Finite Group Databasebeta

← C23×#60037 #19207663

Subgroups — C23×#60037 #19207663

Order 5888.

2228 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2697C213yesmaximal, complemented
2C23×#2679C211yesmaximal, characteristic
3C23×#2576C216yesmaximal, complemented
4C23×#2566C213yesmaximal, complemented
5C23×#2534C213yesmaximal, complemented
6C23×#2460C216yesmaximal, complemented
7C23×#2374C213yesmaximal, complemented
8C23×#2358C213yesmaximal, complemented
9C4×C23×#120C213yesmaximal, complemented
10C24×C4×C23C2211yescharacteristic, complemented
11C2×C23×#125C2213yescomplemented
12C2×C23×#123C2213yes—
13C2×C23×#120C2213yes—
14C2×C23×#120C2213yes—
15C2×C4×D4×C23C2216yes—
16C2×C4×D4×C23C2213yes—
17C22×C22⋊C4×C23C2213yescomplemented
18C23×#401C2212yescomplemented
19C23×#397C2216yescomplemented
20C23×#396C2216yescomplemented
21C23×#396C2216yescomplemented
22C23×#394C2216yes—
23C23×#394C2216yes—
24C23×#390C2213yes—
25C23×#390C2213yes—
26C23×#388C2216yescomplemented
27C23×#388C2216yescomplemented
28C23×#388C2216yescomplemented
29C23×#387C2216yescomplemented
30C23×#387C2216yescomplemented
31C23×#386C2216yes—
32C23×#386C2216yes—
33C23×#386C2213yes—
34C23×#385C2213yescomplemented
35C23×#385C2216yescomplemented
36C23×#385C2216yescomplemented
37C23×#384C2211yescharacteristic, complemented
38C23×#382C2213yescomplemented
39C23×#381C2213yescomplemented
40C23×#380C2213yescomplemented
41C23×#379C2213yescomplemented
42C23×#379C2211yescharacteristic, complemented
43C4×C4⋊C4×C23C2213yescomplemented
44C4×C22⋊C4×C23C2216yescomplemented
45C4×C22⋊C4×C23C2213yescomplemented
46C4×C22⋊C4×C23C2216yescomplemented
47C25×C23C2311yescharacteristic
48C25×C23C2311yescharacteristic
49C22×D4×C23C2316yes—
50C22×D4×C23C2313yes—
51C23×C4×C23—46no—
52C23×C4×C23C2313yes—
53C23×C4×C23C2313yes—
54C23×C4×C23C2313yes—
55C23×C4×C23C2313yes—
56C23×#125—46no—
57C23×#123—212no—
58C23×#120—212no—
59C23×#120—212no—
60C4×D4×C23—46no—
61C4×D4×C23—412no—
62C2×C4⋊C4×C23C2316yes—
63C2×C4⋊C4×C23C2316yes—
64C2×C4⋊C4×C23C2316yes—
65C2×C4⋊C4×C23C2313yes—
66C2×C4⋊C4×C23C2313yes—
67C2×C4⋊C4×C23C2313yes—
68C2×C22⋊C4×C23—46no—
69C2×C22⋊C4×C23C2316yes—
70C2×C22⋊C4×C23C2316yes—
71C2×C22⋊C4×C23C2316yes—
72C2×C22⋊C4×C23C2313yes—
73C2×C22⋊C4×C23C2316yes—
74C2×C22⋊C4×C23C2313yes—
75C2×C22⋊C4×C23C2313yes—
76C2×C22⋊C4×C23C2313yes—
77C2×C22⋊C4×C23C2313yes—
78C2×C22⋊C4×C23C2316yes—
79C2×C22⋊C4×C23C2313yes—
80C2×C22⋊C4×C23C2316yes—
81C2×C42×C23C2313yes—
82C2×C42×C23C2316yes—
83C2×C42×C23C2313yes—
84C2×C42×C23C2311yescharacteristic
85C23×#95C2316yes—
86C23×#95C2316yes—
87C23×#95C2316yes—
88C23×#95C2316yes—
89C23×#95C2316yes—
90C23×#95C2316yes—
91C23×#95C2312yes—
92C24×C23—26no—
93C24×C23—26no—
94C24×C23—43no—
95C24×C23—41no—
96C24×C23—43no—
97C24×C23—41no—
98C24×C23C2413yes—
99C24×C23C2413yes—
100C2×D4×C23—212no—
101C2×D4×C23—212no—
102C2×D4×C23—43no—
103C2×D4×C23—46no—
104C2×D4×C23—43no—
105C2×D4×C23#41112yes—
106C22×C4×C23—26no—
