Finite Group Databasebeta

← C53×#10196 #27217708

Subgroups — C53×#10196 #27217708

Order 13568.

546 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×#3245C211yesmaximal, characteristic
2C53×#3206C211yesmaximal, characteristic, complemented
3C53×#3141C211yesmaximal, characteristic, complemented
4C53×#1987C211yesmaximal, characteristic, complemented
5C53×#1980C211yesmaximal, characteristic, complemented
6C53×#1502C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C53×#532C2211yescharacteristic
9C53×#532C2211yescharacteristic
10C2×C4⋊Q8×C53C2211yescharacteristic
11C53×#492—21no—
12C53×#489—22no—
13C53×#487—21no—
14C53×#487—21no—
15C53×#486—21no—
16C53×#474—21no—
17C53×#474—21no—
18C53×#466—22no—
19C53×#461—22no—
20C53×#459—21no—
21C53×#459—21no—
22C53×#448—22no—
23C53×#431C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×#104×C53—22no—
26C53×#417—22no—
27C4×M16×C53C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C53×#340—22no—
30C53×#327—22no—
31C2×#41×C53—21no—
32C2×#41×C53—21no—
33C22×Q8×C53—21no—
34C2×SD16×C53—24no—
35C2×D8×C53—22no—
36C2×M16×C53D412yes—
37C22×C8×C53D412yes—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53—21no—
41C4⋊Q8×C53D411yescharacteristic
42C4⋊Q8×C53D411yescharacteristic
43C4⋊Q8×C53D412yes—
44#127×C53D411yescharacteristic
45#127×C53D411yescharacteristic
46#124×C53—21no—
47#124×C53—21no—
48#121×C53—21no—
49#121×C53—21no—
50#121×C53—21no—
51#121×C53—21no—
52C4×D4×C53—21no—
53C4×D4×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×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61#107×C53—22no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65#104×C53—44no—
66#103×C53—24no—
67#103×C53—24no—
68#102×C53—24no—
69#102×C53—22no—
70#102×C53—22no—
71C22⋊C8×C53D412yes—
72#97×C53—21no—
73C4×C8×C53—21no—
74C4×C8×C53—21no—
75#95×C53—21no—
76#41×C53—42no—
77#41×C53—42no—
78#41×C53—42no—
79#41×C53—42no—
80C2×Q8×C53—21no—
81C2×Q8×C53—22no—
82C2×Q8×C53—21no—
83C2×Q8×C53—41no—
84C2×Q8×C53—82no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100SD16×C53—84no—
101D8×C53—82no—
102M16×C53—22no—
103M16×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—42no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53—42no—
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C23×C53—41no—
128C23×C53—41no—
129C23×C53#12711yescharacteristic
130Q8×C53—44no—
131Q8×C53—42no—
132Q8×C53—42no—
133Q8×C53—82no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—41no—
137D4×C53—41no—
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—42no—
141D4×C53—81no—
142D4×C53—81no—
143D4×C53—81no—
144D4×C53—81no—
145D4×C53—81no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—81no—
154C2×C4×C53—82no—
155C2×C4×C53—81no—
156C2×C4×C53—82no—
157C2×C4×C53—82no—
158C2×C4×C53#12011yescharacteristic
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#13512yes—
165C2×C4×C53AGL(1,ℤ/8)12yes—
166C8×C53—42no—
167C8×C53—42no—
168C8×C53—42no—
169#10196C5311yesmaximal, characteristic, direct product factor, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53#39211yescharacteristic
179C22×C53#49511yescharacteristic
180C22×C53#49611yescharacteristic
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—82no—
185C4×C53—82no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53#44912yes—
189C4×C53#45511yescharacteristic
190C4×C53#45611yescharacteristic
191C4×C53#46312yes—
192#3245C2×C5311yescharacteristic
193#3206C2×C5311yescharacteristic, complemented
194#3141C2×C5311yescharacteristic, complemented
195#1987C2×C5311yescharacteristic, complemented
196#1980C2×C5311yescharacteristic, complemented
