Finite Group Databasebeta

← C23×#13042 #19160668

Subgroups — C23×#13042 #19160668

Order 5888.

744 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#458C211yesmaximal, characteristic
2C23×#2374C211yesmaximal, characteristic, complemented
3C2×C23×#352C211yesmaximal, characteristic, complemented
4C23×#1792C211yesmaximal, characteristic
5C23×#1790C211yesmaximal, characteristic
6C23×#1393C211yesmaximal, characteristic, complemented
7C23×#1390C211yesmaximal, characteristic, complemented
8C2×C23×H(5,𝔽2)C2211yescharacteristic
9C2×C23×#123—21no—
10C2×C23×#123C2211yescharacteristic
11C23×#458—22no—
12C23×#458C414yescomplemented
13C2×C23×#104—21no—
14C23×#417—21no—
15C2×C23×#100C2211yescharacteristic
16C2×C23×#99—21no—
17C2×C23×#99—21no—
18C2×C23×#99C2211yescharacteristic
19C2×C23×#99C2211yescharacteristic
20C23×#390C2211yescharacteristic
21C23×#388—21no—
22C23×#388—21no—
23C23×#387—21no—
24C23×#386—21no—
25C23×#386C2211yescharacteristic
26C4×C4⋊C4×C23—21no—
27C4×C22⋊C4×C23—21no—
28C4×C22⋊C4×C23—21no—
29C23×#352—22no—
30C23×H(5,𝔽2)—21no—
31C23×H(5,𝔽2)—21no—
32C23×H(5,𝔽2)—21no—
33C23×H(5,𝔽2)—42no—
34C23×H(5,𝔽2)C2×C412yes—
35C2×#41×C23—21no—
36C2×#41×C23—41no—
37C22×D4×C23—21no—
38C22×D4×C23—21no—
39C22×D4×C23—41no—
40C22×D4×C23C2311yescharacteristic
41C23×C4×C23D411yescharacteristic
42C23×C4×C23D411yescharacteristic
43C2×M16×C23—21no—
44C23×#123—21no—
45C23×#123—21no—
46C23×#123—24no—
47C23×#123—42no—
48C23×#123C2×C412yes—
49C23×#123D412yes—
50C2×C4⋊C4×C23—21no—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23—41no—
53C2×C4⋊C4×C23D411yescharacteristic
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—41no—
61C2×C22⋊C4×C23—41no—
62C2×C22⋊C4×C23D411yescharacteristic
63C2×C22⋊C4×C23D411yescharacteristic
64C2×C42×C23—21no—
65C2×C42×C23—21no—
66C2×C42×C23—21no—
67C2×C42×C23D411yescharacteristic
68C23×#104—42no—
69C23×#103—41no—
70C23×#102—41no—
71C23×#100—21no—
72C23×#100—21no—
73C23×#99—21no—
74C23×#99—24no—
75C23×#99—21no—
76C23×#99—21no—
77C23×#99—42no—
78C23×#99C2×C412yes—
79C23×#95—21no—
80C23×#95—21no—
81C23×#95—21no—
82C23×#95—21no—
83C23×#95—21no—
84C23×#95—41no—
85C24×C23—21no—
86C24×C23—21no—
87C24×C23C2×D411yescharacteristic
88C24×C23C2×D411yescharacteristic
89#41×C23—22no—
90#41×C23—22no—
91#41×C23—42no—
92#41×C23—41no—
93#41×C23—42no—
94#41×C23—42no—
95#41×C23—42no—
96#41×C23—41no—
97C2×Q8×C23—21no—
98C2×D4×C23—21no—
99C2×D4×C23—22no—
100C2×D4×C23—22no—
101C2×D4×C23—22no—
102C2×D4×C23—22no—
103C2×D4×C23—41no—
104C2×D4×C23—41no—
105C2×D4×C23—42no—
106C2×D4×C23—42no—
107C2×D4×C23—41no—
108C2×D4×C23—41no—
109C2×D4×C23—41no—
110C2×D4×C23—41no—
111C2×D4×C23—42no—
112C2×D4×C23—42no—
113C2×D4×C23—41no—
114C2×D4×C23—41no—
115C2×D4×C23—41no—
116C2×D4×C23C22×C411yescharacteristic
117C2×D4×C23C2×D411yescharacteristic
118C2×D4×C23#4111yescharacteristic
119C2×D4×C23#4111yescharacteristic
120C22×C4×C23—21no—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—21no—
124C22×C4×C23—21no—
125C22×C4×C23—21no—
126C22×C4×C23—21no—
127C22×C4×C23—21no—
128C22×C4×C23—21no—
129C22×C4×C23—41no—
130C22×C4×C23—42no—
131C22×C4×C23—41no—
132C22×C4×C23—41no—
133C22×C4×C23—41no—
134C22×C4×C23—41no—
135C22×C4×C23—41no—
136C22×C4×C23C22⋊C412yescomplemented
137C22×C4×C23C2×D411yescharacteristic
138M16×C23—41no—
139M16×C23—41no—
140C2×C8×C23—41no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—41no—
144C4⋊C4×C23—42no—
145C4⋊C4×C23—44no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C22⋊C4×C23—21no—
149C22⋊C4×C23—22no—
