Finite Group Databasebeta

← C23×#34530 #19182156

Subgroups — C23×#34530 #19182156

Order 5888.

1800 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#564C211yesmaximal, characteristic
2C23×#2382C213yesmaximal
3C23×#2371C216yesmaximal
4C23×#2370C216yesmaximal
5C23×#2336C216yesmaximal
6C23×#2330C216yesmaximal
7C4×C23×#125C213yesmaximal
8C23×#564C4116yescomplemented
9C2×C23×#125C2213yes—
10C2×C23×#125C2216yes—
11C2×C23×#125C2216yes—
12C23×#389C2213yes—
13C23×#389C2213yes—
14C23×#389C2216yes—
15C23×#389C2216yes—
16C23×#389C2216yes—
17C23×#389C2216yes—
18C23×#387C2216yes—
19C23×#387C2216yes—
20C23×#387C2216yes—
21C23×#387C2216yes—
22C23×#387C2216yes—
23C23×#387C2216yes—
24C23×#387C2216yes—
25C23×#387C2216yes—
26C23×#387C2216yes—
27C23×#387C2216yes—
28C23×#384C2212yes—
29C23×#383C2212yes—
30C23×#382C2213yes—
31C4×C4⋊C4×C23C2216yes—
32C4×C4⋊C4×C23C2213yes—
33C4×C4⋊C4×C23C2216yes—
34C4×C4⋊C4×C23C2213yes—
35C4×C4⋊C4×C23C2216yes—
36C4×C4⋊C4×C23C2216yes—
37C4×C4⋊C4×C23C2216yes—
38C23×#376C2216yes—
39C43×C23C2211yescharacteristic
40C23×#125C2×C4124yes—
41C23×#125C2×C4148yes—
42C23×#125C2×C4148yes—
43C2×C4⋊C4×C23C2316yes—
44C2×C4⋊C4×C23C2313yes—
45C2×C4⋊C4×C23C2313yes—
46C2×C4⋊C4×C23C2316yes—
47C2×C4⋊C4×C23C2316yes—
48C2×C4⋊C4×C23C2316yes—
49C2×C4⋊C4×C23C2316yes—
50C2×C4⋊C4×C23C2313yes—
51C2×C4⋊C4×C23C2316yes—
52C2×C4⋊C4×C23C2313yes—
53C2×C4⋊C4×C23C2316yes—
54C2×C4⋊C4×C23C2316yes—
55C2×C42×C23C2313yes—
56C2×C42×C23C2316yes—
57C2×C42×C23C2313yes—
58C2×C42×C23C2316yes—
59C2×C42×C23C2316yes—
60C2×C42×C23C2316yes—
61C2×C42×C23C2311yescharacteristic
62C2×C42×C23C2312yes—
63C2×C42×C23C2312yes—
64C23×#95C2316yes—
65C23×#95C2316yes—
66C23×#95C2316yes—
67C23×#95C2316yes—
68C23×#95C2316yes—
69C23×#95C2316yes—
70C23×#95C2316yes—
71C23×#95C2316yes—
72C23×#95C2316yes—
73C23×#95C2316yes—
74C22×C4×C23C2416yes—
75C22×C4×C23C2411yescharacteristic
76C22×C4×C23C2416yes—
77C22×C4×C23C2413yes—
78C22×C4×C23C2416yes—
79C22×C4×C23C2416yes—
80C22×C4×C23C2413yes—
81C4⋊C4×C23—212no—
82C4⋊C4×C23—46no—
83C4⋊C4×C23—43no—
84C4⋊C4×C23—46no—
85C4⋊C4×C23—46no—
86C4⋊C4×C23—43no—
87C4⋊C4×C23C22×C4124yes—
88C4⋊C4×C23C22×C4112yes—
89C4⋊C4×C23C22×C4124yes—
90C4⋊C4×C23C22×C4112yes—
91C4⋊C4×C23C22×C4124yes—
92C4⋊C4×C23C22×C4124yes—
93C42×C23—26no—
94C42×C23—212no—
95C42×C23—46no—
96C42×C23—43no—
97C42×C23—46no—
98C42×C23—46no—
99C42×C23C22×C414yes—
100C42×C23C22×C418yes—
101C42×C23C22×C418yes—
102#34530C2311yesmaximal, characteristic, direct product factor, Sylow
103C23×C23C2511yescharacteristic
104C2×C4×C23—23no—
105C2×C4×C23—26no—
106C2×C4×C23—23no—
107C2×C4×C23—26no—
108C2×C4×C23—23no—
109C2×C4×C23—26no—
110C2×C4×C23—23no—
111C2×C4×C23—26no—
112C2×C4×C23—26no—
113C2×C4×C23—26no—
114C2×C4×C23—26no—
115C2×C4×C23—26no—
116C2×C4×C23—26no—
117C2×C4×C23—23no—
118C2×C4×C23—26no—
119C2×C4×C23C23×C416yes—
120C2×C4×C23C23×C4112yes—
121C2×C4×C23C23×C4112yes—
122C2×C4×C232−1+416yes—
123C2×#564C2×C2311yescharacteristic
124#2382C2×C2313yes—
125#2371C2×C2316yes—
126#2370C2×C2316yes—
127#2336C2×C2316yes—
128#2330C2×C2316yes—
129C4×#125C2×C2313yes—
130C22×C23C24×C411yescharacteristic
131C22×C23C2×2−1+413yes—
132C22×C23#58513yes—
133C4×C23—26no—
134C4×C23—212no—
135C4×C23—212no—
136C4×C23—26no—
137C4×C23—46no—
