Finite Group Databasebeta

← C23×#11753 #19159379

Subgroups — C23×#11753 #19159379

Order 5888.

778 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2404C211yesmaximal, characteristic
2C2×C23×#342C211yesmaximal, characteristic, complemented
3C23×#1462C211yesmaximal, characteristic, complemented
4C23×#1351C212yesmaximal, complemented
5C23×#1343C212yesmaximal, complemented
6C22×#41×C23C411yescharacteristic, complemented
7C2×C23×#120—21no—
8C22×C22⋊C4×C23—21no—
9C22×C22⋊C4×C23C2211yescharacteristic
10C23×#407—21no—
11C23×#393C411yescharacteristic, complemented
12C23×#393C2211yescharacteristic
13C23×#392C411yescharacteristic, complemented
14C23×#392C2211yescharacteristic
15C23×#386—21no—
16C23×#386—21no—
17C23×#386—21no—
18C23×#381C411yescharacteristic, complemented
19C23×#342—21no—
20C23×#342C2212yes—
21C23×#323—21no—
22C23×#323C2212yes—
23C25×C23D411yescharacteristic
24C2×#41×C23—41no—
25C2×#41×C23—41no—
26C2×#41×C23—44no—
27C22×Q8×C23C2×C411yescharacteristic, complemented
28C22×D4×C23—21no—
29C22×D4×C23—21no—
30C22×D4×C23—21no—
31C22×D4×C23C2×C411yescharacteristic, complemented
32C23×C4×C23—21no—
33C23×C4×C23—41no—
34C23×C4×C23C2×C411yescharacteristic
35C23×C4×C23D411yescharacteristic
36C2×M16×C23—41no—
37C23×#120—44no—
38C2×C22⋊C4×C23—24no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C22⋊C4×C23—41no—
46C2×C22⋊C4×C23—41no—
47C2×C22⋊C4×C23—41no—
48C2×C22⋊C4×C23—42no—
49C2×C22⋊C4×C23C2×C411yescharacteristic
50C2×C22⋊C4×C23D411yescharacteristic
51C2×C22⋊C4×C23D411yescharacteristic
52C2×C22⋊C4×C23C2311yescharacteristic
53C23×C22⋊C8—22no—
54C23×C22⋊C8—41no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95C2×C411yescharacteristic, complemented
59C23×#95C2×C411yescharacteristic, complemented
60C24×C23—21no—
61C24×C23—21no—
62C24×C23—22no—
63C24×C23—21no—
64C24×C23—21no—
65C24×C23—21no—
66C24×C23—42no—
67C24×C23—81no—
68C24×C23C22⋊C411yescharacteristic
69C24×C23C22⋊C411yescharacteristic
70C24×C23C2×D411yescharacteristic
71#41×C23—44no—
72#41×C23—44no—
73#41×C23—44no—
74#41×C23—44no—
75C2×Q8×C23—42no—
76C2×Q8×C23—41no—
77C2×D4×C23—24no—
78C2×D4×C23—41no—
79C2×D4×C23—42no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—42no—
83C2×D4×C23—42no—
84C2×D4×C23—42no—
85C2×D4×C23—42no—
86C2×D4×C23—41no—
87C2×D4×C23—41no—
88C2×D4×C23—81no—
89C2×D4×C23—81no—
90C22×C4×C23—21no—
91C22×C4×C23—21no—
92C22×C4×C23—21no—
93C22×C4×C23—21no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—42no—
98C22×C4×C23—42no—
99C22×C4×C23—42no—
100C22×C4×C23—42no—
101C22×C4×C23—44no—
102C22×C4×C23—42no—
103C22×C4×C23—81no—
104C22×C4×C23—82no—
105C22×C4×C23—82no—
106C22×C4×C23C22⋊C411yescharacteristic
107C22×C4×C23C22⋊C411yescharacteristic
108C22×C4×C23C22×C411yescharacteristic
109C22×C4×C23C2×D411yescharacteristic
110M16×C23—82no—
111C2×C8×C23—42no—
112C22⋊C4×C23—24no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—42no—
118C22⋊C4×C23—42no—
119C22⋊C4×C23—44no—
120C22⋊C4×C23—42no—
121C22⋊C4×C23—42no—
122C22⋊C4×C23—81no—
123C22⋊C4×C23—81no—
124#11753C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23—22no—
126C23×C23—22no—
127C23×C23—22no—
128C23×C23—22no—
129C23×C23—22no—
130C23×C23—21no—
131C23×C23—21no—
132C23×C23—21no—
133C23×C23—21no—
134C23×C23—21no—
135C23×C23—21no—
136C23×C23—22no—
137C23×C23—41no—
