Finite Group Databasebeta

← C23×#13795 #19161421

Subgroups — C23×#13795 #19161421

Order 5888.

2782 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#117C212yesmaximal
2C23×C4⋊C4×C23C211yesmaximal, characteristic
3C23×C4⋊#95C218yesmaximal, complemented
4C23×C22⋊#95C214yesmaximal
5C24×C4×C23C2212yes—
6C24×C4×C23C2211yescharacteristic
7C2×C23×#117—28no—
8C2×C23×#117—28no—
9C2×C23×#117C4116yescomplemented
10C22×C4⋊C4×C23—28no—
11C22×C4⋊C4×C23C418yescomplemented
12C22×C4⋊C4×C23C2214yes—
13C22×C4⋊C4×C23C2214yes—
14C22×C22⋊C4×C23C2214yes—
15C22×C42×C23C2214yes—
16C2×C23×#95C22116yes—
17C25×C23C2311yescharacteristic
18C23×C4×C23—28no—
19C23×C4×C23—48no—
20C23×C4×C23C2×C418yes—
21C23×C4×C23C2×C414yes—
22C23×C4×C23C2×C418yes—
23C23×C4×C23D418yes—
24C23×C4×C23D414yes—
25C23×C4×C23Q814yes—
26C23×C4×C23C2314yes—
27C23×C4×C23C2318yes—
28C23×C4×C23C2312yes—
29C23×#117—264no—
30C2×C4⋊C4×C23—232no—
31C2×C4⋊C4×C23—28no—
32C2×C4⋊C4×C23—28no—
33C2×C4⋊C4×C23—28no—
34C2×C4⋊C4×C23—28no—
35C2×C4⋊C4×C23C2×C4116yes—
36C2×C4⋊C4×C23C2×C4116yescomplemented
37C2×C22⋊C4×C23—28no—
38C2×C22⋊C4×C23—28no—
39C2×C22⋊C4×C23C2×C4116yes—
40C2×C42×C23—28no—
41C2×C42×C23—28no—
42C2×C42×C23C2×C4116yescomplemented
43C23×#95—416no—
44C24×C23—24no—
45C24×C23—24no—
46C24×C23C22×C414yes—
47C24×C23C22×C412yes—
48C24×C23C2×D412yes—
49C24×C23C2×D414yes—
50C24×C23C2×Q812yes—
51C24×C23C2411yescharacteristic
52C22×C4×C23—216no—
53C22×C4×C23—28no—
54C22×C4×C23—28no—
55C22×C4×C23—28no—
56C22×C4×C23—216no—
57C22×C4×C23—216no—
58C22×C4×C23—216no—
59C22×C4×C23—416no—
60C22×C4×C23—432no—
61C22×C4×C23C42116yes—
62C22×C4×C23C22⋊C4116yes—
63C22×C4×C23C22⋊C4132yes—
64C22×C4×C23C4⋊C4116yes—
65C22×C4×C23C4⋊C4132yes—
66C22×C4×C23C22×C4116yes—
67C22×C4×C23C22×C418yes—
68C22×C4×C23C22×C414yes—
69C22×C4×C23C22×C414yes—
70C22×C4×C23C22×C414yes—
71C22×C4×C23C2×D414yes—
72C22×C4×C23C2×D414yes—
73C22×C4×C23C2×D414yes—
74C22×C4×C23C2×Q814yes—
75C4⋊C4×C23—232no—
76C4⋊C4×C23—232no—
77C22⋊C4×C23—232no—
78C42×C23—232no—
79#13795C2311yesmaximal, characteristic, direct product factor, Sylow
80C23×C23—24no—
81C23×C23—24no—
82C23×C23—216no—
83C23×C23—28no—
84C23×C23—28no—
85C23×C23—28no—
86C23×C23—28no—
87C23×C23C2×C4214yes—
88C23×C23C2×C22⋊C418yes—
89C23×C23C2×C22⋊C414yes—
90C23×C23C2×C4⋊C414yes—
91C23×C23C2×C4⋊C418yes—
92C23×C23C23×C411yescharacteristic
93C23×C23C23×C412yes—
94C23×C23C22×D412yes—
95C23×C23C22×D411yescharacteristic
96C23×C23C22×Q811yescharacteristic
97C23×C23H(5,𝔽2)14yes—
98C23×C232−1+414yes—
99C2×C4×C23—216no—
100C2×C4×C23—28no—
