Finite Group Databasebeta

← C23×#60027 #19207653

Subgroups — C23×#60027 #19207653

Order 5888.

1674 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2787C213yesmaximal, complemented
2C23×#2736C211yesmaximal, characteristic
3C23×#2574C213yesmaximal, complemented
4C23×#2527C213yesmaximal, complemented
5C23×#2515C213yesmaximal, complemented
6C23×#2494C213yesmaximal, complemented
7C23×#2480C213yesmaximal
8C23×#2478C213yesmaximal, complemented
9C23×#2373C213yesmaximal, complemented
10C23×#2342C213yesmaximal
11C23×#2298C213yesmaximal, complemented
12C2×C23×#126C2213yes—
13C2×C23×#117C2213yes—
14C22×C22⋊C4×C23C2213yes—
15C23×#401C2211yescharacteristic, complemented
16C23×#401C2213yescomplemented
17C23×#400C2213yescomplemented
18C23×#398C2213yescomplemented
19C23×#398C2213yescomplemented
20C23×#397C2213yes—
21C23×#397C2213yes—
22C23×#397C2213yes—
23C23×#397C2213yes—
24C23×#396C2213yes—
25C23×#395C2211yescharacteristic, complemented
26C23×#393C2213yes—
27C23×#391C2213yescomplemented
28C23×#389C2213yescomplemented
29C23×#389C2213yescomplemented
30C23×#388C2213yes—
31C23×#388C2213yes—
32C23×#388C2213yes—
33C23×#388C2213yes—
34C23×#388C2213yes—
35C23×#388C2213yes—
36C23×#388C2213yes—
37C23×#388C2213yes—
38C23×#388C2213yes—
39C23×#387C2213yescomplemented
40C23×#387C2213yescomplemented
41C23×#387C2213yescomplemented
42C23×#387C2213yescomplemented
43C23×#387C2213yescomplemented
44C23×#387C2213yescomplemented
45C23×#387C2213yescomplemented
46C23×#387C2213yescomplemented
47C23×#385C2213yes—
48C23×#385C2213yes—
49C23×#385C2213yes—
50C23×#385C2213yes—
51C23×#383C2211yescharacteristic, complemented
52C23×#382C2213yescomplemented
53C23×#381C2213yes—
54C23×#380C2213yes—
55C23×#380C2213yes—
56C23×#379C2211yescharacteristic
57C23×#379C2213yes—
58C4×C4⋊C4×C23C2213yescomplemented
59C4×C4⋊C4×C23C2213yescomplemented
60C4×C4⋊C4×C23C2213yescomplemented
61C4×C22⋊C4×C23C2213yes—
62C4×C22⋊C4×C23C2213yes—
63C4×C22⋊C4×C23C2213yes—
64C4×C22⋊C4×C23C2213yes—
65C23×#376C2213yescomplemented
66C43×C23C2211yescharacteristic, complemented
67C25×C23C2311yescharacteristic
68C22×Q8×C23C2311yescharacteristic
69C23×C4×C23C2313yes—
70C23×C4×C23C2313yes—
71C23×#126—212no—
72C23×#117—212no—
73C2×C4⋊C4×C23C2313yes—
74C2×C4⋊C4×C23C2313yes—
75C2×C4⋊C4×C23C2313yes—
76C2×C4⋊C4×C23C2313yes—
77C2×C4⋊C4×C23C2313yes—
78C2×C4⋊C4×C23C2313yes—
79C2×C4⋊C4×C23C2313yes—
80C2×C4⋊C4×C23C2313yes—
81C2×C4⋊C4×C23C2313yes—
82C2×C4⋊C4×C23C2313yes—
83C2×C4⋊C4×C23C2313yes—
84C2×C22⋊C4×C23—212no—
85C2×C22⋊C4×C23C2313yes—
86C2×C22⋊C4×C23C2313yes—
87C2×C22⋊C4×C23C2313yes—
88C2×C22⋊C4×C23C2313yes—
