Finite Group Databasebeta

← C23×#16470 #19164096

Subgroups — C23×#16470 #19164096

Order 5888.

1438 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#397C211yesmaximal, characteristic
2C2×C23×#397C212yesmaximal
3C2×C23×#385C211yesmaximal, characteristic
4C23×#1442C212yesmaximal, complemented
5C23×#1442C212yesmaximal, complemented
6C23×#1441C212yesmaximal, complemented
7C23×#1441C212yesmaximal, complemented
8C23×#1439C211yesmaximal, characteristic
9C23×C22⋊#95C212yesmaximal
10C24×C4×C23C2211yescharacteristic
11C22×C4⋊C4×C23C2212yes—
12C22×C4⋊C4×C23C2212yes—
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C22⋊C4×C23C2212yes—
16C22×C22⋊C4×C23C2212yes—
17C22×C22⋊C4×C23C2211yescharacteristic
18C22×C22⋊C4×C23C2211yescharacteristic
19C23×#397—28no—
20C23×#397—24no—
21C23×#385—24no—
22C2×C23×#95C2212yes—
23C2×C23×#95C2212yes—
24C2×C23×#95C2212yes—
25C2×C23×#95C2212yes—
26C2×C23×#95C2212yes—
27C2×C23×#95C2212yes—
28C2×C23×#95C2212yes—
29C2×C23×#95C2211yescharacteristic
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2212yes—
33C2×C23×#95C2211yescharacteristic
34C25×C23C2311yescharacteristic
35C23×C4×C23—22no—
36C23×C4×C23—24no—
37C23×C4×C23—42no—
38C23×C4×C23—41no—
39C23×C4×C23—41no—
40C23×C4×C23C2312yes—
41C23×C4×C23C2312yes—
42C23×C4×C23C2312yes—
43C23×C4×C23C2311yescharacteristic
44C23×C4×C23C2312yes—
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2312yes—
47C23×C4×C23C2312yes—
48C2×C4⋊C4×C23—24no—
49C2×C4⋊C4×C23—24no—
50C2×C4⋊C4×C23—22no—
51C2×C4⋊C4×C23—24no—
52C2×C4⋊C4×C23—24no—
53C2×C4⋊C4×C23—42no—
54C2×C4⋊C4×C23—42no—
55C2×C4⋊C4×C23—41no—
56C2×C4⋊C4×C23—41no—
57C2×C4⋊C4×C23—41no—
58C2×C4⋊C4×C23—42no—
59C2×C22⋊C4×C23—22no—
60C2×C22⋊C4×C23—22no—
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—41no—
67C2×C22⋊C4×C23—41no—
68C2×C22⋊C4×C23—42no—
69C2×C22⋊C4×C23D414yes—
70C2×C22⋊C4×C23D414yes—
71C23×#95—24no—
72C23×#95—24no—
73C23×#95—24no—
74C23×#95—24no—
75C23×#95—24no—
76C23×#95—22no—
77C23×#95—24no—
78C23×#95—24no—
79C23×#95—41no—
80C23×#95—42no—
81C23×#95—42no—
82C23×#95—42no—
83C24×C23—22no—
84C24×C23—22no—
85C24×C23—21no—
86C24×C23—21no—
87C24×C23—22no—
88C24×C23—21no—
89C24×C23—21no—
90C24×C23—22no—
91C24×C23C2×D412yes—
92C24×C23C2×D412yes—
93C24×C23#4112yes—
94C24×C23C2411yescharacteristic
95C22×C4×C23—22no—
96C22×C4×C23—22no—
97C22×C4×C23—22no—
98C22×C4×C23—22no—
99C22×C4×C23—22no—
100C22×C4×C23—22no—
101C22×C4×C23—22no—
102C22×C4×C23—22no—
103C22×C4×C23—22no—
104C22×C4×C23—22no—
105C22×C4×C23—21no—
106C22×C4×C23—21no—
107C22×C4×C23—21no—
108C22×C4×C23—22no—
109C22×C4×C23—21no—
110C22×C4×C23—22no—
111C22×C4×C23—22no—
112C22×C4×C23—22no—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—22no—
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—22no—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
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—44no—
143C22×C4×C23—48no—
144C22×C4×C23—48no—
145C22×C4×C23—44no—
146C22×C4×C23—44no—
147C22×C4×C23C2×D412yes—
148C22×C4×C23C2×D414yes—
149C22×C4×C23C2×D412yes—
150C22×C4×C23#4114yes—
151C22×C4×C23#4114yes—
152C4⋊C4×C23—48no—
153C4⋊C4×C23—48no—
154C4⋊C4×C23—44no—
155C4⋊C4×C23—44no—
156C22⋊C4×C23—28no—
157C22⋊C4×C23—44no—
158C22⋊C4×C23—48no—
159C22⋊C4×C23—48no—
160#16470C2311yesmaximal, characteristic, direct product factor, Sylow
161C23×C23—24no—
162C23×C23—22no—
163C23×C23—22no—
164C23×C23—22no—
165C23×C23—22no—
166C23×C23—22no—
167C23×C23—22no—
168C23×C23—24no—
169C23×C23—24no—
170C23×C23—24no—
171C23×C23—24no—
