Finite Group Databasebeta

← C23×#12705 #19160331

Subgroups — C23×#12705 #19160331

Order 5888.

1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3484C211yesmaximal, characteristic
2C23×#2121C214yesmaximal, complemented
3C23×#1783C211yesmaximal, characteristic, complemented
4C23×#1781C211yesmaximal, characteristic, complemented
5C2×C23×H(5,𝔽2)C411yescharacteristic, complemented
6C23×D4×C23C2211yescharacteristic
7C23×#546—24no—
8D42×C23—24no—
9C23×#535—21no—
10C23×#534—21no—
11C23×#534C2212yes—
12C23×#525—21no—
13C2×C23×#120—21no—
14C2×C23×#120C412yescomplemented
15C23×#518C411yescharacteristic, complemented
16C2×C23×#100—21no—
17C2×C23×#100C2212yescomplemented
18C2×C23×#99—21no—
19C2×C23×#99C2212yescomplemented
20C23×#407—21no—
21C23×#379—21no—
22C23×#352—24no—
23C23×AΣL(1,𝔽16)—24no—
24C25×C23C2×C411yescharacteristic
25C25×C23D411yescharacteristic
26C23×H(5,𝔽2)—42no—
27C23×H(5,𝔽2)—42no—
28C2×#41×C23—21no—
29C2×#41×C23—22no—
30C22×D4×C23—21no—
31C22×D4×C23—22no—
32C22×D4×C23—22no—
33C22×D4×C23—22no—
34C22×D4×C23—24no—
35C22×D4×C23—42no—
36C22×D4×C23—42no—
37C22×D4×C23C2×C411yescharacteristic
38C22×D4×C23C2×C412yes—
39C22×D4×C23D411yescharacteristic
40C22×D4×C23D411yescharacteristic
41C22×D4×C23C2311yescharacteristic
42C23×C4×C23D411yescharacteristic
43C2×M16×C23—22no—
44C23×#127—22no—
45C23×#124—22no—
46C23×#123—22no—
47C23×#123—22no—
48C23×#123D414yes—
49C23×C22⋊Q8—22no—
50C23×#121—22no—
51C23×#121—24no—
52C23×#121—24no—
53C23×#121—24no—
54C23×#120—24no—
55C23×#120—22no—
56C23×#120—22no—
57C23×#120—22no—
58C23×#120—22no—
59C23×#120—24no—
60C23×#120D414yes—
61C4×D4×C23—24no—
62C4×D4×C23—24no—
63C23×#117—21no—
64C2×C22⋊C4×C23—22no—
65C2×C22⋊C4×C23—21no—
66C2×C22⋊C4×C23—22no—
67C2×C22⋊C4×C23—22no—
68C2×C22⋊C4×C23—42no—
69C2×C22⋊C4×C23C2×C412yes—
70C23×#100—24no—
71C23×#100—42no—
72C23×#99—24no—
73C23×#99—42no—
74C23×C22⋊C8—42no—
75C24×C23—21no—
76C24×C23—22no—
77C24×C23—22no—
78C24×C23—22no—
79C24×C23—41no—
80C24×C23—41no—
81C24×C23—41no—
82C24×C23—82no—
83C24×C23—82no—
84C24×C23C22⋊C411yescharacteristic
85C24×C23C22⋊C411yescharacteristic
86C24×C23C22⋊C411yescharacteristic
87C24×C23C22⋊C412yescomplemented
88C24×C23C22×C411yescharacteristic
89C24×C23C2×D411yescharacteristic
90#41×C23—42no—
91#41×C23—48no—
92#41×C23—42no—
93C2×Q8×C23—21no—
94C2×D4×C23—22no—
95C2×D4×C23—22no—
96C2×D4×C23—24no—
97C2×D4×C23—22no—
98C2×D4×C23—21no—
99C2×D4×C23—22no—
100C2×D4×C23—22no—
101C2×D4×C23—22no—
102C2×D4×C23—22no—
103C2×D4×C23—24no—
104C2×D4×C23—24no—
105C2×D4×C23—22no—
106C2×D4×C23—22no—
107C2×D4×C23—22no—
108C2×D4×C23—41no—
109C2×D4×C23—41no—
110C2×D4×C23—41no—
111C2×D4×C23—41no—
112C2×D4×C23—416no—
113C2×D4×C23—48no—
114C2×D4×C23—42no—
115C2×D4×C23—42no—
116C2×D4×C23—42no—
117C2×D4×C23—42no—
118C2×D4×C23—84no—
119C2×D4×C23—84no—
120C2×D4×C23C2×D414yes—
121C22×C4×C23—21no—
122C22×C4×C23—22no—
123C22×C4×C23—22no—
124C22×C4×C23—22no—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—41no—
128C22×C4×C23—42no—
129C22×C4×C23C22⋊C411yescharacteristic
130C22×C4×C23C22⋊C412yescomplemented
131C22×C4×C23C2×D411yescharacteristic
132M16×C23—42no—
133M16×C23—42no—
