Finite Group Databasebeta

← C23×#8405 #19156031

Subgroups — C23×#8405 #19156031

Order 5888.

1020 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#543C211yesmaximal, characteristic
2C23×#1872C212yesmaximal, complemented
3C2×C23×#328C212yesmaximal, complemented
4C2×C23×#323C211yesmaximal, characteristic
5C23×#1439C211yesmaximal, characteristic
6C23×#543—24no—
7C23×#543—22no—
8C23×#543C414yescomplemented
9C2×C23×C22⋊Q8—22no—
10C2×C23×C22⋊Q8—21no—
11C2×C23×C22⋊Q8—22no—
12C2×C23×C22⋊Q8C2212yes—
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C22⋊C4×C23—21no—
15C22×C22⋊C4×C23—21no—
16C22×C22⋊C4×C23C2211yescharacteristic
17C2×C23×#103—22no—
18C2×C23×C22⋊C8C2212yes—
19C2×C23×#95—21no—
20C2×C23×#95C2211yescharacteristic
21C23×#340—22no—
22C23×#328—24no—
23C23×#323—22no—
24C25×C23D411yescharacteristic
25C22×Q8×C23—21no—
26C22×Q8×C23D412yes—
27C23×C4×C23—21no—
28C23×C4×C23—21no—
29C23×C4×C23—21no—
30C23×C4×C23—21no—
31C23×C4×C23D411yescharacteristic
32C23×C4×C23C2311yescharacteristic
33C22×C8×C23—22no—
34C23×C22⋊Q8—28no—
35C23×C22⋊Q8—28no—
36C23×C22⋊Q8—24no—
37C23×C22⋊Q8—28no—
38C23×C22⋊Q8—24no—
39C23×C22⋊Q8—22no—
40C23×C22⋊Q8—22no—
41C23×C22⋊Q8—28no—
42C23×C22⋊Q8C2×C414yes—
43C23×C22⋊Q8D414yes—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—22no—
46C2×C4⋊C4×C23—22no—
47C2×C4⋊C4×C23—22no—
48C2×C4⋊C4×C23—42no—
49C2×C4⋊C4×C23D412yes—
50C2×C22⋊C4×C23—22no—
51C2×C22⋊C4×C23—24no—
52C2×C22⋊C4×C23—24no—
53C2×C22⋊C4×C23—22no—
54C2×C22⋊C4×C23—22no—
55C2×C22⋊C4×C23—22no—
56C2×C22⋊C4×C23—42no—
57C2×C22⋊C4×C23—42no—
58C2×C22⋊C4×C23—42no—
59C2×C22⋊C4×C23C2×C412yes—
60C2×C22⋊C4×C23D412yes—
61C23×#103—44no—
62C23×C22⋊C8—22no—
63C23×C22⋊C8—22no—
64C23×#95—22no—
65C23×#95—42no—
66C24×C23—21no—
67C24×C23—21no—
68C24×C23—21no—
69C24×C23—21no—
70C24×C23—41no—
71C24×C23—41no—
72C24×C23—41no—
73C24×C23—41no—
74C24×C23C22⋊C412yes—
75C24×C23SD1612yes—
76C24×C23Q1612yes—
77C24×C23C2×D411yescharacteristic
78C2×Q8×C23—24no—
79C2×Q8×C23—22no—
80C2×Q8×C23—42no—
81C2×Q8×C23—42no—
82C2×Q8×C23—81no—
83C2×Q8×C23—81no—
84C2×Q8×C23C22⋊C414yescomplemented
85C22×C4×C23—22no—
86C22×C4×C23—21no—
87C22×C4×C23—22no—
88C22×C4×C23—24no—
89C22×C4×C23—24no—
90C22×C4×C23—21no—
91C22×C4×C23—22no—
92C22×C4×C23—22no—
93C22×C4×C23—41no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23—41no—
97C22×C4×C23—41no—
98C22×C4×C23—41no—
99C22×C4×C23—41no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—41no—
105C22×C4×C23—41no—
106C22×C4×C23—41no—
107C22×C4×C23—41no—
108C22×C4×C23—41no—
109C22×C4×C23—81no—
110C22×C4×C23—81no—
111C22×C4×C23—81no—
112C22×C4×C23—81no—
113C22×C4×C23C22⋊C412yes—
114C22×C4×C23C22×C411yescharacteristic
115C22×C4×C23C2×D412yes—
116C22×C4×C23C2×D412yes—
117C22×C4×C23C2×D411yescharacteristic
118C2×C8×C23—42no—
119C2×C8×C23—42no—
120C2×C8×C23—42no—
121C4⋊C4×C23—24no—
122C4⋊C4×C23—28no—
123C4⋊C4×C23—24no—
124C4⋊C4×C23—24no—
125C4⋊C4×C23—22no—
126C4⋊C4×C23—24no—
127C4⋊C4×C23—42no—
128C4⋊C4×C23C22⋊C414yescomplemented
129C22⋊C4×C23—28no—
130C22⋊C4×C23—24no—
131C22⋊C4×C23—24no—
132C22⋊C4×C23—24no—
133C22⋊C4×C23—24no—
134C22⋊C4×C23—48no—
135#8405C2311yesmaximal, characteristic, direct product factor, Sylow
136C23×C23—21no—
137C23×C23—21no—
138C23×C23—21no—
139C23×C23—21no—
140C23×C23—22no—
