Finite Group Databasebeta

← C23×#9751 #19157377

Subgroups — C23×#9751 #19157377

Order 5888.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3131C211yesmaximal, characteristic
2C8×C4⋊C4×C23C211yesmaximal, characteristic, complemented
3C23×#1564C211yesmaximal, characteristic, complemented
4C23×#1503C211yesmaximal, characteristic, complemented
5C23×#1502C211yesmaximal, characteristic
6C23×#1481C211yesmaximal, characteristic, complemented
7C23×#1480C211yesmaximal, characteristic
8C23×#533C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#488—21no—
11C23×#488—21no—
12C23×#487C411yescharacteristic, complemented
13C23×#487C411yescharacteristic, complemented
14C23×#486C411yescharacteristic, complemented
15C23×#486C411yescharacteristic, complemented
16C23×#431C2211yescharacteristic
17C23×#431C2211yescharacteristic
18C2×C23×C4⋊C8C2211yescharacteristic
19C23×#416—21no—
20C23×#416—21no—
21C2×C4×C8×C23—21no—
22C2×C4×C8×C23C2211yescharacteristic
23C4×C4⋊C4×C23C2211yescharacteristic
24C23×#336—21no—
25C23×#335—21no—
26C23×#334—21no—
27C23×#328—22no—
28C23×#327—22no—
29C23×Q8⋊C8—22no—
30C23×D4⋊C8—22no—
31C2×#41×C23—21no—
32C22×C8×C23—21no—
33C22×C8×C23D411yescharacteristic
34C22×C8×C23D411yescharacteristic
35C23×C4⋊Q8C2×C411yescharacteristic
36C23×C4⋊Q8C2×C411yescharacteristic
37C23×C4⋊Q8C2×C411yescharacteristic
38C23×#127C2×C411yescharacteristic
39C23×#125—21no—
40C23×#124—21no—
41C23×C22⋊Q8—21no—
42C23×C22⋊Q8—21no—
43C23×#121—21no—
44C23×#121—21no—
45C4×Q8×C23—21no—
46C4×Q8×C23—21no—
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C23×#117—21no—
50C2×C4⋊C4×C23—21no—
51C2×C42×C23—21no—
52C2×C42×C23C2311yescharacteristic
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8D412yes—
57C23×#103—21no—
58C23×#103—21no—
59C23×#103—21no—
60C23×#103—21no—
61C23×#102—21no—
62C23×#102—21no—
63C23×#102—21no—
64C23×#102—21no—
65C23×C22⋊C8—21no—
66C23×C22⋊C8—21no—
67C4×C8×C23—21no—
68C4×C8×C23—21no—
69C4×C8×C23—42no—
70C4×C8×C23C2×C411yescharacteristic
71C4×C8×C23C2×C411yescharacteristic
72C4×C8×C23D411yescharacteristic
73C4×C8×C23D411yescharacteristic
74C23×#95D412yes—
75#41×C23—41no—
76#41×C23—41no—
77#41×C23—41no—
78#41×C23—41no—
79C2×Q8×C23—21no—
80C2×Q8×C23—21no—
81C2×Q8×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—21no—
85C2×D4×C23—41no—
86C22×C4×C23—22no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23#4111yescharacteristic
91C2×C8×C23—21no—
92C2×C8×C23—21no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23—22no—
96C2×C8×C23—41no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C2×C8×C23—41no—
100C2×C8×C23C22⋊C411yescharacteristic
101C2×C8×C23C22⋊C411yescharacteristic
102C2×C8×C23C22⋊C411yescharacteristic
103C2×C8×C23C22⋊C411yescharacteristic
104C4⋊C4×C23—24no—
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—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C42×C23—41no—
117C42×C23—41no—
118C42×C23—41no—
119C42×C23C22×C411yescharacteristic
120C42×C23C2×D411yescharacteristic
121C42×C23C2×D411yescharacteristic
122C42×C23#4111yescharacteristic
123#9751C2311yesmaximal, characteristic, direct product factor, Sylow
124C23×C23—41no—
125C23×C23#12111yescharacteristic
126Q8×C23—41no—
127Q8×C23—41no—
128Q8×C23—42no—
129Q8×C23—81no—
130D4×C23—41no—
131D4×C23—41no—
132D4×C23—42no—
133D4×C23—81no—
134D4×C23—81no—
135D4×C23—81no—
136C2×C4×C23—22no—
137C2×C4×C23—24no—
138C2×C4×C23—41no—
139C2×C4×C23—42no—
140C2×C4×C23—42no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—81no—
145C2×C4×C23—81no—
146C2×C4×C23C2×C22⋊C411yescharacteristic
147C2×C4×C23C4×D411yescharacteristic
148C2×C4×C23C4×D411yescharacteristic
149C2×C4×C23#12111yescharacteristic
150C2×C4×C23#12411yescharacteristic
151C2×C4×C23#12711yescharacteristic
152C2×C4×C23#13511yescharacteristic
153C2×C4×C23#13511yescharacteristic
154C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C23#13711yescharacteristic
156C8×C23—21no—
157C8×C23—21no—
158C8×C23—21no—
159C8×C23—21no—
160C8×C23—42no—
161C8×C23—81no—
162C8×C23—81no—
163#3131C2×C2311yescharacteristic
164C8×C4⋊C4C2×C2311yescharacteristic, complemented
165#1564C2×C2311yescharacteristic, complemented
166#1503C2×C2311yescharacteristic, complemented
167#1502C2×C2311yescharacteristic
168#1481C2×C2311yescharacteristic, complemented
169#1480C2×C2311yescharacteristic
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—81no—
173C22×C23—81no—
174C22×C23—81no—
