Finite Group Databasebeta

← C53×#10058 #27217570

Subgroups — C53×#10058 #27217570

Order 13568.

602 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
1C2×C53×#487C211yesmaximal, characteristic, complemented
2C2×C53×#476C211yesmaximal, characteristic
3C53×#2547C211yesmaximal, characteristic
4C53×#1987C211yesmaximal, characteristic, complemented
5C2×C53×#332C211yesmaximal, characteristic, complemented
6C2×C53×D4⋊C8C211yesmaximal, characteristic
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C53C2211yescharacteristic
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×#124×C53—21no—
11C2×C4×D4×C53C2211yescharacteristic
12C53×#487—22no—
13C53×#487—22no—
14C53×#476—22no—
15C53×#476—22no—
16C2×#106×C53—21no—
17C2×C4⋊C8×C53C2211yescharacteristic
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×#103×C53—22no—
20C2×#102×C53—21no—
21C2×C4×C8×C53C2211yescharacteristic
22C53×#397—22no—
23C53×#391—21no—
24C53×#380—22no—
25C4×C4⋊C4×C53C2211yescharacteristic
26C53×#340—22no—
27C53×#332—22no—
28C53×D4⋊C8—22no—
29C22×Q8×C53—21no—
30C22×Q8×C53—21no—
31C22×D4×C53D411yescharacteristic
32C23×C4×C53—21no—
33C22×C8×C53—21no—
34C22×C8×C53—21no—
35C22×C8×C53—21no—
36C4⋊Q8×C53—21no—
37C4⋊Q8×C53—21no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53—21no—
41C4⋊Q8×C53—21no—
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53D412yes—
44#124×C53—42no—
45#124×C53—42no—
46C4×D4×C53—21no—
47C4×D4×C53—21no—
48C4×D4×C53—21no—
49C4×D4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D412yes—
55C2×C4⋊C4×C53D412yes—
56C2×C22⋊C4×C53—24no—
57C2×C22⋊C4×C53—21no—
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60#106×C53—24no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53D412yes—
65#103×C53—28no—
66#102×C53—24no—
67C4×C8×C53—21no—
68C4×C8×C53D412yes—
69#95×C53—24no—
70#95×C53—21no—
71C24×C53—21no—
72C2×Q8×C53—22no—
73C2×Q8×C53—44no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53SD1612yes—
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53—42no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—42no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C22⋊C4×C53—48no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C42×C53—41no—
121C42×C53—41no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126C23×C53—22no—
127C23×C53—41no—
128C23×C53—42no—
129C23×C53—41no—
130C23×C53—42no—
131C23×C53#12011yescharacteristic
132Q8×C53—44no—
133Q8×C53—82no—
134Q8×C53—81no—
135Q8×C53—81no—
136Q8×C53—81no—
137Q8×C53—81no—
138D4×C53—22no—
139D4×C53—22no—
140D4×C53—41no—
141D4×C53—41no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
154C2×C4×C53—81no—
155C2×C4×C53—81no—
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12711yescharacteristic
162C2×C4×C53C2×SD1611yescharacteristic
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53#13512yes—
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C53#13711yescharacteristic
167C2×C4×C53#13712yes—
168C8×C53—42no—
169C8×C53—42no—
170C8×C53—81no—
171#10058C5311yesmaximal, characteristic, direct product factor, Sylow
172C22×C53—44no—
173C22×C53—42no—
174C22×C53—82no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53#39211yescharacteristic
178C22×C53#45011yescharacteristic
179C22×C53#45212yes—
180C22×C53#45211yescharacteristic
181C22×C53#45412yes—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—82no—
185C4×C53—82no—
186C4×C53—82no—
187C4×C53—81no—
188C4×C53#44811yescharacteristic
189C4×C53#44911yescharacteristic
190C4×C53#46412yes—
191C4×C53#46511yescharacteristic
192C4×C53#46811yescharacteristic
193C4×C53#49511yescharacteristic
194C4×C53#49711yescharacteristic
195C2×#487C2×C5311yescharacteristic, complemented
196C2×#476C2×C5311yescharacteristic
197#2547C2×C5311yescharacteristic
198#1987C2×C5311yescharacteristic, complemented
