Finite Group Databasebeta

← C53×#11773 #27219285

Subgroups — C53×#11773 #27219285

Order 13568.

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

normal subgroups only

This group has 482 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×#2486C211yesmaximal, characteristic
2C4×#99×C53C211yesmaximal, characteristic
3C53×#1466C211yesmaximal, characteristic
4C53×#1344C214yesmaximal, complemented
5C2×#123×C53—21no—
6C2×C4×D4×C53C2211yescharacteristic
7C22×C4⋊C4×C53C411yescharacteristic, complemented
8C53×#432—21no—
9C2×#99×C53—21no—
10C53×#398C412yescomplemented
11C53×#397C2212yes—
12C53×#389—21no—
13C53×#381—21no—
14C53×#380—21no—
15C53×#380C411yescharacteristic, complemented
16C4×C22⋊C4×C53—21no—
17C53×#342C2212yes—
18C53×#323—21no—
19C53×#323C2212yes—
20C22×D4×C53D411yescharacteristic
21C23×C4×C53—21no—
22C23×C4×C53C2×C411yescharacteristic
23C23×C4×C53D411yescharacteristic
24#123×C53—44no—
25C4×D4×C53—41no—
26C4×D4×C53—41no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—41no—
31C2×C4⋊C4×C53—41no—
32C2×C4⋊C4×C53C2×C412yes—
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53C2311yescharacteristic
39C2×C42×C53—41no—
40C2×C42×C53D411yescharacteristic
41C4⋊C8×C53—41no—
42#99×C53—24no—
43C22⋊C8×C53—22no—
44#97×C53—41no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53—41no—
48#95×C53C2×C412yes—
49C24×C53C22⋊C411yescharacteristic
50C24×C53C2×D411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C2×C8×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—44no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—24no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—44no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—82no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#10412yes—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53AGL(1,ℤ/8)12yes—
98D4×C53—44no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—44no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—81no—
120C2×C4×C53—84no—
121C2×C4×C53—82no—
122C2×C4×C53—82no—
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#9911yescharacteristic
125C8×C53—82no—
126#11773C5311yesmaximal, characteristic, direct product factor, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—22no—
130C22×C53—22no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—81no—
139C22×C53C2×#9911yescharacteristic
140C22×C53#41811yescharacteristic
141C22×C53C2×#10411yescharacteristic
142C4×C53—21no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—82no—
148C4×C53—84no—
149C4×C53—82no—
150C4×C53#35212yes—
151#2486C2×C5311yescharacteristic
152C4×#99C2×C5311yescharacteristic
153#1466C2×C5311yescharacteristic
154#1344C2×C5314yescomplemented
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—42no—
158C2×C53—42no—
159C2×C53#150711yescharacteristic
160C2×C53C2×#35211yescharacteristic
161C2×C53#212911yescharacteristic
162C2×#123—21no—
163C2×C4×D4C22×C5311yescharacteristic
164C22×C4⋊C4C4×C5311yescharacteristic, complemented
165#432—21no—
166C2×#99—21no—
167#398C4×C5312yescomplemented
168#397C22×C5312yes—
169#389—21no—
170#381—21no—
171#380—21no—
172#380C4×C5311yescharacteristic, complemented
173C4×C22⋊C4—21no—
174#342C22×C5312yes—
175#323—21no—
176#323C22×C5312yes—
177C53#1177311yescharacteristic, direct product factor, Sylow
178C22×D4D4×C5311yescharacteristic
179C23×C4—21no—
180C23×C4C2×C4×C5311yescharacteristic
181C23×C4D4×C5311yescharacteristic
182#123—44no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×C4×C5312yes—
191C2×C4⋊C4C2×C4×C5311yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4D4×C5311yescharacteristic
196C2×C22⋊C4C23×C5311yescharacteristic
197C2×C42—41no—
198C2×C42D4×C5311yescharacteristic
199C4⋊C8—41no—
200#99—24no—
201C22⋊C8—22no—
202#97—41no—
203#95—22no—
204#95—22no—
205#95—41no—
206#95C2×C4×C5312yes—
207C24C22⋊C4×C5311yescharacteristic
208C24C2×D4×C5311yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C22⋊C4×C5311yescharacteristic
225C22×C4C22⋊C4×C5311yescharacteristic
226C22×C4C22⋊C4×C5311yescharacteristic
227C22×C4C22×C4×C5311yescharacteristic
228C22×C4C2×D4×C5311yescharacteristic
229C2×C8—42no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—24no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C42—21no—
243C42—21no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#104×C5312yes—
254C23C2×C22⋊C4×C5311yescharacteristic
255C23AGL(1,ℤ/8)×C5312yes—
256D4—44no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#99×C5311yescharacteristic
282C2×C4#99×C5311yescharacteristic
283C8—82no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22C2×#99×C5311yescharacteristic
297C22C53×#41811yescharacteristic
298C22C2×#104×C5311yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—84no—
306C4—82no—
307C4C53×#35212yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2C53×#150711yescharacteristic
313C2C2×C53×#35211yescharacteristic
314C2C53×#212911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC22×D4×C53D4
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
4. C53×#1344C2semidirect
7. C22×C4⋊C4×C53C4semidirect
10. C53×#398C4semidirect
15. C53×#380C4semidirect
126. #11773C53direct (N × H)
154. #1344C2×C53semidirect
164. C22×C4⋊C4C4×C53semidirect
167. #398C4×C53semidirect
172. #380C4×C53semidirect
177. C53#11773direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#24862
C4×#99×C532
C53×#14662
C53×#13442
#1177353