Finite Group Databasebeta

← C2×C23×#1390 #19160198

Subgroups — C2×C23×#1390 #19160198

Order 5888.

1202 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#100C211yesmaximal, characteristic
2C22×C23×#99C211yesmaximal, characteristic
3C2×C23×#386C211yesmaximal, characteristic
4C23×#1390C214yesmaximal, direct product factor
5C23×D4×C23C2211yescharacteristic
6C24×C4×C23C411yescharacteristic, complemented
7C22×M16×C23—21no—
8C22×C22⋊C4×C23—21no—
9C22×C22⋊C4×C23—21no—
10C22×C22⋊C4×C23C411yescharacteristic, complemented
11C2×C23×#100—21no—
12C2×C23×#100—22no—
13C2×C23×#100C414yescomplemented
14C2×C23×#100C2212yes—
15C2×C23×#99—21no—
16C2×C23×#99—22no—
17C2×C23×#99C414yescomplemented
18C2×C23×#99C2212yes—
19C23×#386—21no—
20C23×#386—21no—
21C23×#386C412yescomplemented
22C23×#386C2212yes—
23C2×C23×#95—21no—
24C25×C23C2×C411yescharacteristic
25C25×C23D411yescharacteristic
26C22×D4×C23—42no—
27C22×D4×C23—41no—
28C22×D4×C23—41no—
29C22×D4×C23C2×C411yescharacteristic
30C22×D4×C23C2×C412yes—
31C22×D4×C23C2×C412yes—
32C22×D4×C23D411yescharacteristic
33C22×D4×C23D412yes—
34C22×D4×C23D411yescharacteristic
35C22×D4×C23Q812yes—
36C22×D4×C23C2311yescharacteristic
37C23×C4×C23—21no—
38C23×C4×C23—21no—
39C23×C4×C23—21no—
40C23×C4×C23—44no—
41C23×C4×C23—41no—
42C23×C4×C23—41no—
43C23×C4×C23—41no—
44C23×C4×C23C2×C412yescomplemented
45C23×C4×C23D411yescharacteristic
46C2×M16×C23—24no—
47C2×M16×C23—22no—
48C2×M16×C23—42no—
49C2×M16×C23—41no—
50C22×C8×C23—41no—
51C2×C22⋊C4×C23—24no—
52C2×C22⋊C4×C23—22no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—22no—
55C2×C22⋊C4×C23—22no—
56C2×C22⋊C4×C23—44no—
57C2×C22⋊C4×C23—41no—
58C2×C22⋊C4×C23—42no—
59C2×C22⋊C4×C23—42no—
60C2×C22⋊C4×C23C2×C412yescomplemented
61C23×#100—24no—
62C23×#100—22no—
63C23×#100C2×C414yescomplemented
64C23×#99—24no—
65C23×#99—22no—
66C23×#99C2×C414yescomplemented
67C23×#95—22no—
68C23×#95—22no—
69C24×C23—22no—
70C24×C23—21no—
71C24×C23—21no—
72C24×C23—21no—
73C24×C23—22no—
74C24×C23—21no—
75C24×C23—42no—
76C24×C23—42no—
77C24×C23—42no—
78C24×C23—42no—
79C24×C23C4212yes—
80C24×C23C22⋊C412yes—
81C24×C23C22⋊C411yescharacteristic
82C24×C23C22⋊C411yescharacteristic
83C24×C23C22⋊C411yescharacteristic
84C24×C23C4⋊C412yes—
85C24×C23C4⋊C412yes—
86C24×C23C22×C411yescharacteristic
87C24×C23C2×D411yescharacteristic
88C2×D4×C23—42no—
89C2×D4×C23—42no—
90C2×D4×C23—42no—
91C2×D4×C23—42no—
92C2×D4×C23—41no—
93C2×D4×C23—41no—
94C2×D4×C23—42no—
95C2×D4×C23—42no—
96C2×D4×C23—41no—
97C2×D4×C23—41no—
98C2×D4×C23—41no—
99C2×D4×C23—41no—
100C2×D4×C23—42no—
101C2×D4×C23—42no—
102C2×D4×C23—41no—
103C2×D4×C23—41no—
104C2×D4×C23C4212yes—
105C2×D4×C23C22⋊C412yes—
106C2×D4×C23C22⋊C412yes—
107C2×D4×C23C4⋊C412yes—
108C2×D4×C23C4⋊C412yes—
109C2×D4×C23C4⋊C412yes—
110C2×D4×C23C22×C411yescharacteristic
111C2×D4×C23C22×C411yescharacteristic
112C2×D4×C23C2×D411yescharacteristic
113C2×D4×C23C2×Q811yescharacteristic
114C22×C4×C23—22no—
115C22×C4×C23—21no—
116C22×C4×C23—21no—
