Finite Group Databasebeta

← C23×#12687 #19160313

Subgroups — C23×#12687 #19160313

Order 5888.

2508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 2508 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×C23×#120C211yesmaximal, characteristic
2C2×C23×AΣL(1,𝔽16)C214yesmaximal, complemented
3C23×#1783C211yesmaximal, characteristic, complemented
4C23×#1781C211yesmaximal, characteristic, complemented
5C26×C23C411yescharacteristic, complemented
6C23×D4×C23—21no—
7C23×D4×C23C2211yescharacteristic
8C2×C23×#120—21no—
9C2×C23×#120—21no—
10C2×C23×#120—28no—
11C2×C23×#120C412yescomplemented
12C2×C23×#120C2212yes—
13C22×C22⋊C4×C23—21no—
14C22×C22⋊C4×C23C411yescharacteristic, complemented
15C2×C23×#100—21no—
16C2×C23×#100C2212yescomplemented
17C2×C23×#99—21no—
18C2×C23×#99C2212yescomplemented
19C23×#407—21no—
20C23×#379—21no—
21C23×AΣL(1,𝔽16)—28no—
22C25×C23—21no—
23C25×C23—21no—
24C25×C23—24no—
25C25×C23—48no—
26C25×C23—42no—
27C25×C23—42no—
28C25×C23C2×C411yescharacteristic
29C25×C23C2×C412yes—
30C25×C23D411yescharacteristic
31C22×D4×C23—28no—
32C22×D4×C23—22no—
33C22×D4×C23—22no—
34C22×D4×C23—24no—
35C22×D4×C23—42no—
36C22×D4×C23—42no—
37C22×D4×C23—48no—
38C22×D4×C23C2×C411yescharacteristic
39C22×D4×C23D411yescharacteristic
40C22×D4×C23D411yescharacteristic
41C22×D4×C23C2311yescharacteristic
42C23×C4×C23—21no—
43C23×C4×C23D411yescharacteristic
44C2×M16×C23—22no—
45C23×#120—28no—
46C23×#120—28no—
47C23×#120—24no—
48C23×#120—24no—
49C23×#120—24no—
50C23×#120D418yes—
51C2×C22⋊C4×C23—28no—
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—22no—
54C2×C22⋊C4×C23—22no—
55C2×C22⋊C4×C23—24no—
56C2×C22⋊C4×C23—22no—
57C2×C22⋊C4×C23—42no—
58C2×C22⋊C4×C23C2×C412yes—
59C23×#100—24no—
60C23×#100—42no—
61C23×#99—24no—
62C23×#99—42no—
63C23×C22⋊C8—42no—
64C24×C23—24no—
65C24×C23—28no—
66C24×C23—22no—
67C24×C23—22no—
68C24×C23—22no—
69C24×C23—22no—
70C24×C23—24no—
71C24×C23—22no—
72C24×C23—22no—
73C24×C23—22no—
74C24×C23—41no—
75C24×C23—44no—
76C24×C23—48no—
77C24×C23—42no—
78C24×C23—42no—
79C24×C23—44no—
80C24×C23—42no—
81C24×C23—48no—
82C24×C23—432no—
83C24×C23—416no—
84C24×C23—42no—
85C24×C23—44no—
86C24×C23—48no—
87C24×C23—416no—
88C24×C23—432no—
89C24×C23—48no—
90C24×C23—42no—
91C24×C23—42no—
92C24×C23C22⋊C411yescharacteristic
93C24×C23C22⋊C411yescharacteristic
94C24×C23C22⋊C411yescharacteristic
95C24×C23C22⋊C414yescomplemented
96C24×C23C22×C411yescharacteristic
97C24×C23C2×D411yescharacteristic
98C2×D4×C23—28no—
99C2×D4×C23—22no—
100C2×D4×C23—22no—
101C2×D4×C23—22no—
102C2×D4×C23—22no—
103C2×D4×C23—28no—
104C2×D4×C23—22no—
105C2×D4×C23—41no—
106C2×D4×C23—41no—
107C2×D4×C23—41no—
108C2×D4×C23—41no—
109C2×D4×C23—41no—
110C2×D4×C23—41no—
111C2×D4×C23—416no—
112C2×D4×C23—48no—
113C2×D4×C23—48no—
114C2×D4×C23—432no—
115C2×D4×C23C2×D414yes—
116C22×C4×C23—24no—
117C22×C4×C23—22no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—41no—
121C22×C4×C23—44no—
122C22×C4×C23—42no—
123C22×C4×C23C22⋊C411yescharacteristic
124C22×C4×C23C2×D411yescharacteristic
125M16×C23—42no—
126M16×C23—42no—
127C2×C8×C23—42no—
128C22⋊C4×C23—28no—
129C22⋊C4×C23—24no—
130C22⋊C4×C23—22no—
131C22⋊C4×C23—28no—
132C22⋊C4×C23—42no—
133C22⋊C4×C23—42no—
134C22⋊C4×C23—84no—
135C22⋊C4×C23C22⋊C414yescomplemented
136#12687C2311yesmaximal, characteristic, direct product factor, Sylow
137C23×C23—24no—
138C23×C23—21no—
139C23×C23—21no—
140C23×C23—216no—
141C23×C23—28no—
142C23×C23—28no—
143C23×C23—21no—
144C23×C23—24no—
145C23×C23—21no—
146C23×C23—21no—
