Finite Group Databasebeta

← C53×#8461 #27215973

Subgroups — C53×#8461 #27215973

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3507C211yesmaximal, characteristic
2C53×#1985C211yesmaximal, characteristic, complemented
3C53×#1552C211yesmaximal, characteristic, complemented
4C53×#1551C211yesmaximal, characteristic
5C53×#1531C211yesmaximal, characteristic, complemented
6C53×#1530C211yesmaximal, characteristic
7C53×#1467C211yesmaximal, characteristic, complemented
8C53×#559—21no—
9C53×#557—21no—
10C53×#556—21no—
11C53×#556—22no—
12C53×#554—21no—
13C53×#552—22no—
14C53×#539—22no—
15C53×#536C412yes—
16C53×#533C2211yescharacteristic
17C53×#532C2211yescharacteristic
18C53×#529—22no—
19C53×#520C412yes—
20C2×C4×Q8×C53C2211yescharacteristic
21C53×#432C2211yescharacteristic
22C53×#432C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#97×C53C2211yescharacteristic
25C53×#336—22no—
26C53×#333—21no—
27C53×#332—21no—
28C53×#332—21no—
29C53×#331—21no—
30C53×#330—22no—
31C53×#329—22no—
32C53×#324—22no—
33C2×#41×C53C2×C412yes—
34C22×Q8×C53C2×C411yescharacteristic
35C22×C8×C53—21no—
36C22×C8×C53—21no—
37C4⋊Q8×C53—22no—
38C4⋊Q8×C53D411yescharacteristic
39C4⋊Q8×C53D411yescharacteristic
40C4⋊Q8×C53D411yescharacteristic
41#127×C53D411yescharacteristic
42#126×C53—24no—
43#126×C53—24no—
44#125×C53—22no—
45#125×C53—24no—
46#125×C53D412yes—
47#124×C53—22no—
48#124×C53—22no—
49#124×C53D412yes—
50#123×C53—24no—
51#123×C53—24no—
52C22⋊Q8×C53—24no—
53C22⋊Q8×C53—22no—
54C22⋊Q8×C53—22no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59C4×Q8×C53—22no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C4×Q8×C53—22no—
67C4×D4×C53—22no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C4×D4×C53—22no—
71C4×D4×C53—24no—
72#117×C53—24no—
73#117×C53—22no—
74#117×C53C2×C412yes—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53C2×C411yescharacteristic
77C2×C4⋊C4×C53D412yes—
78C2×C42×C53D412yes—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C4⋊C8×C53—21no—
82C4⋊C8×C53—21no—
83C4⋊C8×C53—21no—
84C22⋊C8×C53—21no—
85C22⋊C8×C53—21no—
86#97×C53—21no—
87#97×C53—21no—
88#97×C53—21no—
89#97×C53—21no—
90#41×C53—44no—
91C2×Q8×C53—21no—
92C2×Q8×C53—21no—
93C2×Q8×C53—21no—
94C2×Q8×C53—21no—
95C2×Q8×C53—42no—
96C2×Q8×C53C22⋊C412yes—
97C2×D4×C53—22no—
98C2×D4×C53—21no—
99C2×D4×C53—21no—
100C2×D4×C53C22⋊C412yes—
101C22×C4×C53—24no—
102C22×C4×C53—22no—
103C22×C4×C53C22⋊C412yes—
104C22×C4×C53C22⋊C412yes—
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C2×C8×C53—22no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C2×C8×C53—42no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53—24no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53C22⋊C412yes—
129C4⋊C4×C53C22⋊C412yes—
130C22⋊C4×C53—24no—
131C22⋊C4×C53—24no—
132C22⋊C4×C53—24no—
133C22⋊C4×C53—22no—
134C22⋊C4×C53—22no—
135C22⋊C4×C53—22no—
136C42×C53—24no—
137C42×C53—22no—
138C42×C53—22no—
139C42×C53—22no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53C2×D411yescharacteristic
144C23×C53—22no—
145C23×C53C2×C22⋊C411yescharacteristic
146Q8×C53—42no—
147Q8×C53—42no—
148Q8×C53—42no—
149Q8×C53—41no—
150Q8×C53—41no—
151Q8×C53—81no—
152D4×C53—42no—
153D4×C53—44no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—81no—
157C2×C4×C53—22no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—24no—
166C2×C4×C53—41no—
167C2×C4×C53—44no—
168C2×C4×C53—42no—
169C2×C4×C53—84no—
170C2×C4×C53C2×C22⋊C411yescharacteristic
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53#12011yescharacteristic
173C2×C4×C53#12011yescharacteristic
174C2×C4×C53#12011yescharacteristic
175C2×C4×C53#12011yescharacteristic
176C8×C53—42no—
177C8×C53—81no—
178C8×C53—81no—
179C8×C53—81no—
180#8461C5311yesmaximal, characteristic, direct product factor, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—42no—
184C22×C53—42no—
185C22×C53—42no—
186C22×C53#37911yescharacteristic
187C22×C53#41312yes—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—42no—
