Finite Group Databasebeta

← C23×#9410 #19157036

Subgroups — C23×#9410 #19157036

Order 5888.

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

normal subgroups only

This group has 644 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×#2591C211yesmaximal, characteristic
2C23×#2274C211yesmaximal, characteristic, complemented
3C23×#1507C212yesmaximal
4C23×#1463C211yesmaximal, characteristic, complemented
5C23×#1297C212yesmaximal, complemented
6C2×C23×#121C2211yescharacteristic
7C2×C23×#117—21no—
8C22×C4⋊C4×C23C2211yescharacteristic
9C23×#407C2212yes—
10C23×#400C412yescomplemented
11C23×#394C412yescomplemented
12C23×#388—22no—
13C23×#386—21no—
14C23×#386—21no—
15C4×C4⋊C4×C23—24no—
16C4×C22⋊C4×C23—21no—
17C4×C22⋊C4×C23C2211yescharacteristic
18C4×C22⋊C4×C23C2212yes—
19C23×#327—24no—
20C23×#324—22no—
21C22×D4×C23—21no—
22C22×D4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C23×C4×C23C2311yescharacteristic
26C2×M16×C23—22no—
27C23×#121—22no—
28C23×#121—22no—
29C23×#117—28no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C4⋊C4×C23C2×C412yes—
35C2×C4⋊C4×C23D412yes—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—22no—
41C2×C42×C23—22no—
42C2×C42×C23—22no—
43C2×C42×C23—22no—
44C2×C42×C23—21no—
45C23×C22⋊C8—22no—
46C23×C22⋊C8—22no—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—22no—
50C23×#95C2×C412yes—
51C23×#95D414yes—
52C24×C23—21no—
53C24×C23C2×D411yescharacteristic
54C2×D4×C23—22no—
55C2×D4×C23—44no—
56C2×D4×C23—41no—
57C2×D4×C23—41no—
58C2×D4×C23—81no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—24no—
62C22×C4×C23—24no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—22no—
66C22×C4×C23—21no—
67C22×C4×C23—81no—
68C22×C4×C23—81no—
69C22×C4×C23C22⋊C412yes—
70C22×C4×C23C22⋊C412yes—
71C22×C4×C23C22×C411yescharacteristic
72C22×C4×C23C2×D412yes—
73C22×C4×C23C2×D411yescharacteristic
74C22×C4×C23#4112yes—
75M16×C23—82no—
76C2×C8×C23—42no—
77C4⋊C4×C23—28no—
78C4⋊C4×C23—24no—
79C4⋊C4×C23—24no—
80C4⋊C4×C23—22no—
81C22⋊C4×C23—28no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—44no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—82no—
86C42×C23—28no—
87C42×C23—82no—
88C42×C23—81no—
89C42×C23—82no—
90#9410C2311yesmaximal, characteristic, direct product factor, Sylow
91C23×C23—22no—
92C23×C23—22no—
93C23×C23—41no—
94C23×C23—42no—
95C23×C23—42no—
96C23×C23—41no—
97C23×C23—42no—
98C23×C23C2×C22⋊C411yescharacteristic
99C23×C23#12011yescharacteristic
100C23×C23#12311yescharacteristic
101D4×C23—82no—
102D4×C23—82no—
103D4×C23—82no—
104C2×C4×C23—22no—
105C2×C4×C23—22no—
106C2×C4×C23—24no—
107C2×C4×C23—22no—
108C2×C4×C23—22no—
109C2×C4×C23—28no—
110C2×C4×C23—22no—
111C2×C4×C23—41no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—41no—
115C2×C4×C23—44no—
116C2×C4×C23—42no—
117C2×C4×C23—44no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—41no—
123C2×C4×C23—42no—
124C2×C4×C23—81no—
125C2×C4×C23#10414yes—
126C2×C4×C23C4×D412yes—
127C2×C4×C23#12112yes—
128C8×C23—82no—
129#2591C2×C2311yescharacteristic
130#2274C2×C2311yescharacteristic, complemented
131#1507C2×C2312yes—
132#1463C2×C2311yescharacteristic, complemented
133#1297C2×C2312yescomplemented