107C22×C4×C23—212no—
108C22×C4×C23—26no—
109C22×C4×C23—26no—
110C22×C4×C23—43no—
111C22×C4×C23—46no—
112C22×C4×C23—43no—
113C22×C4×C23—46no—
114C22×C4×C23—43no—
115C22×C4×C23—416no—
116C22×C4×C23C2416yes—
117C22×C4×C23C2413yes—
118C22×C4×C23C2416yes—
119C22×C4×C23C2413yes—
120C22×C4×C23C2411yescharacteristic
121C22×C4×C23C2413yes—
122C22×C4×C23C2413yes—
123C4⋊C4×C23—212no—
124C4⋊C4×C23—212no—
125C4⋊C4×C23—43no—
126C4⋊C4×C23—43no—
127C4⋊C4×C23—43no—
128C4⋊C4×C23—46no—
129C22⋊C4×C23—26no—
130C22⋊C4×C23—26no—
131C22⋊C4×C23—26no—
132C22⋊C4×C23—212no—
133C22⋊C4×C23—212no—
134C22⋊C4×C23—212no—
135C22⋊C4×C23—212no—
136C22⋊C4×C23—46no—
137C22⋊C4×C23—43no—
138C22⋊C4×C23—43no—
139C22⋊C4×C23—43no—
140C22⋊C4×C23—46no—
141C42×C23—26no—
142C42×C23—41no—
143C42×C23—43no—
144C42×C23—46no—
145#60037C2311yesmaximal, characteristic, direct product factor, Sylow
146C23×C23—26no—
147C23×C23—26no—
148C23×C23—26no—
149C23×C23—26no—
150C23×C23—23no—
151C23×C23—23no—
152C23×C23—23no—
153C23×C23—23no—
154C23×C23—412no—
155C23×C23—412no—
156C23×C23—412no—
157C23×C23—44no—
158C23×C23—86no—
159C23×C23—82no—
160C23×C23C2×#4116yes—
161C23×C23C2×#4116yes—
162C23×C23C2511yescharacteristic
163D4×C23—412no—
164D4×C23—424no—
165C2×C4×C23—23no—
166C2×C4×C23—26no—
167C2×C4×C23—23no—
168C2×C4×C23—26no—
169C2×C4×C23—26no—
170C2×C4×C23—26no—
171C2×C4×C23—26no—
172C2×C4×C23—23no—
173C2×C4×C23—26no—
174C2×C4×C23—26no—
175C2×C4×C23—23no—
176C2×C4×C23—26no—
177C2×C4×C23—23no—
178C2×C4×C23—26no—
179C2×C4×C23—412no—
180C2×C4×C23—412no—
181C2×C4×C23—86no—
182C2×C4×C23—86no—
183C2×C4×C23C2×#4116yes—
184C2×C4×C23H(5,𝔽2)16yes—
185#2697C2×C2313yescomplemented
186#2679C2×C2311yescharacteristic
187#2576C2×C2316yescomplemented
188#2566C2×C2313yescomplemented
189#2534C2×C2313yescomplemented
190#2460C2×C2316yescomplemented
191#2374C2×C2313yescomplemented
192#2358C2×C2313yescomplemented
193C4×#120C2×C2313yescomplemented
194C22×C23—212no—
195C22×C23—212no—
196C22×C23—26no—
197C22×C23—26no—
198C22×C23—416no—
199C22×C23—43no—
200C22×C23—43no—
201C22×C23—46no—
202C22×C23—46no—
203C22×C23—43no—
204C22×C23—43no—
205C22×C23—88no—
206C22×C23C22×#4113yes—
207C22×C23C2×H(5,𝔽2)13yes—
208C22×C23#58511yescharacteristic
209C4×C23—26no—
210C4×C23—26no—
211C4×C23—43no—
212C4×C23—46no—
213C4×C23—46no—
214C4×C23—41no—
215C4×C23—43no—
216C24×C4C22×C2311yescharacteristic, complemented
217C2×#125C22×C2313yescomplemented
218C2×#123C22×C2313yes—
219C2×#120C22×C2313yes—
220C2×#120C22×C2313yes—
221C2×C4×D4C22×C2313yes—
222C2×C4×D4C22×C2316yes—
223C22×C22⋊C4C22×C2313yescomplemented
224#401C22×C2312yescomplemented
225#397C22×C2316yescomplemented
226#396C22×C2316yescomplemented
227#396C22×C2316yescomplemented
228#394C22×C2316yes—
229#394C22×C2316yes—
230#390C22×C2313yes—
231#390C22×C2313yes—
232#388C22×C2316yescomplemented
233#388C22×C2316yescomplemented
234#388C22×C2316yescomplemented
235#387C22×C2316yescomplemented
236#387C22×C2316yescomplemented