172C23×C23—24no—
173C23×C23—42no—
174C23×C23—42no—
175C23×C23—42no—
176C23×C23—42no—
177C23×C23—41no—
178C23×C23—42no—
179C23×C23—41no—
180C23×C23—41no—
181C23×C23—42no—
182C23×C23—41no—
183C23×C23#12114yes—
184C23×C23C22×D411yescharacteristic
185C23×C23C22×D411yescharacteristic
186C23×C23C2×#4111yescharacteristic
187C23×C23C2×#4112yes—
188C23×C23C2×#4112yes—
189C23×C23H(5,𝔽2)11yescharacteristic
190C23×C23H(5,𝔽2)11yescharacteristic
191C23×C23H(5,𝔽2)12yes—
192C23×C232−1+411yescharacteristic
193C23×C232−1+411yescharacteristic
194C23×C232−1+412yes—
195C2×C4×C23—24no—
196C2×C4×C23—22no—
197C2×C4×C23—22no—
198C2×C4×C23—24no—
199C2×C4×C23—24no—
200C2×C4×C23—24no—
201C2×C4×C23—24no—
202C2×C4×C23—24no—
203C2×C4×C23—24no—
204C2×C4×C23—24no—
205C2×C4×C23—24no—
206C2×C4×C23—24no—
207C2×C4×C23—24no—
208C2×C4×C23—416no—
209C2×C4×C23—42no—
210C2×C4×C23—42no—
211C2×C4×C23—42no—
212C2×C4×C23—42no—
213C2×C4×C23—42no—
214C2×C4×C23—42no—
215C2×C4×C23—42no—
216C2×C4×C23—42no—
217C2×C4×C23—42no—
218C2×C4×C23—42no—
219C2×C4×C23—42no—
220C2×C4×C23—42no—
221C2×C4×C23—41no—
222C2×C4×C23—41no—
223C2×C4×C23—41no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—41no—
227C2×C4×C23—42no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
231C2×C4×C23—42no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—42no—
237C2×C4×C23—42no—
238C2×C4×C23—42no—
239C2×C4×C23—42no—
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×#397C2×C2311yescharacteristic
249C2×#397C2×C2312yes—
250C2×#385C2×C2311yescharacteristic
251#1442C2×C2312yescomplemented
252#1442C2×C2312yescomplemented
253#1441C2×C2312yescomplemented
254#1441C2×C2312yescomplemented
255#1439C2×C2311yescharacteristic
256C22⋊#95C2×C2312yes—
257C22×C23—24no—
258C22×C23—28no—
259C22×C23—42no—
260C22×C23—44no—
261C22×C23—42no—
262C22×C23—44no—
263C22×C23—44no—
264C22×C23—42no—
265C22×C23—42no—
266C22×C23—42no—
267C22×C23—42no—
268C22×C23C2×#12111yescharacteristic
269C22×C23#52912yes—
270C22×C23#53411yescharacteristic
271C22×C23#53611yescharacteristic
272C22×C23#53711yescharacteristic
273C22×C23#53711yescharacteristic
274C22×C23#53812yes—
275C22×C23#53911yescharacteristic
276C22×C23#53912yes—
277C22×C23#53911yescharacteristic
278C22×C23#53912yes—
279C22×C23#54112yes—
280C22×C23#54212yes—
281C22×C23#54612yes—
282C22×C23#54612yes—
283C22×C23#54712yes—
284C22×C23#54712yes—
285C22×C23#55111yescharacteristic
286C22×C23#55212yes—
287C22×C23#55911yescharacteristic
288C22×C23H(3,𝔽4)11yescharacteristic
289C22×C23#56312yes—
290C22×C23#56311yescharacteristic
291C4×C23—44no—
292C4×C23—42no—
293C4×C23—44no—
294C4×C23—44no—
295C4×C23—42no—
296C4×C23—82no—
297C4×C23—82no—
298C4×C23—82no—
299C24×C4C22×C2311yescharacteristic
300C22×C4⋊C4C22×C2312yes—
301C22×C4⋊C4C22×C2312yes—
302C22×C4⋊C4C22×C2311yescharacteristic
303C22×C4⋊C4C22×C2311yescharacteristic
304C22×C22⋊C4C22×C2312yes—
305C22×C22⋊C4C22×C2312yes—
306C22×C22⋊C4C22×C2311yescharacteristic
307C22×C22⋊C4C22×C2311yescharacteristic
308#397—28no—
309#397—24no—
310#385—24no—
311C2×#95C22×C2312yes—
312C2×#95C22×C2312yes—
313C2×#95C22×C2312yes—
314C2×#95C22×C2312yes—
315C2×#95C22×C2311yescharacteristic
316C2×#95C22×C2312yes—
317C2×#95C22×C2312yes—
318C2×#95C22×C2312yes—
319C2×#95C22×C2312yes—
320C2×#95C22×C2312yes—
321C2×#95C22×C2312yes—
322C2×#95C22×C2311yescharacteristic
323C2×C23—44no—
324C2×C23#242512yes—
325C2×C23#251711yescharacteristic
326C2×C23#261811yescharacteristic
327C2×C23#265212yes—
328C2×C23#268411yescharacteristic
329C2×C23#269512yes—
