Finite Group Databasebeta

← C23×#8401 #19156027

Subgroups — C23×#8401 #19156027

Order 5888.

810 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
1C23×#3459C211yesmaximal, characteristic
2C23×#1955C211yesmaximal, characteristic, complemented
3C4×C23×#100C211yesmaximal, characteristic, complemented
4C23×#1524C212yesmaximal, complemented
5C23×#1483C212yesmaximal, complemented
6C23×#543C411yescharacteristic
7C23×#539—22no—
8C23×#539—21no—
9C23×#538—22no—
10C23×#538C412yes—
11C23×#534—21no—
12C23×#534C411yescharacteristic
13C23×#529—22no—
14C23×#529C2212yes—
15C23×#525—21no—
16C23×#525—22no—
17C2×C4×D4×C23—21no—
18C2×C4×D4×C23C2211yescharacteristic
19C2×C23×C4⋊C8C2211yescharacteristic, complemented
20C2×C23×#100—21no—
21C2×C23×C22⋊C8—21no—
22C2×C23×C22⋊C8—21no—
23C4×M16×C23C2212yescomplemented
24C2×C23×#97C2211yescharacteristic, complemented
25C23×#343—21no—
26C23×#330—22no—
27C23×#329—22no—
28C23×Q8⋊C8—22no—
29C23×D4⋊C8—22no—
30C2×#41×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23D411yescharacteristic
33C23×C4×C23—21no—
34C23×C4×C23D411yescharacteristic
35C23×C4×C23D411yescharacteristic
36C2×M16×C23—22no—
37C22×C8×C23—21no—
38C22×C8×C23—21no—
39C23×#126—22no—
40C23×#126—22no—
41C23×#126—22no—
42C23×#125—22no—
43C23×#125D412yes—
44C23×#124—22no—
45C23×#124D412yes—
46C23×#123—22no—
47C23×#123—22no—
48C23×#123—22no—
49C23×#123—22no—
50C23×#123—22no—
51C23×#123—22no—
52C23×C22⋊Q8—21no—
53C23×C22⋊Q8—22no—
54C23×C22⋊Q8—22no—
55C23×C22⋊Q8—21no—
56C23×C22⋊Q8—22no—
57C23×C22⋊Q8C2×C412yes—
58C23×#121—22no—
59C23×#121—21no—
60C23×#121—22no—
61C23×#121—22no—
62C23×#121—21no—
63C23×#121C2×C412yes—
64C23×#120—22no—
65C23×#120—22no—
66C23×#120—22no—
67C4×Q8×C23D412yes—
68C4×D4×C23—22no—
69C4×D4×C23—22no—
70C4×D4×C23—22no—
71C4×D4×C23—22no—
72C4×D4×C23—22no—
73C4×D4×C23—22no—
74C4×D4×C23—22no—
75C4×D4×C23D412yes—
76C23×#117—21no—
77C23×#117—22no—
78C2×C4⋊C4×C23—21no—
79C2×C4⋊C4×C23D411yescharacteristic
80C2×C22⋊C4×C23—21no—
81C2×C22⋊C4×C23—21no—
82C2×C22⋊C4×C23C2×C411yescharacteristic
83C2×C22⋊C4×C23C2×C411yescharacteristic
84C2×C42×C23—21no—
85C2×C42×C23C2311yescharacteristic
86C23×C4⋊C8—22no—
87C23×#100—24no—
88C23×C22⋊C8—42no—
89C23×C22⋊C8—42no—
90C23×#97—22no—
91C23×#97—22no—
92C4×C8×C23—22no—
93C24×C23—21no—
94C24×C23C22⋊C411yescharacteristic
95C24×C23C22⋊C411yescharacteristic
96#41×C23—81no—
97#41×C23—81no—
98C2×Q8×C23—21no—
99C2×Q8×C23C22⋊C412yes—
100C2×D4×C23—22no—
101C2×D4×C23—22no—
102C2×D4×C23—22no—
103C2×D4×C23—22no—
104C2×D4×C23—21no—
105C2×D4×C23—22no—
106C2×D4×C23—22no—
107C2×D4×C23—41no—
108C2×D4×C23—42no—
109C2×D4×C23—42no—
110C2×D4×C23C22⋊C412yes—
111C22×C4×C23—21no—
112C22×C4×C23—21no—
113C22×C4×C23—21no—
114C22×C4×C23—22no—
115C22×C4×C23—21no—
116C22×C4×C23—21no—
117C22×C4×C23—22no—
118C22×C4×C23—21no—
119C22×C4×C23—21no—
120C22×C4×C23—41no—
121C22×C4×C23—41no—
122C22×C4×C23—41no—
123C22×C4×C23—41no—
124C22×C4×C23—81no—
125C22×C4×C23—81no—
126C22×C4×C23C22⋊C411yescharacteristic
127C22×C4×C23C22⋊C411yescharacteristic
128C22×C4×C23C22×C411yescharacteristic
129C22×C4×C23C2×D411yescharacteristic
130C22×C4×C23C2×D411yescharacteristic
131M16×C23—24no—
132M16×C23—42no—
133C2×C8×C23—22no—
134C2×C8×C23—41no—
135C2×C8×C23—41no—
136C2×C8×C23—42no—
137C2×C8×C23—41no—
138C2×C8×C23—41no—
139C2×C8×C23—41no—
140C4⋊C4×C23—22no—
141C4⋊C4×C23—21no—
