Finite Group Databasebeta

← C2×C4×C23×#107 #19168074

Subgroups — C2×C4×C23×#107 #19168074

Order 5888.

1358 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#107C212yesmaximal
2C22×C4×C8×C23C211yesmaximal, characteristic
3C2×C4×C4⋊C4×C23C212yesmaximal
4C4×C23×#107C218yesmaximal, direct product factor
5C2×C23×#340C212yesmaximal
6C23×C8×C23C2211yescharacteristic
7C23×C8×C23C2211yescharacteristic
8C22×C4⋊C4×C23C2214yes—
9C22×C42×C23—22no—
10C22×C42×C23C2211yescharacteristic
11C2×C23×#107C4116yesdirect product factor
12C2×C23×#107C2218yes—
13C2×C4×C8×C23C418yescomplemented
14C2×C4×C8×C23C2214yes—
15C4×C4⋊C4×C23—24no—
16C4×C4⋊C4×C23C2218yes—
17C2×C23×#95—22no—
18C23×#340C2218yes—
19C23×C4×C23—24no—
20C23×C4×C23D411yescharacteristic
21C23×C4×C23D411yescharacteristic
22C23×C4×C23C2311yescharacteristic
23C22×C8×C23—24no—
24C22×C8×C23C2×C414yes—
25C22×C8×C23C2×C414yes—
26C22×C8×C23C2×C418yes—
27C22×C8×C23C2312yes—
28C22×C8×C23C2312yes—
29C2×C4⋊C4×C23—24no—
30C2×C4⋊C4×C23—28no—
31C2×C4⋊C4×C23C2×C4116yes—
32C2×C4⋊C4×C23C2318yes—
33C2×C42×C23—28no—
34C2×C42×C23—48no—
35C2×C42×C23C2×C414yes—
36C2×C42×C23D412yes—
37C2×C42×C23Q814yes—
38C2×C42×C23C2312yes—
39C23×#107C2×C4132yesdirect product factor
40C4×C8×C23C2×C4116yescomplemented
41C23×#95—28no—
42C24×C23C2×D411yescharacteristic
43C22×C4×C23—24no—
44C22×C4×C23—216no—
45C22×C4×C23—28no—
46C22×C4×C23—48no—
47C22×C4×C23—48no—
48C22×C4×C23C4⋊C414yes—
49C22×C4×C23C4⋊C414yes—
50C22×C4×C23D814yes—
51C22×C4×C23Q1614yes—
52C22×C4×C23C22×C412yes—
53C22×C4×C23C22×C414yes—
54C22×C4×C23C2×D412yes—
55C22×C4×C23C2×D411yescharacteristic
56C22×C4×C23C2×D412yes—
57C22×C4×C23C2×Q812yes—
58C22×C4×C23#4114yes—
59C22×C4×C23C2411yescharacteristic
60C2×C8×C23—28no—
61C2×C8×C23—24no—
62C2×C8×C23C42116yescomplemented
63C2×C8×C23C22×C414yes—
64C2×C8×C23C22×C418yes—
65C2×C8×C23C22×C414yes—
66C4⋊C4×C23—216no—
67C4⋊C4×C23—28no—
68C4⋊C4×C23C22×C4116yes—
69C42×C23—416no—
70C42×C23C4⋊C418yes—
71C42×C23C22×C414yes—
72C42×C23C2×Q814yes—
73C2×C4×#107C2311yesmaximal, characteristic, direct product factor, Sylow
74C23×C23C2×C4⋊C412yes—
75C23×C23C4×D414yes—
76C23×C23C2×D812yes—
77C23×C23C2×Q1612yes—
78C23×C23#13514yes—
79C23×C23C22×D411yescharacteristic
80C2×C4×C23—24no—
81C2×C4×C23—28no—
82C2×C4×C23—216no—
83C2×C4×C23—48no—
84C2×C4×C23—48no—
85C2×C4×C23—432no—
86C2×C4×C23#107116yesdirect product factor
87C2×C4×C23C2×C4214yes—
88C2×C4×C23C2×C4⋊C414yes—
89C2×C4×C23C2×C4⋊C412yes—
90C2×C4×C23C2×C4⋊C414yes—
91C2×C4×C23#11714yes—
92C2×C4×C23C4×D414yes—
93C2×C4×C23C4×Q818yes—
94C2×C4×C23C2×D814yes—
