Finite Group Databasebeta

← C53×#19423 #27226935

Subgroups — C53×#19423 #27226935

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3209C211yesmaximal, characteristic, complemented
2C53×#3077C211yesmaximal, characteristic
3C2×C53×#474C211yesmaximal, characteristic
4C53×#2975C211yesmaximal, characteristic
5C53×#1703C214yesmaximal, complemented
6C53×#1689C214yesmaximal, complemented
7C53×#1570C211yesmaximal, characteristic, complemented
8C53×#1549C211yesmaximal, characteristic, complemented
9C53×#1542C211yesmaximal, characteristic, complemented
10C53×#533C2211yescharacteristic
11C2×C4⋊Q8×C53C2211yescharacteristic
12C2×C4×D4×C53C2211yescharacteristic
13C53×#494C2212yescomplemented
14C53×#487C2212yescomplemented
15C53×#478—22no—
16C53×#476C2212yes—
17C53×#474—22no—
18C53×#474C2212yes—
19C53×#474C2214yes—
20C53×#462—22no—
21C53×#451—22no—
22C53×#436—22no—
23C53×#436C2214yes—
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C53×#428—21no—
27C2×#107×C53—21no—
28C53×#423C2211yescharacteristic
29C2×C4⋊C8×C53C2211yescharacteristic
30C53×#418—21no—
31C2×#102×C53—21no—
32C53×#407—22no—
33C53×#334C2214yescomplemented
34C53×#333C2212yescomplemented
35C53×#332C2212yescomplemented
36C53×#332C2214yescomplemented
37C22×Q8×C53—21no—
38C22×D4×C53D411yescharacteristic
39C23×C4×C53—21no—
40C2×Q16×C53—24no—
41C2×M16×C53—24no—
42C2×M16×C53—21no—
43C22×C8×C53—21no—
44C4⋊Q8×C53—21no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53—21no—
47C4⋊Q8×C53C2311yescharacteristic
48C4⋊Q8×C53C2312yes—
49C4⋊Q8×C53C2311yescharacteristic
50#125×C53—21no—
51C22⋊Q8×C53—21no—
52C22⋊Q8×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53D412yes—
58C4×D4×C53C2312yes—
59#117×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65#107×C53—24no—
66#107×C53—22no—
67#106×C53—22no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53D412yes—
71C4⋊C8×C53C2312yes—
72C4⋊C8×C53C2312yes—
73C4⋊C8×C53C2312yes—
74#103×C53—24no—
75#102×C53—24no—
76#102×C53—22no—
77#102×C53—22no—
78C22⋊C8×C53—24no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53D412yes—
81C4×C8×C53—21no—
82C4×C8×C53C2312yes—
83C24×C53—21no—
84C2×Q8×C53—22no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—41no—
88C2×Q8×C53—82no—
89C2×D4×C53—21no—
90C2×D4×C53—22no—
91C2×D4×C53—22no—
92C2×D4×C53C2×D412yes—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—41no—
97C22×C4×C53—82no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53#4111yescharacteristic
101Q16×C53—84no—
102M16×C53—42no—
103M16×C53—84no—
104C2×C8×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—42no—
108C2×C8×C53C2×D414yes—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53C2×D412yes—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C22⋊C4×C53—41no—
124C42×C53—41no—
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C42×C53C2411yescharacteristic
129C23×C53—22no—
130C23×C53—42no—
131C23×C53—81no—
132C23×C53—81no—
133C23×C53#12111yescharacteristic
134Q8×C53—44no—
135Q8×C53—42no—
136Q8×C53—82no—
137Q8×C53—81no—
138D4×C53—22no—
139D4×C53—22no—
140D4×C53—42no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—44no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—82no—
149C2×C4×C53—81no—
150C2×C4×C53—82no—
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#13712yes—
155C2×C4×C53#13712yes—
156C2×C4×C53C22×D411yescharacteristic
157C2×C4×C53C22×D411yescharacteristic
158C2×C4×C53C2×#4111yescharacteristic
159C8×C53—24no—
160C8×C53—42no—
161C8×C53—42no—
162C8×C53—42no—
163#19423C5311yesmaximal, characteristic, direct product factor, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—84no—
170C22×C53#47012yes—
171C22×C53C2×#12111yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—42no—
176C4×C53—82no—
177C4×C53—81no—
178C4×C53—81no—
179C4×C53#47012yes—
180C4×C53C2×#13711yescharacteristic
181C4×C53C2×#13711yescharacteristic
182C4×C53#57612yes—
183#3209C2×C5311yescharacteristic, complemented
184#3077C2×C5311yescharacteristic
185C2×#474C2×C5311yescharacteristic
186#2975C2×C5311yescharacteristic
187#1703C2×C5314yescomplemented
188#1689C2×C5314yescomplemented
189#1570C2×C5311yescharacteristic, complemented
190#1549C2×C5311yescharacteristic, complemented
191#1542C2×C5311yescharacteristic, complemented
192C2×C53—22no—
193C2×C53—82no—
194C2×C53#304111yescharacteristic