147C23×C23—21no—
148C23×C23—21no—
149C23×C23—21no—
150C23×C23—21no—
151C23×C23—21no—
152C23×C23—21no—
153C23×C23—44no—
154C23×C23—42no—
155C23×C23—48no—
156C23×C23—44no—
157C23×C23—42no—
158C23×C23—48no—
159C23×C23—48no—
160C23×C23—48no—
161C23×C23—416no—
162C23×C23—44no—
163C23×C23—48no—
164C23×C23—44no—
165C23×C23—48no—
166C23×C23—48no—
167C23×C23—416no—
168C23×C23—48no—
169C23×C23—44no—
170C23×C23—42no—
171C23×C23—42no—
172C23×C23—48no—
173C23×C23—48no—
174C23×C23—416no—
175C23×C23—416no—
176C23×C23—464no—
177C23×C23—464no—
178C23×C23—48no—
179C23×C23—416no—
180C23×C23—48no—
181C23×C23—44no—
182C23×C23—42no—
183C23×C23—42no—
184C23×C23—44no—
185C23×C23—82no—
186C23×C23—84no—
187C23×C23#9912yes—
188C23×C23#9912yes—
189C23×C23C2×C22⋊C411yescharacteristic
190C23×C23C2×C22⋊C412yes—
191C23×C23#12012yes—
192D4×C23—44no—
193D4×C23—44no—
194D4×C23—44no—
195D4×C23—416no—
196D4×C23—416no—
197D4×C23—44no—
198C2×C4×C23—21no—
199C2×C4×C23—21no—
200C2×C4×C23—21no—
201C2×C4×C23—24no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—44no—
205C2×C4×C23—48no—
206C2×C4×C23—44no—
207C2×C4×C23—82no—
208C2×C4×C23—84no—
209C2×C4×C23#12012yes—
210C8×C23—82no—
211C22×#120C2×C2311yescharacteristic
212C2×AΣL(1,𝔽16)C2×C2314yescomplemented
213#1783C2×C2311yescharacteristic, complemented
214#1781C2×C2311yescharacteristic, complemented
215C22×C23—24no—
216C22×C23—24no—
217C22×C23—22no—
218C22×C23—22no—
219C22×C23—22no—
220C22×C23—28no—
221C22×C23—22no—
222C22×C23—22no—
223C22×C23—22no—
224C22×C23—42no—
225C22×C23—42no—
226C22×C23—42no—
227C22×C23—42no—
228C22×C23—44no—
229C22×C23—432no—
230C22×C23—416no—
231C22×C23—48no—
232C22×C23—42no—
233C22×C23—42no—
234C22×C23—48no—
235C22×C23—48no—
236C22×C23—416no—
237C22×C23—432no—
238C22×C23—42no—
239C22×C23—48no—
240C22×C23—42no—
241C22×C23—42no—
242C22×C23—44no—
243C22×C23—42no—
244C22×C23—41no—
245C22×C23—81no—
246C22×C23—88no—
247C22×C23—81no—
248C22×C23AΣL(1,𝔽16)14yescomplemented
249C22×C23#37911yescharacteristic
250C22×C23C2×#9911yescharacteristic
251C22×C23C2×#9911yescharacteristic
252C4×C23—22no—
253C4×C23—44no—
254C4×C23—41no—
255C4×C23—162no—
256C26C4×C2311yescharacteristic, complemented
257C23×D4—21no—
258C23×D4C22×C2311yescharacteristic
259C2×#120—21no—
260C2×#120—21no—
261C2×#120—28no—
262C2×#120C4×C2312yescomplemented
263C2×#120C22×C2312yes—
264C22×C22⋊C4—21no—
265C22×C22⋊C4C4×C2311yescharacteristic, complemented
266C2×#100—21no—
267C2×#100C22×C2312yescomplemented
268C2×#99—21no—
269C2×#99C22×C2312yescomplemented
270#407—21no—
271#379—21no—
272AΣL(1,𝔽16)—28no—
273C2×C23—21no—
274C2×C23—24no—
275C2×C23—21no—
276C2×C23—42no—
277C2×C23—48no—
278C2×C23—42no—
279C2×C23—42no—
280C2×C23—81no—
281C2×C23#178111yescharacteristic
282C2×C23C2×AΣL(1,𝔽16)12yes—
283C25—21no—
284C25—21no—
285C25—24no—
286C25—48no—
287C25—42no—
288C25—42no—
289C25C2×C4×C2311yescharacteristic
290C25C2×C4×C2312yes—
291C25D4×C2311yescharacteristic
292C22×D4—28no—
293C22×D4—22no—
294C22×D4—22no—
295C22×D4—24no—
296C22×D4—42no—
297C22×D4—42no—
298C22×D4—48no—
299C22×D4C2×C4×C2311yescharacteristic
300C22×D4D4×C2311yescharacteristic
301C22×D4D4×C2311yescharacteristic
302C22×D4C23×C2311yescharacteristic
303C23×C4—21no—
304C23×C4D4×C2311yescharacteristic
305C2×M16—22no—
306#120—28no—
307#120—28no—
308#120—24no—
309#120—24no—
310#120—24no—
311#120D4×C2318yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4C2×C4×C2312yes—
320#100—24no—
321#100—42no—
322#99—24no—
323#99—42no—
324C22⋊C8—42no—
