Finite Group Databasebeta

← C2×C23×#1386 #19160192

Subgroups — C2×C23×#1386 #19160192

Order 5888.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×M32C211yesmaximal, characteristic
2C2×C23×#427C211yesmaximal, characteristic
3C22×C23×#106C211yesmaximal, characteristic
4C23×#1386C214yesmaximal, direct product factor
5C23×C8×C23C2211yescharacteristic
6C2×C23×#117—21no—
7C22×C4⋊C4×C23—21no—
8C2×C23×M32—22no—
9C2×C23×M32—21no—
10C2×C23×M32C414yescomplemented
11C2×C23×M32C2212yes—
12C22×C16×C23—21no—
13C23×#427—21no—
14C23×#427—21no—
15C23×#427C412yescomplemented
16C23×#427C2212yes—
17C2×C23×#107C412yescomplemented
18C2×C23×#106—21no—
19C2×C23×#106—21no—
20C2×C23×#106—22no—
21C2×C23×#106C414yescomplemented
22C2×C23×#106C2212yes—
23C23×C4×C23—41no—
24C23×C4×C23D411yescharacteristic
25C22×C8×C23C2×C412yes—
26C22×C8×C23C2×C412yes—
27C22×C8×C23C2×C411yescharacteristic
28C22×C8×C23C2×C411yescharacteristic
29C22×C8×C23D411yescharacteristic
30C22×C8×C23D411yescharacteristic
31C22×C8×C23D412yes—
32C22×C8×C23D411yescharacteristic
33C22×C8×C23Q812yes—
34C22×C8×C23C2311yescharacteristic
35C23×#117—22no—
36C23×#117—22no—
37C23×#117—41no—
38C23×#117—41no—
39C2×C4⋊C4×C23—24no—
40C2×C4⋊C4×C23—22no—
41C2×C4⋊C4×C23—22no—
42C2×C4⋊C4×C23—41no—
43C2×C4⋊C4×C23—42no—
44C2×C22⋊C4×C23—41no—
45C2×C42×C23—41no—
46C23×M32—24no—
47C23×M32—22no—
48C23×M32C2×C414yescomplemented
49C2×C16×C23—24no—
50C2×C16×C23—22no—
51C23×#107—22no—
52C23×#107C2×C414yescomplemented
53C23×#106—24no—
54C23×#106—22no—
55C23×#106—22no—
56C23×#106—22no—
57C23×#106C2×C414yescomplemented
58C24×C23D811yescharacteristic
59C22×C4×C23—44no—
60C22×C4×C23—42no—
61C22×C4×C23—42no—
62C22×C4×C23—84no—
63C22×C4×C23C22⋊C412yes—
64C22×C4×C23C22⋊C411yescharacteristic
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C4⋊C412yes—
67C22×C4×C23D811yescharacteristic
68C22×C4×C23SD1612yes—
69C22×C4×C23SD1611yescharacteristic
70C22×C4×C23SD1611yescharacteristic
71C22×C4×C23Q1612yes—
72C22×C4×C23C2×D411yescharacteristic
73C2×C8×C23C4212yes—
74C2×C8×C23C4212yes—
75C2×C8×C23C22⋊C412yes—
76C2×C8×C23C22⋊C411yescharacteristic
77C2×C8×C23C22⋊C411yescharacteristic
78C2×C8×C23C22⋊C412yes—
79C2×C8×C23C22⋊C412yes—
80C2×C8×C23C4⋊C412yes—
81C2×C8×C23C4⋊C412yes—
82C2×C8×C23C4⋊C412yes—
83C2×C8×C23C4⋊C412yes—
84C2×C8×C23C4⋊C412yes—
85C2×C8×C23C22×C411yescharacteristic
86C2×C8×C23C22×C411yescharacteristic
87C2×C8×C23C22×C411yescharacteristic
88C2×C8×C23C2×D411yescharacteristic
89C2×C8×C23C2×D411yescharacteristic
90C2×C8×C23C2×Q811yescharacteristic
91C4⋊C4×C23—24no—
92C4⋊C4×C23—24no—
93C4⋊C4×C23—44no—
94C4⋊C4×C23—42no—
95C4⋊C4×C23—42no—
96C22⋊C4×C23—42no—
97C22⋊C4×C23—42no—
98C42×C23—42no—
99C42×C23—42no—
100C16×C23—24no—
101C2×#1386C2311yesmaximal, characteristic, direct product factor, Sylow
102C23×C23—24no—
103C23×C23#10211yescharacteristic
104C23×C23#10212yes—
105C23×C23#10211yescharacteristic
106C23×C23#10712yes—
107C23×C23C2×D811yescharacteristic
108C2×C4×C23—44no—
109C2×C4×C23—44no—
110C2×C4×C23—84no—
111C2×C4×C23—88no—
112C2×C4×C23—84no—
113C2×C4×C23#9512yes—
114C2×C4×C23#9512yes—