141C23×C23—22no—
142C23×C23—22no—
143C23×C23—22no—
144C23×C23—22no—
145C23×C23—22no—
146C23×C23—42no—
147C23×C23—42no—
148C23×C23—42no—
149C23×C23—42no—
150C23×C23—42no—
151C23×C23—42no—
152C23×C23—41no—
153C23×C23—41no—
154C23×C23—41no—
155C23×C23—41no—
156C23×C23—81no—
157C23×C23—81no—
158C23×C23—81no—
159C23×C23—81no—
160C23×C23—81no—
161C23×C23—81no—
162C23×C23#10314yescomplemented
163C23×C23C2×C22⋊C411yescharacteristic
164C23×C23#12011yescharacteristic
165C23×C23#12011yescharacteristic
166C23×C23C2×SD1611yescharacteristic
167C23×C23C2×Q1611yescharacteristic
168C23×C23#13711yescharacteristic
169C23×C23#13711yescharacteristic
170Q8×C23—44no—
171Q8×C23—82no—
172Q8×C23—84no—
173C2×C4×C23—22no—
174C2×C4×C23—22no—
175C2×C4×C23—24no—
176C2×C4×C23—22no—
177C2×C4×C23—24no—
178C2×C4×C23—41no—
179C2×C4×C23—41no—
180C2×C4×C23—42no—
181C2×C4×C23—41no—
182C2×C4×C23—42no—
183C2×C4×C23—41no—
184C2×C4×C23—42no—
185C2×C4×C23—42no—
186C2×C4×C23—42no—
187C2×C4×C23—42no—
188C2×C4×C23—42no—
189C2×C4×C23—42no—
190C2×C4×C23—42no—
191C2×C4×C23—42no—
192C2×C4×C23—42no—
193C2×C4×C23—42no—
194C2×C4×C23—81no—
195C2×C4×C23—84no—
196C2×C4×C23—84no—
197C2×C4×C23—81no—
198C2×C4×C23—81no—
199C2×C4×C23—81no—
200C2×C4×C23—81no—
201C2×C4×C23—81no—
202C2×C4×C23—81no—
203C2×C4×C23—81no—
204C2×C4×C23—81no—
205C2×C4×C23C2×C22⋊C412yes—
206C2×C4×C23#12012yes—
207C8×C23—82no—
208C2×#543C2×C2311yescharacteristic
209#1872C2×C2312yescomplemented
210C2×#328C2×C2312yescomplemented
211C2×#323C2×C2311yescharacteristic
212#1439C2×C2311yescharacteristic
213C22×C23—21no—
214C22×C23—21no—
215C22×C23—21no—
216C22×C23—21no—
217C22×C23—21no—
218C22×C23—21no—
219C22×C23—21no—
220C22×C23—21no—
221C22×C23—21no—
222C22×C23—21no—
223C22×C23—21no—
224C22×C23—21no—
225C22×C23—24no—
226C22×C23—44no—
227C22×C23—42no—
228C22×C23—42no—
229C22×C23—42no—
230C22×C23—42no—
231C22×C23—42no—
232C22×C23—42no—
233C22×C23—82no—
234C22×C23—82no—
235C22×C23—82no—
236C22×C23#37911yescharacteristic
237C22×C23C2×#10311yescharacteristic
238C22×C23#41911yescharacteristic
239C22×C23#44811yescharacteristic
240C22×C23#44811yescharacteristic
241C22×C23#45111yescharacteristic
242C22×C23#45111yescharacteristic
243C22×C23#45311yescharacteristic
244C22×C23#45511yescharacteristic
245C22×C23B(4,𝔽2)11yescharacteristic
246C22×C23#45811yescharacteristic
247C4×C23—22no—
248C4×C23—22no—
249C4×C23—42no—
250C4×C23—44no—
251C4×C23—44no—
252C4×C23—82no—
253C4×C23—82no—
254#543—24no—
255#543—22no—
256#543C4×C2314yescomplemented
257C2×C22⋊Q8—22no—
258C2×C22⋊Q8—21no—
259C2×C22⋊Q8—22no—
260C2×C22⋊Q8C22×C2312yes—
261C22×C4⋊C4C22×C2311yescharacteristic
262C22×C22⋊C4—21no—
263C22×C22⋊C4—21no—
264C22×C22⋊C4C22×C2311yescharacteristic
265C2×#103—22no—
266C2×C22⋊C8C22×C2312yes—
267C2×#95—21no—
268C2×#95C22×C2311yescharacteristic
269#340—22no—
270#328—24no—
271#323—22no—
272C2×C23—21no—
273C2×C23—21no—
274C2×C23—21no—
275C2×C23—21no—
276C2×C23—44no—
277C2×C23#159611yescharacteristic
278C2×C23#160211yescharacteristic
279C2×C23#160811yescharacteristic
280C2×C23#161411yescharacteristic
281C2×C23#178711yescharacteristic
282C2×C23#179011yescharacteristic
283C2×C23#179211yescharacteristic
284C25D4×C2311yescharacteristic
285C22×Q8—21no—
286C22×Q8D4×C2312yes—
287C23×C4—21no—
