Finite Group Databasebeta

← C2×C23×#1860 #19168488

Subgroups — C2×C23×#1860 #19168488

Order 5888.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#520C211yesmaximal, characteristic
2C2×C23×#428C211yesmaximal, characteristic, complemented
3C22×C23×#104C211yesmaximal, characteristic
4C2×C23×#417C211yesmaximal, characteristic
5C2×C23×#382C211yesmaximal, characteristic, complemented
6C23×#1860C218yesmaximal, direct product factor
7C2×C23×#343C211yesmaximal, characteristic, complemented
8C2×C23×#339C211yesmaximal, characteristic, complemented
9C22×#41×C23C2211yescharacteristic
10C22×M16×C23C2211yescharacteristic
11C22×M16×C23C2211yescharacteristic
12C23×#520—22no—
13C23×#520—22no—
14C23×#520—22no—
15C23×#520C2214yes—
16C2×C4×Q8×C23—21no—
17C2×C4×D4×C23—21no—
18C2×C4×D4×C23—22no—
19C2×C23×#117—21no—
20C2×C23×#117C2211yescharacteristic
21C22×C4⋊C4×C23—21no—
22C22×C4⋊C4×C23C2211yescharacteristic
23C22×C4⋊C4×C23C2211yescharacteristic
24C22×C42×C23C2211yescharacteristic
25C23×#428—22no—
26C23×#428C2214yescomplemented
27C2×C23×#107—21no—
28C2×C23×#106—21no—
29C2×C23×#104C418yescomplemented
30C2×C23×#104C2214yes—
31C23×#417—22no—
32C23×#417C2214yes—
33C2×C23×#103—21no—
34C2×C23×#102—21no—
35C23×#382—22no—
36C23×#382C2214yescomplemented
37C2×C23×#95—22no—
38C23×#343C2214yescomplemented
39C23×#339C2214yescomplemented
40C2×#41×C23—21no—
41C2×#41×C23—21no—
42C2×#41×C23—22no—
43C2×#41×C23—22no—
44C2×#41×C23C2×C414yes—
45C2×#41×C23D412yes—
46C2×#41×C23Q814yes—
47C2×#41×C23C2312yes—
48C22×Q8×C23D411yescharacteristic
49C22×D4×C23—21no—
50C22×D4×C23D411yescharacteristic
51C23×C4×C23—21no—
52C23×C4×C23—21no—
53C23×C4×C23—22no—
54C23×C4×C23—21no—
55C23×C4×C23D412yes—
56C23×C4×C23C2311yescharacteristic
57C2×M16×C23—21no—
58C2×M16×C23—22no—
59C2×M16×C23—21no—
60C2×M16×C23—22no—
61C2×M16×C23—22no—
62C2×M16×C23C2×C414yescomplemented
63C2×M16×C23C2312yes—
64C2×M16×C23C2312yes—
65C22×C8×C23—21no—
66C22×C8×C23—21no—
67C4×Q8×C23—24no—
68C4×Q8×C23—24no—
69C4×D4×C23—28no—
70C4×D4×C23—24no—
71C4×D4×C23—24no—
72C4×D4×C23—28no—
73C23×#117—24no—
74C23×#117—21no—
75C23×#117—21no—
76C23×#117—24no—
77C23×#117D412yes—
78C23×#117C2312yes—
79C2×C4⋊C4×C23—22no—
80C2×C4⋊C4×C23—24no—
81C2×C4⋊C4×C23—21no—
82C2×C4⋊C4×C23—21no—
83C2×C4⋊C4×C23—21no—
84C2×C4⋊C4×C23—21no—
85C2×C4⋊C4×C23—24no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—21no—
88C2×C4⋊C4×C23—21no—
89C2×C4⋊C4×C23—21no—
90C2×C4⋊C4×C23—21no—
91C2×C4⋊C4×C23D412yes—
92C2×C4⋊C4×C23D411yescharacteristic
93C2×C4⋊C4×C23D411yescharacteristic
94C2×C4⋊C4×C23C2312yes—
95C2×C4⋊C4×C23C2312yes—
96C2×C22⋊C4×C23—22no—
97C2×C22⋊C4×C23—21no—
98C2×C42×C23—22no—
99C2×C42×C23—21no—
100C2×C42×C23—21no—
101C2×C42×C23—22no—