138C23×C23—42no—
139C23×C23—41no—
140C23×C23—41no—
141C23×C23—41no—
142C23×C23—41no—
143C23×C23—41no—
144C23×C23—41no—
145C23×C23—41no—
146C23×C23—42no—
147C23×C23—41no—
148C23×C23—41no—
149C23×C23—41no—
150C23×C23—81no—
151C23×C23—81no—
152C23×C23—84no—
153C23×C23—82no—
154C23×C23—82no—
155C23×C23#9911yescharacteristic
156C23×C23#9911yescharacteristic
157C23×C23#10412yes—
158C23×C23C2×C22⋊C411yescharacteristic
159C23×C23AGL(1,ℤ/8)11yescharacteristic
160C23×C23#13711yescharacteristic
161Q8×C23—42no—
162Q8×C23—42no—
163D4×C23—44no—
164D4×C23—42no—
165D4×C23—42no—
166D4×C23—44no—
167D4×C23—84no—
168D4×C23—82no—
169D4×C23—82no—
170C2×C4×C23—22no—
171C2×C4×C23—21no—
172C2×C4×C23—21no—
173C2×C4×C23—21no—
174C2×C4×C23—21no—
175C2×C4×C23—42no—
176C2×C4×C23—41no—
177C2×C4×C23—42no—
178C2×C4×C23—48no—
179C2×C4×C23—42no—
180C2×C4×C23—41no—
181C2×C4×C23—41no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—41no—
185C2×C4×C23—44no—
186C2×C4×C23—41no—
187C2×C4×C23—41no—
188C2×C4×C23—41no—
189C2×C4×C23—41no—
190C2×C4×C23—41no—
191C2×C4×C23—41no—
192C2×C4×C23—81no—
193C2×C4×C23—88no—
194C2×C4×C23—82no—
195C2×C4×C23—82no—
196C8×C23—82no—
197#2404C2×C2311yescharacteristic
198C2×#342C2×C2311yescharacteristic, complemented
199#1462C2×C2311yescharacteristic, complemented
200#1351C2×C2312yescomplemented
201#1343C2×C2312yescomplemented
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—21no—
205C22×C23—21no—
206C22×C23—21no—
207C22×C23—21no—
208C22×C23—21no—
209C22×C23—21no—
210C22×C23—22no—
211C22×C23—21no—
212C22×C23—42no—
213C22×C23—42no—
214C22×C23—42no—
215C22×C23—41no—
216C22×C23—41no—
217C22×C23—44no—
218C22×C23—44no—
219C22×C23—42no—
220C22×C23—42no—
221C22×C23—41no—
222C22×C23—41no—
223C22×C23—41no—
224C22×C23—41no—
225C22×C23—42no—
226C22×C23—81no—
227C22×C23—81no—
228C22×C23—81no—
229C22×C23—81no—
230C22×C23—82no—
231C22×C23—82no—
232C22×C23—81no—
233C22×C23—81no—
234C22×C23AΣL(1,𝔽16)12yes—
235C22×C23C2×#9911yescharacteristic
236C22×C23#41711yescharacteristic
237C22×C23C2×#10411yescharacteristic
238C4×C23—41no—
239C4×C23—42no—
240C4×C23—41no—
241C4×C23—41no—
242C4×C23—41no—
243C4×C23—81no—
244C4×C23—81no—
245C4×C23—84no—
246C22×#41C4×C2311yescharacteristic, complemented
247C2×#120—21no—
248C22×C22⋊C4—21no—
249C22×C22⋊C4C22×C2311yescharacteristic
250#407—21no—
251#393C4×C2311yescharacteristic, complemented
252#393C22×C2311yescharacteristic
253#392C4×C2311yescharacteristic, complemented
254#392C22×C2311yescharacteristic
255#386—21no—
256#386—21no—
257#386—21no—
258#381C4×C2311yescharacteristic, complemented
259#342—21no—
260#342C22×C2312yes—
261#323—21no—
262#323C22×C2312yes—
263C2×C23—21no—
264C2×C23—21no—
265C2×C23—22no—
266C2×C23—41no—
267C2×C23—41no—
268C2×C23—42no—
269C2×C23—41no—
270C2×C23—82no—
271C2×C23#150411yescharacteristic
272C2×C23C2×AΣL(1,𝔽16)11yescharacteristic
273C2×C23#212711yescharacteristic
274C25D4×C2311yescharacteristic
275C2×#41—41no—
276C2×#41—41no—
277C2×#41—44no—
278C22×Q8C2×C4×C2311yescharacteristic, complemented
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2×C4×C2311yescharacteristic, complemented
283C23×C4—21no—
284C23×C4—41no—
285C23×C4C2×C4×C2311yescharacteristic
286C23×C4D4×C2311yescharacteristic
287C2×M16—41no—
