Finite Group Databasebeta

← C53×#11764 #27219276

Subgroups — C53×#11764 #27219276

Order 13568.

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

normal subgroups only

This group has 446 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×#2484C211yesmaximal, characteristic
2C2×C53×#343C211yesmaximal, characteristic
3C53×#1463C211yesmaximal, characteristic
4C53×#1353C212yesmaximal, complemented
5C53×#1346C212yesmaximal, complemented
6C22×M16×C53—21no—
7C2×#117×C53C411yescharacteristic
8C22×C4⋊C4×C53C2211yescharacteristic
9C53×#407—21no—
10C53×#400C411yescharacteristic
11C53×#400C2211yescharacteristic
12C53×#398C411yescharacteristic
13C53×#398C2211yescharacteristic
14C53×#387—21no—
15C53×#387—21no—
16C53×#385—21no—
17C53×#381—21no—
18C53×#381C411yescharacteristic
19C53×#343—21no—
20C53×#343C2212yes—
21C53×#324—21no—
22C53×#324C2212yes—
23C23×C4×C53C2×C411yescharacteristic
24C23×C4×C53D411yescharacteristic
25C23×C4×C53D411yescharacteristic
26C2×M16×C53—24no—
27C2×M16×C53—22no—
28C2×M16×C53—41no—
29C2×M16×C53—42no—
30C2×M16×C53—41no—
31C22×C8×C53—41no—
32#117×C53—41no—
33#117×C53—41no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C42×C53—21no—
47C22⋊C8×C53—22no—
48C22⋊C8×C53—41no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—42no—
73C22×C4×C53—42no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80M16×C53—24no—
81M16×C53—44no—
82M16×C53—42no—
83M16×C53—82no—
84C2×C8×C53—44no—
85C2×C8×C53—42no—
86C2×C8×C53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C42×C53—42no—
100C23×C53—22no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53#9911yescharacteristic
105C23×C53#9911yescharacteristic
106C23×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—81no—
138C2×C4×C53#10412yes—
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C8×C53—44no—
142C8×C53—82no—
143#11764C5311yesmaximal, characteristic, direct product factor, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53#35212yes—
154C22×C53C2×#9911yescharacteristic
155C22×C53#41711yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C4×C53—21no—
158C4×C53—22no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53—81no—
167C4×C53—81no—
168#2484C2×C5311yescharacteristic
169C2×#343C2×C5311yescharacteristic
170#1463C2×C5311yescharacteristic
171#1353C2×C5312yescomplemented
172#1346C2×C5312yescomplemented
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—22no—
176C2×C53—41no—
177C2×C53#150411yescharacteristic
178C2×C53C2×#35211yescharacteristic
179C2×C53#213311yescharacteristic
180C22×M16—21no—
181C2×#117C4×C5311yescharacteristic
182C22×C4⋊C4C22×C5311yescharacteristic
183#407—21no—
184#400C4×C5311yescharacteristic
185#400C22×C5311yescharacteristic
186#398C4×C5311yescharacteristic
187#398C22×C5311yescharacteristic
188#387—21no—
189#387—21no—
190#385—21no—
191#381—21no—
192#381C4×C5311yescharacteristic
193#343—21no—
194#343C22×C5312yes—
195#324—21no—
196#324C22×C5312yes—
197C53#1176411yescharacteristic, direct product factor, Sylow
198C23×C4C2×C4×C5311yescharacteristic
199C23×C4D4×C5311yescharacteristic
200C23×C4D4×C5311yescharacteristic
201C2×M16—24no—
202C2×M16—22no—
203C2×M16—41no—
204C2×M16—42no—
205C2×M16—41no—
206C22×C8—41no—
207#117—41no—
208#117—41no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×C4×C5311yescharacteristic
214C2×C4⋊C4C2×C4×C5311yescharacteristic
215C2×C4⋊C4C2×C4×C5311yescharacteristic
216C2×C4⋊C4D4×C5311yescharacteristic
217C2×C4⋊C4D4×C5311yescharacteristic
218C2×C4⋊C4C23×C5311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C2×C4×C5311yescharacteristic
232#95C2×C4×C5311yescharacteristic
233C24C22⋊C4×C5311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C22⋊C4×C5311yescharacteristic
250C22×C4C22⋊C4×C5311yescharacteristic
251C22×C4C22⋊C4×C5311yescharacteristic
252C22×C4C22×C4×C5311yescharacteristic
253C22×C4C2×D4×C5311yescharacteristic
254C22×C4C2×D4×C5311yescharacteristic
255M16—24no—
256M16—44no—
257M16—42no—
258M16—82no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—42no—
275C23—22no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23#99×C5311yescharacteristic
280C23#99×C5311yescharacteristic
281C23C2×C22⋊C4×C5311yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4#104×C5312yes—
314C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
315C2×C4#137×C5311yescharacteristic
316C8—44no—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22C53×#35212yes—
328C22C2×#99×C5311yescharacteristic
329C22C53×#41711yescharacteristic
330C22C2×#104×C5311yescharacteristic
331C4—21no—
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2C53×#150411yescharacteristic
347C2C2×C53×#35211yescharacteristic
348C2C53×#213311yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC24×C53C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1353C2semidirect
5. C53×#1346C2semidirect
143. #11764C53direct (N × H)
171. #1353C2×C53semidirect
172. #1346C2×C53semidirect
197. C53#11764direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#24842
C2×C53×#3432
C53×#14632
C53×#13532
C53×#13462
#1176453