Finite Group Databasebeta

← C23×#12770 #19160396

Subgroups — C23×#12770 #19160396

Order 5888.

752 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3551C211yesmaximal, characteristic
2C23×#2132C211yesmaximal, characteristic, complemented
3C23×#2123C211yesmaximal, characteristic, complemented
4C23×#2122C211yesmaximal, characteristic, complemented
5C23×#1915C211yesmaximal, characteristic, complemented
6C23×#1914C211yesmaximal, characteristic, complemented
7C23×#1884C211yesmaximal, characteristic, complemented
8C23×#568C2211yescharacteristic
9C23×#563—21no—
10C23×#560C412yescomplemented
11C23×#555—21no—
12C23×#553—21no—
13C23×#552—21no—
14C23×#551C412yescomplemented
15C23×#548—21no—
16C23×#546—21no—
17C23×#540—22no—
18C23×#538—22no—
19C23×#537—21no—
20C23×#536C2211yescharacteristic
21C23×#535C2211yescharacteristic
22C23×#532C2211yescharacteristic
23C23×#529—21no—
24C23×#421—21no—
25C23×#413C2211yescharacteristic
26C23×#413C2211yescharacteristic
27C2×C23×#101—21no—
28C2×C23×#100—21no—
29C23×#410C2211yescharacteristic
30C23×#356—21no—
31C23×#355—21no—
32C23×#352—22no—
33C23×AΣL(1,𝔽16)—22no—
34C23×#344—21no—
35C23×#344—21no—
36C23×#339—21no—
37C2×#41×C23—21no—
38C2×#41×C23D411yescharacteristic
39C2×#41×C23C2311yescharacteristic
40C22×Q8×C23D411yescharacteristic
41C22×D4×C23D411yescharacteristic
42C23×#131—42no—
43C2×M16×C23—21no—
44C2×M16×C23—21no—
45C2×M16×C23—21no—
46C2×M16×C23—21no—
47C2×M16×C23—21no—
48C23×C4⋊Q8—21no—
49C23×C4⋊Q8D411yescharacteristic
50C23×#127—21no—
51C23×#127C2×C411yescharacteristic
52C23×#126—22no—
53C23×#126—22no—
54C23×#126—22no—
55C23×#125—21no—
56C23×#125—22no—
57C23×#124—22no—
58C23×#124—21no—
59C23×#124—21no—
60C23×#124—21no—
61C23×#124C2×C411yescharacteristic
62C23×#124D411yescharacteristic
63C23×#123—22no—
64C23×#123—22no—
65C23×#123—22no—
66C23×#123—22no—
67C23×#123C2×C412yes—
68C23×#123D412yes—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8—22no—
71C23×C22⋊Q8—22no—
72C23×C22⋊Q8—21no—
73C23×C22⋊Q8—21no—
74C23×#121—21no—
75C23×#121—22no—
76C23×#121—22no—
77C23×#121—22no—
78C23×#121—22no—
79C23×#121—21no—
80C23×#121—21no—
81C23×#121—22no—
82C23×#121—21no—
83C23×#121—21no—
84C23×#121—22no—
85C23×#120—22no—
86C23×#120C2×C412yes—
87C23×#120D412yes—
88C4×Q8×C23—21no—
89C4×D4×C23—22no—
90C4×D4×C23—21no—
91C4×D4×C23—22no—
92C4×D4×C23—22no—
93C4×D4×C23—21no—
94C4×D4×C23—21no—
95C23×#117—22no—
96C23×#117—22no—
97C23×#117—21no—
98C23×#117D411yescharacteristic
99C23×#117D411yescharacteristic
100C2×C4⋊C4×C23—21no—
101C2×C22⋊C4×C23—22no—
102C2×C42×C23D411yescharacteristic
103C23×#104—42no—
104C23×#101—21no—
105C23×#101—21no—
106C23×#101—42no—
107C23×#100—21no—
108C23×#100—21no—
109C23×#100—42no—
110C23×#99—21no—
111C23×#99—21no—
112C24×C23C22⋊C412yes—
113#41×C23—41no—
114#41×C23—41no—
115#41×C23—41no—
116#41×C23—41no—
117#41×C23—81no—
118#41×C23—81no—
119C2×Q8×C23—21no—
120C2×Q8×C23—21no—
121C2×Q8×C23—41no—
122C2×Q8×C23—41no—
123C2×Q8×C23C22⋊C411yescharacteristic
124C2×Q8×C23C2×D411yescharacteristic
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—21no—
128C2×D4×C23—22no—
129C2×D4×C23—21no—
130C2×D4×C23—22no—
131C2×D4×C23—21no—
132C2×D4×C23—21no—
133C2×D4×C23—21no—
134C2×D4×C23—41no—
135C2×D4×C23—41no—
136C2×D4×C23C22⋊C411yescharacteristic
137C2×D4×C23C22×C411yescharacteristic
138C2×D4×C23C2×D411yescharacteristic
139C2×D4×C23C2×D411yescharacteristic
140C22×C4×C23—22no—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—21no—
144C22×C4×C23—21no—
145C22×C4×C23C22⋊C412yes—
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D411yescharacteristic
148C22×C4×C23C2×D411yescharacteristic
149M16×C23—22no—
150M16×C23—22no—
151M16×C23—22no—
152M16×C23—41no—
153M16×C23—41no—