89C2×C22⋊C4×C23C2313yes—
90C2×C22⋊C4×C23C2313yes—
91C2×C22⋊C4×C23C2313yes—
92C2×C22⋊C4×C23C2313yes—
93C2×C22⋊C4×C23C2313yes—
94C2×C22⋊C4×C23C2313yes—
95C2×C22⋊C4×C23C2313yes—
96C2×C22⋊C4×C23C2313yes—
97C2×C42×C23C2313yes—
98C2×C42×C23C2313yes—
99C2×C42×C23C2313yes—
100C2×C42×C23C2313yes—
101C2×C42×C23C2311yescharacteristic
102C2×C42×C23C2313yes—
103C2×C42×C23C2313yes—
104C2×C42×C23C2311yescharacteristic
105C23×#95C2313yes—
106C23×#95C2313yes—
107C23×#95C2313yes—
108C23×#95C2313yes—
109C23×#95C2313yes—
110C23×#95C2313yes—
111C23×#95C2313yes—
112C23×#95C2313yes—
113C23×#95C2313yes—
114C23×#95C2313yes—
115C23×#95C2313yes—
116C23×#95C2313yes—
117C23×#95C2313yes—
118C23×#95C2313yes—
119C23×#95C2313yes—
120C23×#95C2313yes—
121C23×#95C2313yes—
122C23×#95C2313yes—
123C23×#95C2313yes—
124C23×#95C2311yescharacteristic
125C24×C23—26no—
126C24×C23—43no—
127C24×C23—41no—
128C24×C23C2413yes—
129C2×Q8×C23—43no—
130C22×C4×C23—26no—
131C22×C4×C23—26no—
132C22×C4×C23—26no—
133C22×C4×C23—43no—
134C22×C4×C23—43no—
135C22×C4×C23—43no—
136C22×C4×C23C2413yes—
137C22×C4×C23C2413yes—
138C22×C4×C23C2411yescharacteristic
139C22×C4×C23C2413yes—
140C22×C4×C23C2413yes—
141C22×C4×C23C2413yes—
142C22×C4×C23C2413yes—
143C22×C4×C23C2413yes—
144C22×C4×C23C2413yes—
145C22×C4×C23C2413yes—
146C4⋊C4×C23—26no—
147C4⋊C4×C23—26no—
148C4⋊C4×C23—26no—
149C4⋊C4×C23—26no—
150C4⋊C4×C23—26no—
151C4⋊C4×C23—43no—
152C4⋊C4×C23—43no—
153C4⋊C4×C23—43no—
154C4⋊C4×C23—43no—
155C4⋊C4×C23—43no—
156C4⋊C4×C23—43no—
157C22⋊C4×C23—26no—
158C22⋊C4×C23—26no—
159C22⋊C4×C23—26no—
160C22⋊C4×C23—26no—
161C22⋊C4×C23—26no—
162C22⋊C4×C23—26no—
163C22⋊C4×C23—43no—
164C22⋊C4×C23—43no—
165C22⋊C4×C23—43no—
166C22⋊C4×C23—43no—
167C22⋊C4×C23—43no—
168C22⋊C4×C23#41112yes—
169C42×C23—26no—
170C42×C23—26no—
171C42×C23—26no—
172C42×C23—43no—
173C42×C23—43no—
174C42×C23—43no—
175C42×C23—41no—
176C42×C23—41no—
177#60027C2311yesmaximal, characteristic, direct product factor, Sylow
178C23×C23—23no—
179C23×C23—23no—
180C23×C23—23no—
181C23×C23—23no—
182C23×C23—23no—
183C23×C23—23no—
184C23×C23—412no—
185C23×C23—86no—
186C23×C23—82no—
187C23×C23C2×#4116yes—
188C23×C23C2511yescharacteristic
189Q8×C23—82no—
190C2×C4×C23—23no—
191C2×C4×C23—23no—
192C2×C4×C23—23no—
193C2×C4×C23—23no—
194C2×C4×C23—23no—
195C2×C4×C23—23no—
196C2×C4×C23—23no—
197C2×C4×C23—23no—
198C2×C4×C23—23no—
199C2×C4×C23—23no—
200C2×C4×C23—23no—
201C2×C4×C23—23no—
202C2×C4×C23—23no—
203C2×C4×C23—23no—
204C2×C4×C23—23no—
205C2×C4×C23—23no—
206C2×C4×C23—23no—
207C2×C4×C23—23no—
208C2×C4×C23—23no—
209C2×C4×C23—23no—
210C2×C4×C23—23no—
211C2×C4×C23—23no—
212C2×C4×C23—23no—