138C4×C23—46no—
139C4×C23—41no—
140#564C4×C23116yescomplemented
141C2×#125C22×C2313yes—
142C2×#125C22×C2316yes—
143C2×#125C22×C2316yes—
144#389C22×C2313yes—
145#389C22×C2313yes—
146#389C22×C2316yes—
147#389C22×C2316yes—
148#389C22×C2316yes—
149#389C22×C2316yes—
150#387C22×C2316yes—
151#387C22×C2316yes—
152#387C22×C2316yes—
153#387C22×C2316yes—
154#387C22×C2316yes—
155#387C22×C2316yes—
156#387C22×C2316yes—
157#387C22×C2316yes—
158#387C22×C2316yes—
159#387C22×C2316yes—
160#384C22×C2312yes—
161#383C22×C2312yes—
162#382C22×C2313yes—
163C4×C4⋊C4C22×C2316yes—
164C4×C4⋊C4C22×C2313yes—
165C4×C4⋊C4C22×C2316yes—
166C4×C4⋊C4C22×C2313yes—
167C4×C4⋊C4C22×C2316yes—
168C4×C4⋊C4C22×C2316yes—
169C4×C4⋊C4C22×C2316yes—
170#376C22×C2316yes—
171C43C22×C2311yescharacteristic
172C2×C23C4×2−1+413yes—
173C2×C23#356313yes—
174C2×C23#356711yescharacteristic
175#125C2×C4×C23124yes—
176#125C2×C4×C23148yes—
177#125C2×C4×C23148yes—
178C2×C4⋊C4C23×C2316yes—
179C2×C4⋊C4C23×C2313yes—
180C2×C4⋊C4C23×C2313yes—
181C2×C4⋊C4C23×C2316yes—
182C2×C4⋊C4C23×C2316yes—
183C2×C4⋊C4C23×C2316yes—
184C2×C4⋊C4C23×C2313yes—
185C2×C4⋊C4C23×C2316yes—
186C2×C4⋊C4C23×C2316yes—
187C2×C4⋊C4C23×C2313yes—
188C2×C4⋊C4C23×C2316yes—
189C2×C4⋊C4C23×C2316yes—
190C2×C42C23×C2313yes—
191C2×C42C23×C2316yes—
192C2×C42C23×C2313yes—
193C2×C42C23×C2316yes—
194C2×C42C23×C2316yes—
195C2×C42C23×C2316yes—
196C2×C42C23×C2311yescharacteristic
197C2×C42C23×C2312yes—
198C2×C42C23×C2312yes—
199#95C23×C2316yes—
200#95C23×C2316yes—
201#95C23×C2316yes—
202#95C23×C2316yes—
203#95C23×C2316yes—
204#95C23×C2316yes—
205#95C23×C2316yes—
206#95C23×C2316yes—
207#95C23×C2316yes—
208#95C23×C2316yes—
209C23#3453011yescharacteristic, direct product factor, Sylow
210C22×C4C24×C2316yes—
211C22×C4C24×C2316yes—
212C22×C4C24×C2311yescharacteristic
213C22×C4C24×C2316yes—
214C22×C4C24×C2313yes—
215C22×C4C24×C2316yes—
216C22×C4C24×C2313yes—
217C4⋊C4—212no—
218C4⋊C4—46no—
219C4⋊C4—43no—
220C4⋊C4—46no—
221C4⋊C4—46no—
222C4⋊C4—43no—
223C4⋊C4C22×C4×C23112yes—
224C4⋊C4C22×C4×C23124yes—
225C4⋊C4C22×C4×C23124yes—
226C4⋊C4C22×C4×C23112yes—
227C4⋊C4C22×C4×C23124yes—
228C4⋊C4C22×C4×C23124yes—
229C42—26no—
230C42—212no—
231C42—46no—
232C42—43no—
233C42—46no—
234C42—46no—
235C42C22×C4×C2314yes—
236C42C22×C4×C2318yes—
237C42C22×C4×C2318yes—
238C23C25×C2311yescharacteristic
239C2×C4—23no—
240C2×C4—26no—
241C2×C4—23no—
242C2×C4—26no—
243C2×C4—23no—
244C2×C4—26no—
245C2×C4—23no—
246C2×C4—26no—
247C2×C4—26no—
248C2×C4—26no—
249C2×C4—26no—
250C2×C4—26no—
251C2×C4—26no—
252C2×C4—23no—
253C2×C4—26no—
254C2×C4C23×C4×C2316yes—
255C2×C4C23×C4×C23112yes—
256C2×C4C23×C4×C23112yes—
257C2×C4C23×2−1+416yes—
258C22C24×C4×C2311yescharacteristic
259C22C2×C23×2−1+413yes—
260C22C23×#58513yes—
261C4—26no—
262C4—212no—
263C4—212no—
264C4—26no—
265C4—46no—
266C4—46no—
267C4—41no—
268C2C4×C23×2−1+413yes—
269C2C23×#356313yes—
270C2C23×#356711yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C23C25
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
8. C23×#564C4semidirect
102. #34530C23direct (N × H)
140. #564C4×C23semidirect
209. C23#34530direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5642
C23×#23822
C23×#23712
C23×#23702
C23×#23362
C23×#23302
C4×C23×#1252
#3453023