Finite Group Databasebeta

← C53×#60099 #27267611

Subgroups — C53×#60099 #27267611

Order 13568.

2738 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#568C212yesmaximal
2C2×C4×#41×C53C211yesmaximal, characteristic
3C53×#2906C2116yesmaximal, complemented
4C53×#2905C216yesmaximal, complemented
5C2×C53×#423C216yesmaximal, complemented
6C22×#41×C53C2211yescharacteristic
7C53×#568—216no—
8C2×#131×C53C22116yes—
9C22×M16×C53C22112yes—
10C4×#41×C53C414yes—
11C4×#41×C53C414yes—
12C4×#41×C53C2218yes—
13C2×C4×Q8×C53C412yes—
14C2×C4×D4×C53C416yes—
15C2×#117×C53C2213yes—
16C22×C42×C53C2213yes—
17C53×#424C22148yescomplemented
18C53×#423C2214yes—
19C53×#423C22112yes—
20C53×#423C22124yescomplemented
21C2×C4⋊C8×C53C22124yescomplemented
22C2×#41×C53C2×C414yes—
23C2×#41×C53D418yes—
24C2×#41×C53Q818yes—
25C2×#41×C53C2314yes—
26C22×Q8×C53C2×C411yescharacteristic
27C22×D4×C53C2×C413yes—
28C23×C4×C53C2×C416yes—
29C23×C4×C53C2313yes—
30#131×C53—264no—
31C2×M16×C53—248no—
32C2×M16×C53C2318yes—
33C2×M16×C53C23124yes—
34C22×C8×C53C23124yes—
35C4×Q8×C53C2×C418yes—
36C4×Q8×C53C2×C418yes—
37C4×D4×C53C2×C4124yes—
38C4×D4×C53C2×C4124yes—
39#117×C53C2×C4112yes—
40#117×C53C23112yes—
41C2×C4⋊C4×C53C2×C416yes—
42C2×C22⋊C4×C53C2×C416yes—
43C2×C42×C53C2×C416yes—
44C2×C42×C53C2×C416yes—
45C2×C42×C53C2×C416yes—
46C2×C42×C53C2311yescharacteristic
47C2×C42×C53C2313yes—
48C2×C42×C53C23112yes—
49C4⋊C8×C53C23116yes—
50C4⋊C8×C53C23148yes—
51C24×C53C22×C413yes—
52#41×C53C4⋊C4132yes—
53#41×C53C2×D4116yes—
54#41×C53C2×Q8116yes—
55C2×Q8×C53C4⋊C418yes—
56C2×Q8×C53C22×C414yes—
57C2×D4×C53C4⋊C4124yes—
58C2×D4×C53C22×C4112yes—
59C22×C4×C53—412no—
60C22×C4×C53C22×C412yes—
61C22×C4×C53C22×C4112yes—
62C22×C4×C53C22×C416yes—
63C22×C4×C53C22×C4112yes—
64C22×C4×C53C22×C413yes—
65C22×C4×C53C22×C416yes—
66C22×C4×C53C2×D4112yes—
67C22×C4×C53C2×Q8112yes—
68C22×C4×C53C2411yescharacteristic
69C22×C4×C53C2416yes—
70M16×C53—216no—
71M16×C53—248no—
72C2×C8×C53—248no—
73C2×C8×C53C24116yes—
74C4⋊C4×C53C22×C4124yes—
75C22⋊C4×C53C22×C4124yes—
76C42×C53C22×C411yescharacteristic
77C42×C53C22×C411yescharacteristic
78C42×C53C22×C416yes—
79C42×C53C22×C4112yes—
80C42×C53C22×C4112yes—
81C42×C53C2412yes—
82C42×C53C2416yes—
83C23×C53—46no—
84C23×C53C2×C4⋊C4112yes—
85C23×C53C23×C411yescharacteristic
86C23×C53C23×C416yes—
87Q8×C53C2×C4⋊C4116yes—
88D4×C53C2×C4⋊C4148yes—
89C2×C4×C53—24no—