213C2×C4×C23—23no—
214C2×C4×C23—23no—
215C2×C4×C23—23no—
216C2×C4×C23—412no—
217C2×C4×C23—86no—
218C2×C4×C23C2×#4116yes—
219C2×C4×C23C2×#4116yes—
220#2787C2×C2313yescomplemented
221#2736C2×C2311yescharacteristic
222#2574C2×C2313yescomplemented
223#2527C2×C2313yescomplemented
224#2515C2×C2313yescomplemented
225#2494C2×C2313yescomplemented
226#2480C2×C2313yes—
227#2478C2×C2313yescomplemented
228#2373C2×C2313yescomplemented
229#2342C2×C2313yes—
230#2298C2×C2313yescomplemented
231C22×C23—26no—
232C22×C23—26no—
233C22×C23—26no—
234C22×C23—43no—
235C22×C23—43no—
236C22×C23—43no—
237C22×C23—43no—
238C22×C23—88no—
239C22×C23C22×#4113yes—
240C22×C23#58513yes—
241C22×C23#58511yescharacteristic
242C4×C23—26no—
243C4×C23—26no—
244C4×C23—43no—
245C4×C23—43no—
246C4×C23—43no—
247C4×C23—43no—
248C4×C23—43no—
249C4×C23—43no—
250C4×C23—41no—
251C4×C23—43no—
252C2×#126C22×C2313yes—
253C2×#117C22×C2313yes—
254C22×C22⋊C4C22×C2313yes—
255#401C22×C2311yescharacteristic, complemented
256#401C22×C2313yescomplemented
257#400C22×C2313yescomplemented
258#398C22×C2313yescomplemented
259#398C22×C2313yescomplemented
260#397C22×C2313yes—
261#397C22×C2313yes—
262#397C22×C2313yes—
263#397C22×C2313yes—
264#396C22×C2313yes—
265#395C22×C2311yescharacteristic, complemented
266#393C22×C2313yes—
267#391C22×C2313yescomplemented
268#389C22×C2313yescomplemented
269#389C22×C2313yescomplemented
270#388C22×C2313yes—
271#388C22×C2313yes—
272#388C22×C2313yes—
273#388C22×C2313yes—
274#388C22×C2313yes—
275#388C22×C2313yes—
276#388C22×C2313yes—
277#388C22×C2313yes—
278#388C22×C2313yes—
279#387C22×C2313yescomplemented
280#387C22×C2313yescomplemented
281#387C22×C2313yescomplemented
282#387C22×C2313yescomplemented
283#387C22×C2313yescomplemented
284#387C22×C2313yescomplemented
285#387C22×C2313yescomplemented
286#387C22×C2313yescomplemented
287#385C22×C2313yes—
288#385C22×C2313yes—
289#385C22×C2313yes—
290#385C22×C2313yes—
291#383C22×C2311yescharacteristic, complemented
292#382C22×C2313yescomplemented
293#381C22×C2313yes—
294#380C22×C2313yes—
295#380C22×C2313yes—
296#379C22×C2313yes—
297#379C22×C2311yescharacteristic
298C4×C4⋊C4C22×C2313yescomplemented
299C4×C4⋊C4C22×C2313yescomplemented
300C4×C4⋊C4C22×C2313yescomplemented
301C4×C22⋊C4C22×C2313yes—
302C4×C22⋊C4C22×C2313yes—
303C4×C22⋊C4C22×C2313yes—
304C4×C22⋊C4C22×C2313yes—
305#376C22×C2313yescomplemented
306C43C22×C2311yescharacteristic, complemented
307C2×C23—46no—
308C2×C23#347513yes—
309C2×C23#352113yes—
310C2×C23#356811yescharacteristic
311C25C23×C2311yescharacteristic
312C22×Q8C23×C2311yescharacteristic
313C23×C4C23×C2313yes—
