Finite Group Databasebeta

← C53×#8463 #27215975

Subgroups — C53×#8463 #27215975

Order 13568.

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3493C211yesmaximal, characteristic
2C53×#1820C211yesmaximal, characteristic, complemented
3C53×#1818C211yesmaximal, characteristic, complemented
4C53×#1516C212yesmaximal, complemented
5C53×#1516C212yesmaximal, complemented
6C53×#556—22no—
7C53×#555—22no—
8C53×#553—22no—
9C53×#552—22no—
10C53×#544C411yescharacteristic, complemented
11C53×#538—22no—
12C53×#537C411yescharacteristic, complemented
13C53×#529—22no—
14C53×#525C2212yes—
15C53×#520C412yescomplemented
16C2×#117×C53C2211yescharacteristic
17C2×C22⋊C8×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2211yescharacteristic
19C53×#381C2212yes—
20C53×#342—22no—
21C53×#336—22no—
22C53×#328—22no—
23C53×#328—22no—
24C53×#327—22no—
25C53×#327—22no—
26C2×#41×C53C2×C412yes—
27C23×C4×C53C2311yescharacteristic
28C22×C8×C53—21no—
29C22×C8×C53—21no—
30C4⋊Q8×C53—22no—
31#126×C53—24no—
32#126×C53—24no—
33#125×C53—22no—
34#125×C53—22no—
35#124×C53—22no—
36#124×C53—24no—
37#123×C53—24no—
38#123×C53—22no—
39C22⋊Q8×C53—22no—
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53D412yes—
42C22⋊Q8×C53D412yes—
43#121×C53—24no—
44#121×C53—22no—
45#121×C53—22no—
46#121×C53D412yes—
47#121×C53D412yes—
48#120×C53—22no—
49C4×Q8×C53—22no—
50C4×Q8×C53—22no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C4×D4×C53—24no—
54C4×D4×C53—22no—
55C4×D4×C53—22no—
56#117×C53—24no—
57#117×C53—22no—
58#117×C53—22no—
59#117×C53—22no—
60#117×C53C2×C412yes—
61C2×C4⋊C4×C53—22no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C4⋊C4×C53C2×C411yescharacteristic
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53D412yes—
66C2×C42×C53D412yes—
67C22⋊C8×C53—22no—
68C22⋊C8×C53—22no—
69#95×C53—22no—
70#95×C53—22no—
71C24×C53C2×D411yescharacteristic
72#41×C53—44no—
73C2×Q8×C53—22no—
74C2×Q8×C53C22⋊C412yes—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53C22⋊C412yes—
80C22×C4×C53—24no—
81C22×C4×C53—22no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53C22⋊C412yes—
85C22×C4×C53C22⋊C412yes—
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D412yes—
90C2×C8×C53—42no—
91C2×C8×C53—42no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53C22⋊C412yes—
103C4⋊C4×C53C22⋊C412yes—
104C22⋊C4×C53—24no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—44no—
113C42×C53—24no—
114C42×C53—22no—
115C42×C53—22no—
116C42×C53—22no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53#12012yes—
123Q8×C53—42no—
124Q8×C53—82no—
125D4×C53—42no—
126D4×C53—44no—
127D4×C53—82no—
128D4×C53—82no—
129D4×C53—82no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—44no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—84no—
143C2×C4×C53—82no—
144C2×C4×C53—82no—
145C2×C4×C53—82no—
146C2×C4×C53C2×C22⋊C412yes—
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C8×C53—81no—
150C8×C53—81no—
151#8463C5311yesmaximal, characteristic, direct product factor, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—82no—
161C22×C53#37911yescharacteristic
162C22×C53#41012yes—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—42no—
167C4×C53—42no—
168C4×C53—42no—
169C4×C53—42no—
170C4×C53—42no—
171C4×C53—162no—
172#3493C2×C5311yescharacteristic
173#1820C2×C5311yescharacteristic, complemented
174#1818C2×C5311yescharacteristic, complemented
175#1516C2×C5312yescomplemented
176#1516C2×C5312yescomplemented
177C2×C53—22no—
178C2×C53—42no—
179C2×C53—82no—
180C2×C53#178211yescharacteristic
181C2×C53#179411yescharacteristic
182C2×C53#179411yescharacteristic
183#556—22no—
184#555—22no—
185#553—22no—
186#552—22no—
187#544C4×C5311yescharacteristic, complemented
188#538—22no—
189#537C4×C5311yescharacteristic, complemented
190#529—22no—
191#525C22×C5312yes—
192#520C4×C5312yescomplemented
193C2×#117C22×C5311yescharacteristic
194C2×C22⋊C8C22×C5311yescharacteristic
195C2×C22⋊C8C22×C5311yescharacteristic
196#381C22×C5312yes—
197#342—22no—
198#336—22no—
199#328—22no—
200#328—22no—
201#327—22no—
202#327—22no—
203C53#846311yescharacteristic, direct product factor, Sylow
204C2×#41C2×C4×C5312yes—
205C23×C4C23×C5311yescharacteristic
206C22×C8—21no—
207C22×C8—21no—
208C4⋊Q8—22no—
209#126—24no—
210#126—24no—
211#125—22no—
212#125—22no—
213#124—22no—
214#124—24no—
215#123—24no—
216#123—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8D4×C5312yes—
220C22⋊Q8D4×C5312yes—
221#121—24no—
222#121—22no—
223#121—22no—
224#121D4×C5312yes—
225#121D4×C5312yes—
226#120—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—24no—
234#117—24no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117C2×C4×C5312yes—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C2×C4×C5311yescharacteristic
241C2×C4⋊C4C2×C4×C5311yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4D4×C5312yes—
244C2×C42D4×C5312yes—
245C22⋊C8—22no—
246C22⋊C8—22no—
247#95—22no—
248#95—22no—
249C24C2×D4×C5311yescharacteristic
250#41—44no—
251C2×Q8—22no—
252C2×Q8C22⋊C4×C5312yes—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊C4×C5312yes—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C22⋊C4×C5312yes—
263C22×C4C22⋊C4×C5312yes—
264C22×C4C22×C4×C5311yescharacteristic
265C22×C4C2×D4×C5311yescharacteristic
266C22×C4C2×D4×C5312yes—
267C22×C4C2×D4×C5312yes—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4C22⋊C4×C5312yes—
281C4⋊C4C22⋊C4×C5312yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—22no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—44no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C5311yescharacteristic
300C23#120×C5312yes—
301Q8—42no—
302Q8—82no—
303D4—42no—
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C2×C22⋊C4×C5312yes—
325C2×C4#120×C5311yescharacteristic
326C2×C4#120×C5311yescharacteristic
327C8—81no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22C53×#37911yescharacteristic
339C22C53×#41012yes—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—162no—
349C2—22no—
350C2—42no—
351C2—82no—
352C2C53×#178211yescharacteristic
353C2C53×#179411yescharacteristic
354C2C53×#179411yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC53×#3493C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1820C2semidirect
3. C53×#1818C2semidirect
4. C53×#1516C2semidirect
5. C53×#1516C2semidirect
10. C53×#544C4semidirect
12. C53×#537C4semidirect
15. C53×#520C4semidirect
151. #8463C53direct (N × H)
173. #1820C2×C53semidirect
174. #1818C2×C53semidirect
175. #1516C2×C53semidirect
176. #1516C2×C53semidirect
187. #544C4×C53semidirect
189. #537C4×C53semidirect
192. #520C4×C53semidirect
203. C53#8463direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34932
C53×#18202
C53×#18182
C53×#15162
#846353