115C2×C4×C23#10211yescharacteristic
116C2×C4×C23#10212yes—
117C2×C4×C23#10211yescharacteristic
118C2×C4×C23#10312yes—
119C2×C4×C23#10312yes—
120C2×C4×C23#10312yes—
121C2×C4×C23#10312yes—
122C2×C4×C23#10612yes—
123C2×C4×C23#10612yes—
124C2×C4×C23#10712yes—
125C2×C4×C23C2×C22⋊C411yescharacteristic
126C2×C4×C23C2×C22⋊C411yescharacteristic
127C2×C4×C23C2×C4⋊C411yescharacteristic
128C2×C4×C23C2×SD1611yescharacteristic
129C2×C4×C23C2×SD1611yescharacteristic
130C2×C4×C23C2×Q1611yescharacteristic
131C8×C23#9512yes—
132C8×C23#9512yes—
133C8×C23C2×C4211yescharacteristic
134C8×C23C2×C22⋊C411yescharacteristic
135C8×C23C2×C4⋊C411yescharacteristic
136C8×C23C2×C4⋊C411yescharacteristic
137C22×M32C2×C2311yescharacteristic
138C2×#427C2×C2311yescharacteristic
139C22×#106C2×C2311yescharacteristic
140#1386C2×C2314yesdirect product factor
141C22×C23—22no—
142C22×C23—22no—
143C22×C23—24no—
144C22×C23—24no—
145C22×C23#34012yes—
146C22×C23#34012yes—
147C22×C23#36014yes—
148C22×C23C2×#10211yescharacteristic
149C22×C23C2×#10211yescharacteristic
150C22×C23C2×#10711yescharacteristic
151C4×C23—84no—
152C4×C23—84no—
153C4×C23#34012yes—
154C4×C23#34012yes—
155C4×C23C2×#9511yescharacteristic
156C4×C23C2×#10311yescharacteristic
157C4×C23C2×#10311yescharacteristic
158C4×C23C2×#10611yescharacteristic
159C23×C8C22×C2311yescharacteristic
160C2×#117—21no—
161C22×C4⋊C4—21no—
162C2×M32—22no—
163C2×M32—21no—
164C2×M32C4×C2314yescomplemented
165C2×M32C22×C2312yes—
166C22×C16—21no—
167#427—21no—
168#427—21no—
169#427C4×C2312yescomplemented
170#427C22×C2312yes—
171C2×#107C4×C2312yescomplemented
172C2×#106—21no—
173C2×#106—22no—
174C2×#106—21no—
175C2×#106C4×C2314yescomplemented
176C2×#106C22×C2312yes—
177C2×C23—21no—
178C2×C23—22no—
179C2×C23—21no—
180C2×C23#138614yesdirect product factor
181C2×C23C2×#34011yescharacteristic
182C2×C23C2×#36012yes—
183C23×C4—41no—
184C23×C4D4×C2311yescharacteristic
185C22×C8C2×C4×C2312yes—
186C22×C8C2×C4×C2312yes—
187C22×C8C2×C4×C2311yescharacteristic
188C22×C8C2×C4×C2311yescharacteristic
189C22×C8D4×C2311yescharacteristic
190C22×C8D4×C2311yescharacteristic
191C22×C8D4×C2312yes—
192C22×C8D4×C2311yescharacteristic
193C22×C8Q8×C2312yes—
194C22×C8C23×C2311yescharacteristic
195#117—22no—
196#117—22no—
197#117—41no—
198#117—41no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C22⋊C4—41no—
205C2×C42—41no—
206M32—24no—
207M32—22no—
208M32C2×C4×C2314yescomplemented
209C2×C16—24no—
210C2×C16—22no—
211#107—22no—
212#107C2×C4×C2314yescomplemented
213#106—24no—
214#106—22no—
215#106—22no—
216#106—22no—
217#106C2×C4×C2314yescomplemented
218C23C2×#138611yescharacteristic, direct product factor, Sylow
219C24D8×C2311yescharacteristic
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—84no—
224C22×C4C22⋊C4×C2312yes—
225C22×C4C22⋊C4×C2311yescharacteristic
226C22×C4C22⋊C4×C2311yescharacteristic
227C22×C4C4⋊C4×C2312yes—
228C22×C4D8×C2311yescharacteristic
229C22×C4SD16×C2311yescharacteristic
230C22×C4SD16×C2312yes—
231C22×C4SD16×C2311yescharacteristic
232C22×C4Q16×C2312yes—
233C22×C4C2×D4×C2311yescharacteristic
