Finite Group Databasebeta

← C53×#19404 #27226916

Subgroups — C53×#19404 #27226916

Order 13568.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3226C211yesmaximal, characteristic
2C53×#3037C211yesmaximal, characteristic
3C2×C53×#462C211yesmaximal, characteristic, complemented
4C53×#2982C211yesmaximal, characteristic, complemented
5C53×#1682C214yesmaximal, complemented
6C53×#1674C214yesmaximal, complemented
7C53×#1571C211yesmaximal, characteristic, complemented
8C53×#1546C211yesmaximal, characteristic, complemented
9C53×#1540C211yesmaximal, characteristic
10C2×#137×C53—21no—
11C22×Q16×C53—21no—
12C53×#532C2211yescharacteristic
13C2×C4⋊Q8×C53C2211yescharacteristic
14C2×C4×Q8×C53C2211yescharacteristic
15C53×#500C2212yes—
16C53×#486C2212yes—
17C53×#480—22no—
18C53×#474—22no—
19C53×#464—22no—
20C53×#462—22no—
21C53×#462C2212yes—
22C53×#462C2214yes—
23C53×#460C2212yes—
24C8×Q8×C53—22no—
25C8×Q8×C53C2214yes—
26C53×#431—22no—
27C53×#431C2211yescharacteristic
28C53×#431C2211yescharacteristic
29C53×#423C2211yescharacteristic
30C2×C4⋊C8×C53C2211yescharacteristic
31C53×#417—21no—
32C2×#103×C53—21no—
33C53×#335C2214yes—
34C53×#332C2212yes—
35C53×#332C2214yes—
36C53×#331C2212yes—
37C2×#41×C53—21no—
38C22×Q8×C53—21no—
39C22×Q8×C53D411yescharacteristic
40#137×C53—44no—
41C2×Q16×C53—24no—
42C2×Q16×C53—22no—
43C2×Q16×C53—44no—
44C2×SD16×C53—22no—
45C2×M16×C53—21no—
46C22×C8×C53—21no—
47C4⋊Q8×C53—21no—
48C4⋊Q8×C53—21no—
49C4⋊Q8×C53—21no—
50C4⋊Q8×C53C2312yes—
51C4⋊Q8×C53C2311yescharacteristic
52#127×C53C2311yescharacteristic
53#124×C53—21no—
54#121×C53—21no—
55#121×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×Q8×C53D412yes—
59C4×Q8×C53C2312yes—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68#107×C53—24no—
69C4⋊C8×C53—24no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yes—
73C4⋊C8×C53C2312yes—
74C4⋊C8×C53C2312yes—
75C4⋊C8×C53C2312yes—
76#103×C53—24no—
77#103×C53—22no—
78#102×C53—24no—
79#102×C53—22no—
80C22⋊C8×C53—21no—
81C22⋊C8×C53D412yes—
82C4×C8×C53—24no—
83C4×C8×C53—21no—
84C4×C8×C53C2312yes—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×Q8×C53—22no—
89C2×Q8×C53—21no—
90C2×Q8×C53—22no—
91C2×Q8×C53—41no—
92C2×Q8×C53—42no—
93C2×Q8×C53—42no—
94C2×Q8×C53D814yes—
95C2×Q8×C53C2×D412yes—
96C2×D4×C53—21no—
97C2×D4×C53—21no—
98C2×D4×C53—21no—
99C2×D4×C53—41no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107Q16×C53—48no—
108Q16×C53—44no—
109SD16×C53—44no—
110M16×C53—42no—
111C2×C8×C53—22no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53—42no—
115C2×C8×C53C2×D414yes—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53C2×D412yes—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C42×C53—22no—
129C42×C53—22no—
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53#4111yescharacteristic
133C42×C53C2411yescharacteristic
134C23×C53—41no—
135C23×C53#12111yescharacteristic
136Q8×C53—22no—
137Q8×C53—24no—
138Q8×C53—42no—
139Q8×C53—41no—
140Q8×C53—42no—
141Q8×C53—41no—
142Q8×C53—82no—
143Q8×C53C2×D814yes—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—42no—
147D4×C53—81no—
148D4×C53—82no—
149C2×C4×C53—22no—
150C2×C4×C53—24no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—81no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53C2×D812yes—
163C2×C4×C53#13512yes—
164C2×C4×C53C22×D411yescharacteristic
165C2×C4×C53C22×D411yescharacteristic
166C2×C4×C53C2×#4111yescharacteristic
167C8×C53—24no—
168C8×C53—42no—
169C8×C53—42no—
170C8×C53—42no—
171#19404C5311yesmaximal, characteristic, direct product factor, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53#46612yes—
178C22×C53C2×#12111yescharacteristic
179C4×C53—21no—
180C4×C53—24no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53—81no—
185C4×C53—82no—
186C4×C53#46612yes—
187C4×C53C22×D811yescharacteristic
188C4×C53C2×#13511yescharacteristic
189C4×C53#57812yes—
190#3226C2×C5311yescharacteristic
191#3037C2×C5311yescharacteristic
192C2×#462C2×C5311yescharacteristic, complemented
193#2982C2×C5311yescharacteristic, complemented
194#1682C2×C5314yescomplemented
195#1674C2×C5314yescomplemented
