Finite Group Databasebeta

← C23×#16599 #19164225

Subgroups — C23×#16599 #19164225

Order 5888.

1550 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#400C211yesmaximal, characteristic, complemented
2C2×C23×#390C211yesmaximal, characteristic
3C23×#1441C211yesmaximal, characteristic, complemented
4C23×#1439C212yesmaximal, complemented
5C23×#1439C212yesmaximal, complemented
6C23×#1439C212yesmaximal, complemented
7C23×#1439C212yesmaximal, complemented
8C23×#1438C212yesmaximal
9C23×#1436C212yesmaximal, complemented
10C23×D4×C23C2211yescharacteristic
11C22×C4⋊C4×C23C2212yes—
12C22×C4⋊C4×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C22⋊C4×C23C2212yes—
16C22×C22⋊C4×C23C2212yes—
17C22×C22⋊C4×C23C2212yes—
18C22×C22⋊C4×C23C2212yes—
19C22×C22⋊C4×C23C2212yes—
20C22×C22⋊C4×C23C2212yes—
21C22×C42×C23C2211yescharacteristic
22C23×#400—24no—
23C23×#390—24no—
24C2×C23×#95C2212yes—
25C2×C23×#95C2212yes—
26C2×C23×#95C2212yes—
27C2×C23×#95C2212yes—
28C2×C23×#95C2211yescharacteristic
29C2×C23×#95C2212yes—
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2211yescharacteristic
33C25×C23C2312yes—
34C22×D4×C23—22no—
35C22×D4×C23—22no—
36C22×D4×C23—41no—
37C22×D4×C23—41no—
38C22×D4×C23—41no—
39C22×D4×C23—41no—
40C22×D4×C23Q814yes—
41C23×C4×C23C2312yes—
42C23×C4×C23C2312yes—
43C23×C4×C23C2312yes—
44C23×C4×C23C2311yescharacteristic
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2312yes—
47C23×C4×C23C2312yes—
48C23×C4×C23C2311yescharacteristic
49C2×C4⋊C4×C23—22no—
50C2×C4⋊C4×C23—24no—
51C2×C4⋊C4×C23—41no—
52C2×C4⋊C4×C23—41no—
53C2×C4⋊C4×C23—42no—
54C2×C4⋊C4×C23—42no—
55C2×C4⋊C4×C23—41no—
56C2×C4⋊C4×C23—41no—
57C2×C4⋊C4×C23—42no—
58C2×C4⋊C4×C23—41no—
59C2×C4⋊C4×C23D414yes—
60C2×C22⋊C4×C23—24no—
61C2×C22⋊C4×C23—24no—
62C2×C22⋊C4×C23—24no—
63C2×C22⋊C4×C23—24no—
64C2×C22⋊C4×C23—24no—
65C2×C22⋊C4×C23—24no—
66C2×C22⋊C4×C23—42no—
67C2×C22⋊C4×C23—42no—
68C2×C22⋊C4×C23—42no—
69C2×C22⋊C4×C23—42no—
70C2×C22⋊C4×C23—42no—
71C2×C22⋊C4×C23—42no—
72C2×C22⋊C4×C23—42no—
73C2×C22⋊C4×C23—42no—
74C2×C22⋊C4×C23—42no—
75C2×C22⋊C4×C23—42no—
76C2×C22⋊C4×C23—42no—
77C2×C22⋊C4×C23—42no—
78C2×C42×C23—22no—
79C2×C42×C23—41no—
80C2×C42×C23—41no—
81C23×#95—24no—
82C23×#95—24no—
83C23×#95—22no—
84C23×#95—24no—
85C23×#95—24no—
86C23×#95—42no—
87C23×#95—41no—
88C23×#95—42no—
89C23×#95—42no—
90C24×C23—22no—
91C24×C23—22no—
92C24×C23—22no—
93C24×C23—22no—
94C24×C23—22no—
95C24×C23—22no—
96C24×C23—22no—
97C24×C23—22no—
98C24×C23—22no—
99C24×C23—22no—
100C24×C23—22no—
101C24×C23—22no—
102C24×C23—22no—
103C24×C23—22no—
104C24×C23C2×Q814yes—
105C24×C23C2411yescharacteristic
106C2×D4×C23—28no—
107C2×D4×C23—44no—
108C2×D4×C23—44no—
109C2×D4×C23—44no—
110C2×D4×C23—44no—
111C2×D4×C23—44no—
112C2×D4×C23—44no—
113C22×C4×C23—21no—
114C22×C4×C23—21no—
115C22×C4×C23—21no—
116C22×C4×C23—21no—
117C22×C4×C23—22no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—22no—
121C22×C4×C23—22no—
122C22×C4×C23—22no—
123C22×C4×C23—22no—
