Finite Group Databasebeta

← C23×#60099 #19207725

Subgroups — C23×#60099 #19207725

Order 5888.

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×C23×#568C212yesmaximal
2C2×C4×#41×C23C211yesmaximal, characteristic
3C23×#2906C2116yesmaximal, complemented
4C23×#2905C216yesmaximal, complemented
5C2×C23×#423C216yesmaximal, complemented
6C22×#41×C23C2211yescharacteristic
7C23×#568—216no—
8C2×C23×#131C22116yes—
9C22×M16×C23C22112yes—
10C4×#41×C23C414yes—
11C4×#41×C23C414yes—
12C4×#41×C23C2218yes—
13C2×C4×Q8×C23C412yes—
14C2×C4×D4×C23C416yes—
15C2×C23×#117C2213yes—
16C22×C42×C23C2213yes—
17C23×#424C22148yescomplemented
18C23×#423C2214yes—
19C23×#423C22112yes—
20C23×#423C22124yescomplemented
21C2×C23×C4⋊C8C22124yescomplemented
22C2×#41×C23C2×C414yes—
23C2×#41×C23D418yes—
24C2×#41×C23Q818yes—
25C2×#41×C23C2314yes—
26C22×Q8×C23C2×C411yescharacteristic
27C22×D4×C23C2×C413yes—
28C23×C4×C23C2×C416yes—
29C23×C4×C23C2313yes—
30C23×#131—264no—
31C2×M16×C23—248no—
32C2×M16×C23C2318yes—
33C2×M16×C23C23124yes—
34C22×C8×C23C23124yes—
35C4×Q8×C23C2×C418yes—
36C4×Q8×C23C2×C418yes—
37C4×D4×C23C2×C4124yes—
38C4×D4×C23C2×C4124yes—
39C23×#117C2×C4112yes—
40C23×#117C23112yes—
41C2×C4⋊C4×C23C2×C416yes—
42C2×C22⋊C4×C23C2×C416yes—
43C2×C42×C23C2×C416yes—
44C2×C42×C23C2×C416yes—
45C2×C42×C23C2×C416yes—
46C2×C42×C23C23112yes—
47C2×C42×C23C2311yescharacteristic
48C2×C42×C23C2313yes—
49C23×C4⋊C8C23116yes—
50C23×C4⋊C8C23148yes—
51C24×C23C22×C413yes—
52#41×C23C4⋊C4132yes—
53#41×C23C2×D4116yes—
54#41×C23C2×Q8116yes—
55C2×Q8×C23C4⋊C418yes—
56C2×Q8×C23C22×C414yes—
57C2×D4×C23C4⋊C4124yes—
58C2×D4×C23C22×C4112yes—
59C22×C4×C23—412no—
60C22×C4×C23C22×C412yes—
61C22×C4×C23C22×C4112yes—
62C22×C4×C23C22×C416yes—
63C22×C4×C23C22×C4112yes—
64C22×C4×C23C22×C413yes—
65C22×C4×C23C22×C416yes—
66C22×C4×C23C2×D4112yes—
67C22×C4×C23C2×Q8112yes—
68C22×C4×C23C2411yescharacteristic
69C22×C4×C23C2416yes—
70M16×C23—216no—
71M16×C23—248no—
72C2×C8×C23—248no—
73C2×C8×C23C24116yes—
74C4⋊C4×C23C22×C4124yes—
75C22⋊C4×C23C22×C4124yes—
76C42×C23C22×C411yescharacteristic
77C42×C23C22×C411yescharacteristic
78C42×C23C22×C416yes—
79C42×C23C22×C4112yes—
80C42×C23C22×C4112yes—
81C42×C23C2412yes—
82C42×C23C2416yes—
83#60099C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—46no—
85C23×C23C2×C4⋊C4112yes—
86C23×C23C23×C416yes—
87C23×C23C23×C411yescharacteristic