154M16×C23—41no—
155M16×C23—41no—
156M16×C23—41no—
157C2×C8×C23—41no—
158C2×C8×C23—41no—
159C2×C8×C23—41no—
160C2×C8×C23—41no—
161C2×C8×C23—41no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—21no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—41no—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—22no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—22no—
179C22⋊C4×C23—21no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—21no—
182C22⋊C4×C23—22no—
183C22⋊C4×C23—22no—
184C22⋊C4×C23—21no—
185C22⋊C4×C23—21no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—41no—
188C22⋊C4×C23C22⋊C412yes—
189C22⋊C4×C23C22⋊C412yes—
190C42×C23—22no—
191C42×C23—21no—
192C42×C23—21no—
193C42×C23—21no—
194C42×C23—41no—
195C42×C23C22⋊C411yescharacteristic
196C42×C23C22⋊C411yescharacteristic
197#12770C2311yesmaximal, characteristic, direct product factor, Sylow
198C23×C23—22no—
199C23×C23—21no—
200C23×C23—42no—
201C23×C23—82no—
202C23×C23C2×C22⋊C411yescharacteristic
203C23×C23C2×C22⋊C411yescharacteristic
204C23×C23#12011yescharacteristic
205Q8×C23—41no—
206Q8×C23—41no—
207Q8×C23—41no—
208Q8×C23—41no—
209Q8×C23—81no—
210D4×C23—41no—
211D4×C23—41no—
212D4×C23—41no—
213D4×C23—41no—
214D4×C23—41no—
215D4×C23—41no—
216D4×C23—81no—
217D4×C23—81no—
218D4×C23—82no—
219D4×C23—82no—
220D4×C23—82no—
221D4×C23—81no—
222D4×C23—81no—
223D4×C23—82no—
224C2×C4×C23—21no—
225C2×C4×C23—21no—
226C2×C4×C23—21no—
227C2×C4×C23—21no—
228C2×C4×C23—21no—
229C2×C4×C23—21no—
230C2×C4×C23—21no—
231C2×C4×C23—22no—
232C2×C4×C23—21no—
233C2×C4×C23—22no—
234C2×C4×C23—22no—
235C2×C4×C23—21no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—42no—
239C2×C4×C23—82no—
240C2×C4×C23—81no—
241C2×C4×C23—82no—
242C2×C4×C23—82no—
243C2×C4×C23—81no—
244C2×C4×C23C2×C22⋊C411yescharacteristic
245C2×C4×C23#12011yescharacteristic
246C2×C4×C23#12011yescharacteristic
247C2×C4×C23#12011yescharacteristic
248C8×C23—42no—
249C8×C23—81no—
250C8×C23—81no—
251#3551C2×C2311yescharacteristic
252#2132C2×C2311yescharacteristic, complemented
253#2123C2×C2311yescharacteristic, complemented
254#2122C2×C2311yescharacteristic, complemented
255#1915C2×C2311yescharacteristic, complemented
256#1914C2×C2311yescharacteristic, complemented
257#1884C2×C2311yescharacteristic, complemented
258C22×C23—21no—
259C22×C23—21no—
260C22×C23—21no—
261C22×C23—41no—
262C22×C23—41no—
263C22×C23—41no—
264C22×C23—41no—
265C22×C23—42no—
266C22×C23—41no—
267C22×C23—42no—
268C22×C23—41no—
269C22×C23—42no—
270C22×C23—162no—
271C22×C23#37911yescharacteristic
272C4×C23—21no—
273C4×C23—21no—
274C4×C23—21no—
275C4×C23—41no—
276C4×C23—42no—
277C4×C23—42no—
278C4×C23—41no—
279C4×C23—41no—
280C4×C23—42no—
281C4×C23—161no—
282C4×C23B(4,𝔽2)11yescharacteristic
283C4×C23#45811yescharacteristic
284#568C22×C2311yescharacteristic
285#563—21no—
286#560C4×C2312yescomplemented
287#555—21no—
288#553—21no—
289#552—21no—
290#551C4×C2312yescomplemented
291#548—21no—
292#546—21no—
293#540—22no—
294#538—22no—
295#537—21no—
296#536C22×C2311yescharacteristic
297#535C22×C2311yescharacteristic
298#532C22×C2311yescharacteristic
299#529—21no—
300#421—21no—
301#413C22×C2311yescharacteristic
302#413C22×C2311yescharacteristic
303C2×#101—21no—
304C2×#100—21no—
305#410C22×C2311yescharacteristic
306#356—21no—
307#355—21no—
308#352—22no—
309AΣL(1,𝔽16)—22no—
310#344—21no—
311#344—21no—
312#339—21no—
313C2×C23—21no—
314C2×C23—41no—
315C2×C23—41no—
316C2×C23—41no—
317C2×C23—81no—
318C2×C23—82no—
319C2×C23#179011yescharacteristic
320C2×#41—21no—
321C2×#41D4×C2311yescharacteristic
322C2×#41C23×C2311yescharacteristic
323C22×Q8D4×C2311yescharacteristic
324C22×D4D4×C2311yescharacteristic
325#131—42no—
326C2×M16—21no—
327C2×M16—21no—