199C2×#332C2×C5311yescharacteristic, complemented
200C2×D4⋊C8C2×C5311yescharacteristic
201#1279C2×C5311yescharacteristic, complemented
202C2×C53—82no—
203C2×C53#163512yes—
204C2×C53#163812yes—
205C2×C53#200011yescharacteristic
206C2×C53#200311yescharacteristic
207C2×C53#201511yescharacteristic
208C2×C4⋊Q8C22×C5311yescharacteristic
209C2×C4⋊Q8C22×C5311yescharacteristic
210C2×#124—21no—
211C2×C4×D4C22×C5311yescharacteristic
212#487—22no—
213#487—22no—
214#476—22no—
215#476—22no—
216C2×#106—21no—
217C2×C4⋊C8C22×C5311yescharacteristic
218C2×C4⋊C8C22×C5311yescharacteristic
219C2×#103—22no—
220C2×#102—21no—
221C2×C4×C8C22×C5311yescharacteristic
222#397—22no—
223#391—21no—
224#380—22no—
225C4×C4⋊C4C22×C5311yescharacteristic
226#340—22no—
227#332—22no—
228D4⋊C8—22no—
229C53#1005811yescharacteristic, direct product factor, Sylow
230C22×Q8—21no—
231C22×Q8—21no—
232C22×D4D4×C5311yescharacteristic
233C23×C4—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8D4×C5312yes—
245#124—42no—
246#124—42no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4D4×C5311yescharacteristic
255C2×C4⋊C4D4×C5312yes—
256C2×C4⋊C4D4×C5312yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—21no—
259C2×C42—21no—
260C2×C42C23×C5311yescharacteristic
261#106—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8D4×C5312yes—
266#103—28no—
267#102—24no—
268C4×C8—21no—
269C4×C8D4×C5312yes—
270#95—24no—
271#95—21no—
272C24—21no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4SD16×C5312yes—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×D4×C5311yescharacteristic
297C22×C4C2×D4×C5311yescharacteristic
298C22×C4C2×D4×C5311yescharacteristic
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—48no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—41no—
322C42—41no—
323C42C2×D4×C5311yescharacteristic
324C42C2×D4×C5311yescharacteristic
325C42C2×D4×C5311yescharacteristic
326C42#41×C5311yescharacteristic
327C23—22no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23#120×C5311yescharacteristic
333Q8—44no—
334Q8—82no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339D4—22no—
340D4—22no—
341D4—41no—
342D4—41no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4#120×C5311yescharacteristic
358C2×C4#120×C5311yescharacteristic
359C2×C4#121×C5311yescharacteristic
360C2×C4#121×C5311yescharacteristic
361C2×C4#121×C5311yescharacteristic
362C2×C4#127×C5311yescharacteristic
363C2×C4C2×SD16×C5311yescharacteristic
364C2×C4#135×C5312yes—
365C2×C4#135×C5311yescharacteristic
366C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
367C2×C4#137×C5312yes—
368C2×C4#137×C5311yescharacteristic
369C8—42no—
370C8—42no—
371C8—81no—
372C22—42no—
373C22—44no—
374C22—81no—
375C22—81no—
376C22—82no—
377C22C53×#39211yescharacteristic
378C22C53×#45011yescharacteristic
379C22C53×#45211yescharacteristic
380C22C53×#45212yes—
381C22C53×#45412yes—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—81no—
387C4—82no—
388C4C53×#44811yescharacteristic
389C4C53×#44911yescharacteristic
390C4C53×#46412yes—
391C4C53×#46511yescharacteristic
392C4C53×#46811yescharacteristic
393C4C53×#49511yescharacteristic
394C4C53×#49711yescharacteristic
395C2—82no—
396C2C53×#163512yes—
397C2C53×#163812yes—
398C2C53×#200011yescharacteristic
399C2C53×#200311yescharacteristic
400C2C53×#201511yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#487C2semidirect
4. C53×#1987C2semidirect
5. C2×C53×#332C2semidirect
7. C53×#1279C2semidirect
171. #10058C53direct (N × H)
195. C2×#487C2×C53semidirect
198. #1987C2×C53semidirect
199. C2×#332C2×C53semidirect
201. #1279C2×C53semidirect
229. C53#10058direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#4872
C2×C53×#4762
C53×#25472
C53×#19872
C2×C53×#3322
C2×C53×D4⋊C82
C53×#12792
#1005853