Finite Group Databasebeta

← C23×#34059 #19181685

Subgroups — C23×#34059 #19181685

Order 5888.

2250 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8×C23C211yesmaximal, characteristic
2C23×#2320C218yesmaximal, complemented
3C23×#2316C212yesmaximal
4C23×#2313C214yesmaximal
5C23×#2310C212yesmaximal
6C23×#2295C218yesmaximal, complemented
7C23×#2283C214yesmaximal
8C4×C23×#117C212yesmaximal
9C23×Q8×C23C2211yescharacteristic
10C2×C4×Q8×C23C4116yescomplemented
11C2×C4×Q8×C23C2214yes—
12C2×C4×Q8×C23C2214yes—
13C2×C23×#117C2214yes—
14C22×C4⋊C4×C23C2212yes—
15C22×C4⋊C4×C23C2211yescharacteristic
16C22×C42×C23C2211yescharacteristic
17C22×C42×C23C2212yes—
18C23×#391C22116yes—
19C23×#389C22116yes—
20C23×#387C22132yes—
21C23×#383C2218yes—
22C23×#382C2218yes—
23C23×#381C2218yes—
24C23×#380C2218yes—
25C4×C4⋊C4×C23C2214yes—
26C4×C4⋊C4×C23C2214yes—
27C4×C4⋊C4×C23C2218yes—
28C4×C22⋊C4×C23C2214yes—
29C4×C22⋊C4×C23C2214yes—
30C23×#376C2214yes—
31C23×#376C2218yes—
32C43×C23C2214yes—
33C22×Q8×C23—28no—
34C22×Q8×C23C2×C418yes—
35C22×Q8×C23C2314yes—
36C23×C4×C23C2312yes—
37C23×C4×C23C2311yescharacteristic
38C23×C4×C23C2311yescharacteristic
39C23×C4×C23C2312yes—
40C23×C4×C23C2311yescharacteristic
41C4×Q8×C23C2×C4132yes—
42C4×Q8×C23C2×C4132yes—
43C23×#117—216no—
44C2×C4⋊C4×C23C2×C4116yes—
45C2×C4⋊C4×C23C2×C418yes—
46C2×C4⋊C4×C23C2318yes—
47C2×C4⋊C4×C23C23116yes—
48C2×C4⋊C4×C23C2318yes—
49C2×C4⋊C4×C23C2314yes—
50C2×C22⋊C4×C23C2318yes—
51C2×C42×C23C2×C4116yes—
52C2×C42×C23C2×C418yes—
53C2×C42×C23C2318yes—
54C2×C42×C23C2318yes—
55C2×C42×C23C2318yes—
56C2×C42×C23C2312yes—
57C2×C42×C23C2314yes—
58C2×C42×C23C2312yes—
59C2×C42×C23C2314yes—
60C23×#95C2318yes—
61C23×#95C2318yes—
62C23×#95C23116yes—
63C23×#95C23116yes—
64C23×#95C23116yes—
65C24×C23C2411yescharacteristic
66C2×Q8×C23—232no—
67C2×Q8×C23—216no—
68C2×Q8×C23C22×C4116yes—
69C22×C4×C23—28no—
70C22×C4×C23—24no—
71C22×C4×C23—42no—
72C22×C4×C23—44no—
73C22×C4×C23C22×C414yes—
74C22×C4×C23C22×C418yes—
75C22×C4×C23C22×C414yes—
76C22×C4×C23C22×C418yes—
77C22×C4×C23C22×C414yes—
78C22×C4×C23#4118yes—
79C22×C4×C23C24116yes—
80C22×C4×C23C2412yes—
81C22×C4×C23C2411yescharacteristic
82C22×C4×C23C2414yes—
83C22×C4×C23C2414yes—
84C22×C4×C23C2412yes—
85C22×C4×C23C2411yescharacteristic
86C4⋊C4×C23—216no—
87C4⋊C4×C23—416no—
88C4⋊C4×C23C22×C4132yes—
89C4⋊C4×C23C22×C4116yes—
90C22⋊C4×C23—216no—
91C42×C23—216no—
92C42×C23—216no—
93C42×C23—48no—
94C42×C23C22×C4116yes—
95C42×C23C22×C418yes—
96C42×C23C22×C4116yes—
