Finite Group Databasebeta

← C53×#9386 #27216898

Subgroups — C53×#9386 #27216898

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2973C211yesmaximal, characteristic, complemented
2C53×#2543C211yesmaximal, characteristic
3C53×#1799C211yesmaximal, characteristic, complemented
4C53×#1519C212yesmaximal, complemented
5C53×#1294C212yesmaximal, complemented
6C2×#121×C53—21no—
7C2×#121×C53C2211yescharacteristic
8C2×C4×D4×C53C2211yescharacteristic
9C53×#435—24no—
10C53×#432—21no—
11C2×#99×C53—21no—
12C53×#407—21no—
13C2×C22⋊C8×C53C2212yes—
14C53×#398C411yescharacteristic, complemented
15C53×#397C411yescharacteristic, complemented
16C53×#394C411yescharacteristic, complemented
17C53×#392C411yescharacteristic, complemented
18C53×#388—21no—
19C53×#386—21no—
20C53×#386—21no—
21C53×#380C2212yes—
22C2×#95×C53C2211yescharacteristic
23C53×#342—21no—
24C53×#327—24no—
25C22×D4×C53—21no—
26C22×D4×C53—21no—
27C22×D4×C53C2×C411yescharacteristic
28C22×D4×C53D411yescharacteristic
29C23×C4×C53—21no—
30C23×C4×C53D411yescharacteristic
31C23×C4×C53C2311yescharacteristic
32C2×M16×C53—22no—
33C22×C8×C53—22no—
34#121×C53—22no—
35#121×C53—22no—
36#121×C53—44no—
37C4×D4×C53—22no—
38C4×D4×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C4⋊C8×C53—22no—
53#99×C53—42no—
54C22⋊C8×C53—22no—
55C22⋊C8×C53—22no—
56C22⋊C8×C53D414yescomplemented
57#97×C53—22no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—22no—
62#95×C53C2×C411yescharacteristic
63#95×C53C2×C411yescharacteristic
64C24×C53—21no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—21no—
69C2×D4×C53—22no—
70C2×D4×C53—21no—
71C2×D4×C53—22no—
72C2×D4×C53—42no—
73C2×D4×C53—42no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—81no—
79C2×D4×C53—81no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53—81no—
89C22×C4×C53—81no—
90C22×C4×C53—81no—
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4112yes—
98M16×C53—82no—
99C2×C8×C53—24no—
100C2×C8×C53—82no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—42no—
104C4⋊C4×C53—82no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—42no—
114C22⋊C4×C53—81no—
115C42×C53—21no—
116C42×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53C2×C22⋊C411yescharacteristic
132C23×C53#12112yes—
133C23×C53AGL(1,ℤ/8)11yescharacteristic
134C23×C53AGL(1,ℤ/8)12yes—
135C23×C53#13711yescharacteristic
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—82no—
139D4×C53—84no—
140D4×C53—82no—
141D4×C53—82no—
142C2×C4×C53—22no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—81no—
164C2×C4×C53—84no—
165C2×C4×C53—82no—
166C2×C4×C53—82no—
167C2×C4×C53—82no—
168C2×C4×C53C4×D412yes—
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53#12311yescharacteristic
171C8×C53—44no—
172#9386C5311yesmaximal, characteristic, direct product factor, Sylow
173C22×C53—21no—
174C22×C53—22no—
175C22×C53—21no—
176C22×C53—22no—
177C22×C53—42no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53#38611yescharacteristic
193C22×C53#41711yescharacteristic
194C22×C53#41811yescharacteristic
195C22×C53#46812yes—
196C22×C53#46912yes—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—22no—
200C4×C53—42no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53—81no—
205C4×C53—81no—
206C4×C53—162no—
207#2973C2×C5311yescharacteristic, complemented
208#2543C2×C5311yescharacteristic
209#1799C2×C5311yescharacteristic, complemented
210#1519C2×C5312yescomplemented
211#1294C2×C5312yescomplemented
212C2×C53—21no—
213C2×C53—22no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—81no—
217C2×C53—161no—
218C2×C53#188411yescharacteristic
219C2×C53#189511yescharacteristic
