Finite Group Databasebeta

← C23×#36691 #19184317

Subgroups — C23×#36691 #19184317

Order 5888.

1474 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
1C23×#2845C212yesmaximal, complemented
2C23×#2844C212yesmaximal, complemented
3C23×#2843C214yesmaximal, complemented
4C23×#2841C218yesmaximal
5C23×#2402C214yesmaximal, complemented
6C23×#2399C214yesmaximal, complemented
7C23×#2396C214yesmaximal
8C23×#2285C211yesmaximal, characteristic
9C23×#2283C212yesmaximal
10C2×C4×Q8×C23C2214yes—
11C2×C23×#117C2212yes—
12C22×C42×C23C2211yescharacteristic
13C23×#400C2218yes—
14C23×#400C2218yes—
15C23×#400C2214yes—
16C23×#399C2214yes—
17C23×#398C2218yes—
18C23×#398C2218yes—
19C23×#398C2218yes—
20C23×#398C2218yes—
21C23×#398C2218yes—
22C23×#396C2218yes—
23C23×#395C2218yes—
24C23×#395C2218yes—
25C23×#395C2214yes—
26C23×#393C2214yes—
27C23×#384C2212yes—
28C23×#383C2214yes—
29C23×#382C2214yes—
30C23×#382C2211yescharacteristic
31C23×#382C2211yescharacteristic
32C23×#381C2214yes—
33C23×#380C2214yes—
34C23×#380C2214yes—
35C4×C4⋊C4×C23C2212yes—
36C4×C4⋊C4×C23C2214yes—
37C4×C4⋊C4×C23C2218yes—
38C4×C4⋊C4×C23C2212yes—
39C4×C4⋊C4×C23C2214yes—
40C4×C22⋊C4×C23C2214yes—
41C23×#376C2214yes—
42C22×Q8×C23C2314yes—
43C23×C4×C23C2311yescharacteristic
44C23×C4×C23C2312yes—
45C4×Q8×C23—48no—
46C23×#117—44no—
47C2×C4⋊C4×C23C2314yes—
48C2×C4⋊C4×C23C2314yes—
49C2×C4⋊C4×C23C2318yes—
50C2×C4⋊C4×C23C2318yes—
51C2×C4⋊C4×C23C2318yes—
52C2×C4⋊C4×C23C2318yes—
53C2×C4⋊C4×C23C2312yes—
54C2×C4⋊C4×C23C2312yes—
55C2×C4⋊C4×C23C2314yes—
56C2×C4⋊C4×C23C2318yes—
57C2×C22⋊C4×C23C2318yes—
58C2×C22⋊C4×C23C2314yes—
59C2×C42×C23Q818yescomplemented
60C2×C42×C23C2314yes—
61C2×C42×C23C2314yes—
62C2×C42×C23C2314yes—
63C2×C42×C23C2311yescharacteristic
64C2×C42×C23C2311yescharacteristic
65C2×C42×C23C2312yes—
66C23×#95C2318yes—
67C23×#95C2318yes—
68C23×#95C2318yes—
69C23×#95C2314yes—
70C23×#95C2314yes—
71C23×#95C2314yes—
72C23×#95C2314yes—
73C23×#95C2318yes—
74C23×#95C2314yes—
75C23×#95C2318yes—
76C23×#95C2314yes—
77C24×C23C2411yescharacteristic
78C2×Q8×C23—44no—
79C2×Q8×C23—48no—
80C22×C4×C23—22no—
81C22×C4×C23—22no—
82C22×C4×C23—44no—
83C22×C4×C23C2×Q814yes—
84C22×C4×C23C2×Q818yes—
85C22×C4×C23C2414yes—
86C22×C4×C23C2414yes—
87C22×C4×C23C2418yes—
88C22×C4×C23C2418yes—
89C22×C4×C23C2411yescharacteristic
90C22×C4×C23C2412yes—
91C22×C4×C23C2412yes—
92C22×C4×C23C2411yescharacteristic
93C4⋊C4×C23—44no—
94C4⋊C4×C23—48no—
95C4⋊C4×C23—44no—
96C4⋊C4×C23—48no—
97C4⋊C4×C23—48no—
98C4⋊C4×C23—48no—
99C4⋊C4×C23—42no—
100C4⋊C4×C23—42no—
101C4⋊C4×C23—48no—