314C23×C4C23×C2313yes—
315#126—212no—
316#117—212no—
317C2×C4⋊C4C23×C2313yes—
318C2×C4⋊C4C23×C2313yes—
319C2×C4⋊C4C23×C2313yes—
320C2×C4⋊C4C23×C2313yes—
321C2×C4⋊C4C23×C2313yes—
322C2×C4⋊C4C23×C2313yes—
323C2×C4⋊C4C23×C2313yes—
324C2×C4⋊C4C23×C2313yes—
325C2×C4⋊C4C23×C2313yes—
326C2×C4⋊C4C23×C2313yes—
327C2×C4⋊C4C23×C2313yes—
328C2×C22⋊C4—212no—
329C2×C22⋊C4C23×C2313yes—
330C2×C22⋊C4C23×C2313yes—
331C2×C22⋊C4C23×C2313yes—
332C2×C22⋊C4C23×C2313yes—
333C2×C22⋊C4C23×C2313yes—
334C2×C22⋊C4C23×C2313yes—
335C2×C22⋊C4C23×C2313yes—
336C2×C22⋊C4C23×C2313yes—
337C2×C22⋊C4C23×C2313yes—
338C2×C22⋊C4C23×C2313yes—
339C2×C22⋊C4C23×C2313yes—
340C2×C22⋊C4C23×C2313yes—
341C2×C42C23×C2313yes—
342C2×C42C23×C2313yes—
343C2×C42C23×C2313yes—
344C2×C42C23×C2313yes—
345C2×C42C23×C2311yescharacteristic
346C2×C42C23×C2313yes—
347C2×C42C23×C2313yes—
348C2×C42C23×C2311yescharacteristic
349#95C23×C2313yes—
350#95C23×C2313yes—
351#95C23×C2313yes—
352#95C23×C2313yes—
353#95C23×C2313yes—
354#95C23×C2313yes—
355#95C23×C2313yes—
356#95C23×C2313yes—
357#95C23×C2313yes—
358#95C23×C2313yes—
359#95C23×C2313yes—
360#95C23×C2313yes—
361#95C23×C2313yes—
362#95C23×C2313yes—
363#95C23×C2313yes—
364#95C23×C2313yes—
365#95C23×C2313yes—
366#95C23×C2313yes—
367#95C23×C2313yes—
368#95C23×C2311yescharacteristic
369C23#6002711yescharacteristic, direct product factor, Sylow
370C24—26no—
371C24—43no—
372C24—41no—
373C24C24×C2313yes—
374C2×Q8—43no—
375C22×C4—26no—
376C22×C4—26no—
377C22×C4—26no—
378C22×C4—43no—
379C22×C4—43no—
380C22×C4—43no—
381C22×C4C24×C2313yes—
382C22×C4C24×C2313yes—
383C22×C4C24×C2313yes—
384C22×C4C24×C2313yes—
385C22×C4C24×C2313yes—
386C22×C4C24×C2311yescharacteristic
387C22×C4C24×C2313yes—
388C22×C4C24×C2313yes—
389C22×C4C24×C2313yes—
390C22×C4C24×C2313yes—
391C4⋊C4—26no—
392C4⋊C4—26no—
393C4⋊C4—26no—
394C4⋊C4—26no—
395C4⋊C4—26no—
396C4⋊C4—43no—
397C4⋊C4—43no—
398C4⋊C4—43no—
399C4⋊C4—43no—
400C4⋊C4—43no—
401C4⋊C4—43no—
402C22⋊C4—26no—
403C22⋊C4—26no—
404C22⋊C4—26no—
405C22⋊C4—26no—
406C22⋊C4—26no—
407C22⋊C4—26no—
408C22⋊C4—43no—
409C22⋊C4—43no—
410C22⋊C4—43no—
411C22⋊C4—43no—
412C22⋊C4—43no—
413C22⋊C4#41×C23112yes—
414C42—26no—
415C42—26no—
416C42—26no—
417C42—43no—
418C42—43no—
419C42—43no—
420C42—41no—
421C42—41no—
422C23—23no—
423C23—23no—
424C23—23no—
425C23—23no—
426C23—23no—
427C23—23no—
428C23—412no—
429C23—86no—
430C23—82no—
431C23C2×#41×C2316yes—
432C23C25×C2311yescharacteristic
433Q8—82no—
434C2×C4—23no—
435C2×C4—23no—
436C2×C4—23no—
437C2×C4—23no—
438C2×C4—23no—
439C2×C4—23no—
440C2×C4—23no—
441C2×C4—23no—
442C2×C4—23no—
443C2×C4—23no—