134C2×C8×C23—42no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23—24no—
137C22⋊C4×C23—24no—
138C22⋊C4×C23—22no—
139C22⋊C4×C23—24no—
140C22⋊C4×C23—22no—
141C22⋊C4×C23—24no—
142C22⋊C4×C23—24no—
143C22⋊C4×C23—42no—
144C22⋊C4×C23—42no—
145C22⋊C4×C23—84no—
146C22⋊C4×C23C22⋊C414yescomplemented
147C42×C23—22no—
148#12705C2311yesmaximal, characteristic, direct product factor, Sylow
149C23×C23—21no—
150C23×C23—21no—
151C23×C23—21no—
152C23×C23—21no—
153C23×C23—21no—
154C23×C23—21no—
155C23×C23—21no—
156C23×C23—21no—
157C23×C23—21no—
158C23×C23—21no—
159C23×C23—22no—
160C23×C23—21no—
161C23×C23—21no—
162C23×C23—42no—
163C23×C23—44no—
164C23×C23—44no—
165C23×C23—44no—
166C23×C23—44no—
167C23×C23—44no—
168C23×C23—42no—
169C23×C23—42no—
170C23×C23—82no—
171C23×C23—82no—
172C23×C23—82no—
173C23×C23—81no—
174C23×C23—81no—
175C23×C23—81no—
176C23×C23—81no—
177C23×C23—82no—
178C23×C23—82no—
179C23×C23—82no—
180C23×C23—82no—
181C23×C23—82no—
182C23×C23—82no—
183C23×C23—82no—
184C23×C23—82no—
185C23×C23#9912yes—
186C23×C23C2×C22⋊C412yes—
187C23×C23C2×C22⋊C411yescharacteristic
188C23×C23#12012yes—
189Q8×C23—42no—
190D4×C23—28no—
191D4×C23—44no—
192D4×C23—44no—
193D4×C23—44no—
194D4×C23—42no—
195D4×C23—44no—
196D4×C23—44no—
197D4×C23—48no—
198D4×C23—42no—
199D4×C23—42no—
200D4×C23—84no—
201D4×C23—84no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—21no—
205C2×C4×C23—21no—
206C2×C4×C23—24no—
207C2×C4×C23—22no—
208C2×C4×C23—44no—
209C2×C4×C23—44no—
210C2×C4×C23—42no—
211C2×C4×C23—42no—
212C2×C4×C23—82no—
213C2×C4×C23—82no—
214C2×C4×C23—82no—
215C2×C4×C23—84no—
216C2×C4×C23#9912yes—
217C2×C4×C23#12012yes—
218C8×C23—82no—
219#3484C2×C2311yescharacteristic
220#2121C2×C2314yescomplemented
221#1783C2×C2311yescharacteristic, complemented
222#1781C2×C2311yescharacteristic, complemented
223C22×C23—22no—
224C22×C23—22no—
225C22×C23—22no—
226C22×C23—22no—
227C22×C23—22no—
228C22×C23—22no—
229C22×C23—41no—
230C22×C23—42no—
231C22×C23—42no—
232C22×C23—42no—
233C22×C23—42no—
234C22×C23—42no—
235C22×C23—41no—
236C22×C23—41no—
237C22×C23—41no—
238C22×C23—44no—
239C22×C23—42no—
240C22×C23—41no—
241C22×C23—42no—
242C22×C23—41no—
243C22×C23—81no—
244C22×C23—81no—
245C22×C23—81no—
246C22×C23—81no—
247C22×C23—81no—
248C22×C23—81no—
249C22×C23—88no—
250C22×C23—88no—
251C22×C23—162no—
252C22×C23—162no—
253C22×C23#37911yescharacteristic
254C22×C23C2×#9911yescharacteristic
255C22×C23C2×#9911yescharacteristic
256C4×C23—22no—
257C4×C23—22no—
258C4×C23—44no—
259C4×C23—42no—
260C4×C23—162no—
261C2×H(5,𝔽2)C4×C2311yescharacteristic, complemented
262C23×D4C22×C2311yescharacteristic
263#546—24no—
264D42—24no—
265#535—21no—
266#534—21no—
267#534C22×C2312yes—
268#525—21no—
269C2×#120—21no—
270C2×#120C4×C2312yescomplemented
271#518C4×C2311yescharacteristic, complemented
272C2×#100—21no—
273C2×#100C22×C2312yescomplemented
274C2×#99—21no—
275C2×#99C22×C2312yescomplemented
276#407—21no—
277#379—21no—
278#352—24no—
279AΣL(1,𝔽16)—24no—
280C2×C23—21no—
281C2×C23—21no—
282C2×C23—21no—
283C2×C23—42no—
284C2×C23—42no—
285C2×C23—42no—
286C2×C23—42no—
287C2×C23—81no—
288C2×C23—82no—
289C2×C23#178111yescharacteristic