330C2×C23#269812yes—
331C2×C23#271211yescharacteristic
332C2×C23#272211yescharacteristic
333C2×C23#272411yescharacteristic
334C2×C23#283511yescharacteristic
335C25C23×C2311yescharacteristic
336C23×C4—22no—
337C23×C4—24no—
338C23×C4—41no—
339C23×C4—41no—
340C23×C4—42no—
341C23×C4C23×C2312yes—
342C23×C4C23×C2312yes—
343C23×C4C23×C2312yes—
344C23×C4C23×C2311yescharacteristic
345C23×C4C23×C2312yes—
346C23×C4C23×C2311yescharacteristic
347C23×C4C23×C2312yes—
348C23×C4C23×C2312yes—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—41no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4—42no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—41no—
368C2×C22⋊C4—41no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4D4×C2314yes—
371C2×C22⋊C4D4×C2314yes—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—41no—
381#95—42no—
382#95—42no—
383#95—42no—
384C23#1647011yescharacteristic, direct product factor, Sylow
385C24—22no—
386C24—22no—
387C24—21no—
388C24—21no—
389C24—22no—
390C24—22no—
391C24—21no—
392C24—21no—
393C24C2×D4×C2312yes—
394C24C2×D4×C2312yes—
395C24#41×C2312yes—
396C24C24×C2311yescharacteristic
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4C2×D4×C2312yes—
450C22×C4C2×D4×C2314yes—
451C22×C4C2×D4×C2312yes—
452C22×C4#41×C2314yes—
453C22×C4#41×C2314yes—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—41no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C23×#12114yes—
485C23C22×D4×C2311yescharacteristic
486C23C22×D4×C2311yescharacteristic
487C23C2×#41×C2312yes—
488C23C2×#41×C2312yes—
489C23C2×#41×C2311yescharacteristic
490C23C23×H(5,𝔽2)11yescharacteristic
491C23C23×H(5,𝔽2)12yes—
492C23C23×H(5,𝔽2)11yescharacteristic
493C23C23×2−1+412yes—
494C23C23×2−1+411yescharacteristic
495C23C23×2−1+411yescharacteristic
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C22—28no—
550C22—24no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22C2×C23×#12111yescharacteristic
561C22C23×#52912yes—
562C22C23×#53411yescharacteristic
563C22C23×#53611yescharacteristic
564C22C23×#53711yescharacteristic
565C22C23×#53711yescharacteristic
566C22C23×#53812yes—
567C22C23×#53912yes—
568C22C23×#53912yes—
569C22C23×#53911yescharacteristic
570C22C23×#53911yescharacteristic
571C22C23×#54112yes—
572C22C23×#54212yes—
573C22C23×#54612yes—
574C22C23×#54612yes—
575C22C23×#54712yes—
576C22C23×#54712yes—
577C22C23×#55111yescharacteristic
578C22C23×#55212yes—
579C22C23×#55911yescharacteristic
580C22C23×H(3,𝔽4)11yescharacteristic
581C22C23×#56312yes—
582C22C23×#56311yescharacteristic
583C4—44no—
584C4—44no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—82no—
590C4—82no—
591C2—44no—
592C2C23×#242512yes—
593C2C23×#251711yescharacteristic
594C2C23×#261811yescharacteristic
595C2C23×#265212yes—
596C2C23×#268411yescharacteristic
597C2C23×#269512yes—
598C2C23×#269812yes—
599C2C23×#271211yescharacteristic
600C2C23×#272211yescharacteristic
601C2C23×#272411yescharacteristic
602C2C23×#283511yescharacteristic
subgroupG/H
center, socleC24×C23C24
commutator, Frattini, cosocleC24C24×C23
prime radicalC25×C23C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1442C2semidirect
5. C23×#1442C2semidirect
6. C23×#1441C2semidirect
7. C23×#1441C2semidirect
160. #16470C23direct (N × H)
251. #1442C2×C23semidirect
252. #1442C2×C23semidirect
253. #1441C2×C23semidirect
254. #1441C2×C23semidirect
384. C23#16470direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3972
C2×C23×#3852
C23×#14422
C23×#14412
C23×#14392
C23×C22⋊#952
#1647023