Finite Group Databasebeta

← C53×#10181 #27217693

Subgroups — C53×#10181 #27217693

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3206C211yesmaximal, characteristic
2C2×C53×#477C211yesmaximal, characteristic, complemented
3C53×#3037C211yesmaximal, characteristic
4C53×#1549C211yesmaximal, characteristic, complemented
5C53×#1480C211yesmaximal, characteristic
6C53×#1461C211yesmaximal, characteristic, complemented
7C53×#1301C211yesmaximal, characteristic, complemented
8C2×#137×C53—21no—
9C53×#532C2211yescharacteristic
10C2×C4⋊Q8×C53C2211yescharacteristic
11C2×C4×Q8×C53C2211yescharacteristic
12C53×#492—21no—
13C53×#487—21no—
14C53×#477—22no—
15C53×#477—22no—
16C53×#464—22no—
17C53×#462—21no—
18C53×#461—22no—
19C53×#460—21no—
20C53×#448—22no—
21C53×#431C2211yescharacteristic
22C53×#431C2211yescharacteristic
23C2×#107×C53—21no—
24C53×#423C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#417—21no—
27C2×#103×C53—21no—
28C53×#332—22no—
29C53×D4⋊C8—22no—
30C53×#324—22no—
31C2×#41×C53—21no—
32C22×Q8×C53—21no—
33C22×Q8×C53D411yescharacteristic
34#137×C53—42no—
35#137×C53—42no—
36C2×Q16×C53—22no—
37C2×SD16×C53—22no—
38C2×SD16×C53—24no—
39C2×M16×C53—21no—
40C22×C8×C53—21no—
41C4⋊Q8×C53—21no—
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53D411yescharacteristic
45C4⋊Q8×C53D412yes—
46#127×C53D411yescharacteristic
47#124×C53—21no—
48#121×C53—21no—
49#121×C53—21no—
50C4×Q8×C53—21no—
51C4×Q8×C53—21no—
52C4×Q8×C53—21no—
53C4×Q8×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62#107×C53—24no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53D412yes—
68C4⋊C8×C53D412yes—
69#103×C53—24no—
70#103×C53—24no—
71#103×C53—22no—
72#102×C53—22no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53D412yes—
75C4×C8×C53—21no—
76C4×C8×C53—21no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53—41no—
89C2×Q8×C53—82no—
90C2×Q8×C53SD1612yescomplemented
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102Q16×C53—44no—
103SD16×C53—44no—
104SD16×C53—84no—
105M16×C53—41no—
106M16×C53—41no—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C42×C53—41no—
126C42×C53—41no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C23×C53—41no—
132C23×C53#12111yescharacteristic
133Q8×C53—22no—
134Q8×C53—22no—
135Q8×C53—44no—
136Q8×C53—41no—
137Q8×C53—41no—
138Q8×C53—41no—
139Q8×C53—41no—
140Q8×C53—82no—
141D4×C53—42no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—42no—
145D4×C53—81no—
146D4×C53—81no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—82no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12711yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53#13711yescharacteristic
169C8×C53—42no—
170C8×C53—42no—
171C8×C53—42no—
172C8×C53—81no—
173#10181C5311yesmaximal, characteristic, direct product factor, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53#39211yescharacteristic
180C22×C53#46511yescharacteristic
181C22×C53#46811yescharacteristic
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—42no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53—82no—
189C4×C53#44811yescharacteristic
190C4×C53#44911yescharacteristic
191C4×C53#45011yescharacteristic
192C4×C53#45211yescharacteristic
193C4×C53#47312yes—
194#3206C2×C5311yescharacteristic
195C2×#477C2×C5311yescharacteristic, complemented
196#3037C2×C5311yescharacteristic
197#1549C2×C5311yescharacteristic, complemented
198#1480C2×C5311yescharacteristic
199#1461C2×C5311yescharacteristic, complemented
200#1301C2×C5311yescharacteristic, complemented
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—161no—
204C2×C53#200011yescharacteristic
