Finite Group Databasebeta

← C53×#11875 #27219387

Subgroups — C53×#11875 #27219387

Order 13568.

536 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2535C211yesmaximal, characteristic
2C53×#1799C211yesmaximal, characteristic, complemented
3C53×#1459C211yesmaximal, characteristic, complemented
4C53×#1344C212yesmaximal, complemented
5C53×#1343C212yesmaximal, complemented
6C2×#121×C53—21no—
7C2×C4×D4×C53C411yescharacteristic, complemented
8C2×C4×D4×C53C2211yescharacteristic
9C2×#99×C53—21no—
10C2×C22⋊C8×C53—21no—
11C53×#397C411yescharacteristic, complemented
12C53×#397C2211yescharacteristic
13C53×#392C411yescharacteristic, complemented
14C53×#392C2211yescharacteristic
15C53×#386—21no—
16C53×#386—21no—
17C53×#382C411yescharacteristic, complemented
18C53×#380—21no—
19C53×#380—21no—
20C53×#342C2212yes—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×D4×C53—21no—
24C22×D4×C53—21no—
25C22×D4×C53C2×C411yescharacteristic
26C22×D4×C53D411yescharacteristic
27C23×C4×C53—21no—
28C23×C4×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C22×C8×C53—41no—
31#121×C53—42no—
32#121×C53—42no—
33C4×D4×C53—41no—
34C4×D4×C53—41no—
35C4×D4×C53—41no—
36C4×D4×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—41no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53C2311yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#99×C53—42no—
51C22⋊C8×C53—22no—
52C22⋊C8×C53—41no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—41no—
57#95×C53C2×C411yescharacteristic
58#95×C53C2×C411yescharacteristic
59C24×C53—21no—
60C24×C53C22⋊C411yescharacteristic
61C24×C53C2×D411yescharacteristic
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—42no—
66C2×D4×C53—42no—
67C2×D4×C53—42no—
68C2×D4×C53—42no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—81no—
72C2×D4×C53—81no—
73C2×D4×C53D812yes—
74C2×D4×C53SD1612yes—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53—81no—
87C22×C4×C53—81no—
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C2×C8×C53—42no—
94C2×C8×C53—82no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—82no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—81no—
109C22⋊C4×C53—81no—
110C42×C53—21no—
111C42×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—22no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53#10212yes—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53C2×D811yescharacteristic
130C23×C53C2×SD1611yescharacteristic
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—84no—
136D4×C53—82no—
137D4×C53—82no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53—84no—
161C2×C4×C53—82no—
162C2×C4×C53—82no—
163C2×C4×C53—82no—
164C2×C4×C53#9911yescharacteristic
165C2×C4×C53#9911yescharacteristic
166C2×C4×C53#10212yes—
167C8×C53—82no—
168#11875C5311yesmaximal, characteristic, direct product factor, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—22no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—42no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53C2×#9911yescharacteristic
191C22×C53C2×#10211yescharacteristic
192C22×C53#42111yescharacteristic
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—22no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—81no—
200C4×C53—81no—
201C4×C53—81no—
202C4×C53—162no—
203#2535C2×C5311yescharacteristic
204#1799C2×C5311yescharacteristic, complemented
205#1459C2×C5311yescharacteristic, complemented
206#1344C2×C5312yescomplemented
207#1343C2×C5312yescomplemented
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—41no—
211C2×C53—42no—
212C2×C53—41no—
213C2×C53—82no—
214C2×C53#151911yescharacteristic
215C2×C53#212011yescharacteristic
216C2×C53#212711yescharacteristic
217C2×#121—21no—
218C2×C4×D4C4×C5311yescharacteristic, complemented
219C2×C4×D4C22×C5311yescharacteristic
220C2×#99—21no—
221C2×C22⋊C8—21no—
222#397C4×C5311yescharacteristic, complemented
223#397C22×C5311yescharacteristic
224#392C4×C5311yescharacteristic, complemented
225#392C22×C5311yescharacteristic
226#386—21no—
227#386—21no—
228#382C4×C5311yescharacteristic, complemented
229#380—21no—
230#380—21no—
231#342C22×C5312yes—
232#324—21no—
233#323C22×C5312yes—
234C53#1187511yescharacteristic, direct product factor, Sylow
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×C4×C5311yescharacteristic
238C22×D4D4×C5311yescharacteristic
239C23×C4—21no—
240C23×C4D4×C5311yescharacteristic
241C23×C4D4×C5311yescharacteristic
242C22×C8—41no—
243#121—42no—
244#121—42no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C2×C4×C5311yescharacteristic
252C2×C4⋊C4D4×C5311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4C2×C4×C5311yescharacteristic
260C2×C22⋊C4C23×C5311yescharacteristic
261C2×C42C2×C4×C5311yescharacteristic
262#99—42no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—41no—
269#95C2×C4×C5311yescharacteristic
270#95C2×C4×C5311yescharacteristic
271C24—21no—
272C24C22⋊C4×C5311yescharacteristic
273C24C2×D4×C5311yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4D8×C5312yes—
286C2×D4SD16×C5312yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C22⋊C4×C5311yescharacteristic
301C22×C4C22⋊C4×C5311yescharacteristic
302C22×C4C22⋊C4×C5311yescharacteristic
303C22×C4C22×C4×C5311yescharacteristic
304C22×C4C2×D4×C5311yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—82no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#102×C5312yes—
340C23C2×C22⋊C4×C5311yescharacteristic
341C23C2×D8×C5311yescharacteristic
342C23C2×SD16×C5311yescharacteristic
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—84no—
348D4—82no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4#99×C5311yescharacteristic
377C2×C4#99×C5311yescharacteristic
378C2×C4#102×C5312yes—
379C8—82no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C2×#99×C5311yescharacteristic
402C22C2×#102×C5311yescharacteristic
403C22C53×#42111yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—162no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—42no—
418C2—41no—
419C2—82no—
420C2C53×#151911yescharacteristic
421C2C53×#212011yescharacteristic
422C2C53×#212711yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC53×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1799C2semidirect
3. C53×#1459C2semidirect
4. C53×#1344C2semidirect
5. C53×#1343C2semidirect
7. C2×C4×D4×C53C4semidirect
11. C53×#397C4semidirect
13. C53×#392C4semidirect
17. C53×#382C4semidirect
168. #11875C53direct (N × H)
204. #1799C2×C53semidirect
205. #1459C2×C53semidirect
206. #1344C2×C53semidirect
207. #1343C2×C53semidirect
218. C2×C4×D4C4×C53semidirect
222. #397C4×C53semidirect
224. #392C4×C53semidirect
228. #382C4×C53semidirect
234. C53#11875direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#25352
C53×#17992
C53×#14592
C53×#13442
C53×#13432
#1187553