325C23#1268711yescharacteristic, direct product factor, Sylow
326C24—24no—
327C24—28no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—41no—
337C24—44no—
338C24—48no—
339C24—42no—
340C24—42no—
341C24—44no—
342C24—42no—
343C24—48no—
344C24—432no—
345C24—416no—
346C24—42no—
347C24—44no—
348C24—48no—
349C24—416no—
350C24—432no—
351C24—48no—
352C24—42no—
353C24—42no—
354C24C22⋊C4×C2311yescharacteristic
355C24C22⋊C4×C2311yescharacteristic
356C24C22⋊C4×C2311yescharacteristic
357C24C22⋊C4×C2314yescomplemented
358C24C22×C4×C2311yescharacteristic
359C24C2×D4×C2311yescharacteristic
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—432no—
377C2×D4C2×D4×C2314yes—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4C22⋊C4×C2311yescharacteristic
386C22×C4C2×D4×C2311yescharacteristic
387M16—42no—
388M16—42no—
389C2×C8—42no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—28no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—84no—
397C22⋊C4C22⋊C4×C2314yescomplemented
398C23—24no—
399C23—21no—
400C23—21no—
401C23—216no—
402C23—28no—
403C23—28no—
404C23—21no—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—44no—
415C23—42no—
416C23—48no—
417C23—44no—
418C23—42no—
419C23—48no—
420C23—48no—
421C23—48no—
422C23—416no—
423C23—48no—
424C23—44no—
425C23—48no—
426C23—44no—
427C23—48no—
428C23—416no—
429C23—48no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—48no—
434C23—48no—
435C23—416no—
436C23—416no—
437C23—464no—
438C23—464no—
439C23—48no—
440C23—416no—
441C23—48no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—82no—
447C23—84no—
448C23C23×#9912yes—
449C23C23×#9912yes—
450C23C2×C22⋊C4×C2311yescharacteristic
451C23C2×C22⋊C4×C2312yes—
452C23C23×#12012yes—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—416no—
457D4—416no—
458D4—44no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4C23×#12012yes—
471C8—82no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—28no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—432no—
487C22—416no—
488C22—48no—
489C22—42no—
490C22—42no—
491C22—48no—
492C22—48no—
493C22—416no—
494C22—432no—
495C22—44no—
496C22—42no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—81no—
503C22—88no—
504C22—81no—
505C22C23×AΣL(1,𝔽16)14yescomplemented
506C22C23×#37911yescharacteristic
507C22C2×C23×#9911yescharacteristic
508C22C2×C23×#9911yescharacteristic
509C4—22no—
510C4—44no—
511C4—41no—
512C4—162no—
513C2—21no—
514C2—24no—
515C2—21no—
516C2—42no—
517C2—48no—
518C2—42no—
519C2—42no—
520C2—81no—
521C2C23×#178111yescharacteristic
522C2C2×C23×AΣL(1,𝔽16)12yes—
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×C23×#120C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C23×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×AΣL(1,𝔽16)C2semidirect
3. C23×#1783C2semidirect
4. C23×#1781C2semidirect
5. C26×C23C4semidirect
11. C2×C23×#120C4semidirect
14. C22×C22⋊C4×C23C4semidirect
16. C2×C23×#100C22semidirect
18. C2×C23×#99C22semidirect
95. C24×C23C22⋊C4semidirect
135. C22⋊C4×C23C22⋊C4semidirect
136. #12687C23direct (N × H)
212. C2×AΣL(1,𝔽16)C2×C23semidirect
213. #1783C2×C23semidirect
214. #1781C2×C23semidirect
248. C22×C23AΣL(1,𝔽16)semidirect
256. C26C4×C23semidirect
262. C2×#120C4×C23semidirect
265. C22×C22⋊C4C4×C23semidirect
267. C2×#100C22×C23semidirect
269. C2×#99C22×C23semidirect
325. C23#12687direct (N × H)
357. C24C22⋊C4×C23semidirect
397. C22⋊C4C22⋊C4×C23semidirect
505. C22C23×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1202
C2×C23×AΣL(1,𝔽16)2
C23×#17832
C23×#17812
#1268723