97C42×C23C22×C418yes—
98#34059C2311yesmaximal, characteristic, direct product factor, Sylow
99C23×C23—22no—
100C23×C23—22no—
101C23×C23C23×C412yes—
102C23×C23C2×#4114yes—
103C23×C23C2511yescharacteristic
104Q8×C23—232no—
105C2×C4×C23—216no—
106C2×C4×C23—28no—
107C2×C4×C23—22no—
108C2×C4×C23—24no—
109C2×C4×C23—28no—
110C2×C4×C23—28no—
111C2×C4×C23—216no—
112C2×C4×C23—216no—
113C2×C4×C23—216no—
114C2×C4×C23—44no—
115C2×C4×C23—48no—
116C2×C4×C23#117116yes—
117C2×C4×C23C23×C412yes—
118C2×C4×C23C23×C414yes—
119C2×C4×C23C23×C418yes—
120C2×C4×C23C23×C418yes—
121C2×C4×C23C23×C414yes—
122C2×C4×C23C23×C412yes—
123C2×C4×C23C2×#4118yes—
124C2×C4×C232−1+414yes—
125C22×C4×Q8C2×C2311yescharacteristic
126#2320C2×C2318yescomplemented
127#2316C2×C2312yes—
128#2313C2×C2314yes—
129#2310C2×C2312yes—
130#2295C2×C2318yescomplemented
131#2283C2×C2314yes—
132C4×#117C2×C2312yes—
133C22×C23—22no—
134C22×C23—22no—
135C22×C23—28no—
136C22×C23C2×#11714yes—
137C22×C23C24×C411yescharacteristic
138C22×C23C22×#4112yes—
139C22×C23C2×2−1+412yes—
140C22×C23#58512yes—
141C4×C23—28no—
142C4×C23—22no—
143C4×C23—24no—
144C4×C23—28no—
145C4×C23—416no—
146C4×C23C2×#11718yes—
147C4×C23#51914yes—
148C23×Q8C22×C2311yescharacteristic
149C2×C4×Q8C4×C23116yescomplemented
150C2×C4×Q8C22×C2314yes—
151C2×C4×Q8C22×C2314yes—
152C2×#117C22×C2314yes—
153C22×C4⋊C4C22×C2312yes—
154C22×C4⋊C4C22×C2311yescharacteristic
155C22×C42C22×C2311yescharacteristic
156C22×C42C22×C2312yes—
157#391C22×C23116yes—
158#389C22×C23116yes—
159#387C22×C23132yes—
160#383C22×C2318yes—
161#382C22×C2318yes—
162#381C22×C2318yes—
163#380C22×C2318yes—
164C4×C4⋊C4C22×C2314yes—
165C4×C4⋊C4C22×C2314yes—
166C4×C4⋊C4C22×C2318yes—
167C4×C22⋊C4C22×C2314yes—
168C4×C22⋊C4C22×C2314yes—
169#376C22×C2314yes—
170#376C22×C2318yes—
171C43C22×C2314yes—
172C2×C23—24no—
173C2×C23C22×#11711yescharacteristic
174C2×C23C2×#51911yescharacteristic
175C2×C23#342811yescharacteristic
176C2×C23#346114yes—
177C22×Q8—28no—
178C22×Q8C2×C4×C2318yes—
179C22×Q8C23×C2314yes—
180C23×C4C23×C2312yes—
181C23×C4C23×C2311yescharacteristic
182C23×C4C23×C2311yescharacteristic
183C23×C4C23×C2312yes—
184C23×C4C23×C2311yescharacteristic
185C4×Q8C2×C4×C23132yes—
186C4×Q8C2×C4×C23132yes—
187#117—216no—
188C2×C4⋊C4C2×C4×C23116yes—
189C2×C4⋊C4C2×C4×C2318yes—
190C2×C4⋊C4C23×C2318yes—
191C2×C4⋊C4C23×C23116yes—
192C2×C4⋊C4C23×C2318yes—
193C2×C4⋊C4C23×C2314yes—
194C2×C22⋊C4C23×C2318yes—
195C2×C42C2×C4×C23116yes—
196C2×C42C2×C4×C2318yes—
197C2×C42C23×C2318yes—
