Finite Group Databasebeta

← C23×#8760 #19156386

Subgroups — C23×#8760 #19156386

Order 5888.

530 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2349C211yesmaximal, characteristic
2C23×#1799C211yesmaximal, characteristic
3C23×#1459C211yesmaximal, characteristic
4C23×#1294C212yesmaximal, complemented
5C23×#1289C212yesmaximal, complemented
6C2×C4×D4×C23—21no—
7C2×C4×D4×C23C2211yescharacteristic
8C2×C23×#99—21no—
9C2×C23×C22⋊C8C2212yes—
10C23×#389—21no—
11C23×#385—21no—
12C23×#385—21no—
13C23×#382C411yescharacteristic, complemented
14C23×#380C411yescharacteristic, complemented
15C23×#380C2212yes—
16C23×#380C2211yescharacteristic
17C4×C4⋊C4×C23C411yescharacteristic, complemented
18C2×C23×#95C411yescharacteristic, complemented
19C2×C23×#95C2211yescharacteristic
20C23×#342—21no—
21C23×#324—21no—
22C23×#323—21no—
23C22×D4×C23D411yescharacteristic
24C23×C4×C23—21no—
25C23×C4×C23—21no—
26C23×C4×C23C2×C411yescharacteristic
27C23×C4×C23C2311yescharacteristic
28C22×C8×C23—22no—
29C4×D4×C23—22no—
30C4×D4×C23—22no—
31C4×D4×C23—42no—
32C4×D4×C23—42no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—22no—
38C2×C4⋊C4×C23D411yescharacteristic
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23D411yescharacteristic
43C2×C22⋊C4×C23D411yescharacteristic
44C2×C42×C23—21no—
45C2×C42×C23C2×C411yescharacteristic
46C23×#99—42no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8—22no—
49C23×#95—21no—
50C23×#95—21no—
51C23×#95—21no—
52C23×#95—22no—
53C23×#95C2×C411yescharacteristic
54C23×#95C2×C411yescharacteristic
55C23×#95C2×C411yescharacteristic
56C23×#95C2×C411yescharacteristic
57C24×C23C22⋊C411yescharacteristic
58C24×C23C2×D411yescharacteristic
59C2×D4×C23—22no—
60C2×D4×C23—22no—
61C2×D4×C23D812yes—
62C2×D4×C23SD1612yes—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23—42no—
72C22×C4×C23—41no—
73C22×C4×C23—41no—
74C22×C4×C23—42no—
75C22×C4×C23—81no—
76C22×C4×C23—81no—
77C22×C4×C23—81no—
78C22×C4×C23—81no—
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22⋊C411yescharacteristic
81C22×C4×C23C22⋊C411yescharacteristic
82C22×C4×C23C22×C411yescharacteristic
83C22×C4×C23C2×D411yescharacteristic
84C22×C4×C23#4112yes—
85C22×C4×C23#4112yes—
86C2×C8×C23—42no—
87C2×C8×C23—82no—
88C4⋊C4×C23—22no—
89C4⋊C4×C23—42no—
90C4⋊C4×C23—42no—
91C4⋊C4×C23—81no—
92C4⋊C4×C23—81no—
93C4⋊C4×C23—82no—
94C22⋊C4×C23—22no—
95C22⋊C4×C23—22no—
96C22⋊C4×C23—42no—
97C22⋊C4×C23—41no—
98C22⋊C4×C23—41no—
99C22⋊C4×C23—42no—
100C22⋊C4×C23—42no—
101C42×C23—21no—
102C42×C23—21no—
103C42×C23—42no—
104#8760C2311yesmaximal, characteristic, direct product factor, Sylow
105C23×C23—21no—
106C23×C23—21no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—41no—
110C23×C23—41no—
111C23×C23#10212yes—
112C23×C23C2×C22⋊C411yescharacteristic
113C23×C23#12312yes—
114C23×C23C2×D811yescharacteristic
115C23×C23C2×SD1611yescharacteristic
116C23×C23#13712yes—
117D4×C23—24no—
118D4×C23—24no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—22no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—41no—
125C2×C4×C23—42no—
126C2×C4×C23—42no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—81no—
142C2×C4×C23—81no—
143C2×C4×C23—82no—
144C2×C4×C23—82no—
145C2×C4×C23—84no—
146C2×C4×C23—82no—
147C2×C4×C23—82no—
148C2×C4×C23—82no—
149C2×C4×C23#10212yes—
150C2×C4×C23#11711yescharacteristic
151C2×C4×C23#11711yescharacteristic
152C2×C4×C23#12312yes—
153C8×C23—82no—
154#2349C2×C2311yescharacteristic
155#1799C2×C2311yescharacteristic
156#1459C2×C2311yescharacteristic
157#1294C2×C2312yescomplemented
158#1289C2×C2312yescomplemented
159C22×C23—21no—
160C22×C23—22no—
161C22×C23—21no—
162C22×C23—22no—
163C22×C23—22no—
164C22×C23—41no—
165C22×C23—41no—
166C22×C23—42no—
167C22×C23—42no—
168C22×C23—42no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23#38111yescharacteristic