142C4⋊C4×C23—21no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—22no—
145C4⋊C4×C23—22no—
146C4⋊C4×C23—22no—
147C4⋊C4×C23—22no—
148C4⋊C4×C23—22no—
149C4⋊C4×C23C22⋊C412yes—
150C4⋊C4×C23C22⋊C412yes—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—22no—
153C22⋊C4×C23—22no—
154C22⋊C4×C23—22no—
155C22⋊C4×C23—22no—
156C22⋊C4×C23—22no—
157C22⋊C4×C23—22no—
158C22⋊C4×C23—22no—
159C22⋊C4×C23—22no—
160C22⋊C4×C23—22no—
161C22⋊C4×C23—22no—
162C22⋊C4×C23—22no—
163C42×C23—22no—
164C42×C23—22no—
165C42×C23C2×D412yes—
166C42×C23C2×D412yes—
167#8401C2311yesmaximal, characteristic, direct product factor, Sylow
168C23×C23—21no—
169C23×C23—21no—
170C23×C23—21no—
171C23×C23—22no—
172C23×C23—21no—
173C23×C23—21no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—41no—
178C23×C23—42no—
179C23×C23—81no—
180C23×C23—81no—
181C23×C23#10412yes—
182C23×C23C2×C22⋊C411yescharacteristic
183Q8×C23—42no—
184Q8×C23—81no—
185D4×C23—44no—
186D4×C23—42no—
187D4×C23—44no—
188D4×C23—42no—
189D4×C23—42no—
190D4×C23—82no—
191D4×C23—81no—
192D4×C23—82no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—21no—
196C2×C4×C23—21no—
197C2×C4×C23—22no—
198C2×C4×C23—22no—
199C2×C4×C23—21no—
200C2×C4×C23—21no—
201C2×C4×C23—22no—
202C2×C4×C23—41no—
203C2×C4×C23—42no—
204C2×C4×C23—41no—
205C2×C4×C23—41no—
206C2×C4×C23—42no—
207C2×C4×C23—41no—
208C2×C4×C23—41no—
209C2×C4×C23—42no—
210C2×C4×C23—41no—
211C2×C4×C23—41no—
212C2×C4×C23—81no—
213C2×C4×C23—81no—
214C2×C4×C23—81no—
215C2×C4×C23—82no—
216C2×C4×C23—81no—
217C2×C4×C23C2×C22⋊C412yes—
218C2×C4×C23#12012yes—
219C2×C4×C23#12011yescharacteristic
220C2×C4×C23#12011yescharacteristic
221C8×C23—42no—
222C8×C23—44no—
223C8×C23—81no—
224#3459C2×C2311yescharacteristic
225#1955C2×C2311yescharacteristic, complemented
226C4×#100C2×C2311yescharacteristic, complemented
227#1524C2×C2312yescomplemented
228#1483C2×C2312yescomplemented
229C22×C23—21no—
230C22×C23—21no—
231C22×C23—22no—
232C22×C23—42no—
233C22×C23—42no—
234C22×C23—41no—
235C22×C23—41no—
236C22×C23—41no—
237C22×C23—42no—
238C22×C23—41no—
239C22×C23—42no—
240C22×C23—42no—
241C22×C23—41no—
242C22×C23—81no—
243C22×C23—81no—
244C22×C23—84no—
245C22×C23#37911yescharacteristic
246C22×C23C2×#10411yescharacteristic
247C22×C23#42111yescharacteristic
248C4×C23—21no—
249C4×C23—22no—
250C4×C23—22no—
251C4×C23—41no—
252C4×C23—41no—
253C4×C23—42no—
254C4×C23—42no—
255C4×C23—42no—
256C4×C23#45311yescharacteristic
257C4×C23#45511yescharacteristic
258#543C4×C2311yescharacteristic
259#539—22no—
260#539—21no—
261#538—22no—
262#538C4×C2312yes—
263#534—21no—
264#534C4×C2311yescharacteristic
265#529—22no—
266#529C22×C2312yes—
267#525—21no—
268#525—22no—
269C2×C4×D4—21no—
270C2×C4×D4C22×C2311yescharacteristic
271C2×C4⋊C8C22×C2311yescharacteristic, complemented
272C2×#100—21no—
273C2×C22⋊C8—21no—
274C2×C22⋊C8—21no—
275C4×M16C22×C2312yescomplemented
276C2×#97C22×C2311yescharacteristic, complemented
277#343—21no—
278#330—22no—
279#329—22no—
280Q8⋊C8—22no—
281D4⋊C8—22no—
282C2×C23—21no—
283C2×C23—21no—
284C2×C23—42no—
285C2×C23—81no—
286C2×C23—82no—
287C2×C23#178911yescharacteristic
288C2×C23#179111yescharacteristic
289C2×C23#179211yescharacteristic
290C2×#41—21no—
291C22×D4—21no—
292C22×D4D4×C2311yescharacteristic
293C23×C4—21no—
294C23×C4D4×C2311yescharacteristic
295C23×C4D4×C2311yescharacteristic
296C2×M16—22no—
297C22×C8—21no—
