Finite Group Databasebeta

← C53×#19403 #27226915

Subgroups — C53×#19403 #27226915

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3227C211yesmaximal, characteristic
2C2×C53×#477C211yesmaximal, characteristic, complemented
3C53×#3037C211yesmaximal, characteristic
4C53×#2982C211yesmaximal, characteristic, complemented
5C53×#1678C214yesmaximal, complemented
6C53×#1672C214yesmaximal, complemented
7C53×#1570C211yesmaximal, characteristic, complemented
8C53×#1548C211yesmaximal, characteristic, complemented
9C53×#1540C211yesmaximal, characteristic
10C2×#137×C53—21no—
11C53×#532C2211yescharacteristic
12C2×C4⋊Q8×C53C2211yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#498C2212yes—
15C53×#486C2212yes—
16C53×#481—22no—
17C53×#477—22no—
18C53×#477C2214yes—
19C53×#476—22no—
20C53×#464—22no—
21C53×#462C2212yes—
22C53×#460C2212yes—
23C8×Q8×C53—22no—
24C8×Q8×C53C2214yes—
25C53×#431—22no—
26C53×#431C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C2×#107×C53—21no—
29C53×#423C2211yescharacteristic
30C2×C4⋊C8×C53C2211yescharacteristic
31C53×#417—21no—
32C2×#103×C53—21no—
33C53×#334C2214yes—
34C53×#333C2214yes—
35C53×#332C2212yes—
36C53×#331C2212yes—
37C2×#41×C53—21no—
38C22×Q8×C53—21no—
39C22×Q8×C53D411yescharacteristic
40#137×C53—44no—
41C2×Q16×C53—22no—
42C2×SD16×C53—22no—
43C2×M16×C53—21no—
44C22×C8×C53—21no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53—21no—
47C4⋊Q8×C53—21no—
48C4⋊Q8×C53C2312yes—
49C4⋊Q8×C53C2311yescharacteristic
50#127×C53C2311yescharacteristic
51#124×C53—21no—
52#121×C53—21no—
53#121×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53D412yes—
57C4×Q8×C53C2312yes—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66#107×C53—24no—
67#106×C53—24no—
68C4⋊C8×C53—24no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D412yes—
72C4⋊C8×C53C2312yes—
73C4⋊C8×C53C2312yes—
74C4⋊C8×C53C2312yes—
75#103×C53—22no—
76#103×C53—24no—
77#102×C53—24no—
78#102×C53—22no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53D412yes—
81C4×C8×C53—24no—
82C4×C8×C53—21no—
83C4×C8×C53C2312yes—
84#41×C53—42no—
85#41×C53—42no—
86C2×Q8×C53—21no—
87C2×Q8×C53—22no—
88C2×Q8×C53—21no—
89C2×Q8×C53—22no—
90C2×Q8×C53—41no—
91C2×Q8×C53—82no—
92C2×Q8×C53SD1614yes—
93C2×Q8×C53C2×D412yes—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—41no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105Q16×C53—44no—
106SD16×C53—44no—
107M16×C53—42no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—42no—
112C2×C8×C53C2×D414yes—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53C2×D412yes—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C42×C53—22no—
126C42×C53—22no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53#4111yescharacteristic
130C42×C53C2411yescharacteristic
131C23×C53—41no—
132C23×C53#12111yescharacteristic
133Q8×C53—22no—
134Q8×C53—24no—
135Q8×C53—41no—
136Q8×C53—41no—
137Q8×C53—82no—
138Q8×C53—82no—
139Q8×C53C2×SD1614yes—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—42no—
143D4×C53—81no—
144D4×C53—82no—
145C2×C4×C53—22no—
146C2×C4×C53—24no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—81no—
153C2×C4×C53—82no—
154C2×C4×C53—82no—
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53C2×SD1612yes—
159C2×C4×C53#13512yes—
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C2×#4111yescharacteristic
163C8×C53—24no—
164C8×C53—42no—
165C8×C53—42no—
166C8×C53—42no—
167#19403C5311yesmaximal, characteristic, direct product factor, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53#46512yes—
174C22×C53C2×#12111yescharacteristic
175C4×C53—21no—
176C4×C53—24no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—42no—
180C4×C53—81no—
181C4×C53—82no—
182C4×C53#46512yes—
183C4×C53C22×SD1611yescharacteristic
184C4×C53C2×#13511yescharacteristic
185C4×C53#57812yes—
186#3227C2×C5311yescharacteristic
187C2×#477C2×C5311yescharacteristic, complemented
188#3037C2×C5311yescharacteristic
189#2982C2×C5311yescharacteristic, complemented
190#1678C2×C5314yescomplemented
191#1672C2×C5314yescomplemented
192#1570C2×C5311yescharacteristic, complemented
