Finite Group Databasebeta

← C53×#11768 #27219280

Subgroups — C53×#11768 #27219280

Order 13568.

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

normal subgroups only

This group has 462 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×#2583C211yesmaximal, characteristic
2C2×C53×#343C211yesmaximal, characteristic
3C53×#1463C211yesmaximal, characteristic
4C53×#1354C212yesmaximal, complemented
5C53×#1346C212yesmaximal, complemented
6C22×M16×C53—21no—
7C2×C22⋊Q8×C53C411yescharacteristic
8C22×C4⋊C4×C53C2211yescharacteristic
9C53×#407—21no—
10C53×#398C411yescharacteristic
11C53×#398C2211yescharacteristic
12C53×#395C411yescharacteristic
13C53×#395C2211yescharacteristic
14C53×#389—21no—
15C53×#389—21no—
16C53×#385—21no—
17C53×#385—21no—
18C4×C22⋊C4×C53C411yescharacteristic
19C53×#343—21no—
20C53×#343C2212yes—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Q8×C53C2×C411yescharacteristic
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53D411yescharacteristic
26C23×C4×C53D411yescharacteristic
27C2×M16×C53—24no—
28C2×M16×C53—22no—
29C2×M16×C53—41no—
30C2×M16×C53—42no—
31C2×M16×C53—41no—
32C22×C8×C53—41no—
33C22⋊Q8×C53—41no—
34C22⋊Q8×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—41no—
41C2×C4⋊C4×C53—41no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53C2311yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C42×C53—21no—
50C22⋊C8×C53—22no—
51C22⋊C8×C53—41no—
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
59C2×Q8×C53—41no—
60C2×Q8×C53—81no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
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—21no—
73C22×C4×C53—21no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82M16×C53—24no—
83M16×C53—44no—
84M16×C53—42no—
85M16×C53—82no—
86C2×C8×C53—44no—
87C2×C8×C53—42no—
88C2×C8×C53—42no—
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—42no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—81no—
99C4⋊C4×C53—81no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C42×C53—81no—
104C23×C53—22no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—41no—
108C23×C53#9911yescharacteristic
109C23×C53#9911yescharacteristic
110C23×C53C2×C22⋊C411yescharacteristic
111Q8×C53—82no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
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—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53#10412yes—
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C8×C53—44no—
148C8×C53—82no—
149#11768C5311yesmaximal, characteristic, direct product factor, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53#35612yes—
160C22×C53C2×#9911yescharacteristic
161C22×C53#41711yescharacteristic
162C22×C53C2×#10411yescharacteristic
163C4×C53—21no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—42no—
167C4×C53—42no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—81no—
173C4×C53—81no—
174#2583C2×C5311yescharacteristic
175C2×#343C2×C5311yescharacteristic
176#1463C2×C5311yescharacteristic
177#1354C2×C5312yescomplemented
178#1346C2×C5312yescomplemented
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—22no—
182C2×C53—41no—
183C2×C53#150411yescharacteristic
184C2×C53#212111yescharacteristic
185C2×C53C2×#35611yescharacteristic
186C22×M16—21no—
187C2×C22⋊Q8C4×C5311yescharacteristic
188C22×C4⋊C4C22×C5311yescharacteristic
189#407—21no—
190#398C4×C5311yescharacteristic
191#398C22×C5311yescharacteristic
192#395C4×C5311yescharacteristic
193#395C22×C5311yescharacteristic
194#389—21no—
195#389—21no—
196#385—21no—
197#385—21no—
198C4×C22⋊C4C4×C5311yescharacteristic
199#343—21no—
200#343C22×C5312yes—
201#324—21no—
202#324C22×C5312yes—
203C53#1176811yescharacteristic, direct product factor, Sylow
204C22×Q8C2×C4×C5311yescharacteristic
205C23×C4C2×C4×C5311yescharacteristic
206C23×C4D4×C5311yescharacteristic
207C23×C4D4×C5311yescharacteristic
208C2×M16—24no—
209C2×M16—22no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—41no—
213C22×C8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×C4×C5311yescharacteristic
224C2×C4⋊C4C2×C4×C5311yescharacteristic
225C2×C4⋊C4D4×C5311yescharacteristic
226C2×C4⋊C4D4×C5311yescharacteristic
227C2×C4⋊C4C23×C5311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C22⋊C8—22no—
232C22⋊C8—41no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95C2×C4×C5311yescharacteristic
238#95C2×C4×C5311yescharacteristic
239C24C22⋊C4×C5311yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C22⋊C4×C5311yescharacteristic
258C22×C4C22⋊C4×C5311yescharacteristic
259C22×C4C22⋊C4×C5311yescharacteristic
260C22×C4C22×C4×C5311yescharacteristic
261C22×C4C2×D4×C5311yescharacteristic
262C22×C4C2×D4×C5311yescharacteristic
263M16—24no—
264M16—44no—
265M16—42no—
266M16—82no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—24no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—81no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23#99×C5311yescharacteristic
290C23#99×C5311yescharacteristic
291C23C2×C22⋊C4×C5311yescharacteristic
292Q8—82no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4#104×C5312yes—
326C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
327C2×C4#137×C5311yescharacteristic
328C8—44no—
329C8—82no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22C53×#35612yes—
340C22C2×#99×C5311yescharacteristic
341C22C53×#41711yescharacteristic
342C22C2×#104×C5311yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2C53×#150411yescharacteristic
359C2C53×#212111yescharacteristic
360C2C2×C53×#35611yescharacteristic
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×#1354C2semidirect
5. C53×#1346C2semidirect
149. #11768C53direct (N × H)
177. #1354C2×C53semidirect
178. #1346C2×C53semidirect
203. C53#11768direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#25832
C2×C53×#3432
C53×#14632
C53×#13542
C53×#13462
#1176853