Finite Group Databasebeta

← C23×#9166 #19156792

Subgroups — C23×#9166 #19156792

Order 5888.

436 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3113C211yesmaximal, characteristic
2C23×#2984C211yesmaximal, characteristic
3C23×#1763C211yesmaximal, characteristic
4C23×#1531C211yesmaximal, characteristic
5C23×#1503C211yesmaximal, characteristic
6C23×#1302C211yesmaximal, characteristic
7C23×#1285C211yesmaximal, characteristic
8C23×#533C2211yescharacteristic
9C2×C4×Q8×C23C2211yescharacteristic
10C23×#483—21no—
11C23×#475C411yescharacteristic, complemented
12C23×#475C411yescharacteristic, complemented
13C23×#462C411yescharacteristic, complemented
14C23×#462C411yescharacteristic, complemented
15C23×#451—21no—
16C23×#446—22no—
17C23×#446—21no—
18C8×Q8×C23—21no—
19C23×#441—21no—
20C23×#441—21no—
21C23×#432—21no—
22C23×#432C2211yescharacteristic
23C23×#432C2211yescharacteristic
24C23×#431—21no—
25C23×#431C2211yescharacteristic
26C2×C23×#103—21no—
27C2×C23×#97C2211yescharacteristic
28C4×C4⋊C4×C23C2211yescharacteristic
29C23×#340—21no—
30C23×#330—22no—
31C23×#328—21no—
32C23×#328—21no—
33C22×Q8×C23D411yescharacteristic
34C2×Q16×C23—21no—
35C2×Q16×C23—21no—
36C22×C8×C23—21no—
37C23×C4⋊Q8C2×C411yescharacteristic
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×#125—21no—
40C23×C22⋊Q8—21no—
41C23×C22⋊Q8—21no—
42C4×Q8×C23—21no—
43C4×Q8×C23—21no—
44C4×Q8×C23—21no—
45C4×Q8×C23—21no—
46C4×Q8×C23C2×C411yescharacteristic
47C4×Q8×C23C2×C411yescharacteristic
48C4×Q8×C23D411yescharacteristic
49C4×Q8×C23D411yescharacteristic
50C23×#117—21no—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23D411yescharacteristic
54C2×C42×C23—21no—
55C2×C42×C23—21no—
56C2×C42×C23C2311yescharacteristic
57C23×#106—21no—
58C23×#106—21no—
59C23×C4⋊C8—22no—
60C23×C4⋊C8—22no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8C2×C411yescharacteristic
64C23×C4⋊C8C2×C411yescharacteristic
65C23×#103—24no—
66C23×#103—21no—
67C23×#103—21no—
68C23×#103—21no—
69C23×#103—21no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8D411yescharacteristic
73C23×C22⋊C8D411yescharacteristic
74C23×#97—22no—
75C23×#97—21no—
76C23×#97—21no—
77C23×#97—21no—
78C23×#97D412yes—
79C4×C8×C23—22no—
80C4×C8×C23—21no—
81C23×#95—21no—
82C2×Q8×C23—21no—
83C2×Q8×C23—22no—
84C2×Q8×C23—21no—
85C2×Q8×C23—21no—
86C2×Q8×C23—41no—
87C2×Q8×C23C22⋊C411yescharacteristic
88C2×Q8×C23C22⋊C411yescharacteristic
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—41no—
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23#4111yescharacteristic
95Q16×C23—42no—
96Q16×C23—42no—
97C2×C8×C23—22no—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—21no—
101C2×C8×C23—21no—
102C2×C8×C23—22no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—41no—
118C4⋊C4×C23C22⋊C411yescharacteristic
119C4⋊C4×C23C22⋊C411yescharacteristic
120C22⋊C4×C23—41no—
121C42×C23—21no—
122C42×C23—21no—
123C42×C23—21no—
124C42×C23—21no—
125C42×C23—41no—
126C42×C23—81no—
127C42×C23C22×C411yescharacteristic
128C42×C23C2×D411yescharacteristic
129C42×C23C2×D411yescharacteristic
130C42×C23#4111yescharacteristic
131#9166C2311yesmaximal, characteristic, direct product factor, Sylow
132C23×C23#12111yescharacteristic
133Q8×C23—21no—
134Q8×C23—21no—
135Q8×C23—21no—
136Q8×C23—42no—
137Q8×C23—41no—
138Q8×C23—41no—
139Q8×C23—81no—
140Q8×C23#10412yes—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—41no—
150C2×C4×C23—42no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—42no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—81no—
157C2×C4×C23—81no—
158C2×C4×C23C2×C22⋊C411yescharacteristic
159C2×C4×C23C4×D411yescharacteristic
160C2×C4×C23C4×D411yescharacteristic
161C2×C4×C23#12011yescharacteristic
162C2×C4×C23#12111yescharacteristic
163C2×C4×C23#12311yescharacteristic
164C8×C23—41no—
165C8×C23—41no—
166C8×C23—42no—
167C8×C23—42no—
168C8×C23—41no—
169C8×C23—41no—
170#3113C2×C2311yescharacteristic
171#2984C2×C2311yescharacteristic
172#1763C2×C2311yescharacteristic
173#1531C2×C2311yescharacteristic
174#1503C2×C2311yescharacteristic
175#1302C2×C2311yescharacteristic
176#1285C2×C2311yescharacteristic
177C22×C23—21no—
