Finite Group Databasebeta

← C23×#9464 #19157090

Subgroups — C23×#9464 #19157090

Order 5888.

610 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2501C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic, complemented
3C23×#1821C211yesmaximal, characteristic, complemented
4C23×#1510C212yesmaximal
5C23×#1284C212yesmaximal, complemented
6C2×C23×#123—21no—
7C2×C23×#123—21no—
8C2×C23×#121C2211yescharacteristic
9C22×C4⋊C4×C23C2211yescharacteristic
10C23×#407C2212yes—
11C23×#399—22no—
12C23×#390C412yescomplemented
13C23×#387—24no—
14C23×#384C412yescomplemented
15C23×#381C2212yes—
16C23×#380—21no—
17C4×C22⋊C4×C23—21no—
18C4×C22⋊C4×C23C2211yescharacteristic
19C23×#343—22no—
20C23×#327—24no—
21C22×D4×C23—21no—
22C22×D4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C23×C4×C23C2311yescharacteristic
26C2×M16×C23—22no—
27C23×#123—44no—
28C23×#123—44no—
29C23×#121—22no—
30C23×#121—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—22no—
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C4⋊C4×C23C2×C412yes—
38C2×C4⋊C4×C23D412yes—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23—21no—
44C2×C42×C23—22no—
45C2×C42×C23C2×C412yes—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—22no—
48C23×#95—22no—
49C23×#95—22no—
50C23×#95—21no—
51C23×#95—22no—
52C23×#95D414yes—
53C24×C23—21no—
54C24×C23C2×D411yescharacteristic
55C2×D4×C23—22no—
56C2×D4×C23—42no—
57C2×D4×C23—42no—
58C2×D4×C23—42no—
59C2×D4×C23—41no—
60C2×D4×C23—41no—
61C22×C4×C23—22no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—24no—
68C22×C4×C23—42no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23C22⋊C412yes—
72C22×C4×C23C22⋊C412yes—
73C22×C4×C23C22×C411yescharacteristic
74C22×C4×C23C2×D411yescharacteristic
75C22×C4×C23C2×D412yes—
76C22×C4×C23#4112yes—
77M16×C23—44no—
78C2×C8×C23—42no—
79C4⋊C4×C23—24no—
80C4⋊C4×C23—24no—
81C4⋊C4×C23—22no—
82C4⋊C4×C23—44no—
83C4⋊C4×C23—44no—
84C4⋊C4×C23—82no—
85C22⋊C4×C23—24no—
86C22⋊C4×C23—44no—
87C22⋊C4×C23—44no—
88C22⋊C4×C23—44no—
89C22⋊C4×C23—42no—
90C42×C23—42no—
91C42×C23—82no—
92#9464C2311yesmaximal, characteristic, direct product factor, Sylow
93C23×C23—22no—
94C23×C23—22no—
95C23×C23—41no—
96C23×C23—41no—
97C23×C23—41no—
98C23×C23—42no—
99C23×C23—41no—
100C23×C23—42no—
101C23×C23C2×C22⋊C411yescharacteristic
102C23×C23#12011yescharacteristic
103C23×C23#12311yescharacteristic
104D4×C23—82no—
105D4×C23—82no—
106D4×C23—82no—
107C2×C4×C23—22no—
108C2×C4×C23—22no—
109C2×C4×C23—24no—
110C2×C4×C23—22no—
111C2×C4×C23—22no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—44no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—44no—
124C2×C4×C23—41no—
125C2×C4×C23—44no—
126C2×C4×C23—81no—
127C2×C4×C23—81no—
128C2×C4×C23C4×D412yes—
129C2×C4×C23#12112yes—
130C8×C23—82no—
131#2501C2×C2311yescharacteristic
132#2313C2×C2311yescharacteristic, complemented
133#1821C2×C2311yescharacteristic, complemented
134#1510C2×C2312yes—
135#1284C2×C2312yescomplemented
136C22×C23—21no—
137C22×C23—21no—
138C22×C23—21no—
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—44no—
144C22×C23—82no—
145C22×C23—82no—
146C22×C23—81no—
147C22×C23—81no—
148C22×C23—81no—
149C22×C23#38611yescharacteristic
150C22×C23#41012yes—
151C22×C23#45312yes—
152C4×C23—22no—
153C4×C23—22no—
154C4×C23—44no—
155C4×C23—42no—
156C4×C23—42no—
157C4×C23—84no—
158C4×C23—81no—
159C2×#123—21no—
160C2×#123—21no—
161C2×#121C22×C2311yescharacteristic
162C22×C4⋊C4C22×C2311yescharacteristic
163#407C22×C2312yes—
164#399—22no—
165#390C4×C2312yescomplemented
166#387—24no—
167#384C4×C2312yescomplemented
168#381C22×C2312yes—
169#380—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4C22×C2311yescharacteristic
172#343—22no—
173#327—24no—
174C2×C23—22no—
175C2×C23—22no—
176C2×C23—41no—
177C2×C23—161no—
178C2×C23#188711yescharacteristic
179C2×C23#189911yescharacteristic
180C2×C23#191211yescharacteristic
181C22×D4—21no—
182C22×D4C2×C4×C2311yescharacteristic
183C23×C4D4×C2311yescharacteristic
184C23×C4D4×C2311yescharacteristic
185C23×C4C23×C2311yescharacteristic
186C2×M16—22no—
187#123—44no—
188#123—44no—
189#121—22no—
190#121—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C2×C4×C2311yescharacteristic
197C2×C4⋊C4C2×C4×C2312yes—
198C2×C4⋊C4D4×C2312yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C42—22no—
205C2×C42C2×C4×C2312yes—
206C22⋊C8—22no—
207C22⋊C8—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95D4×C2314yes—
213C23#946411yescharacteristic, direct product factor, Sylow
214C24—21no—
215C24C2×D4×C2311yescharacteristic
216C2×D4—22no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4C22⋊C4×C2312yes—
233C22×C4C22⋊C4×C2312yes—
234C22×C4C22×C4×C2311yescharacteristic
235C22×C4C2×D4×C2311yescharacteristic
236C22×C4C2×D4×C2312yes—
237C22×C4#41×C2312yes—
238M16—44no—
239C2×C8—42no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—82no—
246C22⋊C4—24no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C42—42no—
252C42—82no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23C2×C22⋊C4×C2311yescharacteristic
262C23C23×#12011yescharacteristic
263C23C23×#12311yescharacteristic
264D4—82no—
265D4—82no—
266D4—82no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—44no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4C4×D4×C2312yes—
289C2×C4C23×#12112yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—82no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22C23×#38611yescharacteristic
305C22C23×#41012yes—
306C22C23×#45312yes—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—42no—
312C4—84no—
313C4—81no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—161no—
318C2C23×#188711yescharacteristic
319C2C23×#189911yescharacteristic
320C2C23×#191211yescharacteristic
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×#2313C2semidirect
3. C23×#1821C2semidirect
5. C23×#1284C2semidirect
12. C23×#390C4semidirect
14. C23×#384C4semidirect
92. #9464C23direct (N × H)
132. #2313C2×C23semidirect
133. #1821C2×C23semidirect
135. #1284C2×C23semidirect
165. #390C4×C23semidirect
167. #384C4×C23semidirect
213. C23#9464direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25012
C23×#23132
C23×#18212
C23×#15102
C23×#12842
#946423