Finite Group Databasebeta

← C23×#9409 #19157035

Subgroups — C23×#9409 #19157035

Order 5888.

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

normal subgroups only

This group has 840 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×#2539C211yesmaximal, characteristic
2C23×#2273C211yesmaximal, characteristic, complemented
3C23×#1507C212yesmaximal, complemented
4C23×#1462C211yesmaximal, characteristic, complemented
5C23×#1297C212yesmaximal, complemented
6C2×C23×#123—21no—
7C2×C23×#121C2211yescharacteristic
8C2×C23×#120—21no—
9C22×C22⋊C4×C23—21no—
10C22×C22⋊C4×C23C2211yescharacteristic
11C23×#407C2212yes—
12C23×#399C412yescomplemented
13C23×#392C412yescomplemented
14C23×#390—22no—
15C4×C22⋊C4×C23—24no—
16C4×C22⋊C4×C23—21no—
17C4×C22⋊C4×C23C2211yescharacteristic
18C4×C22⋊C4×C23C2212yes—
19C23×#327—24no—
20C23×#323—22no—
21C25×C23D411yescharacteristic
22C22×D4×C23—21no—
23C22×D4×C23—22no—
24C22×D4×C23C2×C411yescharacteristic, complemented
25C23×C4×C23—22no—
26C23×C4×C23D411yescharacteristic
27C23×C4×C23C2311yescharacteristic
28C2×M16×C23—22no—
29C23×#123—44no—
30C23×#121—22no—
31C23×#121—22no—
32C23×#120—44no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23C2×C411yescharacteristic, complemented
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—28no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23C2×C412yes—
45C2×C22⋊C4×C23D412yes—
46C2×C42×C23—22no—
47C2×C42×C23—22no—
48C2×C42×C23—21no—
49C23×C22⋊C8—22no—
50C23×C22⋊C8—22no—
51C23×#95—22no—
52C23×#95C2×C412yescomplemented
53C23×#95D414yescomplemented
54C24×C23—21no—
55C24×C23—24no—
56C24×C23—21no—
57C24×C23—21no—
58C24×C23—42no—
59C24×C23—81no—
60C24×C23C22⋊C412yes—
61C24×C23C2×D411yescharacteristic
62C2×D4×C23—22no—
63C2×D4×C23—44no—
64C2×D4×C23—42no—
65C2×D4×C23—42no—
66C2×D4×C23—44no—
67C2×D4×C23—41no—
68C2×D4×C23—41no—
69C2×D4×C23—82no—
70C2×D4×C23—81no—
71C22×C4×C23—22no—
72C22×C4×C23—24no—
73C22×C4×C23—21no—
74C22×C4×C23—28no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—42no—
78C22×C4×C23—81no—
79C22×C4×C23—82no—
80C22×C4×C23—82no—
81C22×C4×C23C22⋊C412yes—
82C22×C4×C23C22×C411yescharacteristic
83C22×C4×C23C2×D412yes—
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23#4112yes—
86M16×C23—82no—
87C2×C8×C23—42no—
88C4⋊C4×C23—22no—
89C4⋊C4×C23—44no—
90C22⋊C4×C23—28no—
91C22⋊C4×C23—28no—
92C22⋊C4×C23—24no—
93C22⋊C4×C23—24no—
94C22⋊C4×C23—24no—
95C22⋊C4×C23—44no—
96C22⋊C4×C23—44no—
97C22⋊C4×C23—42no—
98C42×C23—82no—
99C42×C23—81no—
100C42×C23—82no—
101#9409C2311yesmaximal, characteristic, direct product factor, Sylow
102C23×C23—22no—
103C23×C23—22no—
104C23×C23—24no—
105C23×C23—22no—
106C23×C23—22no—
107C23×C23—22no—
108C23×C23—22no—
109C23×C23—41no—
110C23×C23—42no—
111C23×C23—42no—
112C23×C23—42no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23—42no—
116C23×C23—42no—
117C23×C23—41no—
118C23×C23—42no—
119C23×C23—81no—
120C23×C23—81no—
121C23×C23—88no—
122C23×C23#10414yes—
123C23×C23C2×C22⋊C411yescharacteristic
124C23×C23#12011yescharacteristic
125C23×C23#12311yescharacteristic
126D4×C23—82no—
127D4×C23—88no—
128D4×C23—82no—
129D4×C23—82no—
130C2×C4×C23—28no—
131C2×C4×C23—24no—
132C2×C4×C23—22no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—42no—
136C2×C4×C23—41no—
137C2×C4×C23—44no—
138C2×C4×C23—42no—
139C2×C4×C23—42no—
140C2×C4×C23—42no—
141C2×C4×C23—44no—
142C2×C4×C23—42no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—81no—
147C2×C4×C23—88no—
148C2×C4×C23C4×D412yes—
149C2×C4×C23#12112yes—
150C8×C23—82no—
151#2539C2×C2311yescharacteristic
152#2273C2×C2311yescharacteristic, complemented
153#1507C2×C2312yescomplemented
154#1462C2×C2311yescharacteristic, complemented
155#1297C2×C2312yescomplemented
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—24no—