195C2×C53C2×#47011yescharacteristic
196C2×C53#306911yescharacteristic
197#533C22×C5311yescharacteristic
198C2×C4⋊Q8C22×C5311yescharacteristic
199C2×C4×D4C22×C5311yescharacteristic
200#494C22×C5312yescomplemented
201#487C22×C5312yescomplemented
202#478—22no—
203#476C22×C5312yes—
204#474—22no—
205#474C22×C5312yes—
206#474C22×C5314yes—
207#462—22no—
208#451—22no—
209#436—22no—
210#436C22×C5314yes—
211#431C22×C5311yescharacteristic
212#431C22×C5311yescharacteristic
213#428—21no—
214C2×#107—21no—
215#423C22×C5311yescharacteristic
216C2×C4⋊C8C22×C5311yescharacteristic
217#418—21no—
218C2×#102—21no—
219#407—22no—
220#334C22×C5314yescomplemented
221#333C22×C5312yescomplemented
222#332C22×C5312yescomplemented
223#332C22×C5314yescomplemented
224C53#1942311yescharacteristic, direct product factor, Sylow
225C22×Q8—21no—
226C22×D4D4×C5311yescharacteristic
227C23×C4—21no—
228C2×Q16—24no—
229C2×M16—21no—
230C2×M16—24no—
231C22×C8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C23×C5312yes—
236C4⋊Q8C23×C5311yescharacteristic
237C4⋊Q8C23×C5311yescharacteristic
238#125—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4D4×C5312yes—
246C4×D4C23×C5312yes—
247#117—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4D4×C5311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C42C23×C5311yescharacteristic
253#107—24no—
254#107—22no—
255#106—22no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8D4×C5312yes—
259C4⋊C8C23×C5312yes—
260C4⋊C8C23×C5312yes—
261C4⋊C8C23×C5312yes—
262#103—24no—
263#102—24no—
264#102—22no—
265#102—22no—
266C22⋊C8—21no—
267C22⋊C8—24no—
268C22⋊C8D4×C5312yes—
269C4×C8—21no—
270C4×C8C23×C5312yes—
271C24—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—82no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4C2×D4×C5312yes—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4C2×D4×C5311yescharacteristic
287C22×C4C2×D4×C5311yescharacteristic
288C22×C4#41×C5311yescharacteristic
289Q16—84no—
290M16—42no—
291M16—84no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—42no—
296C2×C8C2×D4×C5314yes—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4C2×D4×C5312yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C42—41no—
313C42C2×D4×C5311yescharacteristic
314C42C2×D4×C5311yescharacteristic
315C42#41×C5311yescharacteristic
316C42C24×C5311yescharacteristic
317C23—22no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23#121×C5311yescharacteristic
322Q8—42no—
323Q8—44no—
324Q8—81no—
325Q8—82no—
326D4—22no—
327D4—22no—
328D4—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4#121×C5311yescharacteristic
340C2×C4#121×C5311yescharacteristic
341C2×C4#121×C5311yescharacteristic
342C2×C4#137×C5312yes—
343C2×C4#137×C5312yes—
344C2×C4C22×D4×C5311yescharacteristic
345C2×C4C22×D4×C5311yescharacteristic
346C2×C4C2×#41×C5311yescharacteristic
347C8—24no—
348C8—42no—
349C8—42no—
350C8—42no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—84no—
357C22C53×#47012yes—
358C22C2×#121×C5311yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4C53×#47012yes—
367C4C2×#137×C5311yescharacteristic
368C4C2×#137×C5311yescharacteristic
369C4C53×#57612yes—
370C2—22no—
371C2—82no—
372C2C53×#304111yescharacteristic
373C2C2×C53×#47011yescharacteristic
374C2C53×#306911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#121
commutator, Frattini, cosocleC42C24×C53
prime radicalC22×D4×C53D4
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
1. C53×#3209C2semidirect
5. C53×#1703C2semidirect
6. C53×#1689C2semidirect
7. C53×#1570C2semidirect
8. C53×#1549C2semidirect
9. C53×#1542C2semidirect
13. C53×#494C22semidirect
14. C53×#487C22semidirect
33. C53×#334C22semidirect
34. C53×#333C22semidirect
35. C53×#332C22semidirect
36. C53×#332C22semidirect
163. #19423C53direct (N × H)
183. #3209C2×C53semidirect
187. #1703C2×C53semidirect
188. #1689C2×C53semidirect
189. #1570C2×C53semidirect
190. #1549C2×C53semidirect
191. #1542C2×C53semidirect
200. #494C22×C53semidirect
201. #487C22×C53semidirect
220. #334C22×C53semidirect
221. #333C22×C53semidirect
222. #332C22×C53semidirect
223. #332C22×C53semidirect
224. C53#19423direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32092
C53×#30772
C2×C53×#4742
C53×#29752
C53×#17032
C53×#16892
C53×#15702
C53×#15492
C53×#15422
#1942353