102C2×C42×C23C2×C414yescomplemented
103C2×C42×C23C2312yes—
104C23×#107—24no—
105C23×#106—24no—
106C23×#104C2×C4116yescomplemented
107C23×#103—24no—
108C23×#102—24no—
109C23×#95—28no—
110C24×C23—21no—
111C24×C23C2×D411yescharacteristic
112#41×C23—24no—
113#41×C23—24no—
114#41×C23—24no—
115#41×C23—24no—
116#41×C23—24no—
117#41×C23—22no—
118#41×C23—22no—
119#41×C23C4⋊C418yes—
120#41×C23C22×C414yes—
121#41×C23C2×Q814yes—
122C2×Q8×C23—21no—
123C2×Q8×C23—22no—
124C2×Q8×C23C4⋊C414yes—
125C2×Q8×C23C2×D412yes—
126C2×D4×C23—24no—
127C2×D4×C23—22no—
128C2×D4×C23—24no—
129C2×D4×C23—22no—
130C2×D4×C23—21no—
131C2×D4×C23—22no—
132C2×D4×C23C4⋊C414yes—
133C2×D4×C23C2×D412yes—
134C22×C4×C23—21no—
135C22×C4×C23—24no—
136C22×C4×C23—22no—
137C22×C4×C23—24no—
138C22×C4×C23—22no—
139C22×C4×C23—24no—
140C22×C4×C23—21no—
141C22×C4×C23—22no—
142C22×C4×C23—21no—
143C22×C4×C23—22no—
144C22×C4×C23—28no—
145C22×C4×C23—22no—
146C22×C4×C23—22no—
147C22×C4×C23—42no—
148C22×C4×C23—42no—
149C22×C4×C23—44no—
150C22×C4×C23—44no—
151C22×C4×C23—41no—
152C22×C4×C23—41no—
153C22×C4×C23—84no—
154C22×C4×C23C22×C412yes—
155C22×C4×C23C2×D414yes—
156C22×C4×C23C2×D411yescharacteristic
157C22×C4×C23C2×D411yescharacteristic
158C22×C4×C23C2×D411yescharacteristic
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×Q812yes—
162C22×C4×C23#4112yes—
163C22×C4×C23#4112yes—
164C22×C4×C23C2411yescharacteristic
165M16×C23—24no—
166M16×C23—22no—
167M16×C23—22no—
168M16×C23—22no—
169M16×C23—42no—
170M16×C23C22×C414yescomplemented
171C2×C8×C23—24no—
172C2×C8×C23—22no—
173C2×C8×C23—22no—
174C2×C8×C23—42no—
175C4⋊C4×C23—28no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—41no—
185C4⋊C4×C23—41no—
186C4⋊C4×C23C2×D412yes—
187C4⋊C4×C23C2×D412yes—
188C22⋊C4×C23—28no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C42×C23—28no—
192C42×C23—24no—
193C42×C23—22no—
194C42×C23—22no—
195C42×C23—22no—
196C42×C23C22×C414yescomplemented
197C2×#1860C2311yesmaximal, characteristic, direct product factor, Sylow
198C23×C23—22no—
199C23×C23—22no—
200C23×C23—22no—
201C23×C23—24no—
202C23×C23—42no—
203C23×C23—42no—
204C23×C23C4×D412yes—
205C23×C23#12011yescharacteristic
206C23×C23#12011yescharacteristic
207C23×C23C22⋊Q812yes—
208C23×C23C22×D411yescharacteristic
209Q8×C23—22no—
210Q8×C23—24no—
211Q8×C23C2×C4⋊C414yes—
212D4×C23—24no—
213D4×C23—24no—
214D4×C23—28no—
215D4×C23—22no—
216D4×C23—24no—
217D4×C23C2×C4⋊C414yes—
218C2×C4×C23—22no—
219C2×C4×C23—24no—
220C2×C4×C23—24no—
221C2×C4×C23—24no—
222C2×C4×C23—28no—
223C2×C4×C23—24no—
224C2×C4×C23—22no—
225C2×C4×C23—24no—
226C2×C4×C23—22no—
227C2×C4×C23—48no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
231C2×C4×C23—44no—
232C2×C4×C23—42no—
233C2×C4×C23—44no—