150C22⋊C4×C23—42no—
151C22⋊C4×C23—42no—
152C22⋊C4×C23—42no—
153C22⋊C4×C23—41no—
154C22⋊C4×C23—44no—
155C22⋊C4×C23—41no—
156C22⋊C4×C23—41no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23—82no—
159C22⋊C4×C23C22⋊C412yescomplemented
160C42×C23—41no—
161C42×C23—42no—
162C42×C23—81no—
163C42×C23—81no—
164#13042C2311yesmaximal, characteristic, direct product factor, Sylow
165C23×C23—21no—
166C23×C23—21no—
167C23×C23—21no—
168C23×C23—21no—
169C23×C23—22no—
170C23×C23—22no—
171C23×C23—41no—
172C23×C23—41no—
173C23×C23—41no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—41no—
178C23×C23—81no—
179C23×C23—81no—
180C23×C23—81no—
181C23×C23—81no—
182C23×C23C2×C22⋊C411yescharacteristic
183C23×C23C4×D411yescharacteristic
184C23×C23#12011yescharacteristic
185C23×C23#12111yescharacteristic
186C23×C23#12311yescharacteristic
187Q8×C23—22no—
188Q8×C23—41no—
189D4×C23—22no—
190D4×C23—41no—
191D4×C23—41no—
192D4×C23—41no—
193D4×C23—41no—
194D4×C23—41no—
195D4×C23—42no—
196D4×C23—42no—
197D4×C23—42no—
198D4×C23—42no—
199D4×C23—42no—
200D4×C23—42no—
201C2×C4×C23—21no—
202C2×C4×C23—21no—
203C2×C4×C23—22no—
204C2×C4×C23—22no—
205C2×C4×C23—22no—
206C2×C4×C23—22no—
207C2×C4×C23—22no—
208C2×C4×C23—22no—
209C2×C4×C23—42no—
210C2×C4×C23—42no—
211C2×C4×C23—41no—
212C2×C4×C23—41no—
213C2×C4×C23—41no—
214C2×C4×C23—41no—
215C2×C4×C23—41no—
216C2×C4×C23—41no—
217C2×C4×C23—41no—
218C2×C4×C23—41no—
219C2×C4×C23—41no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—42no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—41no—
226C2×C4×C23—41no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—41no—
230C2×C4×C23—41no—
231C2×C4×C23—81no—
232C2×C4×C23—81no—
233C2×C4×C23—81no—
234C2×C4×C23—81no—
235C2×C4×C23—81no—
236C2×C4×C23—81no—
237C2×C4×C23—81no—
238C2×C4×C23C4×D411yescharacteristic
239C2×C4×C23#12111yescharacteristic
240C8×C23—81no—
241C2×#458C2×C2311yescharacteristic
242#2374C2×C2311yescharacteristic, complemented
243C2×#352C2×C2311yescharacteristic, complemented
244#1792C2×C2311yescharacteristic
245#1790C2×C2311yescharacteristic
246#1393C2×C2311yescharacteristic, complemented
247#1390C2×C2311yescharacteristic, complemented
248C22×C23—21no—
249C22×C23—21no—
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—21no—
254C22×C23—41no—
255C22×C23—41no—
256C22×C23—41no—
257C22×C23—41no—
258C22×C23—41no—
259C22×C23—41no—
260C22×C23—41no—
261C22×C23—41no—
262C22×C23—42no—
263C22×C23—81no—
264C22×C23—81no—
265C22×C23—82no—
266C22×C23—82no—
267C22×C23—81no—
268C22×C23—81no—
269C22×C23—161no—
270C22×C23—161no—
271C22×C23#38611yescharacteristic
272C22×C23B(4,𝔽2)11yescharacteristic
273C22×C23B(4,𝔽2)11yescharacteristic
274C4×C23—21no—
275C4×C23—21no—
276C4×C23—42no—
277C4×C23—42no—
278C4×C23—42no—
279C4×C23—42no—
280C4×C23—42no—
281C4×C23—81no—
282C4×C23—82no—
283C4×C23—81no—
284C4×C23—161no—
285C2×H(5,𝔽2)C22×C2311yescharacteristic
286C2×#123—21no—
287C2×#123C22×C2311yescharacteristic
288#458—22no—
289#458C4×C2314yescomplemented
290C2×#104—21no—
291#417—21no—
292C2×#100C22×C2311yescharacteristic
293C2×#99—21no—
294C2×#99—21no—
295C2×#99C22×C2311yescharacteristic
296C2×#99C22×C2311yescharacteristic
297#390C22×C2311yescharacteristic
298#388—21no—
299#388—21no—
300#387—21no—
301#386—21no—
302#386C22×C2311yescharacteristic
303C4×C4⋊C4—21no—
304C4×C22⋊C4—21no—
305C4×C22⋊C4—21no—
306#352—22no—
307C2×C23—21no—
308C2×C23—21no—
309C2×C23—41no—
310C2×C23—41no—
311C2×C23—41no—
312C2×C23—41no—
313C2×C23—82no—
314C2×C23#188911yescharacteristic