298C22×C8—21no—
299#126—22no—
300#126—22no—
301#126—22no—
302#125—22no—
303#125D4×C2312yes—
304#124—22no—
305#124D4×C2312yes—
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311#123—22no—
312C22⋊Q8—21no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—21no—
316C22⋊Q8—22no—
317C22⋊Q8C2×C4×C2312yes—
318#121—22no—
319#121—21no—
320#121—22no—
321#121—22no—
322#121—21no—
323#121C2×C4×C2312yes—
324#120—22no—
325#120—22no—
326#120—22no—
327C4×Q8D4×C2312yes—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4D4×C2312yes—
336#117—21no—
337#117—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4D4×C2311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4C2×C4×C2311yescharacteristic
343C2×C22⋊C4C2×C4×C2311yescharacteristic
344C2×C42—21no—
345C2×C42C23×C2311yescharacteristic
346C4⋊C8—22no—
347#100—24no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350#97—22no—
351#97—22no—
352C4×C8—22no—
353C23#840111yescharacteristic, direct product factor, Sylow
354C24—21no—
355C24C22⋊C4×C2311yescharacteristic
356C24C22⋊C4×C2311yescharacteristic
357#41—81no—
358#41—81no—
359C2×Q8—21no—
360C2×Q8C22⋊C4×C2312yes—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4C22⋊C4×C2312yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4C22⋊C4×C2311yescharacteristic
388C22×C4C22⋊C4×C2311yescharacteristic
389C22×C4C22×C4×C2311yescharacteristic
390C22×C4C2×D4×C2311yescharacteristic
391C22×C4C2×D4×C2311yescharacteristic
392M16—24no—
393M16—42no—
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4C22⋊C4×C2312yes—
411C4⋊C4C22⋊C4×C2312yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
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—
424C42—22no—
425C42—22no—
426C42C2×D4×C2312yes—
427C42C2×D4×C2312yes—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—81no—
440C23—81no—
441C23C23×#10412yes—
442C23C2×C22⋊C4×C2311yescharacteristic
443Q8—42no—
444Q8—81no—
445D4—44no—
446D4—42no—
447D4—44no—
448D4—42no—
449D4—42no—
450D4—81no—
451D4—82no—
452D4—82no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4C2×C22⋊C4×C2312yes—
478C2×C4C23×#12012yes—
479C2×C4C23×#12011yescharacteristic
480C2×C4C23×#12011yescharacteristic
481C8—42no—
482C8—44no—
483C8—81no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—84no—
500C22C23×#37911yescharacteristic
501C22C2×C23×#10411yescharacteristic
502C22C23×#42111yescharacteristic
503C4—22no—
504C4—22no—
505C4—21no—
506C4—42no—
507C4—42no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4C23×#45311yescharacteristic
512C4C23×#45511yescharacteristic
513C2—21no—
514C2—21no—
515C2—42no—
516C2—81no—
517C2—82no—
518C2C23×#178911yescharacteristic
519C2C23×#179111yescharacteristic
520C2C23×#179211yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#379
prime radicalC23×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1955C2semidirect
3. C4×C23×#100C2semidirect
4. C23×#1524C2semidirect
5. C23×#1483C2semidirect
19. C2×C23×C4⋊C8C22semidirect
23. C4×M16×C23C22semidirect
24. C2×C23×#97C22semidirect
167. #8401C23direct (N × H)
225. #1955C2×C23semidirect
226. C4×#100C2×C23semidirect
227. #1524C2×C23semidirect
228. #1483C2×C23semidirect
271. C2×C4⋊C8C22×C23semidirect
275. C4×M16C22×C23semidirect
276. C2×#97C22×C23semidirect
353. C23#8401direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34592
C23×#19552
C4×C23×#1002
C23×#15242
C23×#14832
#840123