90C2×C4×C53—224no—
91C2×C4×C53—412no—
92C2×C4×C53—46no—
93C2×C4×C53C2×C4⋊C4112yes—
94C2×C4×C53C2×C4⋊C4124yes—
95C2×C4×C53C23×C412yes—
96C2×C4×C53C23×C4112yes—
97C2×C4×C53C23×C411yescharacteristic
98C2×C4×C53C23×C416yes—
99C2×C4×C53C23×C412yes—
100C2×C4×C53C22×D412yes—
101C2×C4×C53C22×D4112yes—
102C2×C4×C53C22×Q812yes—
103C2×C4×C53C22×Q8112yes—
104C2×C4×C53C2511yescharacteristic
105C8×C53—216no—
106#60099C5311yesmaximal, characteristic, direct product factor, Sylow
107C22×C53—22no—
108C22×C53—212no—
109C22×C53—412no—
110C22×C53C22×C4⋊C412yes—
111C22×C53C22×C4⋊C4112yes—
112C22×C53C24×C411yescharacteristic
113C4×C53—22no—
114C4×C53—212no—
115C4×C53C22×C4⋊C412yes—
116C4×C53C22×C4⋊C4112yes—
117C4×C53#56814yescomplemented
118C4×C53C23×D411yescharacteristic
119C4×C53C23×Q811yescharacteristic
120C2×#568C2×C5312yes—
121C2×C4×#41C2×C5311yescharacteristic
122#2906C2×C53116yescomplemented
123#2905C2×C5316yescomplemented
124C2×#423C2×C5316yescomplemented
125C2×C53—22no—
126C2×C53—212no—
127C2×C53C23×C4⋊C411yescharacteristic
128C2×C53C2×#56812yes—
129C22×#41C22×C5311yescharacteristic
130#568—216no—
131C2×#131C22×C53116yes—
132C22×M16C22×C53112yes—
133C4×#41C4×C5314yes—
134C4×#41C4×C5314yes—
135C4×#41C22×C5318yes—
136C2×C4×Q8C4×C5312yes—
137C2×C4×D4C4×C5316yes—
138C2×#117C22×C5313yes—
139C22×C42C22×C5313yes—
140#424C22×C53148yescomplemented
141#423C22×C5314yes—
142#423C22×C53112yes—
143#423C22×C53124yescomplemented
144C2×C4⋊C8C22×C53124yescomplemented
145C53#6009911yescharacteristic, direct product factor, Sylow
146C2×#41C2×C4×C5314yes—
147C2×#41D4×C5318yes—
148C2×#41Q8×C5318yes—
149C2×#41C23×C5314yes—
150C22×Q8C2×C4×C5311yescharacteristic
151C22×D4C2×C4×C5313yes—
152C23×C4C2×C4×C5316yes—
153C23×C4C23×C5313yes—
154#131—264no—
155C2×M16—248no—
156C2×M16C23×C5318yes—
157C2×M16C23×C53124yes—
158C22×C8C23×C53124yes—
159C4×Q8C2×C4×C5318yes—
160C4×Q8C2×C4×C5318yes—
161C4×D4C2×C4×C53124yes—
162C4×D4C2×C4×C53124yes—
163#117C2×C4×C53112yes—
164#117C23×C53112yes—
165C2×C4⋊C4C2×C4×C5316yes—
166C2×C22⋊C4C2×C4×C5316yes—
167C2×C42C2×C4×C5316yes—
168C2×C42C2×C4×C5316yes—
169C2×C42C2×C4×C5316yes—
170C2×C42C23×C5313yes—
171C2×C42C23×C53112yes—
172C2×C42C23×C5311yescharacteristic
173C4⋊C8C23×C53116yes—
174C4⋊C8C23×C53148yes—
175C24C22×C4×C5313yes—
176#41C4⋊C4×C53132yes—
177#41C2×D4×C53116yes—
178#41C2×Q8×C53116yes—
179C2×Q8C4⋊C4×C5318yes—
