Finite Group Databasebeta

← C53×#7592 #27215104

Subgroups — C53×#7592 #27215104

Order 13568.

846 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C2×C53×#390C211yesmaximal, characteristic
2C53×#1817C212yesmaximal, complemented
3C53×#1270C214yesmaximal, complemented
4C23×D4×C53C411yescharacteristic
5C22×C4⋊C4×C53C411yescharacteristic
6C22×C22⋊C4×C53—22no—
7C22×C42×C53C2211yescharacteristic
8C2×C22⋊C8×C53—22no—
9C53×#390—22no—
10C53×#390C412yes—
11C53×#390C2212yes—
12C53×#336—22no—
13C53×#336C2214yescomplemented
14C25×C53—21no—
15C22×D4×C53—21no—
16C22×D4×C53—22no—
17C22×D4×C53—42no—
18C22×D4×C53—42no—
19C22×D4×C53C814yescomplemented
20C22×D4×C53C2×C412yes—
21C23×C4×C53—21no—
22C23×C4×C53C2×C411yescharacteristic
23C23×C4×C53D412yes—
24C22×C8×C53—24no—
25C2×C4⋊C4×C53—21no—
26C2×C4⋊C4×C53—42no—
27C2×C4⋊C4×C53C2×C412yes—
28C2×C22⋊C4×C53—24no—
29C2×C22⋊C4×C53—24no—
30C2×C22⋊C4×C53—48no—
31C2×C42×C53—42no—
32C2×C42×C53C2×C411yescharacteristic
33C2×C42×C53D412yes—
34C2×C42×C53C2311yescharacteristic
35C22⋊C8×C53—44no—
36C24×C53—24no—
37C24×C53—22no—
38C24×C53—44no—
39C24×C53—44no—
40C24×C53—42no—
41C24×C53—42no—
42C24×C53C22⋊C411yescharacteristic
43C2×D4×C53—24no—
44C2×D4×C53—22no—
45C2×D4×C53—48no—
46C2×D4×C53—42no—
47C2×D4×C53—42no—
48C2×D4×C53—48no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53C2×C814yes—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—24no—
55C22×C4×C53—24no—
56C22×C4×C53—22no—
57C22×C4×C53—42no—
58C22×C4×C53—44no—
59C22×C4×C53—42no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C2×C812yes—
63C22×C4×C53M1612yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C2×C8×C53—44no—
67C2×C8×C53—44no—
68C2×C8×C53—44no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C22⋊C4×C53—48no—
72C22⋊C4×C53—48no—
73C42×C53—41no—
74C42×C53—42no—
75C42×C53—41no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—24no—
79C23×C53—416no—
80C23×C53—48no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53—44no—
84C23×C53—44no—
85C23×C53—84no—
86C23×C53—88no—
87C23×C53—84no—
88C23×C53C22⋊C812yes—
89C23×C53#9912yes—
90C23×C53#10012yes—
91C23×C53C2×C22⋊C411yescharacteristic
92D4×C53—44no—
93D4×C53—48no—
94D4×C53—44no—
95C2×C4×C53—24no—
96C2×C4×C53—24no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—48no—
107C2×C4×C53—42no—
108C2×C4×C53—44no—
109C2×C4×C53—84no—
110C2×C4×C53C22⋊C812yes—
111C2×C4×C53C22×C811yescharacteristic
112C2×C4×C53C2×M1611yescharacteristic
113C2×C4×C53AGL(1,ℤ/8)14yes—
114C8×C53—84no—
115#7592C5311yesmaximal, characteristic, direct product factor, Sylow
116C22×C53—22no—
117C22×C53—24no—
118C22×C53—24no—
119C22×C53—22no—
120C22×C53—44no—
121C22×C53—48no—
122C22×C53—816no—
123C22×C53—82no—
124C22×C53—84no—
125C22×C53—82no—
126C22×C53#32314yes—
127C22×C53C2×C22⋊C811yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53C2×#10011yescharacteristic
130C22×C53#41812yes—
131C22×C53#42112yes—
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—24no—
136C4×C53—42no—
137C4×C53—82no—
138C2×#390C2×C5311yescharacteristic
139#1817C2×C5312yescomplemented
140#1270C2×C5314yescomplemented
141C2×C53—24no—
142C2×C53—84no—
143C2×C53C2×#32311yescharacteristic