444C2×C4—23no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—412no—
461C2×C4—86no—
462C2×C4C2×#41×C2316yes—
463C2×C4C2×#41×C2316yes—
464C22—26no—
465C22—26no—
466C22—26no—
467C22—43no—
468C22—43no—
469C22—43no—
470C22—43no—
471C22—88no—
472C22C22×#41×C2313yes—
473C22C23×#58513yes—
474C22C23×#58511yescharacteristic
475C4—26no—
476C4—26no—
477C4—43no—
478C4—43no—
479C4—43no—
480C4—43no—
481C4—43no—
482C4—43no—
483C4—43no—
484C4—41no—
485C2—46no—
486C2C23×#347513yes—
487C2C23×#352113yes—
488C2C23×#356811yescharacteristic
subgroupG/H
center, socleC23×C23C25
commutator, Frattini, cosocleC23C25×C23
prime radicalC25×C23C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2787C2semidirect
3. C23×#2574C2semidirect
4. C23×#2527C2semidirect
5. C23×#2515C2semidirect
6. C23×#2494C2semidirect
8. C23×#2478C2semidirect
9. C23×#2373C2semidirect
11. C23×#2298C2semidirect
15. C23×#401C22semidirect
16. C23×#401C22semidirect
17. C23×#400C22semidirect
18. C23×#398C22semidirect
19. C23×#398C22semidirect
25. C23×#395C22semidirect
27. C23×#391C22semidirect
28. C23×#389C22semidirect
29. C23×#389C22semidirect
39. C23×#387C22semidirect
40. C23×#387C22semidirect
41. C23×#387C22semidirect
42. C23×#387C22semidirect
43. C23×#387C22semidirect
44. C23×#387C22semidirect
45. C23×#387C22semidirect
46. C23×#387C22semidirect
51. C23×#383C22semidirect
52. C23×#382C22semidirect
58. C4×C4⋊C4×C23C22semidirect
59. C4×C4⋊C4×C23C22semidirect
60. C4×C4⋊C4×C23C22semidirect
65. C23×#376C22semidirect
66. C43×C23C22semidirect
177. #60027C23direct (N × H)
220. #2787C2×C23semidirect
222. #2574C2×C23semidirect
223. #2527C2×C23semidirect
224. #2515C2×C23semidirect
225. #2494C2×C23semidirect
227. #2478C2×C23semidirect
228. #2373C2×C23semidirect
230. #2298C2×C23semidirect
255. #401C22×C23semidirect
256. #401C22×C23semidirect
257. #400C22×C23semidirect
258. #398C22×C23semidirect
259. #398C22×C23semidirect
265. #395C22×C23semidirect
267. #391C22×C23semidirect
268. #389C22×C23semidirect
269. #389C22×C23semidirect
279. #387C22×C23semidirect
280. #387C22×C23semidirect
281. #387C22×C23semidirect
282. #387C22×C23semidirect
283. #387C22×C23semidirect
284. #387C22×C23semidirect
285. #387C22×C23semidirect
286. #387C22×C23semidirect
291. #383C22×C23semidirect
292. #382C22×C23semidirect
298. C4×C4⋊C4C22×C23semidirect
299. C4×C4⋊C4C22×C23semidirect
300. C4×C4⋊C4C22×C23semidirect
305. #376C22×C23semidirect
306. C43C22×C23semidirect
369. C23#60027direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#27872
C23×#27362
C23×#25742
C23×#25272
C23×#25152
C23×#24942
C23×#24802
C23×#24782
C23×#23732
C23×#23422
C23×#22982
#6002723