205C2×C53#200811yescharacteristic
206C2×C53#202011yescharacteristic
207C2×#137—21no—
208#532C22×C5311yescharacteristic
209C2×C4⋊Q8C22×C5311yescharacteristic
210C2×C4×Q8C22×C5311yescharacteristic
211#492—21no—
212#487—21no—
213#477—22no—
214#477—22no—
215#464—22no—
216#462—21no—
217#461—22no—
218#460—21no—
219#448—22no—
220#431C22×C5311yescharacteristic
221#431C22×C5311yescharacteristic
222C2×#107—21no—
223#423C22×C5311yescharacteristic
224C2×C4⋊C8C22×C5311yescharacteristic
225#417—21no—
226C2×#103—21no—
227#332—22no—
228D4⋊C8—22no—
229#324—22no—
230C53#1018111yescharacteristic, direct product factor, Sylow
231C2×#41—21no—
232C22×Q8—21no—
233C22×Q8D4×C5311yescharacteristic
234#137—42no—
235#137—42no—
236C2×Q16—22no—
237C2×SD16—22no—
238C2×SD16—24no—
239C2×M16—21no—
240C22×C8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8D4×C5312yes—
245C4⋊Q8D4×C5311yescharacteristic
246#127D4×C5311yescharacteristic
247#124—21no—
248#121—21no—
249#121—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×D4—21no—
255C4×D4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4D4×C5311yescharacteristic
260C2×C42—21no—
261C2×C42C23×C5311yescharacteristic
262#107—24no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8D4×C5312yes—
268C4⋊C8D4×C5312yes—
269#103—24no—
270#103—22no—
271#103—24no—
272#102—22no—
273C22⋊C8—21no—
274C22⋊C8D4×C5312yes—
275C4×C8—21no—
276C4×C8—21no—
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—82no—
290C2×Q8SD16×C5312yescomplemented
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C2×D4×C5311yescharacteristic
300C22×C4C2×D4×C5311yescharacteristic
301C22×C4#41×C5311yescharacteristic
302Q16—44no—
303SD16—44no—
304SD16—84no—
305M16—41no—
306M16—41no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—41no—
326C42—41no—
327C42C2×D4×C5311yescharacteristic
328C42C2×D4×C5311yescharacteristic
329C42C2×D4×C5311yescharacteristic
330C42C2×D4×C5311yescharacteristic
331C23—41no—
332C23#121×C5311yescharacteristic
333Q8—22no—
334Q8—22no—
335Q8—44no—
336Q8—41no—
337Q8—41no—
338Q8—41no—
339Q8—41no—
340Q8—82no—
341D4—42no—
342D4—41no—
343D4—41no—
344D4—42no—
345D4—81no—
346D4—81no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#120×C5311yescharacteristic
360C2×C4#120×C5311yescharacteristic
361C2×C4#120×C5311yescharacteristic
362C2×C4#121×C5311yescharacteristic
363C2×C4#121×C5311yescharacteristic
364C2×C4#127×C5311yescharacteristic
365C2×C4C2×SD16×C5311yescharacteristic
366C2×C4#135×C5311yescharacteristic
367C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
368C2×C4#137×C5311yescharacteristic
369C8—42no—
370C8—42no—
371C8—42no—
372C8—81no—
373C22—21no—
374C22—21no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C53×#39211yescharacteristic
379C22C53×#46511yescharacteristic
380C22C53×#46811yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4C53×#44811yescharacteristic
389C4C53×#44911yescharacteristic
390C4C53×#45011yescharacteristic
391C4C53×#45211yescharacteristic
392C4C53×#47312yes—
393C2—21no—
394C2—21no—
395C2—161no—
396C2C53×#200011yescharacteristic
397C2C53×#200811yescharacteristic
398C2C53×#202011yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C53×#477C2semidirect
4. C53×#1549C2semidirect
6. C53×#1461C2semidirect
7. C53×#1301C2semidirect
90. C2×Q8×C53SD16semidirect
173. #10181C53direct (N × H)
195. C2×#477C2×C53semidirect
197. #1549C2×C53semidirect
199. #1461C2×C53semidirect
200. #1301C2×C53semidirect
230. C53#10181direct (N × H)
290. C2×Q8SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32062
C2×C53×#4772
C53×#30372
C53×#15492
C53×#14802
C53×#14612
C53×#13012
#1018153