95C2×C4×C23C2×Q1614yes—
96C2×C4×C23C23×C411yescharacteristic
97C2×C4×C23C23×C412yes—
98C2×C4×C23C22×Q811yescharacteristic
99C2×C4×C23C2×#4112yes—
100C8×C23—24no—
101C8×C23C2×C4218yescomplemented
102C22×#107C2×C2312yes—
103C22×C4×C8C2×C2311yescharacteristic
104C2×C4×C4⋊C4C2×C2312yes—
105C4×#107C2×C2318yesdirect product factor
106C2×#340C2×C2312yes—
107C22×C23C4×C4⋊C414yes—
108C22×C23C2×#10714yes—
109C22×C23#42714yes—
110C22×C23C4×D818yes—
111C22×C23C4×Q1618yes—
112C22×C23C22×C4⋊C411yescharacteristic
113C22×C23C2×C4×D412yes—
114C22×C23C22×D811yescharacteristic
115C22×C23C22×Q1611yescharacteristic
116C22×C23C2×#13512yes—
117C4×C23—24no—
118C4×C23—416no—
119C4×C23C4×C4⋊C414yes—
120C4×C23C2×#10718yesdirect product factor
121C4×C23C22×C4211yescharacteristic
122C4×C23C2×#11711yescharacteristic
123C4×C23C2×C4×Q812yes—
124C23×C8C22×C2311yescharacteristic
125C23×C8C22×C2311yescharacteristic
126C22×C4⋊C4C22×C2314yes—
127C22×C42—22no—
128C22×C42C22×C2311yescharacteristic
129C2×#107C4×C23116yesdirect product factor
130C2×#107C22×C2318yes—
131C2×C4×C8C4×C2318yescomplemented
132C2×C4×C8C22×C2314yes—
133C4×C4⋊C4—24no—
134C4×C4⋊C4C22×C2318yes—
135C2×#95—22no—
136#340C22×C2318yes—
137C2×C23C4×#10718yesdirect product factor
138C2×C23C2×C4×C4⋊C411yescharacteristic
139C2×C23C22×#10711yescharacteristic
140C2×C23C2×#42711yescharacteristic
141C2×C23C2×C4×D812yes—
142C2×C23C2×C4×Q1612yes—
143C23×C4—24no—
144C23×C4D4×C2311yescharacteristic
145C23×C4D4×C2311yescharacteristic
146C23×C4C23×C2311yescharacteristic
147C22×C8—24no—
148C22×C8C2×C4×C2314yes—
149C22×C8C2×C4×C2314yes—
150C22×C8C2×C4×C2318yes—
151C22×C8C23×C2312yes—
152C22×C8C23×C2312yes—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4C2×C4×C23116yes—
156C2×C4⋊C4C23×C2318yes—
157C2×C42—28no—
158C2×C42—48no—
159C2×C42C2×C4×C2314yes—
160C2×C42D4×C2312yes—
161C2×C42Q8×C2314yes—
162C2×C42C23×C2312yes—
163#107C2×C4×C23132yesdirect product factor
164C4×C8C2×C4×C23116yescomplemented
165#95—28no—
166C23C2×C4×#10711yescharacteristic, direct product factor, Sylow
167C24C2×D4×C2311yescharacteristic
168C22×C4—24no—
169C22×C4—216no—
170C22×C4—28no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4C4⋊C4×C2314yes—
174C22×C4C4⋊C4×C2314yes—
175C22×C4D8×C2314yes—
176C22×C4Q16×C2314yes—
177C22×C4C22×C4×C2312yes—
178C22×C4C22×C4×C2314yes—
179C22×C4C2×D4×C2312yes—
180C22×C4C2×D4×C2311yescharacteristic
181C22×C4C2×D4×C2312yes—
182C22×C4C2×Q8×C2312yes—
183C22×C4#41×C2314yes—
184C22×C4C24×C2311yescharacteristic
185C2×C8—28no—
186C2×C8—24no—
187C2×C8C42×C23116yescomplemented
