Finite Group Databasebeta

← C53×#10200 #27217712

Subgroups — C53×#10200 #27217712

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3245C211yesmaximal, characteristic
2C53×#3227C211yesmaximal, characteristic, complemented
3C53×#3141C211yesmaximal, characteristic, complemented
4C53×#1981C211yesmaximal, characteristic, complemented
5C53×#1980C211yesmaximal, characteristic, complemented
6C53×#1502C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C53×#532C2211yescharacteristic
9C53×#532C2211yescharacteristic
10C2×C4⋊Q8×C53C2211yescharacteristic
11C53×#498—21no—
12C53×#489—22no—
13C53×#487—21no—
14C53×#486—21no—
15C53×#486—21no—
16C53×#481—22no—
17C53×#476—22no—
18C53×#474—21no—
19C53×#474—21no—
20C53×#466—22no—
21C53×#459—21no—
22C53×#459—21no—
23C53×#431C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×#104×C53—22no—
26C53×#417—22no—
27C2×#103×C53—22no—
28C4×M16×C53C2211yescharacteristic
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#327—22no—
31C2×#41×C53—21no—
32C2×#41×C53—21no—
33C22×Q8×C53—21no—
34C2×D8×C53—22no—
35C2×M16×C53D412yes—
36C22×C8×C53D412yes—
37C4⋊Q8×C53—21no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53D411yescharacteristic
41C4⋊Q8×C53D412yes—
42C4⋊Q8×C53D411yescharacteristic
43#127×C53D411yescharacteristic
44#127×C53D411yescharacteristic
45#124×C53—21no—
46#124×C53—21no—
47#121×C53—21no—
48#121×C53—21no—
49#121×C53—21no—
50#121×C53—21no—
51C4×D4×C53—21no—
52C4×D4×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60#107×C53—22no—
61#106×C53—24no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65#104×C53—44no—
66#103×C53—24no—
67#103×C53—44no—
68#102×C53—24no—
69#102×C53—24no—
70#102×C53—22no—
71#102×C53—22no—
72C22⋊C8×C53D412yes—
73#97×C53—21no—
74C4×C8×C53—21no—
75C4×C8×C53—21no—
76#95×C53—21no—
77#41×C53—42no—
78#41×C53—42no—
79#41×C53—42no—
80#41×C53—42no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—22no—
84C2×Q8×C53—42no—
85C2×Q8×C53—42no—
86C2×Q8×C53—41no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102D8×C53—82no—
103M16×C53—22no—
104M16×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—42no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C42×C53—42no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C23×C53—41no—
129C23×C53—41no—
130C23×C53#12711yescharacteristic
131Q8×C53—44no—
132Q8×C53—42no—
133Q8×C53—42no—
134Q8×C53—81no—
135Q8×C53—81no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—42no—
143D4×C53—81no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53—81no—
147D4×C53—81no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—81no—
156C2×C4×C53—82no—
157C2×C4×C53—81no—
158C2×C4×C53—82no—
159C2×C4×C53—82no—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#13512yes—
167C2×C4×C53AGL(1,ℤ/8)12yes—
168C8×C53—42no—
169C8×C53—42no—
170C8×C53—42no—
171#10200C5311yesmaximal, characteristic, direct product factor, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53#39211yescharacteristic
181C22×C53#49511yescharacteristic
182C22×C53#49611yescharacteristic
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—82no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53#44912yes—
191C4×C53#45311yescharacteristic
192C4×C53#45411yescharacteristic
193C4×C53#46312yes—
194#3245C2×C5311yescharacteristic
195#3227C2×C5311yescharacteristic, complemented
196#3141C2×C5311yescharacteristic, complemented
197#1981C2×C5311yescharacteristic, complemented
198#1980C2×C5311yescharacteristic, complemented
199#1502C2×C5311yescharacteristic, complemented
200#1283C2×C5311yescharacteristic, complemented