102C4⋊C4×C23—44no—
103C22⋊C4×C23—48no—
104C22⋊C4×C23—44no—
105C42×C23—28no—
106C42×C23—28no—
107C42×C23—28no—
108C42×C23C2×Q814yes—
109C42×C23C2×Q814yes—
110C42×C23C2×Q818yes—
111#36691C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—24no—
115C23×C23C22×Q812yes—
116C23×C23C2511yescharacteristic
117Q8×C23—416no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—24no—
121C2×C4×C23—24no—
122C2×C4×C23—28no—
123C2×C4×C23—24no—
124C2×C4×C23—24no—
125C2×C4×C23—28no—
126C2×C4×C23—28no—
127C2×C4×C23—28no—
128C2×C4×C23—28no—
129C2×C4×C23—28no—
130C2×C4×C23—28no—
131C2×C4×C23—28no—
132C2×C4×C23—28no—
133C2×C4×C23—48no—
134C2×C4×C23C22×Q812yes—
135C2×C4×C23C22×Q814yes—
136C2×C4×C23C22×Q814yes—
137C2×C4×C23C22×Q812yes—
138C2×C4×C23H(5,𝔽2)12yes—
139C2×C4×C232−1+412yes—
140#2845C2×C2312yescomplemented
141#2844C2×C2312yescomplemented
142#2843C2×C2314yescomplemented
143#2841C2×C2318yes—
144#2402C2×C2314yescomplemented
145#2399C2×C2314yescomplemented
146#2396C2×C2314yes—
147#2285C2×C2311yescharacteristic
148#2283C2×C2312yes—
149C22×C23—22no—
150C22×C23—24no—
151C22×C23—42no—
152C22×C23—42no—
153C22×C23C23×Q811yescharacteristic
154C22×C23C2×H(5,𝔽2)11yescharacteristic
155C22×C23C2×2−1+411yescharacteristic
156C22×C23#58514yes—
157C4×C23—22no—
158C4×C23—24no—
159C4×C23—24no—
160C4×C23—44no—
161C4×C23—44no—
162C4×C23—48no—
163C4×C23—48no—
164C4×C23#54414yes—
165C2×C4×Q8C22×C2314yes—
166C2×#117C22×C2312yes—
167C22×C42C22×C2311yescharacteristic
168#400C22×C2318yes—
169#400C22×C2318yes—
170#400C22×C2314yes—
171#399C22×C2314yes—
172#398C22×C2318yes—
173#398C22×C2318yes—
174#398C22×C2318yes—
175#398C22×C2318yes—
176#398C22×C2318yes—
177#396C22×C2318yes—
178#395C22×C2318yes—
179#395C22×C2318yes—
180#395C22×C2314yes—
181#393C22×C2314yes—
182#384C22×C2312yes—
183#383C22×C2314yes—
184#382C22×C2311yescharacteristic
185#382C22×C2314yes—
186#382C22×C2311yescharacteristic
187#381C22×C2314yes—
188#380C22×C2314yes—
189#380C22×C2314yes—
190C4×C4⋊C4C22×C2318yes—
191C4×C4⋊C4C22×C2314yes—
192C4×C4⋊C4C22×C2312yes—
193C4×C4⋊C4C22×C2314yes—
194C4×C4⋊C4C22×C2312yes—
195C4×C22⋊C4C22×C2314yes—
196#376C22×C2314yes—
197C2×C23—42no—
198C2×C23C2×#54411yescharacteristic
199C2×C23#345812yes—
200C2×C23#352912yes—
201C2×C23#353112yes—
202C22×Q8C23×C2314yes—
203C23×C4C23×C2311yescharacteristic
204C23×C4C23×C2312yes—
205C4×Q8—48no—
206#117—44no—
207C2×C4⋊C4C23×C2312yes—
208C2×C4⋊C4C23×C2312yes—
209C2×C4⋊C4C23×C2314yes—
210C2×C4⋊C4C23×C2318yes—
211C2×C4⋊C4C23×C2314yes—
212C2×C4⋊C4C23×C2314yes—
213C2×C4⋊C4C23×C2318yes—
214C2×C4⋊C4C23×C2318yes—
