Finite Group Databasebeta

← C23×#9462 #19157088

Subgroups — C23×#9462 #19157088

Order 5888.

700 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2434C211yesmaximal, characteristic
2C23×#2310C211yesmaximal, characteristic
3C23×#1784C211yesmaximal, characteristic
4C23×#1511C212yesmaximal
5C23×#1284C212yesmaximal
6C23×Q8×C23C2211yescharacteristic
7C2×C23×C4⋊Q8C412yescomplemented
8C2×C23×C22⋊Q8C2211yescharacteristic
9C2×C23×#117—21no—
10C2×C23×#101—22no—
11C23×#407C2212yes—
12C23×#395—22no—
13C23×#391—24no—
14C23×#391C412yescomplemented
15C23×#385—21no—
16C23×#385—21no—
17C23×#381—21no—
18C23×#381C2212yes—
19C4×C22⋊C4×C23C2211yescharacteristic
20C23×#328—24no—
21C22×Q8×C23—22no—
22C22×Q8×C23—22no—
23C22×Q8×C23—44no—
24C22×Q8×C23C2×C411yescharacteristic
25C22×Q8×C23C2×C412yes—
26C22×Q8×C23D412yes—
27C23×C4×C23D411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C23×C4×C23C2311yescharacteristic
30C2×M16×C23—22no—
31C23×C4⋊Q8—44no—
32C23×C22⋊Q8—22no—
33C23×C22⋊Q8—22no—
34C23×#117—44no—
35C2×C4⋊C4×C23—22no—
36C2×C4⋊C4×C23—22no—
37C2×C4⋊C4×C23—22no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23C2×C411yescharacteristic
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—21no—
43C2×C42×C23—22no—
44C2×C42×C23C2×C412yes—
45C23×#101—44no—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—22no—
48C23×#95—24no—
49C23×#95—22no—
50C23×#95—22no—
51C23×#95—21no—
52C23×#95—22no—
53C23×#95D414yes—
54C24×C23C2×D411yescharacteristic
55C2×Q8×C23—24no—
56C2×Q8×C23—24no—
57C2×Q8×C23—24no—
58C2×Q8×C23—24no—
59C2×Q8×C23—22no—
60C2×Q8×C23—42no—
61C2×Q8×C23—42no—
62C2×Q8×C23—44no—
63C2×Q8×C23—44no—
64C2×Q8×C23—48no—
65C2×Q8×C23—42no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—24no—
69C22×C4×C23—22no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—24no—
75C22×C4×C23—42no—
76C22×C4×C23—42no—
77C22×C4×C23C22⋊C412yes—
78C22×C4×C23C22⋊C412yes—
79C22×C4×C23C22×C411yescharacteristic
80C22×C4×C23C2×D411yescharacteristic
81C22×C4×C23C2×D412yes—
82C22×C4×C23#4112yes—
83M16×C23—44no—
84C2×C8×C23—42no—
85C4⋊C4×C23—22no—
86C4⋊C4×C23—44no—
87C4⋊C4×C23—44no—
88C4⋊C4×C23—44no—
89C4⋊C4×C23—42no—
90C22⋊C4×C23—24no—
91C22⋊C4×C23—44no—
92C22⋊C4×C23—41no—
93C22⋊C4×C23—41no—
94C42×C23—42no—
95C42×C23—44no—
96#9462C2311yesmaximal, characteristic, direct product factor, Sylow
97C23×C23—22no—
98C23×C23—22no—
99C23×C23—41no—
100C23×C23C2×C22⋊C411yescharacteristic
101C23×C23#12011yescharacteristic
102C23×C23#12311yescharacteristic
103Q8×C23—48no—
104Q8×C23—48no—
105Q8×C23—82no—
106C2×C4×C23—22no—
107C2×C4×C23—22no—
108C2×C4×C23—24no—
109C2×C4×C23—22no—
110C2×C4×C23—22no—
111C2×C4×C23—22no—
112C2×C4×C23—22no—
113C2×C4×C23—24no—
114C2×C4×C23—22no—
115C2×C4×C23—22no—
116C2×C4×C23—22no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23—44no—
124C2×C4×C23—44no—
125C2×C4×C23—42no—
126C2×C4×C23—41no—
127C2×C4×C23—41no—
128C2×C4×C23—42no—
129C2×C4×C23—44no—
130C2×C4×C23#9914yes—
131C2×C4×C23C4×D412yes—
132C2×C4×C23#12112yes—
133C8×C23—82no—
134#2434C2×C2311yescharacteristic