290C25C2×C4×C2311yescharacteristic
291C25D4×C2311yescharacteristic
292H(5,𝔽2)—42no—
293H(5,𝔽2)—42no—
294C2×#41—21no—
295C2×#41—22no—
296C22×D4—21no—
297C22×D4—22no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—24no—
301C22×D4—42no—
302C22×D4—42no—
303C22×D4C2×C4×C2311yescharacteristic
304C22×D4C2×C4×C2312yes—
305C22×D4D4×C2311yescharacteristic
306C22×D4D4×C2311yescharacteristic
307C22×D4C23×C2311yescharacteristic
308C23×C4D4×C2311yescharacteristic
309C2×M16—22no—
310#127—22no—
311#124—22no—
312#123—22no—
313#123—22no—
314#123D4×C2314yes—
315C22⋊Q8—22no—
316#121—22no—
317#121—24no—
318#121—24no—
319#121—24no—
320#120—24no—
321#120—22no—
322#120—22no—
323#120—22no—
324#120—22no—
325#120—24no—
326#120D4×C2314yes—
327C4×D4—24no—
328C4×D4—24no—
329#117—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4C2×C4×C2312yes—
336#100—24no—
337#100—42no—
338#99—24no—
339#99—42no—
340C22⋊C8—42no—
341C23#1270511yescharacteristic, direct product factor, Sylow
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—82no—
350C24—82no—
351C24C22⋊C4×C2311yescharacteristic
352C24C22⋊C4×C2311yescharacteristic
353C24C22⋊C4×C2311yescharacteristic
354C24C22⋊C4×C2312yescomplemented
355C24C22×C4×C2311yescharacteristic
356C24C2×D4×C2311yescharacteristic
357#41—42no—
358#41—48no—
359#41—42no—
360C2×Q8—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4C2×D4×C2314yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4C22⋊C4×C2311yescharacteristic
397C22×C4C22⋊C4×C2312yescomplemented
398C22×C4C2×D4×C2311yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—84no—
413C22⋊C4C22⋊C4×C2314yescomplemented
414C42—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23C23×#9912yes—
452C23C2×C22⋊C4×C2312yes—
453C23C2×C22⋊C4×C2311yescharacteristic
454C23C23×#12012yes—
455Q8—42no—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4C23×#9912yes—
483C2×C4C23×#12012yes—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—88no—
512C22—88no—
513C22—162no—
514C22—162no—
515C22C23×#37911yescharacteristic
516C22C2×C23×#9911yescharacteristic
517C22C2×C23×#9911yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—162no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—82no—
532C2C23×#178111yescharacteristic
subgroupG/H
center, socleC2×C23#1781
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C23×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2121C2semidirect
3. C23×#1783C2semidirect
4. C23×#1781C2semidirect
5. C2×C23×H(5,𝔽2)C4semidirect
14. C2×C23×#120C4semidirect
15. C23×#518C4semidirect
17. C2×C23×#100C22semidirect
19. C2×C23×#99C22semidirect
87. C24×C23C22⋊C4semidirect
130. C22×C4×C23C22⋊C4semidirect
146. C22⋊C4×C23C22⋊C4semidirect
148. #12705C23direct (N × H)
220. #2121C2×C23semidirect
221. #1783C2×C23semidirect
222. #1781C2×C23semidirect
261. C2×H(5,𝔽2)C4×C23semidirect
270. C2×#120C4×C23semidirect
271. #518C4×C23semidirect
273. C2×#100C22×C23semidirect
275. C2×#99C22×C23semidirect
341. C23#12705direct (N × H)
354. C24C22⋊C4×C23semidirect
397. C22×C4C22⋊C4×C23semidirect
413. C22⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#34842
C23×#21212
C23×#17832
C23×#17812
#1270523