198C2×C42C23×C2318yes—
199C2×C42C23×C2318yes—
200C2×C42C23×C2312yes—
201C2×C42C23×C2312yes—
202C2×C42C23×C2314yes—
203C2×C42C23×C2314yes—
204#95C23×C2318yes—
205#95C23×C2318yes—
206#95C23×C23116yes—
207#95C23×C23116yes—
208#95C23×C23116yes—
209C23#3405911yescharacteristic, direct product factor, Sylow
210C24C24×C2311yescharacteristic
211C2×Q8—232no—
212C2×Q8—216no—
213C2×Q8C22×C4×C23116yes—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4C22×C4×C2314yes—
219C22×C4C22×C4×C2318yes—
220C22×C4C22×C4×C2314yes—
221C22×C4C22×C4×C2318yes—
222C22×C4C22×C4×C2314yes—
223C22×C4#41×C2318yes—
224C22×C4C24×C23116yes—
225C22×C4C24×C2311yescharacteristic
226C22×C4C24×C2314yes—
227C22×C4C24×C2312yes—
228C22×C4C24×C2314yes—
229C22×C4C24×C2312yes—
230C22×C4C24×C2311yescharacteristic
231C4⋊C4—216no—
232C4⋊C4—416no—
233C4⋊C4C22×C4×C23132yes—
234C4⋊C4C22×C4×C23116yes—
235C22⋊C4—216no—
236C42—216no—
237C42—216no—
238C42—48no—
239C42C22×C4×C23116yes—
240C42C22×C4×C2318yes—
241C42C22×C4×C23116yes—
242C42C22×C4×C2318yes—
243C23—22no—
244C23—22no—
245C23C23×C4×C2312yes—
246C23C2×#41×C2314yes—
247C23C25×C2311yescharacteristic
248Q8—232no—
249C2×C4—28no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—28no—
253C2×C4—216no—
254C2×C4—28no—
255C2×C4—216no—
256C2×C4—216no—
257C2×C4—216no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4C23×#117116yes—
261C2×C4C23×C4×C2312yes—
262C2×C4C23×C4×C2314yes—
263C2×C4C23×C4×C2318yes—
264C2×C4C23×C4×C2318yes—
265C2×C4C23×C4×C2314yes—
266C2×C4C23×C4×C2312yes—
267C2×C4C2×#41×C2318yes—
268C2×C4C23×2−1+414yes—
269C22—22no—
270C22—22no—
271C22—28no—
272C22C2×C23×#11714yes—
273C22C24×C4×C2311yescharacteristic
274C22C22×#41×C2312yes—
275C22C2×C23×2−1+412yes—
276C22C23×#58512yes—
277C4—28no—
278C4—22no—
279C4—24no—
280C4—28no—
281C4—416no—
282C4C2×C23×#11718yes—
283C4C23×#51914yes—
284C2—24no—
285C2C22×C23×#11711yescharacteristic
286C2C2×C23×#51911yescharacteristic
287C2C23×#342811yescharacteristic
288C2C23×#346114yes—
subgroupG/H
centerC22×C4×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC24×C23C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2320C2semidirect
6. C23×#2295C2semidirect
10. C2×C4×Q8×C23C4semidirect
98. #34059C23direct (N × H)
126. #2320C2×C23semidirect
130. #2295C2×C23semidirect
149. C2×C4×Q8C4×C23semidirect
209. C23#34059direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C232
C23×#23202
C23×#23162
C23×#23132
C23×#23102
C23×#22952
C23×#22832
C4×C23×#1172
#3405923