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—161no—
204C2×C53—161no—
205C2×C53#200611yescharacteristic
206C2×C53#200911yescharacteristic
207C2×C53#201211yescharacteristic
208#532C22×C5311yescharacteristic
209#532C22×C5311yescharacteristic
210C2×C4⋊Q8C22×C5311yescharacteristic
211#498—21no—
212#489—22no—
213#487—21no—
214#486—21no—
215#486—21no—
216#481—22no—
217#476—22no—
218#474—21no—
219#474—21no—
220#466—22no—
221#459—21no—
222#459—21no—
223#431C22×C5311yescharacteristic
224C2×C4⋊C8C22×C5311yescharacteristic
225C2×#104—22no—
226#417—22no—
227C2×#103—22no—
228C4×M16C22×C5311yescharacteristic
229C4×C4⋊C4C22×C5311yescharacteristic
230#327—22no—
231C53#1020011yescharacteristic, direct product factor, Sylow
232C2×#41—21no—
233C2×#41—21no—
234C22×Q8—21no—
235C2×D8—22no—
236C2×M16D4×C5312yes—
237C22×C8D4×C5312yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D4×C5312yes—
242C4⋊Q8D4×C5311yescharacteristic
243C4⋊Q8D4×C5311yescharacteristic
244#127D4×C5311yescharacteristic
245#127D4×C5311yescharacteristic
246#124—21no—
247#124—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251#121—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C42—21no—
260C2×C42C23×C5311yescharacteristic
261#107—22no—
262#106—24no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266#104—44no—
267#103—24no—
268#103—44no—
269#102—24no—
270#102—22no—
271#102—22no—
272#102—24no—
273C22⋊C8D4×C5312yes—
274#97—21no—
275C4×C8—21no—
276C4×C8—21no—
277#95—21no—
278#41—42no—
279#41—42no—
280#41—42no—
281#41—42no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C2×D4×C5311yescharacteristic
301C22×C4C2×D4×C5311yescharacteristic
302C22×C4C2×D4×C5311yescharacteristic
303D8—82no—
304M16—22no—
305M16—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—42no—
325C42C2×D4×C5311yescharacteristic
326C42C2×D4×C5311yescharacteristic
327C42C2×D4×C5311yescharacteristic
328C42#41×C5311yescharacteristic
329C23—41no—
330C23—41no—
331C23#127×C5311yescharacteristic
332Q8—44no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337D4—42no—
338D4—42no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—42no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4#120×C5311yescharacteristic
362C2×C4#120×C5311yescharacteristic
363C2×C4#120×C5311yescharacteristic
364C2×C4#121×C5311yescharacteristic
365C2×C4#121×C5311yescharacteristic
366C2×C4#121×C5311yescharacteristic
367C2×C4#135×C5312yes—
368C2×C4AGL(1,ℤ/8)×C5312yes—
369C8—42no—
370C8—42no—
371C8—42no—
372C22—21no—
373C22—21no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C53×#39211yescharacteristic
381C22C53×#49511yescharacteristic
382C22C53×#49611yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4C53×#44912yes—
391C4C53×#45311yescharacteristic
392C4C53×#45411yescharacteristic
393C4C53×#46312yes—
394C2—21no—
395C2—21no—
396C2—161no—
397C2—161no—
398C2C53×#200611yescharacteristic
399C2C53×#200911yescharacteristic
400C2C53×#201211yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3245C2
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. C53×#3227C2semidirect
3. C53×#3141C2semidirect
4. C53×#1981C2semidirect
5. C53×#1980C2semidirect
6. C53×#1502C2semidirect
7. C53×#1283C2semidirect
171. #10200C53direct (N × H)
195. #3227C2×C53semidirect
196. #3141C2×C53semidirect
197. #1981C2×C53semidirect
198. #1980C2×C53semidirect
199. #1502C2×C53semidirect
200. #1283C2×C53semidirect
231. C53#10200direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32452
C53×#32272
C53×#31412
C53×#19812
C53×#19802
C53×#15022
C53×#12832
#1020053