315C2×C23#219711yescharacteristic
316C2×C23#219811yescharacteristic
317H(5,𝔽2)—21no—
318H(5,𝔽2)—21no—
319H(5,𝔽2)—21no—
320H(5,𝔽2)—42no—
321H(5,𝔽2)C2×C4×C2312yes—
322C2×#41—21no—
323C2×#41—41no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—41no—
327C22×D4C23×C2311yescharacteristic
328C23×C4D4×C2311yescharacteristic
329C23×C4D4×C2311yescharacteristic
330C2×M16—21no—
331#123—21no—
332#123—21no—
333#123—24no—
334#123—42no—
335#123C2×C4×C2312yes—
336#123D4×C2312yes—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4D4×C2311yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—41no—
348C2×C22⋊C4—41no—
349C2×C22⋊C4D4×C2311yescharacteristic
350C2×C22⋊C4D4×C2311yescharacteristic
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42D4×C2311yescharacteristic
355#104—42no—
356#103—41no—
357#102—41no—
358#100—21no—
359#100—21no—
360#99—21no—
361#99—24no—
362#99—21no—
363#99—21no—
364#99—42no—
365#99C2×C4×C2312yes—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—41no—
372C23#1304211yescharacteristic, direct product factor, Sylow
373C24—21no—
374C24—21no—
375C24C2×D4×C2311yescharacteristic
376C24C2×D4×C2311yescharacteristic
377#41—22no—
378#41—22no—
379#41—42no—
380#41—41no—
381#41—42no—
382#41—42no—
383#41—42no—
384#41—41no—
385C2×Q8—21no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4C22×C4×C2311yescharacteristic
405C2×D4C2×D4×C2311yescharacteristic
406C2×D4#41×C2311yescharacteristic
407C2×D4#41×C2311yescharacteristic
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4C22⋊C4×C2312yescomplemented
425C22×C4C2×D4×C2311yescharacteristic
426M16—41no—
427M16—41no—
428C2×C8—41no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—21no—
437C22⋊C4—22no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—44no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—82no—
447C22⋊C4C22⋊C4×C2312yescomplemented
448C42—41no—
449C42—42no—
450C42—81no—
451C42—81no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23C2×C22⋊C4×C2311yescharacteristic
470C23C4×D4×C2311yescharacteristic
471C23C23×#12011yescharacteristic
472C23C23×#12111yescharacteristic
473C23C23×#12311yescharacteristic
474Q8—22no—
475Q8—41no—
476D4—22no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4C4×D4×C2311yescharacteristic
526C2×C4C23×#12111yescharacteristic
527C8—81no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—82no—
546C22—82no—
547C22—81no—
548C22—81no—
549C22—161no—
550C22—161no—
551C22C23×#38611yescharacteristic
552C22C23×B(4,𝔽2)11yescharacteristic
553C22C23×B(4,𝔽2)11yescharacteristic
554C4—21no—
555C4—21no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—81no—
562C4—82no—
563C4—81no—
564C4—161no—
565C2—21no—
566C2—21no—
567C2—41no—
568C2—41no—
569C2—41no—
570C2—41no—
571C2—82no—
572C2C23×#188911yescharacteristic
573C2C23×#219711yescharacteristic
574C2C23×#219811yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×H(5,𝔽2)C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2374C2semidirect
3. C2×C23×#352C2semidirect
6. C23×#1393C2semidirect
7. C23×#1390C2semidirect
12. C23×#458C4semidirect
136. C22×C4×C23C22⋊C4semidirect
159. C22⋊C4×C23C22⋊C4semidirect
164. #13042C23direct (N × H)
242. #2374C2×C23semidirect
243. C2×#352C2×C23semidirect
246. #1393C2×C23semidirect
247. #1390C2×C23semidirect
289. #458C4×C23semidirect
372. C23#13042direct (N × H)
424. C22×C4C22⋊C4×C23semidirect
447. C22⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#4582
C23×#23742
C2×C23×#3522
C23×#17922
C23×#17902
C23×#13932
C23×#13902
#1304223