88Q8×C23C2×C4⋊C4116yes—
89D4×C23C2×C4⋊C4148yes—
90C2×C4×C23—24no—
91C2×C4×C23—224no—
92C2×C4×C23—412no—
93C2×C4×C23—46no—
94C2×C4×C23C2×C4⋊C4112yes—
95C2×C4×C23C2×C4⋊C4124yes—
96C2×C4×C23C23×C412yes—
97C2×C4×C23C23×C4112yes—
98C2×C4×C23C23×C411yescharacteristic
99C2×C4×C23C23×C416yes—
100C2×C4×C23C23×C412yes—
101C2×C4×C23C22×D4112yes—
102C2×C4×C23C22×D412yes—
103C2×C4×C23C22×Q812yes—
104C2×C4×C23C22×Q8112yes—
105C2×C4×C23C2511yescharacteristic
106C8×C23—216no—
107C2×#568C2×C2312yes—
108C2×C4×#41C2×C2311yescharacteristic
109#2906C2×C23116yescomplemented
110#2905C2×C2316yescomplemented
111C2×#423C2×C2316yescomplemented
112C22×C23—22no—
113C22×C23—212no—
114C22×C23—412no—
115C22×C23C22×C4⋊C412yes—
116C22×C23C22×C4⋊C4112yes—
117C22×C23C24×C411yescharacteristic
118C4×C23—22no—
119C4×C23—212no—
120C4×C23C22×C4⋊C412yes—
121C4×C23C22×C4⋊C4112yes—
122C4×C23#56814yescomplemented
123C4×C23C23×D411yescharacteristic
124C4×C23C23×Q811yescharacteristic
125C22×#41C22×C2311yescharacteristic
126#568—216no—
127C2×#131C22×C23116yes—
128C22×M16C22×C23112yes—
129C4×#41C4×C2314yes—
130C4×#41C4×C2314yes—
131C4×#41C22×C2318yes—
132C2×C4×Q8C4×C2312yes—
133C2×C4×D4C4×C2316yes—
134C2×#117C22×C2313yes—
135C22×C42C22×C2313yes—
136#424C22×C23148yescomplemented
137#423C22×C2314yes—
138#423C22×C23112yes—
139#423C22×C23124yescomplemented
140C2×C4⋊C8C22×C23124yescomplemented
141C2×C23—22no—
142C2×C23—212no—
143C2×C23C23×C4⋊C411yescharacteristic
144C2×C23C2×#56812yes—
145C2×#41C2×C4×C2314yes—
146C2×#41D4×C2318yes—
147C2×#41Q8×C2318yes—
148C2×#41C23×C2314yes—
149C22×Q8C2×C4×C2311yescharacteristic
150C22×D4C2×C4×C2313yes—
151C23×C4C2×C4×C2316yes—
152C23×C4C23×C2313yes—
153#131—264no—
154C2×M16—248no—
155C2×M16C23×C2318yes—
156C2×M16C23×C23124yes—
157C22×C8C23×C23124yes—
158C4×Q8C2×C4×C2318yes—
159C4×Q8C2×C4×C2318yes—
160C4×D4C2×C4×C23124yes—
161C4×D4C2×C4×C23124yes—
162#117C2×C4×C23112yes—
163#117C23×C23112yes—
164C2×C4⋊C4C2×C4×C2316yes—
165C2×C22⋊C4C2×C4×C2316yes—
166C2×C42C2×C4×C2316yes—
167C2×C42C2×C4×C2316yes—
168C2×C42C2×C4×C2316yes—
169C2×C42C23×C2313yes—
170C2×C42C23×C23112yes—
171C2×C42C23×C2311yescharacteristic
172C4⋊C8C23×C23116yes—
173C4⋊C8C23×C23148yes—
174C23#6009911yescharacteristic, direct product factor, Sylow
175C24C22×C4×C2313yes—
176#41C4⋊C4×C23132yes—
177#41C2×D4×C23116yes—
178#41C2×Q8×C23116yes—
179C2×Q8C4⋊C4×C2318yes—
