Finite Group Databasebeta

← C53×#9465 #27216977

Subgroups — C53×#9465 #27216977

Order 13568.

554 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2504C211yesmaximal, characteristic
2C53×#2313C211yesmaximal, characteristic, complemented
3C53×#1821C211yesmaximal, characteristic, complemented
4C53×#1510C212yesmaximal
5C53×#1284C212yesmaximal, complemented
6C2×#121×C53C2211yescharacteristic
7C22×C4⋊C4×C53C2211yescharacteristic
8C53×#407C2212yes—
9C53×#397—22no—
10C53×#390C412yescomplemented
11C53×#387—24no—
12C53×#386—21no—
13C53×#386—21no—
14C53×#382C412yescomplemented
15C53×#381C2212yes—
16C53×#380—21no—
17C4×C22⋊C4×C53—21no—
18C4×C22⋊C4×C53C2211yescharacteristic
19C53×#343—22no—
20C53×#327—24no—
21C22×D4×C53—21no—
22C22×D4×C53C2×C411yescharacteristic
23C23×C4×C53D411yescharacteristic
24C23×C4×C53D411yescharacteristic
25C23×C4×C53C2311yescharacteristic
26C2×M16×C53—22no—
27#121×C53—22no—
28#121×C53—22no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C412yes—
34C2×C4⋊C4×C53D412yes—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—21no—
40C2×C42×C53—22no—
41C2×C42×C53C2×C412yes—
42C22⋊C8×C53—22no—
43C22⋊C8×C53—22no—
44#95×C53—22no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—21no—
49#95×C53—22no—
50#95×C53D414yes—
51C24×C53—21no—
52C24×C53C2×D411yescharacteristic
53C2×D4×C53—22no—
54C2×D4×C53—42no—
55C2×D4×C53—44no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—24no—
65C22×C4×C53—42no—
66C22×C4×C53—81no—
67C22×C4×C53—81no—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C22⋊C412yes—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53#4112yes—
74M16×C53—44no—
75C2×C8×C53—42no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—82no—
80C22⋊C4×C53—44no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—82no—
84C22⋊C4×C53—82no—
85C42×C53—42no—
86C42×C53—82no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—41no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—41no—
93C23×C53—42no—
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53#12011yescharacteristic
96C23×C53#12311yescharacteristic
97D4×C53—82no—
98D4×C53—82no—
99D4×C53—82no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—44no—
112C2×C4×C53—42no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—44no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53C4×D412yes—
121C2×C4×C53#12112yes—
122C8×C53—82no—
123#9465C5311yesmaximal, characteristic, direct product factor, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—44no—
132C22×C53—84no—
133C22×C53—81no—
134C22×C53—81no—
135C22×C53—81no—
136C22×C53#38611yescharacteristic
137C22×C53#41012yes—
138C22×C53#45412yes—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—42no—
142C4×C53—42no—
143C4×C53—84no—
144C4×C53—82no—
145C4×C53—81no—
146#2504C2×C5311yescharacteristic
147#2313C2×C5311yescharacteristic, complemented
148#1821C2×C5311yescharacteristic, complemented
149#1510C2×C5312yes—
150#1284C2×C5312yescomplemented
151C2×C53—22no—
152C2×C53—22no—
153C2×C53—41no—
154C2×C53—161no—
155C2×C53#188811yescharacteristic
156C2×C53#189911yescharacteristic
157C2×C53#191111yescharacteristic
158C2×#121C22×C5311yescharacteristic
159C22×C4⋊C4C22×C5311yescharacteristic
160#407C22×C5312yes—
161#397—22no—
162#390C4×C5312yescomplemented
163#387—24no—
164#386—21no—
165#386—21no—
166#382C4×C5312yescomplemented
167#381C22×C5312yes—
168#380—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4C22×C5311yescharacteristic
171#343—22no—
172#327—24no—
173C53#946511yescharacteristic, direct product factor, Sylow
174C22×D4—21no—
175C22×D4C2×C4×C5311yescharacteristic
176C23×C4D4×C5311yescharacteristic
177C23×C4D4×C5311yescharacteristic
178C23×C4C23×C5311yescharacteristic
179C2×M16—22no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C2×C4×C5311yescharacteristic
186C2×C4⋊C4C2×C4×C5312yes—
187C2×C4⋊C4D4×C5312yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—22no—
194C2×C42C2×C4×C5312yes—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95D4×C5314yes—
204C24—21no—
205C24C2×D4×C5311yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4C22⋊C4×C5312yes—
222C22×C4C22⋊C4×C5312yes—
223C22×C4C22×C4×C5311yescharacteristic
224C22×C4C2×D4×C5311yescharacteristic
225C22×C4C2×D4×C5312yes—
226C22×C4#41×C5312yes—
227M16—44no—
228C2×C8—42no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—82no—
233C22⋊C4—44no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C42—42no—
239C42—82no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23C2×C22⋊C4×C5311yescharacteristic
248C23#120×C5311yescharacteristic
249C23#123×C5311yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4C4×D4×C5312yes—
274C2×C4#121×C5312yes—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22C53×#38611yescharacteristic
289C22C53×#41012yes—
290C22C53×#45412yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—84no—
296C4—82no—
297C4—81no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2C53×#188811yescharacteristic
303C2C53×#189911yescharacteristic
304C2C53×#191111yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×#121×C53C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2313C2semidirect
3. C53×#1821C2semidirect
5. C53×#1284C2semidirect
10. C53×#390C4semidirect
14. C53×#382C4semidirect
123. #9465C53direct (N × H)
147. #2313C2×C53semidirect
148. #1821C2×C53semidirect
150. #1284C2×C53semidirect
162. #390C4×C53semidirect
166. #382C4×C53semidirect
173. C53#9465direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#25042
C53×#23132
C53×#18212
C53×#15102
C53×#12842
#946553