Finite Group Databasebeta

← C23×#9722 #19157348

Subgroups — C23×#9722 #19157348

Order 5888.

460 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3167C211yesmaximal, characteristic
2C23×#1986C211yesmaximal, characteristic, complemented
3C23×#1980C211yesmaximal, characteristic
4C4×C23×#107C211yesmaximal, characteristic, complemented
5C23×#1503C211yesmaximal, characteristic, complemented
6C23×#1502C211yesmaximal, characteristic
7C23×#1451C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#498C411yescharacteristic, complemented
11C23×#492C411yescharacteristic, complemented
12C23×#487C411yescharacteristic, complemented
13C23×#486C411yescharacteristic, complemented
14C23×#465—21no—
15C23×#465—21no—
16C4×SD16×C23—21no—
17C4×SD16×C23—21no—
18C23×#431C2211yescharacteristic
19C2×C23×#107—21no—
20C2×C23×#104—22no—
21C4×M16×C23C2211yescharacteristic
22C2×C4×C8×C23C2211yescharacteristic
23C4×C4⋊C4×C23C2211yescharacteristic
24C4×C4⋊C4×C23C2211yescharacteristic
25C23×#340—21no—
26C23×#339—22no—
27C23×#328—22no—
28C23×#327—22no—
29C23×#322—21no—
30C23×#322—21no—
31C2×#41×C23—21no—
32C2×SD16×C23—21no—
33C2×SD16×C23—21no—
34C2×M16×C23D412yes—
35C22×C8×C23D411yescharacteristic
36C22×C8×C23D411yescharacteristic
37C23×C4⋊Q8C2×C411yescharacteristic
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×C4⋊Q8C2×C411yescharacteristic
40C23×#127C2×C411yescharacteristic
41C23×#125—21no—
42C23×#124—21no—
43C23×C22⋊Q8—21no—
44C23×C22⋊Q8—21no—
45C23×#121—21no—
46C23×#121—21no—
47C4×Q8×C23—21no—
48C4×Q8×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C23×#117—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C42×C23—21no—
55C2×C42×C23—21no—
56C2×C42×C23C2311yescharacteristic
57C23×#107—24no—
58C23×#106—21no—
59C23×#106—21no—
60C23×C4⋊C8—21no—
61C23×#104—44no—
62C23×#103—21no—
63C23×#103—21no—
64C23×#102—21no—
65C23×#102—21no—
66C23×C22⋊C8D412yes—
67C23×#97—21no—
68C4×C8×C23—21no—
69C4×C8×C23—21no—
70C4×C8×C23C2×C411yescharacteristic
71C4×C8×C23C2×C411yescharacteristic
72C4×C8×C23D411yescharacteristic
73C4×C8×C23D411yescharacteristic
74C23×#95—21no—
75C23×#95—21no—
76#41×C23—42no—
77#41×C23—42no—
78C2×Q8×C23—21no—
79C2×Q8×C23—21no—
80C2×Q8×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—41no—
85C22×C4×C23—41no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91SD16×C23—42no—
92SD16×C23—42no—
93M16×C23—22no—
94M16×C23—22no—
95C2×C8×C23—22no—
96C2×C8×C23—22no—
97C2×C8×C23—22no—
98C2×C8×C23—41no—
99C2×C8×C23C22⋊C411yescharacteristic
100C2×C8×C23C22⋊C411yescharacteristic
101C2×C8×C23C22⋊C411yescharacteristic
102C2×C8×C23C22⋊C411yescharacteristic
103C4⋊C4×C23—22no—
104C4⋊C4×C23—22no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—41no—
118C42×C23—41no—
119C42×C23—42no—
120C42×C23—42no—
121C42×C23C22×C411yescharacteristic
122C42×C23C2×D411yescharacteristic
123C42×C23#4111yescharacteristic
124C42×C23#4111yescharacteristic
125#9722C2311yesmaximal, characteristic, direct product factor, Sylow
126C23×C23—41no—
127C23×C23#12711yescharacteristic
128Q8×C23—41no—
129Q8×C23—41no—
130Q8×C23—42no—
131Q8×C23—81no—
132D4×C23—41no—
133D4×C23—41no—
134D4×C23—42no—
135D4×C23—81no—
136D4×C23—81no—
137D4×C23—81no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—41no—
141C2×C4×C23—42no—
142C2×C4×C23—42no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—81no—
147C2×C4×C23—81no—
148C2×C4×C23—81no—
149C2×C4×C23—81no—
150C2×C4×C23—82no—
151C2×C4×C23C2×C22⋊C411yescharacteristic
152C2×C4×C23C4×D411yescharacteristic
153C2×C4×C23C4×D411yescharacteristic
154C2×C4×C23#12111yescharacteristic
155C2×C4×C23#12111yescharacteristic
156C2×C4×C23#12411yescharacteristic
157C2×C4×C23AGL(1,ℤ/8)12yes—
158C2×C4×C23#13712yes—
159C8×C23—21no—
160C8×C23—21no—
161C8×C23—21no—
162C8×C23—42no—
163C8×C23—42no—
164C8×C23#10412yescomplemented
165#3167C2×C2311yescharacteristic
166#1986C2×C2311yescharacteristic, complemented
167#1980C2×C2311yescharacteristic
168C4×#107C2×C2311yescharacteristic, complemented
169#1503C2×C2311yescharacteristic, complemented
170#1502C2×C2311yescharacteristic
171#1451C2×C2311yescharacteristic, complemented
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—81no—
175C22×C23—81no—
176C22×C23—81no—
177C22×C23#39011yescharacteristic
178C22×C23#49611yescharacteristic
179C22×C23#49711yescharacteristic
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—82no—