215C2×C4⋊C4C23×C2318yes—
216C2×C4⋊C4C23×C2318yes—
217C2×C22⋊C4C23×C2314yes—
218C2×C22⋊C4C23×C2318yes—
219C2×C42Q8×C2318yescomplemented
220C2×C42C23×C2314yes—
221C2×C42C23×C2314yes—
222C2×C42C23×C2314yes—
223C2×C42C23×C2312yes—
224C2×C42C23×C2311yescharacteristic
225C2×C42C23×C2311yescharacteristic
226#95C23×C2318yes—
227#95C23×C2318yes—
228#95C23×C2318yes—
229#95C23×C2314yes—
230#95C23×C2314yes—
231#95C23×C2314yes—
232#95C23×C2314yes—
233#95C23×C2318yes—
234#95C23×C2314yes—
235#95C23×C2318yes—
236#95C23×C2314yes—
237C23#3669111yescharacteristic, direct product factor, Sylow
238C24C24×C2311yescharacteristic
239C2×Q8—44no—
240C2×Q8—48no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4C2×Q8×C2314yes—
245C22×C4C2×Q8×C2318yes—
246C22×C4C24×C2318yes—
247C22×C4C24×C2314yes—
248C22×C4C24×C2318yes—
249C22×C4C24×C2314yes—
250C22×C4C24×C2311yescharacteristic
251C22×C4C24×C2312yes—
252C22×C4C24×C2312yes—
253C22×C4C24×C2311yescharacteristic
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—48no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C22⋊C4—44no—
265C22⋊C4—48no—
266C42—28no—
267C42—28no—
268C42—28no—
269C42C2×Q8×C2314yes—
270C42C2×Q8×C2314yes—
271C42C2×Q8×C2318yes—
272C23—21no—
273C23—21no—
274C23—24no—
275C23C22×Q8×C2312yes—
276C23C25×C2311yescharacteristic
277Q8—416no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—48no—
294C2×C4C22×Q8×C2312yes—
295C2×C4C22×Q8×C2314yes—
296C2×C4C22×Q8×C2314yes—
297C2×C4C22×Q8×C2312yes—
298C2×C4C23×H(5,𝔽2)12yes—
299C2×C4C23×2−1+412yes—
300C22—22no—
301C22—24no—
302C22—42no—
303C22—42no—
304C22C23×Q8×C2311yescharacteristic
305C22C2×C23×H(5,𝔽2)11yescharacteristic
306C22C2×C23×2−1+411yescharacteristic
307C22C23×#58514yes—
308C4—22no—
309C4—24no—
310C4—24no—
311C4—44no—
312C4—48no—
313C4—44no—
314C4—48no—
315C4C23×#54414yes—
316C2—42no—
317C2C2×C23×#54411yescharacteristic
318C2C23×#345812yes—
319C2C23×#352912yes—
320C2C23×#353112yes—
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC24×C23C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2845C2semidirect
2. C23×#2844C2semidirect
3. C23×#2843C2semidirect
5. C23×#2402C2semidirect
6. C23×#2399C2semidirect
59. C2×C42×C23Q8semidirect
111. #36691C23direct (N × H)
140. #2845C2×C23semidirect
141. #2844C2×C23semidirect
142. #2843C2×C23semidirect
144. #2402C2×C23semidirect
145. #2399C2×C23semidirect
219. C2×C42Q8×C23semidirect
237. C23#36691direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28452
C23×#28442
C23×#28432
C23×#28412
C23×#24022
C23×#23992
C23×#23962
C23×#22852
C23×#22832
#3669123