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—21no—
138C22×C23—21no—
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—44no—
142C22×C23—84no—
143C22×C23—81no—
144C22×C23—81no—
145C22×C23—81no—
146C22×C23#38611yescharacteristic
147C22×C23C2×#10412yes—
148C22×C23#45412yes—
149C4×C23—24no—
150C4×C23—22no—
151C4×C23—22no—
152C4×C23—48no—
153C4×C23—42no—
154C4×C23—82no—
155C4×C23—81no—
156C2×#121C22×C2311yescharacteristic
157C2×#117—21no—
158C22×C4⋊C4C22×C2311yescharacteristic
159#407C22×C2312yes—
160#400C4×C2312yescomplemented
161#394C4×C2312yescomplemented
162#388—22no—
163#386—21no—
164#386—21no—
165C4×C4⋊C4—24no—
166C4×C22⋊C4—21no—
167C4×C22⋊C4C22×C2311yescharacteristic
168C4×C22⋊C4C22×C2312yes—
169#327—24no—
170#324—22no—
171C2×C23—22no—
172C2×C23—22no—
173C2×C23—41no—
174C2×C23—161no—
175C2×C23#188811yescharacteristic
176C2×C23#189611yescharacteristic
177C2×C23#191311yescharacteristic
178C22×D4—21no—
179C22×D4C2×C4×C2311yescharacteristic
180C23×C4D4×C2311yescharacteristic
181C23×C4D4×C2311yescharacteristic
182C23×C4C23×C2311yescharacteristic
183C2×M16—22no—
184#121—22no—
185#121—22no—
186#117—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C2×C4×C2311yescharacteristic
191C2×C4⋊C4C2×C4×C2312yes—
192C2×C4⋊C4D4×C2312yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95C2×C4×C2312yes—
208#95D4×C2314yes—
209C23#941011yescharacteristic, direct product factor, Sylow
210C24—21no—
211C24C2×D4×C2311yescharacteristic
212C2×D4—22no—
213C2×D4—44no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—81no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4C22⋊C4×C2312yes—
228C22×C4C22⋊C4×C2312yes—
229C22×C4C22×C4×C2311yescharacteristic
230C22×C4C2×D4×C2312yes—
231C22×C4C2×D4×C2311yescharacteristic
232C22×C4#41×C2312yes—
233M16—82no—
234C2×C8—42no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C22⋊C4—28no—
240C22⋊C4—42no—
241C22⋊C4—44no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C42—28no—
245C42—82no—
246C42—81no—
247C42—82no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—42no—
255C23C2×C22⋊C4×C2311yescharacteristic
256C23C23×#12011yescharacteristic
257C23C23×#12311yescharacteristic
258D4—82no—
259D4—82no—
260D4—82no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4C23×#10414yes—
283C2×C4C4×D4×C2312yes—
284C2×C4C23×#12112yes—
285C8—82no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22C23×#38611yescharacteristic
299C22C2×C23×#10412yes—
300C22C23×#45412yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—48no—
305C4—42no—
306C4—82no—
307C4—81no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—161no—
312C2C23×#188811yescharacteristic
313C2C23×#189611yescharacteristic
314C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
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
2. C23×#2274C2semidirect
4. C23×#1463C2semidirect
5. C23×#1297C2semidirect
10. C23×#400C4semidirect
11. C23×#394C4semidirect
90. #9410C23direct (N × H)
130. #2274C2×C23semidirect
132. #1463C2×C23semidirect
133. #1297C2×C23semidirect
160. #400C4×C23semidirect
161. #394C4×C23semidirect
209. C23#9410direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25912
C23×#22742
C23×#15072
C23×#14632
C23×#12972
#941023