188C2×C8C22×C4×C2314yes—
189C2×C8C22×C4×C2314yes—
190C2×C8C22×C4×C2318yes—
191C4⋊C4—216no—
192C4⋊C4—28no—
193C4⋊C4C22×C4×C23116yes—
194C42—416no—
195C42C4⋊C4×C2318yes—
196C42C22×C4×C2314yes—
197C42C2×Q8×C2314yes—
198C23C2×C4⋊C4×C2312yes—
199C23C4×D4×C2314yes—
200C23C2×D8×C2312yes—
201C23C2×Q16×C2312yes—
202C23C23×#13514yes—
203C23C22×D4×C2311yescharacteristic
204C2×C4—24no—
205C2×C4—28no—
206C2×C4—216no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—432no—
210C2×C4C23×#107116yesdirect product factor
211C2×C4C2×C42×C2314yes—
212C2×C4C2×C4⋊C4×C2314yes—
213C2×C4C2×C4⋊C4×C2312yes—
214C2×C4C2×C4⋊C4×C2314yes—
215C2×C4C23×#11714yes—
216C2×C4C4×D4×C2314yes—
217C2×C4C4×Q8×C2318yes—
218C2×C4C2×D8×C2314yes—
219C2×C4C2×Q16×C2314yes—
220C2×C4C23×C4×C2311yescharacteristic
221C2×C4C23×C4×C2312yes—
222C2×C4C22×Q8×C2311yescharacteristic
223C2×C4C2×#41×C2312yes—
224C8—24no—
225C8C2×C42×C2318yescomplemented
226C22C4×C4⋊C4×C2314yes—
227C22C2×C23×#10714yes—
228C22C23×#42714yes—
229C22C4×D8×C2318yes—
230C22C4×Q16×C2318yes—
231C22C22×C4⋊C4×C2311yescharacteristic
232C22C2×C4×D4×C2312yes—
233C22C22×D8×C2311yescharacteristic
234C22C22×Q16×C2311yescharacteristic
235C22C2×C23×#13512yes—
236C4—24no—
237C4—416no—
238C4C4×C4⋊C4×C2314yes—
239C4C2×C23×#10718yesdirect product factor
240C4C22×C42×C2311yescharacteristic
241C4C2×C23×#11711yescharacteristic
242C4C2×C4×Q8×C2312yes—
243C2C4×C23×#10718yesdirect product factor
244C2C2×C4×C4⋊C4×C2311yescharacteristic
245C2C22×C23×#10711yescharacteristic
246C2C2×C23×#42711yescharacteristic
247C2C2×C4×D8×C2312yes—
248C2C2×C4×Q16×C2312yes—
subgroupG/H
centerC23×C4×C23D4
commutatorC4C22×C42×C23
Frattini, cosocleC22×C4C24×C23
socle, prime radicalC24×C23C2×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C4×C23×#107C2direct (N × H)
11. C2×C23×#107C4direct (N × H)
13. C2×C4×C8×C23C4semidirect
39. C23×#107C2×C4direct (N × H)
40. C4×C8×C23C2×C4semidirect
62. C2×C8×C23C42semidirect
73. C2×C4×#107C23direct (N × H)
86. C2×C4×C23#107direct (N × H)
101. C8×C23C2×C42semidirect
105. C4×#107C2×C23direct (N × H)
120. C4×C23C2×#107direct (N × H)
129. C2×#107C4×C23direct (N × H)
131. C2×C4×C8C4×C23semidirect
137. C2×C23C4×#107direct (N × H)
163. #107C2×C4×C23direct (N × H)
164. C4×C8C2×C4×C23semidirect
166. C23C2×C4×#107direct (N × H)
187. C2×C8C42×C23semidirect
210. C2×C4C23×#107direct (N × H)
225. C8C2×C42×C23semidirect
239. C4C2×C23×#107direct (N × H)
243. C2C4×C23×#107direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1072
C22×C4×C8×C232
C2×C4×C4⋊C4×C232
C4×C23×#1072
C2×C23×#3402
C2×C4×#10723