234C2×C8C42×C2312yes—
235C2×C8C42×C2312yes—
236C2×C8C22⋊C4×C2312yes—
237C2×C8C22⋊C4×C2312yes—
238C2×C8C22⋊C4×C2312yes—
239C2×C8C22⋊C4×C2311yescharacteristic
240C2×C8C22⋊C4×C2311yescharacteristic
241C2×C8C4⋊C4×C2312yes—
242C2×C8C4⋊C4×C2312yes—
243C2×C8C4⋊C4×C2312yes—
244C2×C8C4⋊C4×C2312yes—
245C2×C8C4⋊C4×C2312yes—
246C2×C8C22×C4×C2311yescharacteristic
247C2×C8C22×C4×C2311yescharacteristic
248C2×C8C22×C4×C2311yescharacteristic
249C2×C8C2×D4×C2311yescharacteristic
250C2×C8C2×D4×C2311yescharacteristic
251C2×C8C2×Q8×C2311yescharacteristic
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—42no—
260C42—42no—
261C16—24no—
262C23—24no—
263C23C23×#10211yescharacteristic
264C23C23×#10212yes—
265C23C23×#10211yescharacteristic
266C23C23×#10712yes—
267C23C2×D8×C2311yescharacteristic
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—84no—
271C2×C4—88no—
272C2×C4—84no—
273C2×C4C23×#9512yes—
274C2×C4C23×#9512yes—
275C2×C4C23×#10211yescharacteristic
276C2×C4C23×#10212yes—
277C2×C4C23×#10211yescharacteristic
278C2×C4C23×#10312yes—
279C2×C4C23×#10312yes—
280C2×C4C23×#10312yes—
281C2×C4C23×#10312yes—
282C2×C4C23×#10612yes—
283C2×C4C23×#10612yes—
284C2×C4C23×#10712yes—
285C2×C4C2×C22⋊C4×C2311yescharacteristic
286C2×C4C2×C22⋊C4×C2311yescharacteristic
287C2×C4C2×C4⋊C4×C2311yescharacteristic
288C2×C4C2×SD16×C2311yescharacteristic
289C2×C4C2×SD16×C2311yescharacteristic
290C2×C4C2×Q16×C2311yescharacteristic
291C8C23×#9512yes—
292C8C23×#9512yes—
293C8C2×C42×C2311yescharacteristic
294C8C2×C22⋊C4×C2311yescharacteristic
295C8C2×C4⋊C4×C2311yescharacteristic
296C8C2×C4⋊C4×C2311yescharacteristic
297C22—22no—
298C22—22no—
299C22—24no—
300C22—24no—
301C22C23×#34012yes—
302C22C23×#34012yes—
303C22C23×#36014yes—
304C22C2×C23×#10211yescharacteristic
305C22C2×C23×#10211yescharacteristic
306C22C2×C23×#10711yescharacteristic
307C4—84no—
308C4—84no—
309C4C23×#34012yes—
310C4C23×#34012yes—
311C4C2×C23×#9511yescharacteristic
312C4C2×C23×#10311yescharacteristic
313C4C2×C23×#10311yescharacteristic
314C4C2×C23×#10611yescharacteristic
315C2—21no—
316C2—22no—
317C2—21no—
318C2C23×#138614yesdirect product factor
319C2C2×C23×#34011yescharacteristic
320C2C2×C23×#36012yes—
subgroupG/H
center, socleC23×C23#102
commutatorC8C2×C42×C23
Frattini, cosocleC22×C8C23×C23
prime radicalC24×C23D8
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ C23×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1386C2direct (N × H)
10. C2×C23×M32C4semidirect
15. C23×#427C4semidirect
17. C2×C23×#107C4semidirect
21. C2×C23×#106C4semidirect
48. C23×M32C2×C4semidirect
52. C23×#107C2×C4semidirect
57. C23×#106C2×C4semidirect
101. C2×#1386C23direct (N × H)
140. #1386C2×C23direct (N × H)
164. C2×M32C4×C23semidirect
169. #427C4×C23semidirect
171. C2×#107C4×C23semidirect
175. C2×#106C4×C23semidirect
180. C2×C23#1386direct (N × H)
208. M32C2×C4×C23semidirect
212. #107C2×C4×C23semidirect
217. #106C2×C4×C23semidirect
218. C23C2×#1386direct (N × H)
318. C2C23×#1386direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×M322
C2×C23×#4272
C22×C23×#1062
C23×#13862
C2×#138623