197#1502C2×C5311yescharacteristic, complemented
198#1283C2×C5311yescharacteristic, complemented
199C2×C53—21no—
200C2×C53—21no—
201C2×C53—161no—
202C2×C53—161no—
203C2×C53#200711yescharacteristic
204C2×C53#200811yescharacteristic
205C2×C53#201211yescharacteristic
206#532C22×C5311yescharacteristic
207#532C22×C5311yescharacteristic
208C2×C4⋊Q8C22×C5311yescharacteristic
209#492—21no—
210#489—22no—
211#487—21no—
212#487—21no—
213#486—21no—
214#474—21no—
215#474—21no—
216#466—22no—
217#461—22no—
218#459—21no—
219#459—21no—
220#448—22no—
221#431C22×C5311yescharacteristic
222C2×C4⋊C8C22×C5311yescharacteristic
223C2×#104—22no—
224#417—22no—
225C4×M16C22×C5311yescharacteristic
226C4×C4⋊C4C22×C5311yescharacteristic
227#340—22no—
228#327—22no—
229C53#1019611yescharacteristic, direct product factor, Sylow
230C2×#41—21no—
231C2×#41—21no—
232C22×Q8—21no—
233C2×SD16—24no—
234C2×D8—22no—
235C2×M16D4×C5312yes—
236C22×C8D4×C5312yes—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8D4×C5312yes—
241C4⋊Q8D4×C5311yescharacteristic
242C4⋊Q8D4×C5311yescharacteristic
243#127D4×C5311yescharacteristic
244#127D4×C5311yescharacteristic
245#124—21no—
246#124—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C42—21no—
259C2×C42C23×C5311yescharacteristic
260#107—22no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264#104—44no—
265#103—24no—
266#103—24no—
267#102—24no—
268#102—22no—
269#102—22no—
270C22⋊C8D4×C5312yes—
271#97—21no—
272C4×C8—21no—
273C4×C8—21no—
274#95—21no—
275#41—42no—
276#41—42no—
277#41—42no—
278#41—42no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—41no—
283C2×Q8—82no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C2×D4×C5311yescharacteristic
297C22×C4C2×D4×C5311yescharacteristic
298C22×C4C2×D4×C5311yescharacteristic
299SD16—84no—
300D8—82no—
301M16—22no—
302M16—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—42no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—42no—
322C42C2×D4×C5311yescharacteristic
323C42C2×D4×C5311yescharacteristic
324C42C2×D4×C5311yescharacteristic
325C42#41×C5311yescharacteristic
326C23—41no—
327C23—41no—
328C23#127×C5311yescharacteristic
329Q8—44no—
330Q8—42no—
331Q8—42no—
332Q8—82no—
333D4—42no—
334D4—42no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#120×C5311yescharacteristic
358C2×C4#120×C5311yescharacteristic
359C2×C4#120×C5311yescharacteristic
360C2×C4#121×C5311yescharacteristic
361C2×C4#121×C5311yescharacteristic
362C2×C4#121×C5311yescharacteristic
363C2×C4#135×C5312yes—
364C2×C4AGL(1,ℤ/8)×C5312yes—
365C8—42no—
366C8—42no—
367C8—42no—
368C22—21no—
369C22—21no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22C53×#39211yescharacteristic
377C22C53×#49511yescharacteristic
378C22C53×#49611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—82no—
384C4—81no—
385C4—81no—
386C4C53×#44912yes—
387C4C53×#45511yescharacteristic
388C4C53×#45611yescharacteristic
389C4C53×#46312yes—
390C2—21no—
391C2—21no—
392C2—161no—
393C2—161no—
394C2C53×#200711yescharacteristic
395C2C53×#200811yescharacteristic
396C2C53×#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×#3206C2semidirect
3. C53×#3141C2semidirect
4. C53×#1987C2semidirect
5. C53×#1980C2semidirect
6. C53×#1502C2semidirect
7. C53×#1283C2semidirect
169. #10196C53direct (N × H)
193. #3206C2×C53semidirect
194. #3141C2×C53semidirect
195. #1987C2×C53semidirect
196. #1980C2×C53semidirect
197. #1502C2×C53semidirect
198. #1283C2×C53semidirect
229. C53#10196direct (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×#32062
C53×#31412
C53×#19872
C53×#19802
C53×#15022
C53×#12832
#1019653