193#1548C2×C5311yescharacteristic, complemented
194#1540C2×C5311yescharacteristic
195C2×C53—22no—
196C2×C53—161no—
197C2×C53#304411yescharacteristic
198C2×C53C2×#46511yescharacteristic
199C2×C53#306811yescharacteristic
200C2×#137—21no—
201#532C22×C5311yescharacteristic
202C2×C4⋊Q8C22×C5311yescharacteristic
203C2×C4×Q8C22×C5311yescharacteristic
204#498C22×C5312yes—
205#486C22×C5312yes—
206#481—22no—
207#477—22no—
208#477C22×C5314yes—
209#476—22no—
210#464—22no—
211#462C22×C5312yes—
212#460C22×C5312yes—
213C8×Q8—22no—
214C8×Q8C22×C5314yes—
215#431—22no—
216#431C22×C5311yescharacteristic
217#431C22×C5311yescharacteristic
218C2×#107—21no—
219#423C22×C5311yescharacteristic
220C2×C4⋊C8C22×C5311yescharacteristic
221#417—21no—
222C2×#103—21no—
223#334C22×C5314yes—
224#333C22×C5314yes—
225#332C22×C5312yes—
226#331C22×C5312yes—
227C53#1940311yescharacteristic, direct product factor, Sylow
228C2×#41—21no—
229C22×Q8—21no—
230C22×Q8D4×C5311yescharacteristic
231#137—44no—
232C2×Q16—22no—
233C2×SD16—22no—
234C2×M16—21no—
235C22×C8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8C23×C5312yes—
240C4⋊Q8C23×C5311yescharacteristic
241#127C23×C5311yescharacteristic
242#124—21no—
243#121—21no—
244#121—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8D4×C5312yes—
248C4×Q8C23×C5312yes—
249C4×D4—21no—
250C4×D4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4D4×C5311yescharacteristic
255C2×C42—21no—
256C2×C42C23×C5311yescharacteristic
257#107—24no—
258#106—24no—
259C4⋊C8—24no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8D4×C5312yes—
263C4⋊C8C23×C5312yes—
264C4⋊C8C23×C5312yes—
265C4⋊C8C23×C5312yes—
266#103—24no—
267#103—22no—
268#102—24no—
269#102—22no—
270C22⋊C8—21no—
271C22⋊C8D4×C5312yes—
272C4×C8—24no—
273C4×C8—21no—
274C4×C8C23×C5312yes—
275#41—42no—
276#41—42no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8SD16×C5314yes—
284C2×Q8C2×D4×C5312yes—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4C2×D4×C5311yescharacteristic
294C22×C4C2×D4×C5311yescharacteristic
295C22×C4#41×C5311yescharacteristic
296Q16—44no—
297SD16—44no—
298M16—42no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8C2×D4×C5314yes—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4C2×D4×C5312yes—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—22no—
317C42—22no—
318C42C2×D4×C5311yescharacteristic
319C42C2×D4×C5311yescharacteristic
320C42#41×C5311yescharacteristic
321C42C24×C5311yescharacteristic
322C23—41no—
323C23#121×C5311yescharacteristic
324Q8—22no—
325Q8—24no—
326Q8—41no—
327Q8—41no—
328Q8—82no—
329Q8—82no—
330Q8C2×SD16×C5314yes—
331D4—41no—
332D4—41no—
333D4—42no—
334D4—81no—
335D4—82no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#121×C5311yescharacteristic
347C2×C4#121×C5311yescharacteristic
348C2×C4#121×C5311yescharacteristic
349C2×C4C2×SD16×C5312yes—
350C2×C4#135×C5312yes—
351C2×C4C22×D4×C5311yescharacteristic
352C2×C4C22×D4×C5311yescharacteristic
353C2×C4C2×#41×C5311yescharacteristic
354C8—24no—
355C8—42no—
356C8—42no—
357C8—42no—
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22C53×#46512yes—
364C22C2×#121×C5311yescharacteristic
365C4—21no—
366C4—24no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—81no—
371C4—82no—
372C4C53×#46512yes—
373C4C22×SD16×C5311yescharacteristic
374C4C2×#135×C5311yescharacteristic
375C4C53×#57812yes—
376C2—22no—
377C2—161no—
378C2C53×#304411yescharacteristic
379C2C2×C53×#46511yescharacteristic
380C2C53×#306811yescharacteristic
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
2. C2×C53×#477C2semidirect
4. C53×#2982C2semidirect
5. C53×#1678C2semidirect
6. C53×#1672C2semidirect
7. C53×#1570C2semidirect
8. C53×#1548C2semidirect
167. #19403C53direct (N × H)
187. C2×#477C2×C53semidirect
189. #2982C2×C53semidirect
190. #1678C2×C53semidirect
191. #1672C2×C53semidirect
192. #1570C2×C53semidirect
193. #1548C2×C53semidirect
227. C53#19403direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32272
C2×C53×#4772
C53×#30372
C53×#29822
C53×#16782
C53×#16722
C53×#15702
C53×#15482
C53×#15402
#1940353