C53C2×D412yes—
87C22×C4×C53—22no—
88C22×C4×C53—21no—
89C22×C4×C53—24no—
90C22×C4×C53—24no—
91C22×C4×C53—21no—
92C22×C4×C53—44no—
93C22×C4×C53C22×C411yescharacteristic
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—24no—
107C22⋊C4×C53—44no—
108C22⋊C4×C53—44no—
109C22⋊C4×C53C22⋊C412yes—
110C22⋊C4×C53C22⋊C412yes—
111C42×C53—22no—
112C42×C53—21no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53C22⋊C411yescharacteristic
116C42×C53C22⋊C411yescharacteristic
117C42×C53C22⋊C412yes—
118C23×C53—24no—
119C23×C53—21no—
120C23×C53—22no—
121C23×C53—42no—
122C23×C53—82no—
123C23×C53—82no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12014yes—
126Q8×C53—44no—
127Q8×C53—41no—
128D4×C53—42no—
129D4×C53—42no—
130D4×C53—42no—
131D4×C53—41no—
132D4×C53—42no—
133D4×C53—48no—
134D4×C53—42no—
135D4×C53—84no—
136D4×C53—84no—
137D4×C53—84no—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—44no—
148C2×C4×C53—84no—
149C2×C4×C53—84no—
150C2×C4×C53—84no—
151C2×C4×C53—164no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C2×C22⋊C411yescharacteristic
154#12727C5311yesmaximal, characteristic, direct product factor, Sylow
155C22×C53—24no—
156C22×C53—41no—
157C22×C53—44no—
158C22×C53—42no—
159C22×C53—44no—
160C22×C53—41no—
161C22×C53—42no—
162C22×C53—84no—
163C22×C53—162no—
164C22×C53#37911yescharacteristic
165C22×C53#41012yes—
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53—41no—
174C4×C53—164no—
175#3508C2×C5311yescharacteristic
176#2127C2×C5312yescomplemented
177C2×#353C2×C5312yescomplemented
178#1828C2×C5312yescomplemented
179C2×C53—21no—
180C2×C53—22no—
181C2×C53—44no—
182C2×C53—81no—
183C2×C53—82no—
184C2×C53#178211yescharacteristic
185#559—22no—
186#550—22no—
187#546—24no—
188#540—22no—
189#535C4×C5311yescharacteristic, complemented
190#535C22×C5311yescharacteristic
191#533C4×C5311yescharacteristic, complemented
192#532—22no—
193C2×#127C22×C5311yescharacteristic
194#518C4×C5311yescharacteristic, complemented
195#518C22×C5311yescharacteristic
196C4×#41C4×C5311yescharacteristic, complemented
197C2×#99C22×C5314yes—
198#354—22no—
199#353—22no—
200#353—22no—
201#353—22no—
202#342—24no—
203C53#1272711yescharacteristic, direct product factor, Sylow
204C2×#41—24no—
205C2×#41C2×C4×C5311yescharacteristic
206C22×D4—22no—
207C22×D4C2×C4×C5311yescharacteristic
208C22×D4C23×C5311yescharacteristic
209C4⋊Q8—21no—
210#127—22no—
211#127—21no—
212#127D4×C5312yes—
213#127D4×C5312yes—
214#127D4×C5312yes—
215#125—21no—
216#124—24no—
217#124—21no—
218#124D4×C5312yes—
219#123—24no—
220#123—24no—
221C22⋊Q8—22no—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—22no—
226#121—22no—
227#120—22no—
228#120—22no—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—21no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238#117—21no—
239#117C2×C4×C5311yescharacteristic
240#117C2×C4×C5311yescharacteristic
241#117C2×C4×C5311yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4D4×C5314yes—
245C2×C42—21no—
246C2×C42C2×C4×C5311yescharacteristic
247#99—24no—
248#99—24no—
249C24—21no—
250C24C2×D4×C5312yes—
251#41—42no—
252#41—88no—
253C2×Q8—22no—
254C2×Q8C22⋊C4×C5311yescharacteristic
255C2×D4—21no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C2×D4C22⋊C4×C5312yes—
265C2×D4C22⋊C4×C5311yescharacteristic
266C2×D4C2×D4×C5312yes—
267C2×D4C2×D4×C5312yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—44no—
274C22×C4C22×C4×C5311yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—21no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4C22⋊C4×C5312yes—
291C22⋊C4C22⋊C4×C5312yes—
292C42—22no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22⋊C4×C5311yescharacteristic
297C42C22⋊C4×C5311yescharacteristic
298C42C22⋊C4×C5312yes—
299C23—24no—
300C23—22no—
301C23—21no—
302C23—42no—
303C23—82no—
304C23—82no—
305C23C2×C22⋊C4×C5311yescharacteristic
306C23#120×C5314yes—
307Q8—41no—
308Q8—44no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—41no—
313D4—42no—
314D4—48no—
315D4—42no—
316D4—84no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—164no—
333C2×C4C2×C22⋊C4×C5311yescharacteristic
334C2×C4C2×C22⋊C4×C5311yescharacteristic
335C22—24no—
336C22—41no—
337C22—44no—
338C22—42no—
339C22—44no—
340C22—41no—
341C22—42no—
342C22—84no—
343C22—162no—
344C22C53×#37911yescharacteristic
345C22C53×#41012yes—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—164no—
355C2—21no—
356C2—22no—
357C2—44no—
358C2—82no—
359C2—81no—
360C2C53×#178211yescharacteristic
subgroupG/H
center, socleC2×C53#1782
commutatorC22×C4C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC53×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C23×C53 ◁ C53×#518 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2127C2semidirect
3. C2×C53×#353C2semidirect
4. C53×#1828C2semidirect
9. C53×#535C4semidirect
11. C53×#533C4semidirect
14. C53×#518C4semidirect
16. C4×#41×C53C4semidirect
154. #12727C53direct (N × H)
176. #2127C2×C53semidirect
177. C2×#353C2×C53semidirect
178. #1828C2×C53semidirect
189. #535C4×C53semidirect
191. #533C4×C53semidirect
194. #518C4×C53semidirect
196. C4×#41C4×C53semidirect
203. C53#12727direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35082
C53×#21272
C2×C53×#3532
C53×#18282
#1272753