237#386C22×C2316yes—
238#386C22×C2316yes—
239#386C22×C2313yes—
240#385C22×C2316yescomplemented
241#385C22×C2316yescomplemented
242#385C22×C2313yescomplemented
243#384C22×C2311yescharacteristic, complemented
244#382C22×C2313yescomplemented
245#381C22×C2313yescomplemented
246#380C22×C2313yescomplemented
247#379C22×C2313yescomplemented
248#379C22×C2311yescharacteristic, complemented
249C4×C4⋊C4C22×C2313yescomplemented
250C4×C22⋊C4C22×C2313yescomplemented
251C4×C22⋊C4C22×C2316yescomplemented
252C4×C22⋊C4C22×C2316yescomplemented
253C2×C23—46no—
254C2×C23—46no—
255C2×C23#348113yes—
256C2×C23#350513yes—
257C2×C23#353211yescharacteristic
258C25C23×C2311yescharacteristic
259C25C23×C2311yescharacteristic
260C22×D4C23×C2316yes—
261C22×D4C23×C2313yes—
262C23×C4—46no—
263C23×C4C23×C2313yes—
264C23×C4C23×C2313yes—
265C23×C4C23×C2313yes—
266C23×C4C23×C2313yes—
267#125—46no—
268#123—212no—
269#120—212no—
270#120—212no—
271C4×D4—46no—
272C4×D4—412no—
273C2×C4⋊C4C23×C2316yes—
274C2×C4⋊C4C23×C2316yes—
275C2×C4⋊C4C23×C2313yes—
276C2×C4⋊C4C23×C2316yes—
277C2×C4⋊C4C23×C2313yes—
278C2×C4⋊C4C23×C2313yes—
279C2×C22⋊C4—46no—
280C2×C22⋊C4C23×C2316yes—
281C2×C22⋊C4C23×C2316yes—
282C2×C22⋊C4C23×C2316yes—
283C2×C22⋊C4C23×C2313yes—
284C2×C22⋊C4C23×C2313yes—
285C2×C22⋊C4C23×C2316yes—
286C2×C22⋊C4C23×C2313yes—
287C2×C22⋊C4C23×C2316yes—
288C2×C22⋊C4C23×C2313yes—
289C2×C22⋊C4C23×C2316yes—
290C2×C22⋊C4C23×C2313yes—
291C2×C22⋊C4C23×C2313yes—
292C2×C42C23×C2313yes—
293C2×C42C23×C2316yes—
294C2×C42C23×C2313yes—
295C2×C42C23×C2311yescharacteristic
296#95C23×C2316yes—
297#95C23×C2316yes—
298#95C23×C2316yes—
299#95C23×C2316yes—
300#95C23×C2316yes—
301#95C23×C2316yes—
302#95C23×C2312yes—
303C23#6003711yescharacteristic, direct product factor, Sylow
304C24—26no—
305C24—26no—
306C24—43no—
307C24—41no—
308C24—43no—
309C24—41no—
310C24C24×C2313yes—
311C24C24×C2313yes—
312C2×D4—212no—
313C2×D4—212no—
314C2×D4—43no—
315C2×D4—46no—
316C2×D4—43no—
317C2×D4#41×C23112yes—
318C22×C4—26no—
319C22×C4—212no—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—43no—
323C22×C4—46no—
324C22×C4—43no—
325C22×C4—46no—
326C22×C4—43no—
327C22×C4—416no—
328C22×C4C24×C2316yes—
329C22×C4C24×C2316yes—
330C22×C4C24×C2311yescharacteristic
331C22×C4C24×C2313yes—
332C22×C4C24×C2313yes—
333C22×C4C24×C2313yes—
334C22×C4C24×C2313yes—
335C4⋊C4—212no—
336C4⋊C4—212no—
337C4⋊C4—43no—
338C4⋊C4—43no—
339C4⋊C4—43no—
340C4⋊C4—46no—
341C22⋊C4—26no—
342C22⋊C4—26no—
343C22⋊C4—26no—
344C22⋊C4—212no—
345C22⋊C4—212no—
346C22⋊C4—212no—
347C22⋊C4—212no—
348C22⋊C4—46no—
349C22⋊C4—43no—
350C22⋊C4—43no—
351C22⋊C4—46no—
352C22⋊C4—43no—
353C42—26no—
354C42—46no—
355C42—41no—
356C42—43no—
357C23—26no—
358C23—26no—
359C23—26no—
360C23—26no—
361C23—23no—
362C23—23no—
363C23—23no—
364C23—23no—
365C23—412no—
366C23—412no—
367C23—412no—
368C23—44no—
369C23—86no—
370C23—82no—
371C23C2×#41×C2316yes—
372C23C2×#41×C2316yes—
373C23C25×C2311yescharacteristic
374D4—412no—
375D4—424no—