234C2×C4×C23—44no—
235C2×C4×C23—42no—
236C2×C4×C23—82no—
237C2×C4×C23—88no—
238C2×C4×C23—82no—
239C2×C4×C23—82no—
240C2×C4×C23C2×C4⋊C412yes—
241C2×C4×C23C2×C4⋊C412yes—
242C2×C4×C23C4×D412yes—
243C2×C4×C23C4×D412yes—
244C2×C4×C23C4×D412yes—
245C2×C4×C23#12011yescharacteristic
246C2×C4×C23#12011yescharacteristic
247C2×C4×C23C22⋊Q812yes—
248C2×C4×C23C22⋊Q812yes—
249C2×C4×C23C22⋊Q812yes—
250C2×C4×C23#12312yes—
251C2×C4×C23#12312yes—
252C2×C4×C23C23×C411yescharacteristic
253C2×C4×C23C22×D411yescharacteristic
254C2×C4×C23C22×D411yescharacteristic
255C2×C4×C23C22×Q811yescharacteristic
256C2×C4×C23C2×#4111yescharacteristic
257C2×C4×C23C2×#4111yescharacteristic
258C8×C23—24no—
259C8×C23—42no—
260C2×#520C2×C2311yescharacteristic
261C2×#428C2×C2311yescharacteristic, complemented
262C22×#104C2×C2311yescharacteristic
263C2×#417C2×C2311yescharacteristic
264C2×#382C2×C2311yescharacteristic, complemented
265#1860C2×C2318yesdirect product factor
266C2×#343C2×C2311yescharacteristic, complemented
267C2×#339C2×C2311yescharacteristic, complemented
268C22×C23—24no—
269C22×C23—24no—
270C22×C23—21no—
271C22×C23—21no—
272C22×C23—21no—
273C22×C23—21no—
274C22×C23—24no—
275C22×C23—48no—
276C22×C23—42no—
277C22×C23—42no—
278C22×C23#38512yes—
279C22×C23#38512yes—
280C22×C23#45412yes—
281C22×C23#45512yes—
282C22×C23C2×C4×D411yescharacteristic
283C22×C23C2×#12011yescharacteristic
284C22×C23C2×C22⋊Q811yescharacteristic
285C4×C23—24no—
286C4×C23—44no—
287C4×C23—44no—
288C4×C23—42no—
289C4×C23—42no—
290C4×C23—82no—
291C4×C23#38512yes—
292C4×C23#38512yes—
293C4×C23C22×C4⋊C411yescharacteristic
294C4×C23C2×C4×D411yescharacteristic
295C4×C23C2×C22⋊Q811yescharacteristic
296C4×C23C2×#12311yescharacteristic
297C22×#41C22×C2311yescharacteristic
298C22×M16C22×C2311yescharacteristic
299C22×M16C22×C2311yescharacteristic
300#520—22no—
301#520—22no—
302#520—22no—
303#520C22×C2314yes—
304C2×C4×Q8—21no—
305C2×C4×D4—21no—
306C2×C4×D4—22no—
307C2×#117—21no—
308C2×#117C22×C2311yescharacteristic
309C22×C4⋊C4—21no—
310C22×C4⋊C4C22×C2311yescharacteristic
311C22×C4⋊C4C22×C2311yescharacteristic
312C22×C42C22×C2311yescharacteristic
313#428—22no—
314#428C22×C2314yescomplemented
315C2×#107—21no—
316C2×#106—21no—
317C2×#104C4×C2318yescomplemented
318C2×#104C22×C2314yes—
319#417—22no—
320#417C22×C2314yes—
321C2×#103—21no—
322C2×#102—21no—
323#382—22no—
324#382C22×C2314yescomplemented
325C2×#95—22no—
326#343C22×C2314yescomplemented
327#339C22×C2314yescomplemented
328C2×C23—21no—
329C2×C23—21no—
330C2×C23—22no—
331C2×C23—44no—
332C2×C23#186014yesdirect product factor
333C2×C23C2×#38511yescharacteristic
334C2×C23C2×#45511yescharacteristic
335C2×C23C2×#45411yescharacteristic
