Finite Group Databasebeta

← C53×#10328 #27217840

Subgroups — C53×#10328 #27217840

Order 13568.

698 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3192C211yesmaximal, characteristic
2C53×#2504C211yesmaximal, characteristic, complemented
3C53×#2501C211yesmaximal, characteristic, complemented
4C53×#1821C211yesmaximal, characteristic, complemented
5C53×#1507C211yesmaximal, characteristic, complemented
6C53×#1507C211yesmaximal, characteristic, complemented
7C53×#1297C211yesmaximal, characteristic, complemented
8C2×#123×C53—22no—
9C2×#121×C53C2211yescharacteristic
10C2×#121×C53C2211yescharacteristic
11C22×C4⋊C4×C53C2211yescharacteristic
12C53×#481—21no—
13C53×#481—21no—
14C53×#469—22no—
15C53×#447—21no—
16C53×#447—21no—
17C53×#407C2211yescharacteristic
18C53×#407C2211yescharacteristic
19C53×#399—22no—
20C53×#397—22no—
21C53×#390—21no—
22C53×#390—21no—
23C53×#386—22no—
24C53×#384—21no—
25C53×#382—21no—
26C4×C22⋊C4×C53C2211yescharacteristic
27C4×C22⋊C4×C53C2211yescharacteristic
28C53×#343—22no—
29C53×#327—22no—
30C53×#327—22no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53D411yescharacteristic
34C22×D4×C53D411yescharacteristic
35C23×C4×C53D412yes—
36C23×C4×C53C2311yescharacteristic
37C2×D8×C53—22no—
38C2×M16×C53—21no—
39C2×M16×C53D412yes—
40#123×C53—48no—
41#121×C53—21no—
42#121×C53—21no—
43#121×C53—21no—
44#121×C53—21no—
45#121×C53—21no—
46#121×C53—21no—
47#121×C53—21no—
48#121×C53—21no—
49C2×C4⋊C4×C53—24no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53—24no—
59C2×C22⋊C4×C53—24no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C42×C53—21no—
65C2×C42×C53—21no—
66#106×C53—22no—
67#102×C53—22no—
68#102×C53—22no—
69C22⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73#95×C53—24no—
74#95×C53D412yes—
75#95×C53D412yes—
76C24×C53—21no—
77C24×C53—21no—
78C24×C53C2×D411yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C2×D4×C53—44no—
86C2×D4×C53—44no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C22×C4×C53—22no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—42no—
99C22×C4×C53—44no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106D8×C53—82no—
107M16×C53—42no—
108M16×C53—42no—
109C2×C8×C53—22no—
110C2×C8×C53—41no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—48no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—48no—
122C22⋊C4×C53—42no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—84no—
128C42×C53—42no—
129C42×C53—42no—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53#12011yescharacteristic
142C23×C53#12011yescharacteristic
143C23×C53#12111yescharacteristic
144D4×C53—42no—
145D4×C53—42no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—82no—
149D4×C53—82no—
150D4×C53—82no—
151D4×C53—82no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—24no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—44no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—44no—
170C2×C4×C53—42no—
171C2×C4×C53—82no—
172C2×C4×C53#12011yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12711yescharacteristic
176C8×C53—42no—
177C8×C53—81no—
178#10328C5311yesmaximal, characteristic, direct product factor, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—44no—
187C22×C53—84no—
188C22×C53—84no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53#39211yescharacteristic
196C22×C53#45312yes—
197C22×C53#45412yes—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—44no—
202C4×C53—82no—
203C4×C53—82no—
204C4×C53—81no—
205C4×C53—81no—
206C4×C53#45812yes—
207#3192C2×C5311yescharacteristic
208#2504C2×C5311yescharacteristic, complemented
209#2501C2×C5311yescharacteristic, complemented
