Finite Group Databasebeta

← C53×#10188 #27217700

Subgroups — C53×#10188 #27217700

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3227C211yesmaximal, characteristic, complemented
2C2×C53×#474C211yesmaximal, characteristic, complemented
3C53×#3036C211yesmaximal, characteristic
4C53×#1548C211yesmaximal, characteristic, complemented
5C53×#1480C211yesmaximal, characteristic, complemented
6C53×#1460C211yesmaximal, characteristic, complemented
7C53×#1301C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C53—21no—
9C53×#532C2211yescharacteristic
10C2×C4⋊Q8×C53C2211yescharacteristic
11C2×C4×D4×C53C2211yescharacteristic
12C53×#498—21no—
13C53×#486—21no—
14C53×#481—22no—
15C53×#476—22no—
16C53×#474—22no—
17C53×#474—22no—
18C53×#463—22no—
19C53×#461—21no—
20C53×#459—21no—
21C53×#431C2211yescharacteristic
22C53×#431C2211yescharacteristic
23C2×#107×C53—21no—
24C53×#423C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#417—21no—
27C2×#102×C53—21no—
28C53×#333—22no—
29C53×D4⋊C8—22no—
30C53×#323—22no—
31C2×#41×C53—21no—
32C22×Q8×C53—21no—
33C22×D4×C53D411yescharacteristic
34C23×C4×C53—21no—
35AGL(1,ℤ/8)×C53—42no—
36AGL(1,ℤ/8)×C53—42no—
37C2×SD16×C53—22no—
38C2×D8×C53—22no—
39C2×M16×C53—21no—
40C22×C8×C53—21no—
41C4⋊Q8×C53—21no—
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53D411yescharacteristic
45C4⋊Q8×C53D412yes—
46#127×C53D411yescharacteristic
47#124×C53—21no—
48#121×C53—21no—
49#121×C53—21no—
50C4×D4×C53—21no—
51C4×D4×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×C53D411yescharacteristic
59C2×C22⋊C4×C53—21no—
60C2×C42×C53C2311yescharacteristic
61#107×C53—24no—
62#106×C53—24no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53D412yes—
68C4⋊C8×C53D412yes—
69#103×C53—22no—
70#102×C53—24no—
71#102×C53—24no—
72#102×C53—22no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53D412yes—
75C4×C8×C53—21no—
76C4×C8×C53—21no—
77C24×C53—21no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82C2×Q8×C53—21no—
83C2×Q8×C53—22no—
84C2×Q8×C53—41no—
85C2×Q8×C53—82no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C2×D4×C53SD1612yes—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—82no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105SD16×C53—44no—
106D8×C53—44no—
107M16×C53—41no—
108M16×C53—41no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C23×C53—22no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53#12111yescharacteristic
138Q8×C53—41no—
139Q8×C53—41no—
140Q8×C53—82no—
141Q8×C53—82no—
142D4×C53—22no—
143D4×C53—22no—
144D4×C53—42no—
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149D4×C53—42no—
150D4×C53—81no—
151D4×C53—81no—
152C2×C4×C53—22no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—82no—
162C2×C4×C53—81no—
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12711yescharacteristic
169C2×C4×C53C2×SD1611yescharacteristic
170C2×C4×C53#13511yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C53#13711yescharacteristic
173C8×C53—42no—
174C8×C53—42no—
175C8×C53—42no—
176C8×C53—81no—
177#10188C5311yesmaximal, characteristic, direct product factor, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—42no—
181C22×C53—82no—
182C22×C53—82no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53#39211yescharacteristic
189C22×C53#46511yescharacteristic
190C22×C53#46811yescharacteristic
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—81no—
195C4×C53—81no—
196C4×C53—82no—
197C4×C53#44811yescharacteristic
198C4×C53#44911yescharacteristic
199C4×C53#45011yescharacteristic
200C4×C53#45211yescharacteristic
201C4×C53#47212yes—
202#3227C2×C5311yescharacteristic, complemented
203C2×#474C2×C5311yescharacteristic, complemented
204#3036C2×C5311yescharacteristic
205#1548C2×C5311yescharacteristic, complemented
206#1480C2×C5311yescharacteristic, complemented
207#1460C2×C5311yescharacteristic, complemented
