Finite Group Databasebeta

← C53×#9055 #27216567

Subgroups — C53×#9055 #27216567

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3167C211yesmaximal, characteristic
2C53×#2935C211yesmaximal, characteristic
3C53×#1986C211yesmaximal, characteristic, complemented
4C53×#1502C211yesmaximal, characteristic
5C53×#1481C211yesmaximal, characteristic, complemented
6C53×#1279C211yesmaximal, characteristic, complemented
7C53×#1278C211yesmaximal, characteristic, complemented
8C2×#131×C53—21no—
9C53×#533C2211yescharacteristic
10C53×#532C2211yescharacteristic
11C53×#498C411yescharacteristic, complemented
12C53×#492C411yescharacteristic, complemented
13C53×#487C411yescharacteristic, complemented
14C53×#486C411yescharacteristic, complemented
15C53×#465—22no—
16C4×SD16×C53—22no—
17C53×#436—21no—
18C53×#435—22no—
19C8×D4×C53—21no—
20C53×#431C2211yescharacteristic
21C2×C4⋊C8×C53C2211yescharacteristic
22C53×#408—21no—
23C4×M16×C53C2211yescharacteristic
24C2×C4×C8×C53C2211yescharacteristic
25C4×C4⋊C4×C53C2211yescharacteristic
26C53×#339—21no—
27C53×#339—21no—
28C53×#327—22no—
29C53×Q8⋊C8—21no—
30C53×Q8⋊C8—21no—
31C2×#41×C53—21no—
32C2×SD16×C53—22no—
33#131×C53—42no—
34#131×C53—42no—
35C2×M16×C53—22no—
36C2×M16×C53D411yescharacteristic
37C2×M16×C53D411yescharacteristic
38C22×C8×C53—22no—
39C22×C8×C53—21no—
40C22×C8×C53—21no—
41C4⋊Q8×C53C2×C411yescharacteristic
42C4⋊Q8×C53C2×C411yescharacteristic
43C4⋊Q8×C53C2×C411yescharacteristic
44#127×C53C2×C411yescharacteristic
45#125×C53—21no—
46#124×C53—21no—
47C22⋊Q8×C53—21no—
48C22⋊Q8×C53—21no—
49#121×C53—21no—
50#121×C53D412yes—
51C4×Q8×C53—21no—
52C4×Q8×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53D412yes—
55#117×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C42×C53—21no—
58C2×C42×C53C2311yescharacteristic
59#106×C53—22no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53D411yescharacteristic
63C4⋊C8×C53D411yescharacteristic
64#103×C53—22no—
65#102×C53—22no—
66C22⋊C8×C53—22no—
67C22⋊C8×C53—22no—
68C22⋊C8×C53—21no—
69#97×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73C4×C8×C53C2×C411yescharacteristic
74C4×C8×C53C2×C411yescharacteristic
75#95×C53—21no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53C22⋊C412yes—
87C22×C4×C53—22no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
92SD16×C53—44no—
93M16×C53—21no—
94M16×C53—21no—
95M16×C53—21no—
96M16×C53—44no—
97M16×C53SD1612yescomplemented
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—21no—
104C2×C8×C53—42no—
105C2×C8×C53—44no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53C22⋊C412yes—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53—41no—
125C42×C53C22×C411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129C23×C53—22no—
130C23×C53#12111yescharacteristic
131Q8×C53—42no—
132Q8×C53—41no—
133Q8×C53—41no—
134Q8×C53—41no—
135Q8×C53—41no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×C4×C53C2×SD1611yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53#13711yescharacteristic
164C8×C53—22no—
165C8×C53—22no—
166C8×C53—41no—
167C8×C53—41no—
168C8×C53—41no—
169C8×C53—41no—
170C8×C53—81no—
171#9055C5311yesmaximal, characteristic, direct product factor, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#38611yescharacteristic
178C22×C53#46111yescharacteristic
179C22×C53#46311yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53—82no—
185C4×C53—81no—
186C4×C53—81no—
187C4×C53C4×SD1611yescharacteristic
188C4×C53#44011yescharacteristic
189C4×C53#44311yescharacteristic
190C4×C53#44411yescharacteristic
191C4×C53#45011yescharacteristic
192C4×C53#45211yescharacteristic
193#3167C2×C5311yescharacteristic
194#2935C2×C5311yescharacteristic
195#1986C2×C5311yescharacteristic, complemented
196#1502C2×C5311yescharacteristic
197#1481C2×C5311yescharacteristic, complemented
198#1279C2×C5311yescharacteristic, complemented
199#1278C2×C5311yescharacteristic, complemented
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—82no—
203C2×C53#187611yescharacteristic
204C2×C53#187711yescharacteristic
205C2×C53#188411yescharacteristic
206C2×#131—21no—
207#533C22×C5311yescharacteristic
208#532C22×C5311yescharacteristic