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—44no—
165C22×C23—44no—
166C22×C23—44no—
167C22×C23—44no—
168C22×C23—42no—
169C22×C23—42no—
170C22×C23—82no—
171C22×C23—82no—
172C22×C23—82no—
173C22×C23—84no—
174C22×C23—81no—
175C22×C23—81no—
176C22×C23—81no—
177C22×C23#38611yescharacteristic
178C22×C23C2×#10412yes—
179C22×C23#45312yes—
180C4×C23—22no—
181C4×C23—22no—
182C4×C23—48no—
183C4×C23—44no—
184C4×C23—42no—
185C4×C23—81no—
186C2×#123—21no—
187C2×#121C22×C2311yescharacteristic
188C2×#120—21no—
189C22×C22⋊C4—21no—
190C22×C22⋊C4C22×C2311yescharacteristic
191#407C22×C2312yes—
192#399C4×C2312yescomplemented
193#392C4×C2312yescomplemented
194#390—22no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—24no—
197C4×C22⋊C4C22×C2311yescharacteristic
198C4×C22⋊C4C22×C2312yes—
199#327—24no—
200#323—22no—
201C2×C23—22no—
202C2×C23—22no—
203C2×C23—41no—
204C2×C23—44no—
205C2×C23—161no—
206C2×C23#188711yescharacteristic
207C2×C23#189611yescharacteristic
208C2×C23#191311yescharacteristic
209C25D4×C2311yescharacteristic
210C22×D4—21no—
211C22×D4—22no—
212C22×D4C2×C4×C2311yescharacteristic, complemented
213C23×C4—22no—
214C23×C4D4×C2311yescharacteristic
215C23×C4C23×C2311yescharacteristic
216C2×M16—22no—
217#123—44no—
218#121—22no—
219#121—22no—
220#120—44no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4C2×C4×C2311yescharacteristic, complemented
223C2×C22⋊C4—28no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4C2×C4×C2312yes—
233C2×C22⋊C4D4×C2312yes—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239#95—22no—
240#95C2×C4×C2312yescomplemented
241#95D4×C2314yescomplemented
242C23#940911yescharacteristic, direct product factor, Sylow
243C24—21no—
244C24—24no—
245C24—21no—
246C24—21no—
247C24—42no—
248C24—81no—
249C24C22⋊C4×C2312yes—
250C24C2×D4×C2311yescharacteristic
251C2×D4—22no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—82no—
259C2×D4—81no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C22⋊C4×C2312yes—
271C22×C4C22×C4×C2311yescharacteristic
272C22×C4C2×D4×C2312yes—
273C22×C4C2×D4×C2311yescharacteristic
274C22×C4#41×C2312yes—
275M16—82no—
276C2×C8—42no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C42—82no—
288C42—81no—
289C42—82no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—42no—
307C23—81no—
308C23—81no—
309C23—88no—
310C23C23×#10414yes—
311C23C2×C22⋊C4×C2311yescharacteristic
312C23C23×#12011yescharacteristic
313C23C23×#12311yescharacteristic
314D4—82no—
315D4—88no—
316D4—82no—
317D4—82no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—88no—
336C2×C4C4×D4×C2312yes—
337C2×C4C23×#12112yes—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—84no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22C23×#38611yescharacteristic
361C22C2×C23×#10412yes—
362C22C23×#45312yes—
363C4—22no—
364C4—22no—
365C4—48no—
366C4—44no—
367C4—42no—
368C4—81no—
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—161no—
374C2C23×#188711yescharacteristic
375C2C23×#189611yescharacteristic
376C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#2539C2
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×#2273C2semidirect
3. C23×#1507C2semidirect
4. C23×#1462C2semidirect
5. C23×#1297C2semidirect
12. C23×#399C4semidirect
13. C23×#392C4semidirect
24. C22×D4×C23C2×C4semidirect
34. C2×C4⋊C4×C23C2×C4semidirect
52. C23×#95C2×C4semidirect
53. C23×#95D4semidirect
101. #9409C23direct (N × H)
152. #2273C2×C23semidirect
153. #1507C2×C23semidirect
154. #1462C2×C23semidirect
155. #1297C2×C23semidirect
192. #399C4×C23semidirect
193. #392C4×C23semidirect
212. C22×D4C2×C4×C23semidirect
222. C2×C4⋊C4C2×C4×C23semidirect
240. #95C2×C4×C23semidirect
241. #95D4×C23semidirect
242. C23#9409direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25392
C23×#22732
C23×#15072
C23×#14622
C23×#12972
#940923