Finite Group Databasebeta

← C53×#12187 #27219699

Subgroups — C53×#12187 #27219699

Order 13568.

508 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2760C211yesmaximal, characteristic, complemented
2C53×#2022C211yesmaximal, characteristic, complemented
3C53×#1601C211yesmaximal, characteristic, complemented
4C53×#1595C211yesmaximal, characteristic
5C53×#1351C211yesmaximal, characteristic, complemented
6C53×#1343C211yesmaximal, characteristic, complemented
7C53×#1317C211yesmaximal, characteristic, complemented
8C53×#534C2211yescharacteristic
9C2×#123×C53—21no—
10C53×#448—21no—
11C53×#447—21no—
12C2×#99×C53—21no—
13C53×#401—21no—
14C53×#393—21no—
15C53×#393C2211yescharacteristic
16C53×#392C2211yescharacteristic
17C53×#386—21no—
18C53×#386—21no—
19C53×#386—21no—
20C53×#381—21no—
21C53×#381C2211yescharacteristic
22C53×#342C2211yescharacteristic
23C53×#327—21no—
24C53×#327—21no—
25C53×#323C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C22×Q8×C53D411yescharacteristic
28C22×D4×C53—21no—
29C22×D4×C53—21no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53D411yescharacteristic
33C2×SD16×C53—41no—
34C2×D8×C53—41no—
35#123×C53—41no—
36#123×C53—42no—
37#123×C53—42no—
38#121×C53—21no—
39#121×C53—21no—
40#120×C53—41no—
41C2×C4⋊C4×C53—41no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—41no—
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53C2311yescharacteristic
51#103×C53—41no—
52#102×C53—41no—
53#99×C53—42no—
54C22⋊C8×C53—21no—
55C22⋊C8×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53D411yescharacteristic
64#95×C53D411yescharacteristic
65C24×C53—21no—
66C24×C53—21no—
67C24×C53C2×D411yescharacteristic
68C2×Q8×C53—41no—
69C2×Q8×C53—81no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—41no—
73C2×D4×C53—42no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—42no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—81no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—42no—
96C22×C4×C53—81no—
97C22×C4×C53—81no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100SD16×C53—82no—
101D8×C53—82no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—44no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—44no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—81no—
115C22⋊C4×C53—81no—
116C22⋊C4×C53—81no—
117C22⋊C4×C53—81no—
118C22⋊C4×C53—81no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—22no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—81no—
134C23×C53—81no—
135C23×C53#12011yescharacteristic
136C23×C53#13511yescharacteristic
137C23×C53AGL(1,ℤ/8)11yescharacteristic
138Q8×C53—42no—
139Q8×C53—81no—
140D4×C53—42no—
141D4×C53—81no—
142D4×C53—81no—
143D4×C53—81no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53—82no—
147D4×C53—82no—
148D4×C53—81no—
149D4×C53—81no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—81no—
175C2×C4×C53—81no—
176C2×C4×C53—84no—
177C2×C4×C53—81no—
178C8×C53—81no—
179C8×C53—81no—
180#12187C5311yesmaximal, characteristic, direct product factor, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—42no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—82no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53—161no—
202C22×C53#44911yescharacteristic
203C22×C53#45311yescharacteristic
204C22×C53B(4,𝔽2)11yescharacteristic
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—81no—
208C4×C53—81no—
209C4×C53—81no—
210C4×C53—82no—
211C4×C53—81no—
212C4×C53—81no—
213C4×C53—81no—
214#2760C2×C5311yescharacteristic, complemented
215#2022C2×C5311yescharacteristic, complemented
216#1601C2×C5311yescharacteristic, complemented
217#1595C2×C5311yescharacteristic
218#1351C2×C5311yescharacteristic, complemented
219#1343C2×C5311yescharacteristic, complemented
220#1317C2×C5311yescharacteristic, complemented
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—41no—
224C2×C53—41no—
225C2×C53—82no—
226C2×C53—161no—
227C2×C53#160011yescharacteristic
228C2×C53#219711yescharacteristic
229C2×C53#220011yescharacteristic
230#534C22×C5311yescharacteristic
231C2×#123—21no—
232#448—21no—
233#447—21no—
234C2×#99—21no—
235#401—21no—
236#393—21no—
237#393C22×C5311yescharacteristic
238#392C22×C5311yescharacteristic
239#386—21no—
240#386—21no—
241#386—21no—
242#381—21no—
243#381C22×C5311yescharacteristic
244#342C22×C5311yescharacteristic
245#327—21no—
246#327—21no—
247#323C22×C5311yescharacteristic
248#323C22×C5311yescharacteristic
249C53#1218711yescharacteristic, direct product factor, Sylow
250C22×Q8D4×C5311yescharacteristic
251C22×D4—21no—
252C22×D4—21no—
253C22×D4D4×C5311yescharacteristic
254C23×C4—21no—
255C23×C4D4×C5311yescharacteristic
256C2×SD16—41no—
257C2×D8—41no—
258#123—42no—
259#123—41no—
260#123—42no—
261#121—21no—
262#121—21no—
263#120—41no—
264C2×C4⋊C4—41no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4D4×C5311yescharacteristic
273C2×C22⋊C4C23×C5311yescharacteristic
274#103—41no—
275#102—41no—
276#99—42no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95D4×C5311yescharacteristic
287#95D4×C5311yescharacteristic
288C24—21no—
289C24—21no—
290C24C2×D4×C5311yescharacteristic
291C2×Q8—41no—
292C2×Q8—81no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—81no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4C2×D4×C5311yescharacteristic
322C22×C4C2×D4×C5311yescharacteristic
323SD16—82no—
324D8—82no—
325C2×C8—41no—
326C2×C8—41no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—44no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—44no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—81no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—81no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—81no—
357C23—81no—
358C23#120×C5311yescharacteristic
359C23#135×C5311yescharacteristic
360C23AGL(1,ℤ/8)×C5311yescharacteristic
361Q8—42no—
362Q8—81no—
363D4—42no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—82no—
370D4—82no—
371D4—81no—
372D4—81no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—84no—
400C2×C4—81no—
401C8—81no—
402C8—81no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—82no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—161no—
424C22C53×#44911yescharacteristic
425C22C53×#45311yescharacteristic
426C22C53×B(4,𝔽2)11yescharacteristic
427C4—41no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4—81no—
434C4—81no—
435C4—81no—
436C2—21no—
437C2—21no—
438C2—41no—
439C2—41no—
440C2—82no—
441C2—161no—
442C2C53×#160011yescharacteristic
443C2C53×#219711yescharacteristic
444C2C53×#220011yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC53×#1595C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2760C2semidirect
2. C53×#2022C2semidirect
3. C53×#1601C2semidirect
5. C53×#1351C2semidirect
6. C53×#1343C2semidirect
7. C53×#1317C2semidirect
180. #12187C53direct (N × H)
214. #2760C2×C53semidirect
215. #2022C2×C53semidirect
216. #1601C2×C53semidirect
218. #1351C2×C53semidirect
219. #1343C2×C53semidirect
220. #1317C2×C53semidirect
249. C53#12187direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27602
C53×#20222
C53×#16012
C53×#15952
C53×#13512
C53×#13432
C53×#13172
#1218753