135#2310C2×C2311yescharacteristic
136#1784C2×C2311yescharacteristic
137#1511C2×C2312yes—
138#1284C2×C2312yes—
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—21no—
145C22×C23—21no—
146C22×C23—44no—
147C22×C23#38611yescharacteristic
148C22×C23C2×#9912yes—
149C22×C23#45612yes—
150C4×C23—22no—
151C4×C23—24no—
152C4×C23—22no—
153C4×C23—44no—
154C4×C23—42no—
155C4×C23—84no—
156C4×C23—81no—
157C4×C23—81no—
158C23×Q8C22×C2311yescharacteristic
159C2×C4⋊Q8C4×C2312yescomplemented
160C2×C22⋊Q8C22×C2311yescharacteristic
161C2×#117—21no—
162C2×#101—22no—
163#407C22×C2312yes—
164#395—22no—
165#391—24no—
166#391C4×C2312yescomplemented
167#385—21no—
168#385—21no—
169#381—21no—
170#381C22×C2312yes—
171C4×C22⋊C4C22×C2311yescharacteristic
172#328—24no—
173C2×C23—22no—
174C2×C23—22no—
175C2×C23—41no—
176C2×C23#188611yescharacteristic
177C2×C23#189811yescharacteristic
178C2×C23#191211yescharacteristic
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—44no—
182C22×Q8C2×C4×C2311yescharacteristic
183C22×Q8C2×C4×C2312yes—
184C22×Q8D4×C2312yes—
185C23×C4D4×C2311yescharacteristic
186C23×C4D4×C2311yescharacteristic
187C23×C4C23×C2311yescharacteristic
188C2×M16—22no—
189C4⋊Q8—44no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192#117—44no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C4×C2311yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C42—22no—
202C2×C42C2×C4×C2312yes—
203#101—44no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206#95—24no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95D4×C2314yes—
212C23#946211yescharacteristic, direct product factor, Sylow
213C24C2×D4×C2311yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊C4×C2312yes—
237C22×C4C22⋊C4×C2312yes—
238C22×C4C22×C4×C2311yescharacteristic
239C22×C4C2×D4×C2311yescharacteristic
240C22×C4C2×D4×C2312yes—
241C22×C4#41×C2312yes—
242M16—44no—
243C2×C8—42no—
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C22⋊C4—24no—
250C22⋊C4—44no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C42—42no—
254C42—44no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23C2×C22⋊C4×C2311yescharacteristic
259C23C23×#12011yescharacteristic
260C23C23×#12311yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—82no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4C23×#9914yes—
289C2×C4C4×D4×C2312yes—
290C2×C4C23×#12112yes—
291C8—82no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22C23×#38611yescharacteristic
301C22C2×C23×#9912yes—
302C22C23×#45612yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—84no—
309C4—81no—
310C4—81no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2C23×#188611yescharacteristic
315C2C23×#189811yescharacteristic
316C2C23×#191211yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×C23×C4⋊Q8C4semidirect
14. C23×#391C4semidirect
96. #9462C23direct (N × H)
159. C2×C4⋊Q8C4×C23semidirect
166. #391C4×C23semidirect
212. C23#9462direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24342
C23×#23102
C23×#17842
C23×#15112
C23×#12842
#946223