178C22×C23—21no—
179C22×C23#38611yescharacteristic
180C22×C23#47111yescharacteristic
181C22×C23#47311yescharacteristic
182C4×C23—21no—
183C4×C23—22no—
184C4×C23—21no—
185C4×C23—21no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—41no—
189C4×C23—41no—
190C4×C23—41no—
191C4×C23—82no—
192C4×C23—81no—
193C4×C23—81no—
194C4×C23C2×#10411yescharacteristic
195C4×C23#42111yescharacteristic
196#533C22×C2311yescharacteristic
197C2×C4×Q8C22×C2311yescharacteristic
198#483—21no—
199#475C4×C2311yescharacteristic, complemented
200#475C4×C2311yescharacteristic, complemented
201#462C4×C2311yescharacteristic, complemented
202#462C4×C2311yescharacteristic, complemented
203#451—21no—
204#446—21no—
205#446—22no—
206C8×Q8—21no—
207#441—21no—
208#441—21no—
209#432—21no—
210#432C22×C2311yescharacteristic
211#432C22×C2311yescharacteristic
212#431—21no—
213#431C22×C2311yescharacteristic
214C2×#103—21no—
215C2×#97C22×C2311yescharacteristic
216C4×C4⋊C4C22×C2311yescharacteristic
217#340—21no—
218#330—22no—
219#328—21no—
220#328—21no—
221C2×C23—21no—
222C2×C23—21no—
223C2×C23#187111yescharacteristic
224C2×C23#190511yescharacteristic
225C2×C23#190811yescharacteristic
226C22×Q8D4×C2311yescharacteristic
227C2×Q16—21no—
228C2×Q16—21no—
229C22×C8—21no—
230C4⋊Q8C2×C4×C2311yescharacteristic
231C4⋊Q8C2×C4×C2311yescharacteristic
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8C2×C4×C2311yescharacteristic
240C4×Q8C2×C4×C2311yescharacteristic
241C4×Q8D4×C2311yescharacteristic
242C4×Q8D4×C2311yescharacteristic
243#117—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4D4×C2311yescharacteristic
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C23×C2311yescharacteristic
250#106—21no—
251#106—21no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8C2×C4×C2311yescharacteristic
257C4⋊C8C2×C4×C2311yescharacteristic
258#103—24no—
259#103—21no—
260#103—21no—
261#103—21no—
262#103—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8D4×C2311yescharacteristic
266C22⋊C8D4×C2311yescharacteristic
267#97—22no—
268#97—21no—
269#97—21no—
270#97—21no—
271#97D4×C2312yes—
272C4×C8—22no—
273C4×C8—21no—
274#95—21no—
275C23#916611yescharacteristic, direct product factor, Sylow
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8C22⋊C4×C2311yescharacteristic
282C2×Q8C22⋊C4×C2311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4C2×D4×C2311yescharacteristic
287C22×C4C2×D4×C2311yescharacteristic
288C22×C4#41×C2311yescharacteristic
289Q16—42no—
290Q16—42no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4C22⋊C4×C2311yescharacteristic
313C4⋊C4C22⋊C4×C2311yescharacteristic
314C22⋊C4—41no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—81no—
321C42C22×C4×C2311yescharacteristic
322C42C2×D4×C2311yescharacteristic
323C42C2×D4×C2311yescharacteristic
324C42#41×C2311yescharacteristic
325C23C23×#12111yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333Q8C23×#10412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C2×C22⋊C4×C2311yescharacteristic
352C2×C4C4×D4×C2311yescharacteristic
353C2×C4C4×D4×C2311yescharacteristic
354C2×C4C23×#12011yescharacteristic
355C2×C4C23×#12111yescharacteristic
356C2×C4C23×#12311yescharacteristic
357C8—42no—
358C8—41no—
359C8—41no—
360C8—42no—
361C8—41no—
362C8—41no—
363C22—21no—
364C22—21no—
365C22C23×#38611yescharacteristic
366C22C23×#47111yescharacteristic
367C22C23×#47311yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—82no—
378C4—81no—
379C4—81no—
380C4C2×C23×#10411yescharacteristic
381C4C23×#42111yescharacteristic
382C2—21no—
383C2—21no—
384C2C23×#187111yescharacteristic
385C2C23×#190511yescharacteristic
386C2C23×#190811yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×C23#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
11. C23×#475C4semidirect
12. C23×#475C4semidirect
13. C23×#462C4semidirect
14. C23×#462C4semidirect
131. #9166C23direct (N × H)
199. #475C4×C23semidirect
200. #475C4×C23semidirect
201. #462C4×C23semidirect
202. #462C4×C23semidirect
275. C23#9166direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31132
C23×#29842
C23×#17632
C23×#15312
C23×#15032
C23×#13022
C23×#12852
#916623