117C22×C4×C23—24no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—44no—
121C22×C4×C23—44no—
122C22×C4×C23—42no—
123C22×C4×C23—42no—
124C22×C4×C23—48no—
125C22×C4×C23—44no—
126C22×C4×C23—41no—
127C22×C4×C23—41no—
128C22×C4×C23—42no—
129C22×C4×C23—44no—
130C22×C4×C23—44no—
131C22×C4×C23—42no—
132C22×C4×C23—42no—
133C22×C4×C23—42no—
134C22×C4×C23—42no—
135C22×C4×C23—82no—
136C22×C4×C23—82no—
137C22×C4×C23C22⋊C411yescharacteristic
138C22×C4×C23C22⋊C412yes—
139C22×C4×C23C4⋊C412yes—
140C22×C4×C23C2×D411yescharacteristic
141M16×C23—24no—
142M16×C23—44no—
143M16×C23—42no—
144C2×C8×C23—44no—
145C2×C8×C23—42no—
146C22⋊C4×C23—24no—
147C22⋊C4×C23—44no—
148C22⋊C4×C23—44no—
149C22⋊C4×C23—44no—
150C22⋊C4×C23—42no—
151C22⋊C4×C23—42no—
152C22⋊C4×C23—42no—
153C2×#1390C2311yesmaximal, characteristic, direct product factor, Sylow
154C23×C23—22no—
155C23×C23—22no—
156C23×C23—22no—
157C23×C23—22no—
158C23×C23—22no—
159C23×C23—22no—
160C23×C23—22no—
161C23×C23—22no—
162C23×C23—21no—
163C23×C23—21no—
164C23×C23—21no—
165C23×C23—21no—
166C23×C23—21no—
167C23×C23—21no—
168C23×C23—22no—
169C23×C23—22no—
170C23×C23—21no—
171C23×C23—21no—
172C23×C23—41no—
173C23×C23—41no—
174C23×C23—44no—
175C23×C23—41no—
176C23×C23—44no—
177C23×C23—41no—
178C23×C23—42no—
179C23×C23—42no—
180C23×C23—48no—
181C23×C23—41no—
182C23×C23—41no—
183C23×C23—41no—
184C23×C23—44no—
185C23×C23—44no—
186C23×C23—41no—
187C23×C23—42no—
188C23×C23—42no—
189C23×C23—42no—
190C23×C23—42no—
191C23×C23—81no—
192C23×C23—81no—
193C23×C23—84no—
194C23×C23#9512yes—
195C23×C23#9512yes—
196C23×C23#9512yes—
197C23×C23#9911yescharacteristic
198C23×C23#9912yes—
199C23×C23#9911yescharacteristic
200C23×C23C2×C4211yescharacteristic
201C23×C23C2×C22⋊C411yescharacteristic
202C23×C23C2×C22⋊C411yescharacteristic
203C23×C23C2×C4⋊C411yescharacteristic
204C23×C23C2×C4⋊C411yescharacteristic
205D4×C23—42no—
206D4×C23—42no—
207D4×C23—42no—
208D4×C23—42no—
209D4×C23—42no—
210D4×C23—42no—
211D4×C23—41no—
212D4×C23—41no—
213D4×C23—41no—
214D4×C23—41no—
215C2×C4×C23—22no—
216C2×C4×C23—22no—
217C2×C4×C23—22no—
218C2×C4×C23—22no—
219C2×C4×C23—21no—
220C2×C4×C23—21no—
221C2×C4×C23—21no—
222C2×C4×C23—21no—
223C2×C4×C23—24no—
224C2×C4×C23—42no—
225C2×C4×C23—44no—
226C2×C4×C23—48no—
227C2×C4×C23—44no—
228C2×C4×C23—44no—
229C2×C4×C23—42no—
230C2×C4×C23—44no—
231C2×C4×C23—44no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—84no—
236C2×C4×C23—82no—
237C2×C4×C23—82no—
238C2×C4×C23—88no—
239C2×C4×C23#9512yes—
240C2×C4×C23C2×C22⋊C411yescharacteristic
241C2×C4×C23C2×C4⋊C411yescharacteristic
242C8×C23—44no—
243C22×#100C2×C2311yescharacteristic
244C22×#99C2×C2311yescharacteristic
245C2×#386C2×C2311yescharacteristic
246#1390C2×C2314yesdirect product factor
247C22×C23—21no—
248C22×C23—21no—
249C22×C23—21no—
250C22×C23—21no—
251C22×C23—24no—
252C22×C23—24no—
253C22×C23—21no—
254C22×C23—22no—
255C22×C23—22no—
256C22×C23—21no—
257C22×C23—21no—
258C22×C23—21no—
259C22×C23—42no—
