Finite Group Databasebeta

← C53×#10185 #27217697

Subgroups — C53×#10185 #27217697

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 416 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×#1546C211yesmaximal, characteristic, complemented
5C53×#1480C211yesmaximal, characteristic
6C53×#1461C211yesmaximal, characteristic, complemented
7C53×#1301C211yesmaximal, characteristic, complemented
8C2×#137×C53—21no—
9C22×Q16×C53—21no—
10C53×#532C2211yescharacteristic
11C2×C4⋊Q8×C53C2211yescharacteristic
12C2×C4×Q8×C53C2211yescharacteristic
13C53×#500—21no—
14C53×#486—21no—
15C53×#480—22no—
16C53×#474—22no—
17C53×#464—22no—
18C53×#462—22no—
19C53×#462—21no—
20C53×#462—22no—
21C53×#460—21no—
22C53×#431C2211yescharacteristic
23C53×#431C2211yescharacteristic
24C53×#423C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#417—21no—
27C2×#103×C53—21no—
28C53×#332—22no—
29C53×D4⋊C8—22no—
30C53×#324—22no—
31C2×#41×C53—21no—
32C22×Q8×C53—21no—
33C22×Q8×C53D411yescharacteristic
34#137×C53—42no—
35#137×C53—42no—
36C2×Q16×C53—24no—
37C2×Q16×C53—22no—
38C2×Q16×C53—42no—
39C2×Q16×C53—42no—
40C2×SD16×C53—22no—
41C2×M16×C53—21no—
42C22×C8×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—21no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53D411yescharacteristic
47C4⋊Q8×C53D412yes—
48#127×C53D411yescharacteristic
49#124×C53—21no—
50#121×C53—21no—
51#121×C53—21no—
52C4×Q8×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64#107×C53—24no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53D412yes—
70C4⋊C8×C53D412yes—
71#103×C53—24no—
72#103×C53—22no—
73#102×C53—24no—
74#102×C53—22no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53D412yes—
77C4×C8×C53—21no—
78C4×C8×C53—21no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82#41×C53—41no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—22no—
88C2×Q8×C53—21no—
89C2×Q8×C53—21no—
90C2×Q8×C53—41no—
91C2×Q8×C53—41no—
92C2×Q8×C53—41no—
93C2×Q8×C53—41no—
94C2×Q8×C53—41no—
95C2×Q8×C53D812yes—
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—41no—
111M16×C53—41no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53—22no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53—41no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C42×C53—41no—
131C42×C53—41no—
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53C2×D411yescharacteristic
136C23×C53—41no—
137C23×C53#12111yescharacteristic
138Q8×C53—22no—
139Q8×C53—22no—
140Q8×C53—41no—
141Q8×C53—41no—
142Q8×C53—42no—
143Q8×C53—41no—
144Q8×C53—42no—
145Q8×C53—41no—
146Q8×C53—81no—
147Q8×C53—81no—
148D4×C53—42no—
149D4×C53—41no—
150D4×C53—41no—
151D4×C53—42no—
152D4×C53—81no—
153D4×C53—81no—
154C2×C4×C53—22no—
155C2×C4×C53—22no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53—81no—
167C2×C4×C53#12011yescharacteristic
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12711yescharacteristic
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53#13511yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C8×C53—42no—
178C8×C53—42no—
179C8×C53—42no—
180C8×C53—81no—
181#10185C5311yesmaximal, characteristic, direct product factor, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#39211yescharacteristic
188C22×C53#46611yescharacteristic
189C22×C53#46911yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—42no—
194C4×C53—81no—
195C4×C53—82no—
196C4×C53—81no—
197C4×C53#44711yescharacteristic
198C4×C53#44711yescharacteristic
199C4×C53#44911yescharacteristic
200C4×C53#44911yescharacteristic
201C4×C53#47312yes—
202#3226C2×C5311yescharacteristic
203#3037C2×C5311yescharacteristic
204C2×#462C2×C5311yescharacteristic, complemented
205#1546C2×C5311yescharacteristic, complemented
206#1480C2×C5311yescharacteristic
207#1461C2×C5311yescharacteristic, complemented