175C22×C23#39011yescharacteristic
176C22×C23#46311yescharacteristic
177C22×C23#46411yescharacteristic
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—44no—
182C4×C23—81no—
183C4×C23—81no—
184C4×C23—81no—
185C4×C23#41611yescharacteristic
186C4×C23#41711yescharacteristic
187C4×C23#44312yes—
188C4×C23#46311yescharacteristic
189C4×C23#46411yescharacteristic
190#533C22×C2311yescharacteristic
191#532C22×C2311yescharacteristic
192#488—21no—
193#488—21no—
194#487C4×C2311yescharacteristic, complemented
195#487C4×C2311yescharacteristic, complemented
196#486C4×C2311yescharacteristic, complemented
197#486C4×C2311yescharacteristic, complemented
198#431C22×C2311yescharacteristic
199#431C22×C2311yescharacteristic
200C2×C4⋊C8C22×C2311yescharacteristic
201#416—21no—
202#416—21no—
203C2×C4×C8—21no—
204C2×C4×C8C22×C2311yescharacteristic
205C4×C4⋊C4C22×C2311yescharacteristic
206#336—21no—
207#335—21no—
208#334—21no—
209#328—22no—
210#327—22no—
211Q8⋊C8—22no—
212D4⋊C8—22no—
213C2×C23—21no—
214C2×C23—21no—
215C2×C23—161no—
216C2×C23#197411yescharacteristic
217C2×C23#197611yescharacteristic
218C2×C23#198311yescharacteristic
219C2×#41—21no—
220C22×C8—21no—
221C22×C8D4×C2311yescharacteristic
222C22×C8D4×C2311yescharacteristic
223C4⋊Q8C2×C4×C2311yescharacteristic
224C4⋊Q8C2×C4×C2311yescharacteristic
225C4⋊Q8C2×C4×C2311yescharacteristic
226#127C2×C4×C2311yescharacteristic
227#125—21no—
228#124—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—21no—
232#121—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—21no—
236C4×D4—21no—
237#117—21no—
238C2×C4⋊C4—21no—
239C2×C42—21no—
240C2×C42C23×C2311yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8D4×C2312yes—
245#103—21no—
246#103—21no—
247#103—21no—
248#103—21no—
249#102—21no—
250#102—21no—
251#102—21no—
252#102—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—42no—
258C4×C8C2×C4×C2311yescharacteristic
259C4×C8C2×C4×C2311yescharacteristic
260C4×C8D4×C2311yescharacteristic
261C4×C8D4×C2311yescharacteristic
262#95D4×C2312yes—
263C23#975111yescharacteristic, direct product factor, Sylow
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4C2×D4×C2311yescharacteristic
278C22×C4C2×D4×C2311yescharacteristic
279C22×C4#41×C2311yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8C22⋊C4×C2311yescharacteristic
290C2×C8C22⋊C4×C2311yescharacteristic
291C2×C8C22⋊C4×C2311yescharacteristic
292C2×C8C22⋊C4×C2311yescharacteristic
293C4⋊C4—24no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42C22×C4×C2311yescharacteristic
309C42C2×D4×C2311yescharacteristic
310C42C2×D4×C2311yescharacteristic
311C42#41×C2311yescharacteristic
312C23—41no—
313C23C23×#12111yescharacteristic
314Q8—41no—
315Q8—41no—
316Q8—42no—
317Q8—81no—
318D4—41no—
319D4—41no—
320D4—42no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4C2×C22⋊C4×C2311yescharacteristic
335C2×C4C4×D4×C2311yescharacteristic
336C2×C4C4×D4×C2311yescharacteristic
337C2×C4C23×#12111yescharacteristic
338C2×C4C23×#12411yescharacteristic
339C2×C4C23×#12711yescharacteristic
340C2×C4C23×#13511yescharacteristic
341C2×C4C23×#13511yescharacteristic
342C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
343C2×C4C23×#13711yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—42no—
349C8—81no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C23×#39011yescharacteristic
357C22C23×#46311yescharacteristic
358C22C23×#46411yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—44no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4C23×#41611yescharacteristic
367C4C23×#41711yescharacteristic
368C4C23×#44312yes—
369C4C23×#46311yescharacteristic
370C4C23×#46411yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2C23×#197411yescharacteristic
375C2C23×#197611yescharacteristic
376C2C23×#198311yescharacteristic
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. C8×C4⋊C4×C23C2semidirect
3. C23×#1564C2semidirect
4. C23×#1503C2semidirect
6. C23×#1481C2semidirect
12. C23×#487C4semidirect
13. C23×#487C4semidirect
14. C23×#486C4semidirect
15. C23×#486C4semidirect
123. #9751C23direct (N × H)
164. C8×C4⋊C4C2×C23semidirect
165. #1564C2×C23semidirect
166. #1503C2×C23semidirect
168. #1481C2×C23semidirect
194. #487C4×C23semidirect
195. #487C4×C23semidirect
196. #486C4×C23semidirect
197. #486C4×C23semidirect
263. C23#9751direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31312
C8×C4⋊C4×C232
C23×#15642
C23×#15032
C23×#15022
C23×#14812
C23×#14802
#975123