376C2×C4—23no—
377C2×C4—26no—
378C2×C4—23no—
379C2×C4—26no—
380C2×C4—26no—
381C2×C4—26no—
382C2×C4—26no—
383C2×C4—23no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—23no—
387C2×C4—26no—
388C2×C4—23no—
389C2×C4—26no—
390C2×C4—412no—
391C2×C4—412no—
392C2×C4—86no—
393C2×C4—86no—
394C2×C4C2×#41×C2316yes—
395C2×C4C23×H(5,𝔽2)16yes—
396C22—212no—
397C22—212no—
398C22—26no—
399C22—26no—
400C22—416no—
401C22—43no—
402C22—43no—
403C22—46no—
404C22—46no—
405C22—43no—
406C22—43no—
407C22—88no—
408C22C22×#41×C2313yes—
409C22C2×C23×H(5,𝔽2)13yes—
410C22C23×#58511yescharacteristic
411C4—26no—
412C4—26no—
413C4—46no—
414C4—43no—
415C4—46no—
416C4—41no—
417C4—43no—
418C2—46no—
419C2—46no—
420C2C23×#348113yes—
421C2C23×#350513yes—
422C2C23×#353211yescharacteristic
subgroupG/H
center, socleC23×C23C25
commutator, Frattini, cosocleC23C25×C23
prime radicalC23×#2679C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2697C2semidirect
3. C23×#2576C2semidirect
4. C23×#2566C2semidirect
5. C23×#2534C2semidirect
6. C23×#2460C2semidirect
7. C23×#2374C2semidirect
8. C23×#2358C2semidirect
9. C4×C23×#120C2semidirect
10. C24×C4×C23C22semidirect
11. C2×C23×#125C22semidirect
17. C22×C22⋊C4×C23C22semidirect
18. C23×#401C22semidirect
19. C23×#397C22semidirect
20. C23×#396C22semidirect
21. C23×#396C22semidirect
26. C23×#388C22semidirect
27. C23×#388C22semidirect
28. C23×#388C22semidirect
29. C23×#387C22semidirect
30. C23×#387C22semidirect
34. C23×#385C22semidirect
35. C23×#385C22semidirect
36. C23×#385C22semidirect
37. C23×#384C22semidirect
38. C23×#382C22semidirect
39. C23×#381C22semidirect
40. C23×#380C22semidirect
41. C23×#379C22semidirect
42. C23×#379C22semidirect
43. C4×C4⋊C4×C23C22semidirect
44. C4×C22⋊C4×C23C22semidirect
45. C4×C22⋊C4×C23C22semidirect
46. C4×C22⋊C4×C23C22semidirect
145. #60037C23direct (N × H)
185. #2697C2×C23semidirect
187. #2576C2×C23semidirect
188. #2566C2×C23semidirect
189. #2534C2×C23semidirect
190. #2460C2×C23semidirect
191. #2374C2×C23semidirect
192. #2358C2×C23semidirect
193. C4×#120C2×C23semidirect
216. C24×C4C22×C23semidirect
217. C2×#125C22×C23semidirect
223. C22×C22⋊C4C22×C23semidirect
224. #401C22×C23semidirect
225. #397C22×C23semidirect
226. #396C22×C23semidirect
227. #396C22×C23semidirect
232. #388C22×C23semidirect
233. #388C22×C23semidirect
234. #388C22×C23semidirect
235. #387C22×C23semidirect
236. #387C22×C23semidirect
240. #385C22×C23semidirect
241. #385C22×C23semidirect
242. #385C22×C23semidirect
243. #384C22×C23semidirect
244. #382C22×C23semidirect
245. #381C22×C23semidirect
246. #380C22×C23semidirect
247. #379C22×C23semidirect
248. #379C22×C23semidirect
249. C4×C4⋊C4C22×C23semidirect
250. C4×C22⋊C4C22×C23semidirect
251. C4×C22⋊C4C22×C23semidirect
252. C4×C22⋊C4C22×C23semidirect
303. C23#60037direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26972
C23×#26792
C23×#25762
C23×#25662
C23×#25342
C23×#24602
C23×#23742
C23×#23582
C4×C23×#1202
#6003723