124C22×C4×C23—22no—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—22no—
128C22×C4×C23—22no—
129C22×C4×C23—22no—
130C22×C4×C23—22no—
131C22×C4×C23—22no—
132C22×C4×C23—21no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—22no—
137C22×C4×C23—22no—
138C22×C4×C23—22no—
139C22×C4×C23—22no—
140C22×C4×C23—22no—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—22no—
144C22×C4×C23—22no—
145C22×C4×C23—22no—
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—22no—
150C22×C4×C23—22no—
151C22×C4×C23—22no—
152C22×C4×C23—22no—
153C22×C4×C23—22no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23—22no—
157C22×C4×C23—22no—
158C22×C4×C23C2×D412yes—
159C22×C4×C23C2×D412yes—
160C22×C4×C23C2×D412yes—
161C22×C4×C23C2×Q812yes—
162C22×C4×C23#4114yes—
163C22×C4×C23#4114yes—
164C22×C4×C23#4112yes—
165C4⋊C4×C23—44no—
166C4⋊C4×C23—44no—
167C4⋊C4×C23—48no—
168C4⋊C4×C23—44no—
169C22⋊C4×C23—48no—
170C22⋊C4×C23—48no—
171C22⋊C4×C23—48no—
172C22⋊C4×C23—48no—
173C22⋊C4×C23—48no—
174C22⋊C4×C23—48no—
175C42×C23—44no—
176#16599C2311yesmaximal, characteristic, direct product factor, Sylow
177C23×C23—24no—
178C23×C23—24no—
179C23×C23—24no—
180C23×C23—24no—
181C23×C23—24no—
182C23×C23—24no—
183C23×C23—24no—
184C23×C23—42no—
185C23×C23—42no—
186C23×C23—42no—
187C23×C23—42no—
188C23×C23—42no—
189C23×C23—42no—
190C23×C23—42no—
191C23×C23—42no—
192C23×C23—42no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—42no—
197C23×C23—42no—
198C23×C23—42no—
199C23×C23—42no—
200C23×C23—42no—
201C23×C23—42no—
202C23×C23—42no—
203C23×C23—42no—
204C23×C23—42no—
205C23×C23—42no—
206C23×C23—42no—
207C23×C23—42no—
208C23×C23—42no—
209C23×C23—42no—
210C23×C23—42no—
211C23×C23—42no—
212C23×C23C22×D411yescharacteristic
213C23×C23C22×Q811yescharacteristic
214C23×C23C2×#4111yescharacteristic
215C23×C23C2×#4112yes—
216C23×C23C2×#4112yes—
217C23×C23H(5,𝔽2)11yescharacteristic
218C23×C23H(5,𝔽2)11yescharacteristic
219C23×C23H(5,𝔽2)11yescharacteristic
220C23×C23H(5,𝔽2)11yescharacteristic
221C23×C23H(5,𝔽2)12yes—
222C23×C23H(5,𝔽2)12yes—
223D4×C23—416no—
224C2×C4×C23—22no—
225C2×C4×C23—22no—
226C2×C4×C23—22no—
227C2×C4×C23—22no—
228C2×C4×C23—22no—
229C2×C4×C23—22no—
230C2×C4×C23—24no—
231C2×C4×C23—22no—
232C2×C4×C23—22no—
233C2×C4×C23—24no—
234C2×C4×C23—24no—
235C2×C4×C23—24no—
236C2×C4×C23—24no—
237C2×C4×C23—24no—
238C2×C4×C23—24no—
239C2×C4×C23—24no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—42no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—42no—
248C2×C4×C23—42no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—42no—
254C2×C4×C23—42no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—42no—
258C2×C4×C23—42no—
259C2×C4×C23—42no—
260C2×C4×C23—42no—
261C2×C4×C23—42no—
262C2×C4×C23—42no—
263C2×C4×C23—42no—
264C2×C4×C23—42no—
265C2×C4×C23—42no—
266C2×C4×C23—42no—
267C2×C4×C23—42no—
268C2×C4×C23—42no—
269C2×C4×C23—42no—
270C2×C4×C23—42no—
271C2×C4×C23—42no—
272C2×C4×C23—42no—
273C2×C4×C23—42no—
274C2×C4×C23—42no—
275C2×C4×C23C22⋊Q814yes—