210#1821C2×C5311yescharacteristic, complemented
211#1507C2×C5311yescharacteristic, complemented
212#1507C2×C5311yescharacteristic, complemented
213#1297C2×C5311yescharacteristic, complemented
214C2×C53—22no—
215C2×C53—22no—
216C2×C53—41no—
217C2×C53—161no—
218C2×C53—161no—
219C2×C53#200211yescharacteristic
220C2×C53#200311yescharacteristic
221C2×C53#202211yescharacteristic
222C2×#123—22no—
223C2×#121C22×C5311yescharacteristic
224C2×#121C22×C5311yescharacteristic
225C22×C4⋊C4C22×C5311yescharacteristic
226#481—21no—
227#481—21no—
228#469—22no—
229#447—21no—
230#447—21no—
231#407C22×C5311yescharacteristic
232#407C22×C5311yescharacteristic
233#399—22no—
234#397—22no—
235#390—21no—
236#390—21no—
237#386—22no—
238#384—21no—
239#382—21no—
240C4×C22⋊C4C22×C5311yescharacteristic
241C4×C22⋊C4C22×C5311yescharacteristic
242#343—22no—
243#327—22no—
244#327—22no—
245C53#1032811yescharacteristic, direct product factor, Sylow
246C22×D4—21no—
247C22×D4—21no—
248C22×D4D4×C5311yescharacteristic
249C22×D4D4×C5311yescharacteristic
250C23×C4D4×C5312yes—
251C23×C4C23×C5311yescharacteristic
252C2×D8—22no—
253C2×M16—21no—
254C2×M16D4×C5312yes—
255#123—48no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4D4×C5311yescharacteristic
272C2×C4⋊C4D4×C5311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—24no—
279C2×C42—21no—
280C2×C42—21no—
281#106—22no—
282#102—22no—
283#102—22no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288#95—24no—
289#95D4×C5312yes—
290#95D4×C5312yes—
291C24—21no—
292C24—21no—
293C24C2×D4×C5311yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4C2×D4×C5311yescharacteristic
316C22×C4C2×D4×C5311yescharacteristic
317C22×C4C2×D4×C5311yescharacteristic
318C22×C4C2×D4×C5311yescharacteristic
319C22×C4C2×D4×C5311yescharacteristic
320C22×C4#41×C5311yescharacteristic
321D8—82no—
322M16—42no—
323M16—42no—
324C2×C8—22no—
325C2×C8—41no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—48no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C22⋊C4—24no—
336C22⋊C4—48no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—84no—
343C42—42no—
344C42—42no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23#120×C5311yescharacteristic
357C23#120×C5311yescharacteristic
358C23#121×C5311yescharacteristic
359D4—42no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—82no—
387C2×C4#120×C5311yescharacteristic
388C2×C4#121×C5311yescharacteristic
389C2×C4#121×C5311yescharacteristic
390C2×C4#127×C5311yescharacteristic
391C8—42no—
392C8—81no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—44no—
401C22—84no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22C53×#39211yescharacteristic
410C22C53×#45312yes—
411C22C53×#45412yes—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—44no—
416C4—82no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4C53×#45812yes—
421C2—22no—
422C2—22no—
423C2—41no—
424C2—161no—
425C2—161no—
426C2C53×#200211yescharacteristic
427C2C53×#200311yescharacteristic
428C2C53×#202211yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC23×C4C23×C53
prime radicalC53×#3192C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2504C2semidirect
3. C53×#2501C2semidirect
4. C53×#1821C2semidirect
5. C53×#1507C2semidirect
6. C53×#1507C2semidirect
7. C53×#1297C2semidirect
178. #10328C53direct (N × H)
208. #2504C2×C53semidirect
209. #2501C2×C53semidirect
210. #1821C2×C53semidirect
211. #1507C2×C53semidirect
212. #1507C2×C53semidirect
213. #1297C2×C53semidirect
245. C53#10328direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31922
C53×#25042
C53×#25012
C53×#18212
C53×#15072
C53×#12972
#1032853