144C2×C53#147812yes—
145C2×C53#151012yes—
146C2×C53#153311yescharacteristic
147C2×C53#155511yescharacteristic
148C23×D4C4×C5311yescharacteristic
149C22×C4⋊C4C4×C5311yescharacteristic
150C22×C22⋊C4—22no—
151C22×C42C22×C5311yescharacteristic
152C2×C22⋊C8—22no—
153#390—22no—
154#390C4×C5312yes—
155#390C22×C5312yes—
156#336—22no—
157#336C22×C5314yescomplemented
158C53#759211yescharacteristic, direct product factor, Sylow
159C25—21no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4C8×C5314yescomplemented
165C22×D4C2×C4×C5312yes—
166C23×C4—21no—
167C23×C4C2×C4×C5311yescharacteristic
168C23×C4D4×C5312yes—
169C22×C8—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×C4×C5312yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—48no—
176C2×C42—42no—
177C2×C42C2×C4×C5311yescharacteristic
178C2×C42D4×C5312yes—
179C2×C42C23×C5311yescharacteristic
180C22⋊C8—44no—
181C24—24no—
182C24—22no—
183C24—44no—
184C24—44no—
185C24—42no—
186C24—42no—
187C24C22⋊C4×C5311yescharacteristic
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—48no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—48no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4C2×C8×C5314yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4C22⋊C4×C5312yes—
206C22×C4C22⋊C4×C5311yescharacteristic
207C22×C4C2×C8×C5312yes—
208C22×C4M16×C5312yes—
209C22×C4C22×C4×C5311yescharacteristic
210C22×C4C2×D4×C5312yes—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—44no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C42—41no—
219C42—42no—
220C42—41no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—416no—
225C23—48no—
226C23—42no—
227C23—42no—
228C23—44no—
229C23—44no—
230C23—88no—
231C23—84no—
232C23—84no—
233C23C22⋊C8×C5312yes—
234C23#99×C5312yes—
235C23#100×C5312yes—
236C23C2×C22⋊C4×C5311yescharacteristic
237D4—44no—
238D4—48no—
239D4—44no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—48no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—84no—
255C2×C4C22⋊C8×C5312yes—
256C2×C4C22×C8×C5311yescharacteristic
257C2×C4C2×M16×C5311yescharacteristic
258C2×C4AGL(1,ℤ/8)×C5314yes—
259C8—84no—
260C22—24no—
261C22—24no—
262C22—22no—
263C22—22no—
264C22—44no—
265C22—48no—
266C22—816no—
267C22—84no—
268C22—82no—
269C22—82no—
270C22C53×#32314yes—
271C22C2×C22⋊C8×C5311yescharacteristic
272C22C2×#99×C5311yescharacteristic
273C22C2×#100×C5311yescharacteristic
274C22C53×#41812yes—
275C22C53×#42112yes—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—24no—
280C4—42no—
281C4—82no—
282C2—24no—
283C2—84no—
284C2C2×C53×#32311yescharacteristic
285C2C53×#147812yes—
286C2C53×#151012yes—
287C2C53×#153311yescharacteristic
288C2C53×#155511yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC2×C4C22×C8×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC23×D4×C53C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1817C2semidirect
3. C53×#1270C2semidirect
13. C53×#336C22semidirect
19. C22×D4×C53C8semidirect
115. #7592C53direct (N × H)
139. #1817C2×C53semidirect
140. #1270C2×C53semidirect
157. #336C22×C53semidirect
158. C53#7592direct (N × H)
164. C22×D4C8×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#3902
C53×#18172
C53×#12702
#759253