276C2×#400C2×C2311yescharacteristic, complemented
277C2×#390C2×C2311yescharacteristic
278#1441C2×C2311yescharacteristic, complemented
279#1439C2×C2312yescomplemented
280#1439C2×C2312yescomplemented
281#1439C2×C2312yescomplemented
282#1439C2×C2312yescomplemented
283#1438C2×C2312yes—
284#1436C2×C2312yescomplemented
285C22×C23—44no—
286C22×C23—44no—
287C22×C23—44no—
288C22×C23—44no—
289C22×C23—44no—
290C22×C23—44no—
291C22×C23—44no—
292C22×C23—82no—
293C22×C23—82no—
294C22×C23—82no—
295C22×C23—82no—
296C22×C23—82no—
297C22×C23—82no—
298C22×C23—82no—
299C22×C23—82no—
300C22×C23C2×C22⋊Q811yescharacteristic
301C22×C23#52912yes—
302C22×C23#53411yescharacteristic
303C22×C23#53511yescharacteristic
304C22×C23#53811yescharacteristic
305C22×C23#53811yescharacteristic
306C22×C23#53812yes—
307C22×C23#53812yes—
308C22×C23#53812yes—
309C22×C23#54012yes—
310C22×C23#54012yes—
311C22×C23#54311yescharacteristic
312C22×C23#54311yescharacteristic
313C22×C23#54612yes—
314C22×C23#55012yes—
315C22×C23#55312yes—
316C22×C23#55412yes—
317C22×C23#55412yes—
318C22×C23#55511yescharacteristic
319C22×C23#55911yescharacteristic
320C22×C23#56011yescharacteristic
321C22×C23#56012yes—
322C22×C23H(3,𝔽4)11yescharacteristic
323C4×C23—24no—
324C4×C23—44no—
325C4×C23—42no—
326C4×C23—42no—
327C4×C23—44no—
328C4×C23—82no—
329C4×C23—82no—
330C4×C23—82no—
331C23×D4C22×C2311yescharacteristic
332C22×C4⋊C4C22×C2311yescharacteristic
333C22×C4⋊C4C22×C2312yes—
334C22×C4⋊C4C22×C2311yescharacteristic
335C22×C4⋊C4C22×C2311yescharacteristic
336C22×C22⋊C4C22×C2312yes—
337C22×C22⋊C4C22×C2312yes—
338C22×C22⋊C4C22×C2312yes—
339C22×C22⋊C4C22×C2312yes—
340C22×C22⋊C4C22×C2312yes—
341C22×C22⋊C4C22×C2312yes—
342C22×C42C22×C2311yescharacteristic
343#400—24no—
344#390—24no—
345C2×#95C22×C2312yes—
346C2×#95C22×C2311yescharacteristic
347C2×#95C22×C2312yes—
348C2×#95C22×C2312yes—
349C2×#95C22×C2312yes—
350C2×#95C22×C2312yes—
351C2×#95C22×C2312yes—
352C2×#95C22×C2312yes—
353C2×#95C22×C2311yescharacteristic
354C2×C23—84no—
355C2×C23#245412yes—
356C2×C23#259311yescharacteristic
357C2×C23#262611yescharacteristic
358C2×C23#264812yes—
359C2×C23#270911yescharacteristic
360C2×C23#271412yes—
361C2×C23#273311yescharacteristic
362C2×C23#278012yes—
363C2×C23#278411yescharacteristic
364C2×C23#280411yescharacteristic
365C2×C23#282811yescharacteristic
366C25C23×C2312yes—
367C22×D4—22no—
368C22×D4—22no—
369C22×D4—41no—
370C22×D4—41no—
371C22×D4—41no—
372C22×D4—41no—
373C22×D4Q8×C2314yes—
374C23×C4C23×C2312yes—
375C23×C4C23×C2312yes—
376C23×C4C23×C2312yes—
377C23×C4C23×C2311yescharacteristic
378C23×C4C23×C2311yescharacteristic
379C23×C4C23×C2312yes—
380C23×C4C23×C2312yes—
381C23×C4C23×C2311yescharacteristic
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—41no—
385C2×C4⋊C4—41no—
386C2×C4⋊C4—42no—
387C2×C4⋊C4—42no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—41no—
390C2×C4⋊C4—42no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4D4×C2314yes—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4—42no—
