Finite Group Databasebeta

← C23×#8761 #19156387

Subgroups — C23×#8761 #19156387

Order 5888.

530 conjugacy classes of subgroups.

The subgroup lattice has 372 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—44no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—22no—
37C2×C4⋊C4×C23D411yescharacteristic
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23D411yescharacteristic
42C2×C22⋊C4×C23D411yescharacteristic
43C2×C42×C23—21no—
44C2×C42×C23C2×C411yescharacteristic
45C23×#99—42no—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—22no—
48C23×#95—21no—
49C23×#95—21no—
50C23×#95—21no—
51C23×#95—22no—
52C23×#95C2×C411yescharacteristic
53C23×#95C2×C411yescharacteristic
54C23×#95C2×C411yescharacteristic
55C23×#95C2×C411yescharacteristic
56C24×C23C22⋊C411yescharacteristic
57C24×C23C2×D411yescharacteristic
58C2×D4×C23—22no—
59C2×D4×C23—22no—
60C2×D4×C23SD1612yes—
61C2×D4×C23Q1612yes—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—42no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23—42no—
72C22×C4×C23—42no—
73C22×C4×C23—81no—
74C22×C4×C23—81no—
75C22×C4×C23—81no—
76C22×C4×C23—81no—
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22×C411yescharacteristic
81C22×C4×C23C2×D411yescharacteristic
82C22×C4×C23#4112yes—
83C22×C4×C23#4112yes—
84C2×C8×C23—42no—
85C2×C8×C23—82no—
86C4⋊C4×C23—22no—
87C4⋊C4×C23—42no—
88C4⋊C4×C23—42no—
89C4⋊C4×C23—81no—
90C4⋊C4×C23—81no—
91C4⋊C4×C23—82no—
92C22⋊C4×C23—22no—
93C22⋊C4×C23—22no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—42no—
97C22⋊C4×C23—42no—
98C42×C23—21no—
99C42×C23—21no—
100C42×C23—42no—
101#8761C2311yesmaximal, characteristic, direct product factor, Sylow
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23#10312yes—
109C23×C23C2×C22⋊C411yescharacteristic
110C23×C23#12312yes—
111C23×C23C2×SD1611yescharacteristic
112C23×C23C2×Q1611yescharacteristic
113C23×C23AGL(1,ℤ/8)12yes—
114D4×C23—24no—
115D4×C23—24no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—22no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—41no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—41no—
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—81no—
139C2×C4×C23—81no—
140C2×C4×C23—84no—
141C2×C4×C23—84no—
142C2×C4×C23—82no—
143C2×C4×C23—82no—
144C2×C4×C23—82no—
145C2×C4×C23#10312yes—
146C2×C4×C23#11711yescharacteristic
147C2×C4×C23#11711yescharacteristic
148C2×C4×C23#12312yes—
149C8×C23—82no—
150#2349C2×C2311yescharacteristic
151#1799C2×C2311yescharacteristic
152#1459C2×C2311yescharacteristic
153#1294C2×C2312yescomplemented
154#1289C2×C2312yescomplemented
155C22×C23—21no—
156C22×C23—22no—
157C22×C23—21no—
158C22×C23—22no—
159C22×C23—22no—
160C22×C23—42no—
161C22×C23—41no—
162C22×C23—41no—
163C22×C23—42no—
164C22×C23—42no—
165C22×C23—41no—
166C22×C23—41no—
167C22×C23#38111yescharacteristic
168C22×C23C2×#10311yescharacteristic
169C22×C23#41811yescharacteristic
170C22×C23#48112yes—
171C22×C23#48412yes—
172C4×C23—22no—
173C4×C23—22no—
174C4×C23—21no—
175C4×C23—21no—
176C4×C23—42no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—81no—
180C4×C23—81no—
181C4×C23—81no—
182C4×C23—162no—
183C2×C4×D4—21no—
184C2×C4×D4C22×C2311yescharacteristic
185C2×#99—21no—
186C2×C22⋊C8C22×C2312yes—
187#389—21no—
188#385—21no—
189#385—21no—
190#382C4×C2311yescharacteristic, complemented
191#380C4×C2311yescharacteristic, complemented
192#380C22×C2312yes—
193#380C22×C2311yescharacteristic
194C4×C4⋊C4C4×C2311yescharacteristic, complemented
195C2×#95C4×C2311yescharacteristic, complemented
196C2×#95C22×C2311yescharacteristic
197#342—21no—
198#324—21no—
199#323—21no—
200C2×C23—21no—
201C2×C23—22no—
202C2×C23—21no—
203C2×C23—22no—
204C2×C23—42no—
205C2×C23#182411yescharacteristic
206C2×C23#182711yescharacteristic
207C2×C23#182811yescharacteristic
208C22×D4D4×C2311yescharacteristic
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C2×C4×C2311yescharacteristic
212C23×C4C23×C2311yescharacteristic
213C22×C8—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4D4×C2311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4D4×C2311yescharacteristic
227C2×C22⋊C4D4×C2311yescharacteristic
228C2×C42—21no—
229C2×C42C2×C4×C2311yescharacteristic
230#99—42no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95C2×C4×C2311yescharacteristic
238#95C2×C4×C2311yescharacteristic
239#95C2×C4×C2311yescharacteristic
240#95C2×C4×C2311yescharacteristic
241C23#876111yescharacteristic, direct product factor, Sylow
242C24C22⋊C4×C2311yescharacteristic
243C24C2×D4×C2311yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4SD16×C2312yes—
247C2×D4Q16×C2312yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊C4×C2311yescharacteristic
264C22×C4C22⋊C4×C2311yescharacteristic
265C22×C4C22⋊C4×C2311yescharacteristic
266C22×C4C22×C4×C2311yescharacteristic
267C22×C4C2×D4×C2311yescharacteristic
268C22×C4#41×C2312yes—
269C22×C4#41×C2312yes—
270C2×C8—42no—
271C2×C8—82no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—82no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C42—21no—
285C42—21no—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23C23×#10312yes—
294C23C2×C22⋊C4×C2311yescharacteristic
295C23C23×#12312yes—
296C23C2×SD16×C2311yescharacteristic
297C23C2×Q16×C2311yescharacteristic
298C23C23×AGL(1,ℤ/8)12yes—
299D4—24no—
300D4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4C23×#10312yes—
331C2×C4C23×#11711yescharacteristic
332C2×C4C23×#11711yescharacteristic
333C2×C4C23×#12312yes—
334C8—82no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22C23×#38111yescharacteristic
348C22C2×C23×#10311yescharacteristic
349C22C23×#41811yescharacteristic
350C22C23×#48112yes—
351C22C23×#48412yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—162no—
363C2—21no—
364C2—22no—
365C2—21no—
366C2—22no—
367C2—42no—
368C2C23×#182411yescharacteristic
369C2C23×#182711yescharacteristic
370C2C23×#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
101. #8761C23direct (N × H)
153. #1294C2×C23semidirect
154. #1289C2×C23semidirect
190. #382C4×C23semidirect
191. #380C4×C23semidirect
194. C4×C4⋊C4C4×C23semidirect
195. C2×#95C4×C23semidirect
241. C23#8761direct (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
#876123