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41C2×C4×C2314yes—
341C2×#41D4×C2312yes—
342C2×#41Q8×C2314yes—
343C2×#41C23×C2312yes—
344C22×Q8D4×C2311yescharacteristic
345C22×D4—21no—
346C22×D4D4×C2311yescharacteristic
347C23×C4—21no—
348C23×C4—21no—
349C23×C4—22no—
350C23×C4—21no—
351C23×C4D4×C2312yes—
352C23×C4C23×C2311yescharacteristic
353C2×M16—21no—
354C2×M16—22no—
355C2×M16—21no—
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C2×C4×C2314yescomplemented
359C2×M16C23×C2312yes—
360C2×M16C23×C2312yes—
361C22×C8—21no—
362C22×C8—21no—
363C4×Q8—24no—
364C4×Q8—24no—
365C4×D4—28no—
366C4×D4—24no—
367C4×D4—24no—
368C4×D4—28no—
369#117—24no—
370#117—21no—
371#117—21no—
372#117—24no—
373#117D4×C2312yes—
374#117C23×C2312yes—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4D4×C2312yes—
388C2×C4⋊C4D4×C2311yescharacteristic
389C2×C4⋊C4D4×C2311yescharacteristic
390C2×C4⋊C4C23×C2312yes—
391C2×C4⋊C4C23×C2312yes—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—21no—
394C2×C42—22no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42C2×C4×C2314yescomplemented
399C2×C42C23×C2312yes—
400#107—24no—
401#106—24no—
402#104C2×C4×C23116yescomplemented
403#103—24no—
404#102—24no—
405#95—28no—
406C23C2×#186011yescharacteristic, direct product factor, Sylow
407C24—21no—
408C24C2×D4×C2311yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41C4⋊C4×C2318yes—
417#41C22×C4×C2314yes—
418#41C2×Q8×C2314yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8C4⋊C4×C2314yes—
422C2×Q8C2×D4×C2312yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4C4⋊C4×C2314yes—
430C2×D4C2×D4×C2312yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—84no—
451C22×C4C22×C4×C2312yes—
452C22×C4C2×D4×C2314yes—
453C22×C4C2×D4×C2311yescharacteristic
454C22×C4C2×D4×C2311yescharacteristic
455C22×C4C2×D4×C2311yescharacteristic
456C22×C4C2×D4×C2311yescharacteristic
457C22×C4C2×D4×C2311yescharacteristic
458C22×C4C2×Q8×C2312yes—
459C22×C4#41×C2312yes—
460C22×C4#41×C2312yes—
461C22×C4C24×C2311yescharacteristic
462M16—24no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—42no—
467M16C22×C4×C2314yescomplemented
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—42no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4C2×D4×C2312yes—
484C4⋊C4C2×D4×C2312yes—
485C22⋊C4—28no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C42—28no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42C22×C4×C2314yescomplemented
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23C4×D4×C2312yes—
501C23C23×#12011yescharacteristic
502C23C23×#12011yescharacteristic
503C23C23×C22⋊Q812yes—
504C23C22×D4×C2311yescharacteristic
505Q8—24no—
506Q8—22no—
507Q8C2×C4⋊C4×C2314yes—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—24no—