208#1301C2×C5311yescharacteristic, complemented
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—161no—
212C2×C53#199911yescharacteristic
213C2×C53#200911yescharacteristic
214C2×C53#202011yescharacteristic
215C2×#137—21no—
216C22×Q16—21no—
217#532C22×C5311yescharacteristic
218C2×C4⋊Q8C22×C5311yescharacteristic
219C2×C4×Q8C22×C5311yescharacteristic
220#500—21no—
221#486—21no—
222#480—22no—
223#474—22no—
224#464—22no—
225#462—22no—
226#462—21no—
227#462—22no—
228#460—21no—
229#431C22×C5311yescharacteristic
230#431C22×C5311yescharacteristic
231#423C22×C5311yescharacteristic
232C2×C4⋊C8C22×C5311yescharacteristic
233#417—21no—
234C2×#103—21no—
235#332—22no—
236D4⋊C8—22no—
237#324—22no—
238C53#1018511yescharacteristic, direct product factor, Sylow
239C2×#41—21no—
240C22×Q8—21no—
241C22×Q8D4×C5311yescharacteristic
242#137—42no—
243#137—42no—
244C2×Q16—24no—
245C2×Q16—22no—
246C2×Q16—42no—
247C2×Q16—42no—
248C2×SD16—22no—
249C2×M16—21no—
250C22×C8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8D4×C5312yes—
255C4⋊Q8D4×C5311yescharacteristic
256#127D4×C5311yescharacteristic
257#124—21no—
258#121—21no—
259#121—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×D4—21no—
265C4×D4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4D4×C5311yescharacteristic
270C2×C42—21no—
271C2×C42C23×C5311yescharacteristic
272#107—24no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8D4×C5312yes—
278C4⋊C8D4×C5312yes—
279#103—24no—
280#103—22no—
281#102—22no—
282#102—24no—
283C22⋊C8—21no—
284C22⋊C8D4×C5312yes—
285C4×C8—21no—
286C4×C8—21no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8D8×C5312yes—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C2×D4×C5311yescharacteristic
313C22×C4C2×D4×C5311yescharacteristic
314C22×C4#41×C5311yescharacteristic
315Q16—48no—
316Q16—44no—
317SD16—44no—
318M16—41no—
319M16—41no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—41no—
339C42—41no—
340C42C2×D4×C5311yescharacteristic
341C42C2×D4×C5311yescharacteristic
342C42C2×D4×C5311yescharacteristic
343C42C2×D4×C5311yescharacteristic
344C23—41no—
345C23#121×C5311yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—41no—
349Q8—41no—
350Q8—42no—
351Q8—41no—
352Q8—42no—
353Q8—41no—
354Q8—81no—
355Q8—81no—
356D4—42no—
357D4—41no—
358D4—41no—
359D4—42no—
360D4—81no—
361D4—81no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4#120×C5311yescharacteristic
376C2×C4#120×C5311yescharacteristic
377C2×C4#120×C5311yescharacteristic
378C2×C4#121×C5311yescharacteristic
379C2×C4#121×C5311yescharacteristic
380C2×C4#127×C5311yescharacteristic
381C2×C4C2×D8×C5311yescharacteristic
382C2×C4#135×C5311yescharacteristic
383C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
384C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
385C8—42no—
386C8—42no—
387C8—42no—
388C8—81no—
389C22—21no—
390C22—21no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22C53×#39211yescharacteristic
395C22C53×#46611yescharacteristic
396C22C53×#46911yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4C53×#44711yescharacteristic
405C4C53×#44711yescharacteristic
406C4C53×#44911yescharacteristic
407C4C53×#44911yescharacteristic
408C4C53×#47312yes—
409C2—21no—
410C2—21no—
411C2—161no—
412C2C53×#199911yescharacteristic
413C2C53×#200911yescharacteristic
414C2C53×#202011yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#532C22
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
3. C2×C53×#462C2semidirect
4. C53×#1546C2semidirect
6. C53×#1461C2semidirect
7. C53×#1301C2semidirect
181. #10185C53direct (N × H)
204. C2×#462C2×C53semidirect
205. #1546C2×C53semidirect
207. #1461C2×C53semidirect
208. #1301C2×C53semidirect
238. C53#10185direct (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×#15462
C53×#14802
C53×#14612
C53×#13012
#1018553