328C2×M16—21no—
329C2×M16—21no—
330C2×M16—21no—
331C4⋊Q8—21no—
332C4⋊Q8D4×C2311yescharacteristic
333#127—21no—
334#127C2×C4×C2311yescharacteristic
335#126—22no—
336#126—22no—
337#126—22no—
338#125—21no—
339#125—22no—
340#124—21no—
341#124—22no—
342#124—21no—
343#124—21no—
344#124C2×C4×C2311yescharacteristic
345#124D4×C2311yescharacteristic
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350#123C2×C4×C2312yes—
351#123D4×C2312yes—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357#121—22no—
358#121—22no—
359#121—21no—
360#121—22no—
361#121—22no—
362#121—21no—
363#121—21no—
364#121—22no—
365#121—21no—
366#121—21no—
367#121—22no—
368#120—22no—
369#120C2×C4×C2312yes—
370#120D4×C2312yes—
371C4×Q8—21no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—21no—
377C4×D4—21no—
378#117—22no—
379#117—22no—
380#117—21no—
381#117D4×C2311yescharacteristic
382#117D4×C2311yescharacteristic
383C2×C4⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C42D4×C2311yescharacteristic
386#104—42no—
387#101—21no—
388#101—21no—
389#101—42no—
390#100—21no—
391#100—21no—
392#100—42no—
393#99—21no—
394#99—21no—
395C23#1277011yescharacteristic, direct product factor, Sylow
396C24C22⋊C4×C2312yes—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—81no—
402#41—81no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—41no—
406C2×Q8—41no—
407C2×Q8C22⋊C4×C2311yescharacteristic
408C2×Q8C2×D4×C2311yescharacteristic
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4C22⋊C4×C2311yescharacteristic
421C2×D4C22×C4×C2311yescharacteristic
422C2×D4C2×D4×C2311yescharacteristic
423C2×D4C2×D4×C2311yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4C22⋊C4×C2312yes—
430C22×C4C2×D4×C2311yescharacteristic
431C22×C4C2×D4×C2311yescharacteristic
432C22×C4C2×D4×C2311yescharacteristic
433M16—22no—
434M16—22no—
435M16—22no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—21no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4C22⋊C4×C2312yes—
473C22⋊C4C22⋊C4×C2312yes—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42C22⋊C4×C2311yescharacteristic
480C42C22⋊C4×C2311yescharacteristic
481C23—22no—
482C23—21no—
483C23—42no—
484C23—82no—
485C23C2×C22⋊C4×C2311yescharacteristic
486C23C2×C22⋊C4×C2311yescharacteristic
487C23C23×#12011yescharacteristic
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—81no—
500D4—81no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—81no—
506D4—81no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4C2×C22⋊C4×C2311yescharacteristic
528C2×C4C23×#12011yescharacteristic
529C2×C4C23×#12011yescharacteristic
530C2×C4C23×#12011yescharacteristic
531C8—42no—
532C8—81no—
533C8—81no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—42no—
546C22—162no—
547C22C23×#37911yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—161no—
558C4C23×B(4,𝔽2)11yescharacteristic
559C4C23×#45811yescharacteristic
560C2—21no—
561C2—41no—
562C2—41no—
563C2—41no—
564C2—82no—
565C2—81no—
566C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×D4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3551C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2132C2semidirect
3. C23×#2123C2semidirect
4. C23×#2122C2semidirect
5. C23×#1915C2semidirect
6. C23×#1914C2semidirect
7. C23×#1884C2semidirect
10. C23×#560C4semidirect
14. C23×#551C4semidirect
197. #12770C23direct (N × H)
252. #2132C2×C23semidirect
253. #2123C2×C23semidirect
254. #2122C2×C23semidirect
255. #1915C2×C23semidirect
256. #1914C2×C23semidirect
257. #1884C2×C23semidirect
286. #560C4×C23semidirect
290. #551C4×C23semidirect
395. C23#12770direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35512
C23×#21322
C23×#21232
C23×#21222
C23×#19152
C23×#19142
C23×#18842
#1277023