512D4—22no—
513D4C2×C4⋊C4×C2314yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—88no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4C2×C4⋊C4×C2312yes—
537C2×C4C2×C4⋊C4×C2312yes—
538C2×C4C4×D4×C2312yes—
539C2×C4C4×D4×C2312yes—
540C2×C4C4×D4×C2312yes—
541C2×C4C23×#12011yescharacteristic
542C2×C4C23×#12011yescharacteristic
543C2×C4C23×C22⋊Q812yes—
544C2×C4C23×C22⋊Q812yes—
545C2×C4C23×C22⋊Q812yes—
546C2×C4C23×#12312yes—
547C2×C4C23×#12312yes—
548C2×C4C23×C4×C2311yescharacteristic
549C2×C4C22×D4×C2311yescharacteristic
550C2×C4C22×D4×C2311yescharacteristic
551C2×C4C22×Q8×C2311yescharacteristic
552C2×C4C2×#41×C2311yescharacteristic
553C2×C4C2×#41×C2311yescharacteristic
554C8—24no—
555C8—42no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22C23×#38512yes—
567C22C23×#38512yes—
568C22C23×#45412yes—
569C22C23×#45512yes—
570C22C2×C4×D4×C2311yescharacteristic
571C22C2×C23×#12011yescharacteristic
572C22C2×C23×C22⋊Q811yescharacteristic
573C4—24no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4C23×#38512yes—
580C4C23×#38512yes—
581C4C22×C4⋊C4×C2311yescharacteristic
582C4C2×C4×D4×C2311yescharacteristic
583C4C2×C23×C22⋊Q811yescharacteristic
584C4C2×C23×#12311yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2C23×#186014yesdirect product factor
590C2C2×C23×#38511yescharacteristic
591C2C2×C23×#45511yescharacteristic
592C2C2×C23×#45411yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC2×C4C23×C4×C23
Frattini, cosocleC22×C4C24×C23
prime radicalC22×#41×C23C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#428C2semidirect
5. C2×C23×#382C2semidirect
6. C23×#1860C2direct (N × H)
7. C2×C23×#343C2semidirect
8. C2×C23×#339C2semidirect
26. C23×#428C22semidirect
29. C2×C23×#104C4semidirect
36. C23×#382C22semidirect
38. C23×#343C22semidirect
39. C23×#339C22semidirect
62. C2×M16×C23C2×C4semidirect
102. C2×C42×C23C2×C4semidirect
106. C23×#104C2×C4semidirect
170. M16×C23C22×C4semidirect
196. C42×C23C22×C4semidirect
197. C2×#1860C23direct (N × H)
261. C2×#428C2×C23semidirect
264. C2×#382C2×C23semidirect
265. #1860C2×C23direct (N × H)
266. C2×#343C2×C23semidirect
267. C2×#339C2×C23semidirect
314. #428C22×C23semidirect
317. C2×#104C4×C23semidirect
324. #382C22×C23semidirect
326. #343C22×C23semidirect
327. #339C22×C23semidirect
332. C2×C23#1860direct (N × H)
358. C2×M16C2×C4×C23semidirect
398. C2×C42C2×C4×C23semidirect
402. #104C2×C4×C23semidirect
406. C23C2×#1860direct (N × H)
467. M16C22×C4×C23semidirect
493. C42C22×C4×C23semidirect
589. C2C23×#1860direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5202
C2×C23×#4282
C22×C23×#1042
C2×C23×#4172
C2×C23×#3822
C23×#18602
C2×C23×#3432
C2×C23×#3392
C2×#186023