209#498C4×C5311yescharacteristic, complemented
210#492C4×C5311yescharacteristic, complemented
211#487C4×C5311yescharacteristic, complemented
212#486C4×C5311yescharacteristic, complemented
213#465—22no—
214C4×SD16—22no—
215#436—21no—
216#435—22no—
217C8×D4—21no—
218#431C22×C5311yescharacteristic
219C2×C4⋊C8C22×C5311yescharacteristic
220#408—21no—
221C4×M16C22×C5311yescharacteristic
222C2×C4×C8C22×C5311yescharacteristic
223C4×C4⋊C4C22×C5311yescharacteristic
224#339—21no—
225#339—21no—
226#327—22no—
227Q8⋊C8—21no—
228Q8⋊C8—21no—
229C53#905511yescharacteristic, direct product factor, Sylow
230C2×#41—21no—
231C2×SD16—22no—
232#131—42no—
233#131—42no—
234C2×M16—22no—
235C2×M16D4×C5311yescharacteristic
236C2×M16D4×C5311yescharacteristic
237C22×C8—22no—
238C22×C8—21no—
239C22×C8—21no—
240C4⋊Q8C2×C4×C5311yescharacteristic
241C4⋊Q8C2×C4×C5311yescharacteristic
242C4⋊Q8C2×C4×C5311yescharacteristic
243#127C2×C4×C5311yescharacteristic
244#125—21no—
245#124—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—21no—
249#121D4×C5312yes—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—21no—
253C4×D4D4×C5312yes—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C42—21no—
257C2×C42C23×C5311yescharacteristic
258#106—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8D4×C5311yescharacteristic
262C4⋊C8D4×C5311yescharacteristic
263#103—22no—
264#102—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—21no—
268#97—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8C2×C4×C5311yescharacteristic
273C4×C8C2×C4×C5311yescharacteristic
274#95—21no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C22⋊C4×C5312yes—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4C2×D4×C5311yescharacteristic
289C22×C4C2×D4×C5311yescharacteristic
290C22×C4#41×C5311yescharacteristic
291SD16—44no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—44no—
296M16SD16×C5312yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—44no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C22⋊C4×C5312yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42C22×C4×C5311yescharacteristic
325C42C2×D4×C5311yescharacteristic
326C42C2×D4×C5311yescharacteristic
327C42#41×C5311yescharacteristic
328C23—22no—
329C23#121×C5311yescharacteristic
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C2×C22⋊C4×C5311yescharacteristic
354C2×C4C4×D4×C5311yescharacteristic
355C2×C4C4×D4×C5311yescharacteristic
356C2×C4#120×C5311yescharacteristic
357C2×C4#121×C5311yescharacteristic
358C2×C4#123×C5311yescharacteristic
359C2×C4C2×SD16×C5311yescharacteristic
360C2×C4#135×C5311yescharacteristic
361C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
362C2×C4#137×C5311yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8—41no—
367C8—41no—
368C8—41no—
369C8—81no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22C53×#38611yescharacteristic
376C22C53×#46111yescharacteristic
377C22C53×#46311yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—82no—
383C4—81no—
384C4—81no—
385C4C4×SD16×C5311yescharacteristic
386C4C53×#44011yescharacteristic
387C4C53×#44311yescharacteristic
388C4C53×#44411yescharacteristic
389C4C53×#45011yescharacteristic
390C4C53×#45211yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2C53×#187611yescharacteristic
395C2C53×#187711yescharacteristic
396C2C53×#188411yescharacteristic
subgroupG/H
centerC2×C4×C53#120
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
socleC22×C53#386
prime radicalC53×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1986C2semidirect
5. C53×#1481C2semidirect
6. C53×#1279C2semidirect
7. C53×#1278C2semidirect
11. C53×#498C4semidirect
12. C53×#492C4semidirect
13. C53×#487C4semidirect
14. C53×#486C4semidirect
97. M16×C53SD16semidirect
171. #9055C53direct (N × H)
195. #1986C2×C53semidirect
197. #1481C2×C53semidirect
198. #1279C2×C53semidirect
199. #1278C2×C53semidirect
209. #498C4×C53semidirect
210. #492C4×C53semidirect
211. #487C4×C53semidirect
212. #486C4×C53semidirect
229. C53#9055direct (N × H)
296. M16SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#31672
C53×#29352
C53×#19862
C53×#15022
C53×#14812
C53×#12792
C53×#12782
#905553