288C23×C4—21no—
289C23×C4—21no—
290C23×C4—21no—
291C23×C4D4×C2311yescharacteristic
292C23×C4C23×C2311yescharacteristic
293C22×C8—22no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—24no—
297C22⋊Q8—28no—
298C22⋊Q8—24no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—28no—
302C22⋊Q8C2×C4×C2314yes—
303C22⋊Q8D4×C2314yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4D4×C2312yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4C2×C4×C2312yes—
320C2×C22⋊C4D4×C2312yes—
321#103—44no—
322C22⋊C8—22no—
323C22⋊C8—22no—
324#95—22no—
325#95—42no—
326C23#840511yescharacteristic, direct product factor, Sylow
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—41no—
332C24—41no—
333C24—41no—
334C24—41no—
335C24C22⋊C4×C2312yes—
336C24SD16×C2312yes—
337C24Q16×C2312yes—
338C24C2×D4×C2311yescharacteristic
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8C22⋊C4×C2314yescomplemented
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4C22⋊C4×C2312yes—
375C22×C4C22×C4×C2311yescharacteristic
376C22×C4C2×D4×C2312yes—
377C22×C4C2×D4×C2312yes—
378C22×C4C2×D4×C2311yescharacteristic
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—42no—
389C4⋊C4C22⋊C4×C2314yescomplemented
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—48no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23C23×#10314yescomplemented
423C23C2×C22⋊C4×C2311yescharacteristic
424C23C23×#12011yescharacteristic
425C23C23×#12011yescharacteristic
426C23C2×SD16×C2311yescharacteristic
427C23C2×Q16×C2311yescharacteristic
428C23C23×#13711yescharacteristic
429C23C23×#13711yescharacteristic
430Q8—44no—
431Q8—82no—
432Q8—84no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—81no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4C2×C22⋊C4×C2312yes—
466C2×C4C23×#12012yes—
467C8—82no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—24no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22C23×#37911yescharacteristic
492C22C2×C23×#10311yescharacteristic
493C22C23×#41911yescharacteristic
494C22C23×#44811yescharacteristic
495C22C23×#44811yescharacteristic
496C22C23×#45111yescharacteristic
497C22C23×#45111yescharacteristic
498C22C23×#45311yescharacteristic
499C22C23×#45511yescharacteristic
500C22C23×B(4,𝔽2)11yescharacteristic
501C22C23×#45811yescharacteristic
502C4—22no—
503C4—22no—
504C4—44no—
505C4—42no—
506C4—44no—
507C4—82no—
508C4—82no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—44no—
514C2C23×#159611yescharacteristic
515C2C23×#160211yescharacteristic
516C2C23×#160811yescharacteristic
517C2C23×#161411yescharacteristic
518C2C23×#178711yescharacteristic
519C2C23×#179011yescharacteristic
520C2C23×#179211yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC25×C23D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1872C2semidirect
3. C2×C23×#328C2semidirect
8. C23×#543C4semidirect
84. C2×Q8×C23C22⋊C4semidirect
128. C4⋊C4×C23C22⋊C4semidirect
135. #8405C23direct (N × H)
162. C23×C23#103semidirect
209. #1872C2×C23semidirect
210. C2×#328C2×C23semidirect
256. #543C4×C23semidirect
326. C23#8405direct (N × H)
345. C2×Q8C22⋊C4×C23semidirect
389. C4⋊C4C22⋊C4×C23semidirect
422. C23C23×#103semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5432
C23×#18722
C2×C23×#3282
C2×C23×#3232
C23×#14392
#840523