Finite Group Databasebeta

← C53×#11453 #27218965

Subgroups — C53×#11453 #27218965

Order 13568.

562 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2315C211yesmaximal, characteristic
2C2×C53×#323C212yesmaximal
3C53×#1326C214yesmaximal, complemented
4C22×C22⋊C4×C53C411yescharacteristic
5C22×C22⋊C4×C53C411yescharacteristic
6C22×C22⋊C4×C53C2211yescharacteristic
7C2×C22⋊C8×C53—22no—
8C53×#385—22no—
9C53×#385C412yes—
10C53×#385C2212yes—
11C53×#380—22no—
12C53×#323—22no—
13C53×#323C2214yes—
14C25×C53C2×C411yescharacteristic
15C23×C4×C53—21no—
16C23×C4×C53C2×C412yes—
17C23×C4×C53D412yes—
18C22×C8×C53—42no—
19C2×C4⋊C4×C53—22no—
20C2×C4⋊C4×C53—22no—
21C2×C22⋊C4×C53—21no—
22C2×C22⋊C4×C53—21no—
23C2×C22⋊C4×C53—21no—
24C2×C22⋊C4×C53—22no—
25C2×C22⋊C4×C53—42no—
26C2×C22⋊C4×C53—42no—
27C2×C22⋊C4×C53C814yescomplemented
28C2×C22⋊C4×C53C2×C412yes—
29C2×C22⋊C4×C53C2×C411yescharacteristic
30C2×C22⋊C4×C53D412yes—
31C2×C22⋊C4×C53C2311yescharacteristic
32C22⋊C8×C53—24no—
33C22⋊C8×C53—42no—
34#95×C53—22no—
35#95×C53—22no—
36C24×C53—22no—
37C24×C53—22no—
38C24×C53—82no—
39C24×C53C22⋊C411yescharacteristic
40C24×C53C22⋊C411yescharacteristic
41C24×C53C2×C812yes—
42C24×C53M1612yes—
43C24×C53C22×C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—24no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—42no—
50C22×C4×C53—82no—
51C22×C4×C53—82no—
52C22×C4×C53—82no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C2×D412yes—
55C2×C8×C53—44no—
56C2×C8×C53—84no—
57C4⋊C4×C53—82no—
58C4⋊C4×C53—82no—
59C22⋊C4×C53—24no—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—41no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53C2×C814yescomplemented
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—24no—
72C23×C53—24no—
73C23×C53—42no—
74C23×C53—42no—
75C23×C53—42no—
76C23×C53—82no—
77C23×C53—82no—
78C23×C53—82no—
79C23×C53—82no—
80C23×C53—84no—
81C23×C53C22⋊C812yes—
82C23×C53C22⋊C812yes—
83C23×C53#9912yes—
84C23×C53#10112yes—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53C22×C811yescharacteristic
87C23×C53C2×M1611yescharacteristic
88C2×C4×C53—24no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—44no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—82no—
100C2×C4×C53—84no—
101C2×C4×C53—84no—
102C8×C53—84no—
103#11453C5311yesmaximal, characteristic, direct product factor, Sylow
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—84no—
120C22×C53—84no—
121C22×C53#32414yes—
122C22×C53C2×C22⋊C811yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53C2×#10111yescharacteristic
125C4×C53—44no—
126C4×C53—42no—
127C4×C53—42no—
128C4×C53—82no—
129#2315C2×C5311yescharacteristic
130C2×#323C2×C5312yes—
131#1326C2×C5314yescomplemented
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—24no—
135C2×C53—42no—
136C2×C53—41no—
137C2×C53—41no—
138C2×C53C2×#32411yescharacteristic
139C2×C53#212112yes—
140C22×C22⋊C4C4×C5311yescharacteristic
141C22×C22⋊C4C4×C5311yescharacteristic
142C22×C22⋊C4C22×C5311yescharacteristic
143C2×C22⋊C8—22no—
144#385—22no—
145#385C4×C5312yes—
146#385C22×C5312yes—
147#380—22no—
148#323—22no—
149#323C22×C5314yes—
150C53#1145311yescharacteristic, direct product factor, Sylow
151C25C2×C4×C5311yescharacteristic
152C23×C4—21no—
153C23×C4C2×C4×C5312yes—
154C23×C4D4×C5312yes—
155C22×C8—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4C8×C5314yescomplemented
165C2×C22⋊C4C2×C4×C5312yes—
166C2×C22⋊C4C2×C4×C5311yescharacteristic
167C2×C22⋊C4D4×C5312yes—
168C2×C22⋊C4C23×C5311yescharacteristic
169C22⋊C8—24no—
170C22⋊C8—42no—
171#95—22no—
172#95—22no—
173C24—22no—
174C24—22no—
175C24—82no—
176C24C22⋊C4×C5311yescharacteristic
177C24C22⋊C4×C5311yescharacteristic
178C24C2×C8×C5312yes—
179C24M16×C5312yes—
180C24C22×C4×C5311yescharacteristic
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C22⋊C4×C5312yes—
191C22×C4C2×D4×C5312yes—
192C2×C8—44no—
193C2×C8—84no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C22⋊C4—24no—
197C22⋊C4—22no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4C2×C8×C5314yescomplemented
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—24no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—82no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—84no—
218C23C22⋊C8×C5312yes—
219C23C22⋊C8×C5312yes—
220C23#99×C5312yes—
221C23#101×C5312yes—
222C23C2×C22⋊C4×C5311yescharacteristic
223C23C22×C8×C5311yescharacteristic
224C23C2×M16×C5311yescharacteristic
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—84no—
239C8—84no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—21no—
245C22—21no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—84no—
256C22—84no—
257C22C53×#32414yes—
258C22C2×C22⋊C8×C5311yescharacteristic
259C22C2×#99×C5311yescharacteristic
260C22C2×#101×C5311yescharacteristic
261C4—44no—
262C4—42no—
263C4—42no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—24no—
268C2—42no—
269C2—41no—
270C2—41no—
271C2C2×C53×#32411yescharacteristic
272C2C53×#212112yes—
subgroupG/H
center, socleC22×C53C2×#99
commutatorC23C22×C8×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC25×C53C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1326C2semidirect
27. C2×C22⋊C4×C53C8semidirect
66. C22⋊C4×C53C2×C8semidirect
103. #11453C53direct (N × H)
131. #1326C2×C53semidirect
150. C53#11453direct (N × H)
164. C2×C22⋊C4C8×C53semidirect
203. C22⋊C4C2×C8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#23152
C2×C53×#3232
C53×#13262
#1145353