101C2×C4×C23—216no—
102C2×C4×C23—232no—
103C2×C4×C23—432no—
104C2×C4×C23—48no—
105C2×C4×C23—416no—
106C2×C4×C23—48no—
107C2×C4×C23—416no—
108C2×C4×C23#95164yescomplemented
109C2×C4×C23C2×C4218yes—
110C2×C4×C23C2×C22⋊C418yes—
111C2×C4×C23C2×C22⋊C418yes—
112C2×C4×C23C2×C22⋊C418yes—
113C2×C4×C23C2×C4⋊C418yes—
114C2×C4×C23C2×C4⋊C418yes—
115C2×C4×C23C2×C4⋊C418yes—
116C22×#117C2×C2312yes—
117C23×C4⋊C4C2×C2311yescharacteristic
118C4⋊#95C2×C2318yescomplemented
119C22⋊#95C2×C2314yes—
120C22×C23—24no—
121C22×C23—24no—
122C22×C23—216no—
123C22×C23—28no—
124C22×C23—216no—
125C22×C23—28no—
126C22×C23C2×#9518yes—
127C22×C23C22×C4211yescharacteristic
128C22×C23C22×C22⋊C411yescharacteristic
129C22×C23C22×C22⋊C412yes—
130C22×C23C22×C4⋊C411yescharacteristic
131C22×C23C22×C4⋊C412yes—
132C22×C23#51812yes—
133C22×C23#51912yes—
134C22×C23#52018yes—
135C22×C23#53514yes—
136C22×C23#53614yes—
137C22×C23#53714yes—
138C22×C23#54414yes—
139C4×C23—48no—
140C4×C23—416no—
141C4×C23C2×#95116yes—
142C24×C4C22×C2312yes—
143C24×C4C22×C2311yescharacteristic
144C2×#117—28no—
145C2×#117—28no—
146C2×#117C4×C23116yescomplemented
147C22×C4⋊C4—28no—
148C22×C4⋊C4C4×C2318yescomplemented
149C22×C4⋊C4C22×C2314yes—
150C22×C4⋊C4C22×C2314yes—
151C22×C22⋊C4C22×C2314yes—
152C22×C42C22×C2314yes—
153C2×#95C22×C23116yes—
154C2×C23—28no—
155C2×C23C22×#9511yescharacteristic
156C2×C23#227412yes—
157C2×C23#230911yescharacteristic
158C2×C23#231011yescharacteristic
159C2×C23#231114yes—
160C2×C23#231612yes—
161C2×C23#231714yes—
162C25C23×C2311yescharacteristic
163C23×C4—28no—
164C23×C4—48no—
165C23×C4C2×C4×C2318yes—
166C23×C4C2×C4×C2314yes—
167C23×C4C2×C4×C2318yes—
168C23×C4D4×C2318yes—
169C23×C4D4×C2314yes—
170C23×C4Q8×C2314yes—
171C23×C4C23×C2318yes—
172C23×C4C23×C2314yes—
173C23×C4C23×C2312yes—
174#117—264no—
175C2×C4⋊C4—232no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4C2×C4×C23116yes—
181C2×C4⋊C4C2×C4×C23116yescomplemented
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4C2×C4×C23116yes—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42C2×C4×C23116yescomplemented
188#95—416no—
189C23#1379511yescharacteristic, direct product factor, Sylow
190C24—24no—
191C24—24no—
192C24C22×C4×C2314yes—
193C24C22×C4×C2312yes—
194C24C2×D4×C2312yes—
195C24C2×D4×C2314yes—
196C24C2×Q8×C2312yes—
197C24C24×C2311yescharacteristic
198C22×C4—216no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—28no—
202C22×C4—216no—
203C22×C4—216no—
204C22×C4—216no—
205C22×C4—416no—
206C22×C4—432no—
207C22×C4C42×C23116yes—
208C22×C4C22⋊C4×C23116yes—