180C2×Q8C22×C4×C2314yes—
181C2×D4C4⋊C4×C23124yes—
182C2×D4C22×C4×C23112yes—
183C22×C4—412no—
184C22×C4C22×C4×C2312yes—
185C22×C4C22×C4×C23112yes—
186C22×C4C22×C4×C2316yes—
187C22×C4C22×C4×C23112yes—
188C22×C4C22×C4×C2313yes—
189C22×C4C22×C4×C2316yes—
190C22×C4C2×D4×C23112yes—
191C22×C4C2×Q8×C23112yes—
192C22×C4C24×C2311yescharacteristic
193C22×C4C24×C2316yes—
194M16—216no—
195M16—248no—
196C2×C8—248no—
197C2×C8C24×C23116yes—
198C4⋊C4C22×C4×C23124yes—
199C22⋊C4C22×C4×C23124yes—
200C42C22×C4×C2311yescharacteristic
201C42C22×C4×C2311yescharacteristic
202C42C22×C4×C2316yes—
203C42C22×C4×C23112yes—
204C42C22×C4×C23112yes—
205C42C24×C2312yes—
206C42C24×C2316yes—
207C23—46no—
208C23C2×C4⋊C4×C23112yes—
209C23C23×C4×C2316yes—
210C23C23×C4×C2311yescharacteristic
211Q8C2×C4⋊C4×C23116yes—
212D4C2×C4⋊C4×C23148yes—
213C2×C4—24no—
214C2×C4—224no—
215C2×C4—412no—
216C2×C4—46no—
217C2×C4C2×C4⋊C4×C23112yes—
218C2×C4C2×C4⋊C4×C23124yes—
219C2×C4C23×C4×C2312yes—
220C2×C4C23×C4×C23112yes—
221C2×C4C23×C4×C2312yes—
222C2×C4C23×C4×C2311yescharacteristic
223C2×C4C23×C4×C2316yes—
224C2×C4C22×D4×C23112yes—
225C2×C4C22×D4×C2312yes—
226C2×C4C22×Q8×C2312yes—
227C2×C4C22×Q8×C23112yes—
228C2×C4C25×C2311yescharacteristic
229C8—216no—
230C22—22no—
231C22—212no—
232C22—412no—
233C22C22×C4⋊C4×C2312yes—
234C22C22×C4⋊C4×C23112yes—
235C22C24×C4×C2311yescharacteristic
236C4—22no—
237C4—212no—
238C4C22×C4⋊C4×C2312yes—
239C4C22×C4⋊C4×C23112yes—
240C4C23×#56814yescomplemented
241C4C23×D4×C2311yescharacteristic
242C4C23×Q8×C2311yescharacteristic
243C2—22no—
244C2—212no—
245C2C23×C4⋊C4×C2311yescharacteristic
246C2C2×C23×#56812yes—
subgroupG/H
centerC2×C4×C23C25
commutatorC22C24×C4×C23
Frattini, cosocleC2×C4C25×C23
socleC22×C23C24×C4
prime radicalC22×#41×C23C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#2906C2semidirect
4. C23×#2905C2semidirect
5. C2×C23×#423C2semidirect
17. C23×#424C22semidirect
20. C23×#423C22semidirect
21. C2×C23×C4⋊C8C22semidirect
83. #60099C23direct (N × H)
109. #2906C2×C23semidirect
110. #2905C2×C23semidirect
111. C2×#423C2×C23semidirect
122. C4×C23#568semidirect
136. #424C22×C23semidirect
139. #423C22×C23semidirect
140. C2×C4⋊C8C22×C23semidirect
174. C23#60099direct (N × H)
240. C4C23×#568semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5682
C2×C4×#41×C232
C23×#29062
C23×#29052
C2×C23×#4232
#6009923