208#1301C2×C5311yescharacteristic, complemented
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—82no—
212C2×C53—161no—
213C2×C53#200011yescharacteristic
214C2×C53#200911yescharacteristic
215C2×C53#201811yescharacteristic
216C2×AGL(1,ℤ/8)—21no—
217#532C22×C5311yescharacteristic
218C2×C4⋊Q8C22×C5311yescharacteristic
219C2×C4×D4C22×C5311yescharacteristic
220#498—21no—
221#486—21no—
222#481—22no—
223#476—22no—
224#474—22no—
225#474—22no—
226#463—22no—
227#461—21no—
228#459—21no—
229#431C22×C5311yescharacteristic
230#431C22×C5311yescharacteristic
231C2×#107—21no—
232#423C22×C5311yescharacteristic
233C2×C4⋊C8C22×C5311yescharacteristic
234#417—21no—
235C2×#102—21no—
236#333—22no—
237D4⋊C8—22no—
238#323—22no—
239C53#1018811yescharacteristic, direct product factor, Sylow
240C2×#41—21no—
241C22×Q8—21no—
242C22×D4D4×C5311yescharacteristic
243C23×C4—21no—
244AGL(1,ℤ/8)—42no—
245AGL(1,ℤ/8)—42no—
246C2×SD16—22no—
247C2×D8—22no—
248C2×M16—21no—
249C22×C8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8D4×C5312yes—
254C4⋊Q8D4×C5311yescharacteristic
255#127D4×C5311yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4D4×C5311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C23×C5311yescharacteristic
270#107—24no—
271#106—24no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8D4×C5312yes—
277C4⋊C8D4×C5312yes—
278#103—22no—
279#102—24no—
280#102—22no—
281#102—24no—
282C22⋊C8—21no—
283C22⋊C8D4×C5312yes—
284C4×C8—21no—
285C4×C8—21no—
286C24—21no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4SD16×C5312yes—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4C2×D4×C5311yescharacteristic
312C22×C4C2×D4×C5311yescharacteristic
313C22×C4#41×C5311yescharacteristic
314SD16—44no—
315D8—44no—
316M16—41no—
317M16—41no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C42C2×D4×C5311yescharacteristic
337C42C2×D4×C5311yescharacteristic
338C42C2×D4×C5311yescharacteristic
339C42C2×D4×C5311yescharacteristic
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—81no—
345C23—81no—
346C23#121×C5311yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—82no—
350Q8—82no—
351D4—22no—
352D4—22no—
353D4—42no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—81no—
360D4—81no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4#120×C5311yescharacteristic
373C2×C4#120×C5311yescharacteristic
374C2×C4#120×C5311yescharacteristic
375C2×C4#121×C5311yescharacteristic
376C2×C4#121×C5311yescharacteristic
377C2×C4#127×C5311yescharacteristic
378C2×C4C2×SD16×C5311yescharacteristic
379C2×C4#135×C5311yescharacteristic
380C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
381C2×C4#137×C5311yescharacteristic
382C8—42no—
383C8—42no—
384C8—42no—
385C8—81no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—82no—
390C22—82no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22C53×#39211yescharacteristic
397C22C53×#46511yescharacteristic
398C22C53×#46811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—82no—
405C4C53×#44811yescharacteristic
406C4C53×#44911yescharacteristic
407C4C53×#45011yescharacteristic
408C4C53×#45211yescharacteristic
409C4C53×#47212yes—
410C2—21no—
411C2—21no—
412C2—82no—
413C2—161no—
414C2C53×#200011yescharacteristic
415C2C53×#200911yescharacteristic
416C2C53×#201811yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3036C2
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
1. C53×#3227C2semidirect
2. C2×C53×#474C2semidirect
4. C53×#1548C2semidirect
5. C53×#1480C2semidirect
6. C53×#1460C2semidirect
7. C53×#1301C2semidirect
177. #10188C53direct (N × H)
202. #3227C2×C53semidirect
203. C2×#474C2×C53semidirect
205. #1548C2×C53semidirect
206. #1480C2×C53semidirect
207. #1460C2×C53semidirect
208. #1301C2×C53semidirect
239. C53#10188direct (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×#4742
C53×#30362
C53×#15482
C53×#14802
C53×#14602
C53×#13012
#1018853