209C22×C4C22⋊C4×C23132yes—
210C22×C4C4⋊C4×C23116yes—
211C22×C4C4⋊C4×C23132yes—
212C22×C4C22×C4×C23116yes—
213C22×C4C22×C4×C2318yes—
214C22×C4C22×C4×C2314yes—
215C22×C4C22×C4×C2314yes—
216C22×C4C22×C4×C2314yes—
217C22×C4C2×D4×C2314yes—
218C22×C4C2×D4×C2314yes—
219C22×C4C2×D4×C2314yes—
220C22×C4C2×Q8×C2314yes—
221C4⋊C4—232no—
222C4⋊C4—232no—
223C22⋊C4—232no—
224C42—232no—
225C23—24no—
226C23—24no—
227C23—216no—
228C23—28no—
229C23—28no—
230C23—28no—
231C23—28no—
232C23C2×C42×C2314yes—
233C23C2×C22⋊C4×C2318yes—
234C23C2×C22⋊C4×C2314yes—
235C23C2×C4⋊C4×C2314yes—
236C23C2×C4⋊C4×C2318yes—
237C23C23×C4×C2311yescharacteristic
238C23C23×C4×C2312yes—
239C23C22×D4×C2312yes—
240C23C22×D4×C2311yescharacteristic
241C23C22×Q8×C2311yescharacteristic
242C23C23×H(5,𝔽2)14yes—
243C23C23×2−1+414yes—
244C2×C4—216no—
245C2×C4—28no—
246C2×C4—216no—
247C2×C4—232no—
248C2×C4—432no—
249C2×C4—416no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—416no—
253C2×C4C23×#95164yescomplemented
254C2×C4C2×C42×C2318yes—
255C2×C4C2×C22⋊C4×C2318yes—
256C2×C4C2×C22⋊C4×C2318yes—
257C2×C4C2×C22⋊C4×C2318yes—
258C2×C4C2×C4⋊C4×C2318yes—
259C2×C4C2×C4⋊C4×C2318yes—
260C2×C4C2×C4⋊C4×C2318yes—
261C22—28no—
262C22—24no—
263C22—24no—
264C22—216no—
265C22—28no—
266C22—216no—
267C22C2×C23×#9518yes—
268C22C22×C42×C2311yescharacteristic
269C22C22×C22⋊C4×C2311yescharacteristic
270C22C22×C22⋊C4×C2312yes—
271C22C22×C4⋊C4×C2311yescharacteristic
272C22C22×C4⋊C4×C2312yes—
273C22C23×#51812yes—
274C22C23×#51912yes—
275C22C23×#52018yes—
276C22C23×#53514yes—
277C22C23×#53614yes—
278C22C23×#53714yes—
279C22C23×#54414yes—
280C4—48no—
281C4—416no—
282C4C2×C23×#95116yes—
283C2—28no—
284C2C22×C23×#9511yescharacteristic
285C2C23×#227412yes—
286C2C23×#230911yescharacteristic
287C2C23×#231011yescharacteristic
288C2C23×#231114yes—
289C2C23×#231612yes—
290C2C23×#231714yes—
subgroupG/H
center, socleC24×C23C24
commutatorC22C22×C42×C23
Frattini, cosocleC24C24×C23
prime radicalC25×C23C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×C4⋊#95C2semidirect
9. C2×C23×#117C4semidirect
11. C22×C4⋊C4×C23C4semidirect
36. C2×C4⋊C4×C23C2×C4semidirect
42. C2×C42×C23C2×C4semidirect
79. #13795C23direct (N × H)
108. C2×C4×C23#95semidirect
118. C4⋊#95C2×C23semidirect
146. C2×#117C4×C23semidirect
148. C22×C4⋊C4C4×C23semidirect
181. C2×C4⋊C4C2×C4×C23semidirect
187. C2×C42C2×C4×C23semidirect
189. C23#13795direct (N × H)
253. C2×C4C23×#95semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1172
C23×C4⋊C4×C232
C23×C4⋊#952
C23×C22⋊#952
#1379523