260C22×C23—42no—
261C22×C23—41no—
262C22×C23—41no—
263C22×C23—41no—
264C22×C23—41no—
265C22×C23—42no—
266C22×C23—42no—
267C22×C23—44no—
268C22×C23—44no—
269C22×C23—41no—
270C22×C23—41no—
271C22×C23—41no—
272C22×C23—41no—
273C22×C23—41no—
274C22×C23—41no—
275C22×C23—41no—
276C22×C23—41no—
277C22×C23—44no—
278C22×C23—44no—
279C22×C23—44no—
280C22×C23—44no—
281C22×C23—81no—
282C22×C23—81no—
283C22×C23—82no—
284C22×C23—82no—
285C22×C23#34212yes—
286C22×C23#34212yes—
287C22×C23AΣL(1,𝔽16)12yes—
288C22×C23#35212yes—
289C22×C23C2×#9511yescharacteristic
290C22×C23C2×#9911yescharacteristic
291C22×C23C2×#9911yescharacteristic
292C4×C23—21no—
293C4×C23—22no—
294C4×C23—21no—
295C4×C23—44no—
296C4×C23—82no—
297C4×C23—84no—
298C23×D4C22×C2311yescharacteristic
299C24×C4C4×C2311yescharacteristic, complemented
300C22×M16—21no—
301C22×C22⋊C4—21no—
302C22×C22⋊C4—21no—
303C22×C22⋊C4C4×C2311yescharacteristic, complemented
304C2×#100—21no—
305C2×#100—22no—
306C2×#100C4×C2314yescomplemented
307C2×#100C22×C2312yes—
308C2×#99—21no—
309C2×#99—22no—
310C2×#99C4×C2314yescomplemented
311C2×#99C22×C2312yes—
312#386—21no—
313#386—21no—
314#386C4×C2312yescomplemented
315#386C22×C2312yes—
316C2×#95—21no—
317C2×C23—21no—
318C2×C23—21no—
319C2×C23—22no—
320C2×C23—41no—
321C2×C23—42no—
322C2×C23—41no—
323C2×C23—41no—
324C2×C23—42no—
325C2×C23—41no—
326C2×C23#139014yesdirect product factor
327C2×C23C2×#34211yescharacteristic
328C2×C23C2×AΣL(1,𝔽16)11yescharacteristic
329C2×C23C2×#35211yescharacteristic
330C25C2×C4×C2311yescharacteristic
331C25D4×C2311yescharacteristic
332C22×D4—41no—
333C22×D4—42no—
334C22×D4—41no—
335C22×D4C2×C4×C2311yescharacteristic
336C22×D4C2×C4×C2312yes—
337C22×D4C2×C4×C2312yes—
338C22×D4D4×C2311yescharacteristic
339C22×D4D4×C2312yes—
340C22×D4D4×C2311yescharacteristic
341C22×D4Q8×C2312yes—
342C22×D4C23×C2311yescharacteristic
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—44no—
347C23×C4—41no—
348C23×C4—41no—
349C23×C4—41no—
350C23×C4C2×C4×C2312yescomplemented
351C23×C4D4×C2311yescharacteristic
352C2×M16—24no—
353C2×M16—22no—
354C2×M16—42no—
355C2×M16—41no—
356C22×C8—41no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—44no—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4C2×C4×C2312yescomplemented
367#100—24no—
368#100—22no—
369#100C2×C4×C2314yescomplemented
370#99—24no—
371#99—22no—
372#99C2×C4×C2314yescomplemented
373#95—22no—
374#95—22no—
375C23C2×#139011yescharacteristic, direct product factor, Sylow
376C24—22no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—21no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C42×C2312yes—
387C24C22⋊C4×C2311yescharacteristic
388C24C22⋊C4×C2312yes—
389C24C22⋊C4×C2311yescharacteristic
390C24C22⋊C4×C2311yescharacteristic
391C24C4⋊C4×C2312yes—
392C24C4⋊C4×C2312yes—
393C24C22×C4×C2311yescharacteristic
394C24C2×D4×C2311yescharacteristic
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C42×C2312yes—
412C2×D4C22⋊C4×C2312yes—
413C2×D4C22⋊C4×C2312yes—