196#1571C2×C5311yescharacteristic, complemented
197#1546C2×C5311yescharacteristic, complemented
198#1540C2×C5311yescharacteristic
199C2×C53—22no—
200C2×C53—161no—
201C2×C53#304411yescharacteristic
202C2×C53C2×#46611yescharacteristic
203C2×C53#306911yescharacteristic
204C2×#137—21no—
205C22×Q16—21no—
206#532C22×C5311yescharacteristic
207C2×C4⋊Q8C22×C5311yescharacteristic
208C2×C4×Q8C22×C5311yescharacteristic
209#500C22×C5312yes—
210#486C22×C5312yes—
211#480—22no—
212#474—22no—
213#464—22no—
214#462—22no—
215#462C22×C5312yes—
216#462C22×C5314yes—
217#460C22×C5312yes—
218C8×Q8—22no—
219C8×Q8C22×C5314yes—
220#431—22no—
221#431C22×C5311yescharacteristic
222#431C22×C5311yescharacteristic
223#423C22×C5311yescharacteristic
224C2×C4⋊C8C22×C5311yescharacteristic
225#417—21no—
226C2×#103—21no—
227#335C22×C5314yes—
228#332C22×C5312yes—
229#332C22×C5314yes—
230#331C22×C5312yes—
231C53#1940411yescharacteristic, direct product factor, Sylow
232C2×#41—21no—
233C22×Q8—21no—
234C22×Q8D4×C5311yescharacteristic
235#137—44no—
236C2×Q16—24no—
237C2×Q16—22no—
238C2×Q16—44no—
239C2×SD16—22no—
240C2×M16—21no—
241C22×C8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8C23×C5312yes—
246C4⋊Q8C23×C5311yescharacteristic
247#127C23×C5311yescharacteristic
248#124—21no—
249#121—21no—
250#121—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8D4×C5312yes—
254C4×Q8C23×C5312yes—
255C4×D4—21no—
256C4×D4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4D4×C5311yescharacteristic
261C2×C42—21no—
262C2×C42C23×C5311yescharacteristic
263#107—24no—
264C4⋊C8—24no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8D4×C5312yes—
268C4⋊C8C23×C5312yes—
269C4⋊C8C23×C5312yes—
270C4⋊C8C23×C5312yes—
271#103—24no—
272#103—22no—
273#102—24no—
274#102—22no—
275C22⋊C8—21no—
276C22⋊C8D4×C5312yes—
277C4×C8—24no—
278C4×C8—21no—
279C4×C8C23×C5312yes—
280#41—42no—
281#41—42no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8D8×C5314yes—
290C2×Q8C2×D4×C5312yes—
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—48no—
303Q16—44no—
304SD16—44no—
305M16—42no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C2×C8C2×D4×C5314yes—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4C2×D4×C5312yes—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—22no—
324C42—22no—
325C42C2×D4×C5311yescharacteristic
326C42C2×D4×C5311yescharacteristic
327C42#41×C5311yescharacteristic
328C42C24×C5311yescharacteristic
329C23—41no—
330C23#121×C5311yescharacteristic
331Q8—22no—
332Q8—24no—
333Q8—42no—
334Q8—41no—
335Q8—42no—
336Q8—41no—
337Q8—82no—
338Q8C2×D8×C5314yes—
339D4—41no—
340D4—41no—
341D4—42no—
342D4—81no—
343D4—82no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4#121×C5311yescharacteristic
355C2×C4#121×C5311yescharacteristic
356C2×C4#121×C5311yescharacteristic
357C2×C4C2×D8×C5312yes—
358C2×C4#135×C5312yes—
359C2×C4C22×D4×C5311yescharacteristic
360C2×C4C22×D4×C5311yescharacteristic
361C2×C4C2×#41×C5311yescharacteristic
362C8—24no—
363C8—42no—
364C8—42no—
365C8—42no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C53×#46612yes—
372C22C2×#121×C5311yescharacteristic
373C4—21no—
374C4—24no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—81no—
379C4—82no—
380C4C53×#46612yes—
381C4C22×D8×C5311yescharacteristic
382C4C2×#135×C5311yescharacteristic
383C4C53×#57812yes—
384C2—22no—
385C2—161no—
386C2C53×#304411yescharacteristic
387C2C2×C53×#46611yescharacteristic
388C2C53×#306911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#121
commutator, Frattini, cosocleC42C24×C53
prime radicalC53×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C53×#462C2semidirect
4. C53×#2982C2semidirect
5. C53×#1682C2semidirect
6. C53×#1674C2semidirect
7. C53×#1571C2semidirect
8. C53×#1546C2semidirect
171. #19404C53direct (N × H)
192. C2×#462C2×C53semidirect
193. #2982C2×C53semidirect
194. #1682C2×C53semidirect
195. #1674C2×C53semidirect
196. #1571C2×C53semidirect
197. #1546C2×C53semidirect
231. C53#19404direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32262
C53×#30372
C2×C53×#4622
C53×#29822
C53×#16822
C53×#16742
C53×#15712
C53×#15462
C53×#15402
#1940453