220C2×C53#191311yescharacteristic
221C2×#121—21no—
222C2×#121C22×C5311yescharacteristic
223C2×C4×D4C22×C5311yescharacteristic
224#435—24no—
225#432—21no—
226C2×#99—21no—
227#407—21no—
228C2×C22⋊C8C22×C5312yes—
229#398C4×C5311yescharacteristic, complemented
230#397C4×C5311yescharacteristic, complemented
231#394C4×C5311yescharacteristic, complemented
232#392C4×C5311yescharacteristic, complemented
233#388—21no—
234#386—21no—
235#386—21no—
236#380C22×C5312yes—
237C2×#95C22×C5311yescharacteristic
238#342—21no—
239#327—24no—
240C53#938611yescharacteristic, direct product factor, Sylow
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×C4×C5311yescharacteristic
244C22×D4D4×C5311yescharacteristic
245C23×C4—21no—
246C23×C4D4×C5311yescharacteristic
247C23×C4C23×C5311yescharacteristic
248C2×M16—22no—
249C22×C8—22no—
250#121—22no—
251#121—22no—
252#121—44no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C2×C4×C5311yescharacteristic
258C2×C4⋊C4C2×C4×C5311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4C2×C4×C5311yescharacteristic
266C2×C22⋊C4D4×C5311yescharacteristic
267C2×C42D4×C5311yescharacteristic
268C4⋊C8—22no—
269#99—42no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8D4×C5314yescomplemented
273#97—22no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95C2×C4×C5311yescharacteristic
279#95C2×C4×C5311yescharacteristic
280C24—21no—
281C24C22⋊C4×C5311yescharacteristic
282C24C2×D4×C5311yescharacteristic
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—81no—
295C2×D4—81no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4C22⋊C4×C5311yescharacteristic
308C22×C4C22⋊C4×C5311yescharacteristic
309C22×C4C22⋊C4×C5311yescharacteristic
310C22×C4C22×C4×C5311yescharacteristic
311C22×C4C2×D4×C5312yes—
312C22×C4C2×D4×C5311yescharacteristic
313C22×C4#41×C5312yes—
314M16—82no—
315C2×C8—24no—
316C2×C8—82no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—81no—
331C42—21no—
332C42—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C2×C22⋊C4×C5311yescharacteristic
348C23#121×C5312yes—
349C23AGL(1,ℤ/8)×C5311yescharacteristic
350C23AGL(1,ℤ/8)×C5312yes—
351C23#137×C5311yescharacteristic
352D4—42no—
353D4—42no—
354D4—82no—
355D4—84no—
356D4—82no—
357D4—82no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4C4×D4×C5312yes—
385C2×C4#120×C5311yescharacteristic
386C2×C4#123×C5311yescharacteristic
387C8—44no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—22no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22C53×#38611yescharacteristic
408C22C53×#41711yescharacteristic
409C22C53×#41811yescharacteristic
410C22C53×#46812yes—
411C22C53×#46912yes—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—162no—
422C2—21no—
423C2—22no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—161no—
428C2C53×#188411yescharacteristic
429C2C53×#189511yescharacteristic
430C2C53×#191311yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC53×#2543C2
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
1. C53×#2973C2semidirect
3. C53×#1799C2semidirect
4. C53×#1519C2semidirect
5. C53×#1294C2semidirect
14. C53×#398C4semidirect
15. C53×#397C4semidirect
16. C53×#394C4semidirect
17. C53×#392C4semidirect
56. C22⋊C8×C53D4semidirect
172. #9386C53direct (N × H)
207. #2973C2×C53semidirect
209. #1799C2×C53semidirect
210. #1519C2×C53semidirect
211. #1294C2×C53semidirect
229. #398C4×C53semidirect
230. #397C4×C53semidirect
231. #394C4×C53semidirect
232. #392C4×C53semidirect
240. C53#9386direct (N × H)
272. C22⋊C8D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#29732
C53×#25432
C53×#17992
C53×#15192
C53×#12942
#938653