172C22×C23C2×#10211yescharacteristic
173C22×C23#41911yescharacteristic
174C22×C23#48012yes—
175C22×C23#48312yes—
176C4×C23—22no—
177C4×C23—22no—
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—42no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23—81no—
184C4×C23—81no—
185C4×C23—81no—
186C4×C23—162no—
187C2×C4×D4—21no—
188C2×C4×D4C22×C2311yescharacteristic
189C2×#99—21no—
190C2×C22⋊C8C22×C2312yes—
191#389—21no—
192#385—21no—
193#385—21no—
194#382C4×C2311yescharacteristic, complemented
195#380C4×C2311yescharacteristic, complemented
196#380C22×C2312yes—
197#380C22×C2311yescharacteristic
198C4×C4⋊C4C4×C2311yescharacteristic, complemented
199C2×#95C4×C2311yescharacteristic, complemented
200C2×#95C22×C2311yescharacteristic
201#342—21no—
202#324—21no—
203#323—21no—
204C2×C23—21no—
205C2×C23—22no—
206C2×C23—21no—
207C2×C23—22no—
208C2×C23—42no—
209C2×C23#182311yescharacteristic
210C2×C23#182711yescharacteristic
211C2×C23#182811yescharacteristic
212C22×D4D4×C2311yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4C2×C4×C2311yescharacteristic
216C23×C4C23×C2311yescharacteristic
217C22×C8—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—42no—
221C4×D4—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4D4×C2311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4D4×C2311yescharacteristic
232C2×C22⋊C4D4×C2311yescharacteristic
233C2×C42—21no—
234C2×C42C2×C4×C2311yescharacteristic
235#99—42no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95C2×C4×C2311yescharacteristic
243#95C2×C4×C2311yescharacteristic
244#95C2×C4×C2311yescharacteristic
245#95C2×C4×C2311yescharacteristic
246C23#876011yescharacteristic, direct product factor, Sylow
247C24C22⋊C4×C2311yescharacteristic
248C24C2×D4×C2311yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4D8×C2312yes—
252C2×D4SD16×C2312yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4C22⋊C4×C2311yescharacteristic
270C22×C4C22⋊C4×C2311yescharacteristic
271C22×C4C22⋊C4×C2311yescharacteristic
272C22×C4C22×C4×C2311yescharacteristic
273C22×C4C2×D4×C2311yescharacteristic
274C22×C4#41×C2312yes—
275C22×C4#41×C2312yes—
276C2×C8—42no—
277C2×C8—82no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—82no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C42—21no—
292C42—21no—
293C42—42no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C23×#10212yes—
301C23C2×C22⋊C4×C2311yescharacteristic
302C23C23×#12312yes—
303C23C2×D8×C2311yescharacteristic
304C23C2×SD16×C2311yescharacteristic
305C23C23×#13712yes—
306D4—24no—
307D4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—84no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4C23×#10212yes—
339C2×C4C23×#11711yescharacteristic
340C2×C4C23×#11711yescharacteristic
341C2×C4C23×#12312yes—
342C8—82no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22C23×#38111yescharacteristic
356C22C2×C23×#10211yescharacteristic
357C22C23×#41911yescharacteristic
358C22C23×#48012yes—
359C22C23×#48312yes—
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—162no—
371C2—21no—
372C2—22no—
373C2—21no—
374C2—22no—
375C2—42no—
376C2C23×#182311yescharacteristic
377C2C23×#182711yescharacteristic
378C2C23×#182811yescharacteristic
subgroupG/H
center, socleC22×C23#381
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC22×D4×C23D4
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
4. C23×#1294C2semidirect
5. C23×#1289C2semidirect
13. C23×#382C4semidirect
14. C23×#380C4semidirect
17. C4×C4⋊C4×C23C4semidirect
18. C2×C23×#95C4semidirect
104. #8760C23direct (N × H)
157. #1294C2×C23semidirect
158. #1289C2×C23semidirect
194. #382C4×C23semidirect
195. #380C4×C23semidirect
198. C4×C4⋊C4C4×C23semidirect
199. C2×#95C4×C23semidirect
246. C23#8760direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23492
C23×#17992
C23×#14592
C23×#12942
C23×#12892
#876023