288#120—44no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4C2×C4×C2311yescharacteristic
301C2×C22⋊C4D4×C2311yescharacteristic
302C2×C22⋊C4D4×C2311yescharacteristic
303C2×C22⋊C4C23×C2311yescharacteristic
304C22⋊C8—22no—
305C22⋊C8—41no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95C2×C4×C2311yescharacteristic, complemented
310#95C2×C4×C2311yescharacteristic, complemented
311C23#1175311yescharacteristic, direct product factor, Sylow
312C24—21no—
313C24—21no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—42no—
319C24—81no—
320C24C22⋊C4×C2311yescharacteristic
321C24C22⋊C4×C2311yescharacteristic
322C24C2×D4×C2311yescharacteristic
323#41—44no—
324#41—44no—
325#41—44no—
326#41—44no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×D4—24no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—81no—
341C2×D4—81no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—81no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4C22⋊C4×C2311yescharacteristic
359C22×C4C22⋊C4×C2311yescharacteristic
360C22×C4C22×C4×C2311yescharacteristic
361C22×C4C2×D4×C2311yescharacteristic
362M16—82no—
363C2×C8—42no—
364C22⋊C4—24no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—81no—
375C22⋊C4—81no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—41no—
389C23—42no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—81no—
402C23—81no—
403C23—84no—
404C23—82no—
405C23—82no—
406C23C23×#9911yescharacteristic
407C23C23×#9911yescharacteristic
408C23C23×#10412yes—
409C23C2×C22⋊C4×C2311yescharacteristic
410C23C23×AGL(1,ℤ/8)11yescharacteristic
411C23C23×#13711yescharacteristic
412Q8—42no—
413Q8—42no—
414D4—44no—
415D4—42no—
416D4—42no—
417D4—44no—
418D4—84no—
419D4—82no—
420D4—82no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—48no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—88no—
445C2×C4—82no—
446C2×C4—82no—
447C8—82no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—82no—
477C22—82no—
478C22—81no—
479C22—81no—
480C22C23×AΣL(1,𝔽16)12yes—
481C22C2×C23×#9911yescharacteristic
482C22C23×#41711yescharacteristic
483C22C2×C23×#10411yescharacteristic
484C4—41no—
485C4—42no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—81no—
490C4—81no—
491C4—84no—
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—41no—
497C2—42no—
498C2—41no—
499C2—82no—
500C2C23×#150411yescharacteristic
501C2C2×C23×AΣL(1,𝔽16)11yescharacteristic
502C2C23×#212711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#2404C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#342C2semidirect
3. C23×#1462C2semidirect
4. C23×#1351C2semidirect
5. C23×#1343C2semidirect
6. C22×#41×C23C4semidirect
11. C23×#393C4semidirect
13. C23×#392C4semidirect
18. C23×#381C4semidirect
27. C22×Q8×C23C2×C4semidirect
31. C22×D4×C23C2×C4semidirect
58. C23×#95C2×C4semidirect
59. C23×#95C2×C4semidirect
124. #11753C23direct (N × H)
198. C2×#342C2×C23semidirect
199. #1462C2×C23semidirect
200. #1351C2×C23semidirect
201. #1343C2×C23semidirect
246. C22×#41C4×C23semidirect
251. #393C4×C23semidirect
253. #392C4×C23semidirect
258. #381C4×C23semidirect
278. C22×Q8C2×C4×C23semidirect
282. C22×D4C2×C4×C23semidirect
309. #95C2×C4×C23semidirect
310. #95C2×C4×C23semidirect
311. C23#11753direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24042
C2×C23×#3422
C23×#14622
C23×#13512
C23×#13432
#1175323