180C2×Q8C22×C4×C5314yes—
181C2×D4C4⋊C4×C53124yes—
182C2×D4C22×C4×C53112yes—
183C22×C4—412no—
184C22×C4C22×C4×C5312yes—
185C22×C4C22×C4×C53112yes—
186C22×C4C22×C4×C5316yes—
187C22×C4C22×C4×C53112yes—
188C22×C4C22×C4×C5313yes—
189C22×C4C22×C4×C5316yes—
190C22×C4C2×D4×C53112yes—
191C22×C4C2×Q8×C53112yes—
192C22×C4C24×C5311yescharacteristic
193C22×C4C24×C5316yes—
194M16—216no—
195M16—248no—
196C2×C8—248no—
197C2×C8C24×C53116yes—
198C4⋊C4C22×C4×C53124yes—
199C22⋊C4C22×C4×C53124yes—
200C42C22×C4×C5311yescharacteristic
201C42C22×C4×C5311yescharacteristic
202C42C22×C4×C5316yes—
203C42C22×C4×C53112yes—
204C42C22×C4×C53112yes—
205C42C24×C5312yes—
206C42C24×C5316yes—
207C23—46no—
208C23C2×C4⋊C4×C53112yes—
209C23C23×C4×C5316yes—
210C23C23×C4×C5311yescharacteristic
211Q8C2×C4⋊C4×C53116yes—
212D4C2×C4⋊C4×C53148yes—
213C2×C4—24no—
214C2×C4—224no—
215C2×C4—412no—
216C2×C4—46no—
217C2×C4C2×C4⋊C4×C53112yes—
218C2×C4C2×C4⋊C4×C53124yes—
219C2×C4C23×C4×C5312yes—
220C2×C4C23×C4×C53112yes—
221C2×C4C23×C4×C5311yescharacteristic
222C2×C4C23×C4×C5316yes—
223C2×C4C23×C4×C5312yes—
224C2×C4C22×D4×C53112yes—
225C2×C4C22×D4×C5312yes—
226C2×C4C22×Q8×C5312yes—
227C2×C4C22×Q8×C53112yes—
228C2×C4C25×C5311yescharacteristic
229C8—216no—
230C22—22no—
231C22—212no—
232C22—412no—
233C22C22×C4⋊C4×C5312yes—
234C22C22×C4⋊C4×C53112yes—
235C22C24×C4×C5311yescharacteristic
236C4—22no—
237C4—212no—
238C4C22×C4⋊C4×C5312yes—
239C4C22×C4⋊C4×C53112yes—
240C4C53×#56814yescomplemented
241C4C23×D4×C5311yescharacteristic
242C4C23×Q8×C5311yescharacteristic
243C2—22no—
244C2—212no—
245C2C23×C4⋊C4×C5311yescharacteristic
246C2C2×C53×#56812yes—
subgroupG/H
centerC2×C4×C53C25
commutatorC22C24×C4×C53
Frattini, cosocleC2×C4C25×C53
socleC22×C53C24×C4
prime radicalC22×#41×C53C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#2906C2semidirect
4. C53×#2905C2semidirect
5. C2×C53×#423C2semidirect
17. C53×#424C22semidirect
20. C53×#423C22semidirect
21. C2×C4⋊C8×C53C22semidirect
106. #60099C53direct (N × H)
117. C4×C53#568semidirect
122. #2906C2×C53semidirect
123. #2905C2×C53semidirect
124. C2×#423C2×C53semidirect
140. #424C22×C53semidirect
143. #423C22×C53semidirect
144. C2×C4⋊C8C22×C53semidirect
145. C53#60099direct (N × H)
240. C4C53×#568semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5682
C2×C4×#41×C532
C53×#29062
C53×#29052
C2×C53×#4232
#6009953