411C2×C42—22no—
412C2×C42—41no—
413C2×C42—41no—
414#95—22no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—42no—
420#95—41no—
421#95—42no—
422#95—42no—
423C23#1659911yescharacteristic, direct product factor, Sylow
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24C2×Q8×C2314yes—
439C24C24×C2311yescharacteristic
440C2×D4—28no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4C2×D4×C2312yes—
493C22×C4C2×D4×C2312yes—
494C22×C4C2×D4×C2312yes—
495C22×C4C2×Q8×C2312yes—
496C22×C4#41×C2314yes—
497C22×C4#41×C2314yes—
498C22×C4#41×C2312yes—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C42—44no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C22×D4×C2311yescharacteristic
546C23C22×Q8×C2311yescharacteristic
547C23C2×#41×C2312yes—
548C23C2×#41×C2312yes—
549C23C2×#41×C2311yescharacteristic
550C23C23×H(5,𝔽2)12yes—
551C23C23×H(5,𝔽2)11yescharacteristic
552C23C23×H(5,𝔽2)12yes—
553C23C23×H(5,𝔽2)11yescharacteristic
554C23C23×H(5,𝔽2)11yescharacteristic
555C23C23×H(5,𝔽2)11yescharacteristic
556D4—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4C23×C22⋊Q814yes—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C2×C23×C22⋊Q811yescharacteristic
625C22C23×#52912yes—
626C22C23×#53411yescharacteristic
627C22C23×#53511yescharacteristic
628C22C23×#53812yes—
629C22C23×#53812yes—
630C22C23×#53812yes—
631C22C23×#53811yescharacteristic
632C22C23×#53811yescharacteristic
633C22C23×#54012yes—
634C22C23×#54012yes—
635C22C23×#54311yescharacteristic
636C22C23×#54311yescharacteristic
637C22C23×#54612yes—
638C22C23×#55012yes—
639C22C23×#55312yes—
640C22C23×#55412yes—
641C22C23×#55412yes—
642C22C23×#55511yescharacteristic
643C22C23×#55911yescharacteristic
644C22C23×#56011yescharacteristic
645C22C23×#56012yes—
646C22C23×H(3,𝔽4)11yescharacteristic
647C4—24no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—44no—
652C4—82no—
653C4—82no—
654C4—82no—
655C2—84no—
656C2C23×#245412yes—
657C2C23×#259311yescharacteristic
658C2C23×#262611yescharacteristic
659C2C23×#264812yes—
660C2C23×#270911yescharacteristic
661C2C23×#271412yes—
662C2C23×#273311yescharacteristic
663C2C23×#278012yes—
664C2C23×#278411yescharacteristic
665C2C23×#280411yescharacteristic
666C2C23×#282811yescharacteristic
subgroupG/H
center, socleC24×C23C24
commutator, Frattini, cosocleC24C24×C23
prime radicalC23×D4×C23C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×#400C2semidirect
3. C23×#1441C2semidirect
4. C23×#1439C2semidirect
5. C23×#1439C2semidirect
6. C23×#1439C2semidirect
7. C23×#1439C2semidirect
9. C23×#1436C2semidirect
176. #16599C23direct (N × H)
276. C2×#400C2×C23semidirect
278. #1441C2×C23semidirect
279. #1439C2×C23semidirect
280. #1439C2×C23semidirect
281. #1439C2×C23semidirect
282. #1439C2×C23semidirect
284. #1436C2×C23semidirect
423. C23#16599direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4002
C2×C23×#3902
C23×#14412
C23×#14392
C23×#14382
C23×#14362
#1659923