184C4×C23—82no—
185C4×C23—81no—
186C4×C23—81no—
187C4×C23—81no—
188C4×C23C2×#10411yescharacteristic
189C4×C23#42111yescharacteristic
190C4×C23#44012yes—
191C4×C23#46812yes—
192#533C22×C2311yescharacteristic
193#532C22×C2311yescharacteristic
194#498C4×C2311yescharacteristic, complemented
195#492C4×C2311yescharacteristic, complemented
196#487C4×C2311yescharacteristic, complemented
197#486C4×C2311yescharacteristic, complemented
198#465—21no—
199#465—21no—
200C4×SD16—21no—
201C4×SD16—21no—
202#431C22×C2311yescharacteristic
203C2×#107—21no—
204C2×#104—22no—
205C4×M16C22×C2311yescharacteristic
206C2×C4×C8C22×C2311yescharacteristic
207C4×C4⋊C4C22×C2311yescharacteristic
208C4×C4⋊C4C22×C2311yescharacteristic
209#340—21no—
210#339—22no—
211#328—22no—
212#327—22no—
213#322—21no—
214#322—21no—
215C2×C23—21no—
216C2×C23—21no—
217C2×C23—161no—
218C2×C23#197311yescharacteristic
219C2×C23#197611yescharacteristic
220C2×C23#198011yescharacteristic
221C2×#41—21no—
222C2×SD16—21no—
223C2×SD16—21no—
224C2×M16D4×C2312yes—
225C22×C8D4×C2311yescharacteristic
226C22×C8D4×C2311yescharacteristic
227C4⋊Q8C2×C4×C2311yescharacteristic
228C4⋊Q8C2×C4×C2311yescharacteristic
229C4⋊Q8C2×C4×C2311yescharacteristic
230#127C2×C4×C2311yescharacteristic
231#125—21no—
232#124—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235#121—21no—
236#121—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—21no—
240C4×D4—21no—
241#117—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42C23×C2311yescharacteristic
247#107—24no—
248#106—21no—
249#106—21no—
250C4⋊C8—21no—
251#104—44no—
252#103—21no—
253#103—21no—
254#102—21no—
255#102—21no—
256C22⋊C8D4×C2312yes—
257#97—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8C2×C4×C2311yescharacteristic
261C4×C8C2×C4×C2311yescharacteristic
262C4×C8D4×C2311yescharacteristic
263C4×C8D4×C2311yescharacteristic
264#95—21no—
265#95—21no—
266C23#972211yescharacteristic, direct product factor, Sylow
267#41—42no—
268#41—42no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×D4×C2311yescharacteristic
280C22×C4C2×D4×C2311yescharacteristic
281C22×C4C2×D4×C2311yescharacteristic
282SD16—42no—
283SD16—42no—
284M16—22no—
285M16—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8C22⋊C4×C2311yescharacteristic
291C2×C8C22⋊C4×C2311yescharacteristic
292C2×C8C22⋊C4×C2311yescharacteristic
293C2×C8C22⋊C4×C2311yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—41no—
310C42—42no—
311C42—42no—
312C42C22×C4×C2311yescharacteristic
313C42C2×D4×C2311yescharacteristic
314C42#41×C2311yescharacteristic
315C42#41×C2311yescharacteristic
316C23—41no—
317C23C23×#12711yescharacteristic
318Q8—41no—
319Q8—41no—
320Q8—42no—
321Q8—81no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4C2×C22⋊C4×C2311yescharacteristic
342C2×C4C4×D4×C2311yescharacteristic
343C2×C4C4×D4×C2311yescharacteristic
344C2×C4C23×#12111yescharacteristic
345C2×C4C23×#12111yescharacteristic
346C2×C4C23×#12411yescharacteristic
347C2×C4C23×AGL(1,ℤ/8)12yes—
348C2×C4C23×#13712yes—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—42no—
353C8—42no—
354C8C23×#10412yescomplemented
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22C23×#39011yescharacteristic
361C22C23×#49611yescharacteristic
362C22C23×#49711yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4C2×C23×#10411yescharacteristic
372C4C23×#42111yescharacteristic
373C4C23×#44012yes—
374C4C23×#46812yes—
375C2—21no—
376C2—21no—
377C2—161no—
378C2C23×#197311yescharacteristic
379C2C23×#197611yescharacteristic
380C2C23×#198011yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#390
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1986C2semidirect
4. C4×C23×#107C2semidirect
5. C23×#1503C2semidirect
7. C23×#1451C2semidirect
10. C23×#498C4semidirect
11. C23×#492C4semidirect
12. C23×#487C4semidirect
13. C23×#486C4semidirect
125. #9722C23direct (N × H)
164. C8×C23#104semidirect
166. #1986C2×C23semidirect
168. C4×#107C2×C23semidirect
169. #1503C2×C23semidirect
171. #1451C2×C23semidirect
194. #498C4×C23semidirect
195. #492C4×C23semidirect
196. #487C4×C23semidirect
197. #486C4×C23semidirect
266. C23#9722direct (N × H)
354. C8C23×#104semidirect

Maximal subgroups

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

subgroupindex
C23×#31672
C23×#19862
C23×#19802
C4×C23×#1072
C23×#15032
C23×#15022
C23×#14512
#972223