414C2×D4C4⋊C4×C2312yes—
415C2×D4C4⋊C4×C2312yes—
416C2×D4C4⋊C4×C2312yes—
417C2×D4C22×C4×C2311yescharacteristic
418C2×D4C22×C4×C2311yescharacteristic
419C2×D4C2×D4×C2311yescharacteristic
420C2×D4C2×Q8×C2311yescharacteristic
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22⋊C4×C2312yes—
445C22×C4C22⋊C4×C2311yescharacteristic
446C22×C4C4⋊C4×C2312yes—
447C22×C4C2×D4×C2311yescharacteristic
448M16—24no—
449M16—44no—
450M16—42no—
451C2×C8—44no—
452C2×C8—42no—
453C22⋊C4—24no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—41no—
479C23—41no—
480C23—44no—
481C23—41no—
482C23—44no—
483C23—41no—
484C23—42no—
485C23—48no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—44no—
490C23—41no—
491C23—44no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—84no—
498C23—81no—
499C23—81no—
500C23C23×#9512yes—
501C23C23×#9512yes—
502C23C23×#9512yes—
503C23C23×#9912yes—
504C23C23×#9911yescharacteristic
505C23C23×#9911yescharacteristic
506C23C2×C42×C2311yescharacteristic
507C23C2×C22⋊C4×C2311yescharacteristic
508C23C2×C22⋊C4×C2311yescharacteristic
509C23C2×C4⋊C4×C2311yescharacteristic
510C23C2×C4⋊C4×C2311yescharacteristic
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—88no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4C23×#9512yes—
546C2×C4C2×C22⋊C4×C2311yescharacteristic
547C2×C4C2×C4⋊C4×C2311yescharacteristic
548C8—44no—
549C22—24no—
550C22—24no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22C23×#34212yes—
588C22C23×#34212yes—
589C22C23×AΣL(1,𝔽16)12yes—
590C22C23×#35212yes—
591C22C2×C23×#9511yescharacteristic
592C22C2×C23×#9911yescharacteristic
593C22C2×C23×#9911yescharacteristic
594C4—21no—
595C4—21no—
596C4—22no—
597C4—44no—
598C4—82no—
599C4—84no—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—41no—
604C2—41no—
605C2—41no—
606C2—42no—
607C2—41no—
608C2—42no—
609C2C23×#139014yesdirect product factor
610C2C2×C23×#34211yescharacteristic
611C2C2×C23×AΣL(1,𝔽16)11yescharacteristic
612C2C2×C23×#35211yescharacteristic
subgroupG/H
center, socleC23×C23#99
commutatorC23C2×C42×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C23 ◁ C23×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1390C2direct (N × H)
6. C24×C4×C23C4semidirect
10. C22×C22⋊C4×C23C4semidirect
13. C2×C23×#100C4semidirect
17. C2×C23×#99C4semidirect
21. C23×#386C4semidirect
44. C23×C4×C23C2×C4semidirect
60. C2×C22⋊C4×C23C2×C4semidirect
63. C23×#100C2×C4semidirect
66. C23×#99C2×C4semidirect
153. C2×#1390C23direct (N × H)
246. #1390C2×C23direct (N × H)
299. C24×C4C4×C23semidirect
303. C22×C22⋊C4C4×C23semidirect
306. C2×#100C4×C23semidirect
310. C2×#99C4×C23semidirect
314. #386C4×C23semidirect
326. C2×C23#1390direct (N × H)
350. C23×C4C2×C4×C23semidirect
366. C2×C22⋊C4C2×C4×C23semidirect
369. #100C2×C4×C23semidirect
372. #99C2×C4×C23semidirect
375. C23C2×#1390direct (N × H)
609. C2C23×#1390direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1002
C22×C23×#992
C2×C23×#3862
C23×#13902
C2×#139023