194C4×C53—42no—
195C4×C53—42no—
196C4×C53—42no—
197C4×C53—42no—
198C4×C53#45411yescharacteristic
199C4×C53#45611yescharacteristic
200#3507C2×C5311yescharacteristic
201#1985C2×C5311yescharacteristic, complemented
202#1552C2×C5311yescharacteristic, complemented
203#1551C2×C5311yescharacteristic
204#1531C2×C5311yescharacteristic, complemented
205#1530C2×C5311yescharacteristic
206#1467C2×C5311yescharacteristic, complemented
207C2×C53—22no—
208C2×C53—82no—
209C2×C53#178511yescharacteristic
210C2×C53#179311yescharacteristic
211C2×C53#179411yescharacteristic
212#559—21no—
213#557—21no—
214#556—21no—
215#556—22no—
216#554—21no—
217#552—22no—
218#539—22no—
219#536C4×C5312yes—
220#533C22×C5311yescharacteristic
221#532C22×C5311yescharacteristic
222#529—22no—
223#520C4×C5312yes—
224C2×C4×Q8C22×C5311yescharacteristic
225#432C22×C5311yescharacteristic
226#432C22×C5311yescharacteristic
227C2×C4⋊C8C22×C5311yescharacteristic
228C2×#97C22×C5311yescharacteristic
229#336—22no—
230#333—21no—
231#332—21no—
232#332—21no—
233#331—21no—
234#330—22no—
235#329—22no—
236#324—22no—
237C53#846111yescharacteristic, direct product factor, Sylow
238C2×#41C2×C4×C5312yes—
239C22×Q8C2×C4×C5311yescharacteristic
240C22×C8—21no—
241C22×C8—21no—
242C4⋊Q8—22no—
243C4⋊Q8D4×C5311yescharacteristic
244C4⋊Q8D4×C5311yescharacteristic
245C4⋊Q8D4×C5311yescharacteristic
246#127D4×C5311yescharacteristic
247#126—24no—
248#126—24no—
249#125—22no—
250#125—24no—
251#125D4×C5312yes—
252#124—22no—
253#124—22no—
254#124D4×C5312yes—
255#123—24no—
256#123—24no—
257C22⋊Q8—24no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262#121—21no—
263#121—21no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—21no—
276C4×D4—21no—
277#117—24no—
278#117—22no—
279#117C2×C4×C5312yes—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C2×C4×C5311yescharacteristic
282C2×C4⋊C4D4×C5312yes—
283C2×C42D4×C5312yes—
284C2×C42C23×C5311yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291#97—21no—
292#97—21no—
293#97—21no—
294#97—21no—
295#41—44no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×Q8C22⋊C4×C5312yes—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C22⋊C4×C5312yes—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4C22⋊C4×C5312yes—
309C22×C4C22⋊C4×C5312yes—
310C22×C4C22×C4×C5311yescharacteristic
311C22×C4C2×D4×C5311yescharacteristic
312C22×C4C2×D4×C5311yescharacteristic
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4C22⋊C4×C5312yes—
334C4⋊C4C22⋊C4×C5312yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42C2×D4×C5311yescharacteristic
346C42C2×D4×C5311yescharacteristic
347C42C2×D4×C5311yescharacteristic
348C42C2×D4×C5311yescharacteristic
349C23—22no—
350C23C2×C22⋊C4×C5311yescharacteristic
351Q8—42no—
352Q8—42no—
353Q8—42no—
354Q8—41no—
355Q8—41no—
356Q8—81no—
357D4—42no—
358D4—44no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4C2×C22⋊C4×C5311yescharacteristic
376C2×C4C2×C22⋊C4×C5311yescharacteristic
377C2×C4#120×C5311yescharacteristic
378C2×C4#120×C5311yescharacteristic
379C2×C4#120×C5311yescharacteristic
380C2×C4#120×C5311yescharacteristic
381C8—42no—
382C8—81no—
383C8—81no—
384C8—81no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22C53×#37911yescharacteristic
391C22C53×#41312yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4C53×#45411yescharacteristic
403C4C53×#45611yescharacteristic
404C2—22no—
405C2—82no—
406C2C53×#178511yescharacteristic
407C2C53×#179311yescharacteristic
408C2C53×#179411yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1985C2semidirect
3. C53×#1552C2semidirect
5. C53×#1531C2semidirect
7. C53×#1467C2semidirect
180. #8461C53direct (N × H)
201. #1985C2×C53semidirect
202. #1552C2×C53semidirect
204. #1531C2×C53semidirect
206. #1467C2×C53semidirect
237. C53#8461direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35072
C53×#19852
C53×#15522
C53×#15512
C53×#15312
C53×#15302
C53×#14672
#846153