Finite Group Databasebeta

← C23×#13351 #19160977

Subgroups — C23×#13351 #19160977

Order 5888.

710 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3200C211yesmaximal, characteristic, complemented
2C23×#3028C211yesmaximal, characteristic, complemented
3C23×#3027C211yesmaximal, characteristic
4C23×#2129C211yesmaximal, characteristic, complemented
5C23×#2128C211yesmaximal, characteristic, complemented
6C23×#1794C211yesmaximal, characteristic, complemented
7C23×#1391C211yesmaximal, characteristic, complemented
8C23×#585C2211yescharacteristic
9C23×#536—21no—
10C23×#536C2211yescharacteristic
11C23×#535—21no—
12C23×#535C2211yescharacteristic
13C23×#485—21no—
14C23×#482—21no—
15C23×#468—21no—
16C23×#465—21no—
17C23×#458—22no—
18C23×#458—22no—
19C23×B(4,𝔽2)—22no—
20C23×B(4,𝔽2)—22no—
21C23×#450—21no—
22C23×#448—21no—
23C23×#421—21no—
24C23×#416—21no—
25C23×#410—21no—
26C23×#410—21no—
27C23×#410C2211yescharacteristic
28C23×#410C2211yescharacteristic
29C23×#410C2211yescharacteristic
30C23×#408C2211yescharacteristic
31C23×#354—22no—
32C23×#353—22no—
33C23×2−1+4—21no—
34C23×2−1+4—41no—
35C23×H(5,𝔽2)—21no—
36C23×H(5,𝔽2)—21no—
37C23×H(5,𝔽2)—41no—
38C23×H(5,𝔽2)D412yes—
39C2×#41×C23—21no—
40C2×#41×C23—21no—
41C2×#41×C23—21no—
42C2×#41×C23—42no—
43C2×#41×C23C2311yescharacteristic
44C22×Q8×C23D411yescharacteristic, complemented
45C22×D4×C23D411yescharacteristic
46C2×SD16×C23—21no—
47C2×SD16×C23—21no—
48C2×M16×C23D411yescharacteristic, complemented
49C22×C8×C23D411yescharacteristic, complemented
50C23×C4⋊Q8—21no—
51C23×C4⋊Q8—41no—
52C23×#127—21no—
53C23×#127—41no—
54C23×#124—21no—
55C23×#124—21no—
56C23×#124—41no—
57C23×#124—41no—
58C23×#123—24no—
59C23×#123—21no—
60C23×#123—21no—
61C23×C22⋊Q8—21no—
62C23×C22⋊Q8—21no—
63C23×C22⋊Q8—42no—
64C23×#121—21no—
65C23×#121—21no—
66C23×#121—42no—
67C23×#120—24no—
68C23×#120—21no—
69C23×#120—21no—
70C23×#117—21no—
71C23×#117—21no—
72C23×#117—21no—
73C23×#117D411yescharacteristic
74C23×#117D411yescharacteristic, complemented
75C23×#107—21no—
76C23×#106—21no—
77C23×#104—42no—
78C23×#103—21no—
79C23×#103—21no—
80C23×#103—41no—
81C23×#102—21no—
82C23×#102—21no—
83C23×#102—41no—
84C23×#99—24no—
85C23×#99—24no—
86C23×#99—21no—
87C23×#99—21no—
88C23×#99—21no—
89C23×#99—21no—
90C23×#99D412yes—
91C23×#99D412yescomplemented
92C23×C22⋊C8—21no—
93C23×C22⋊C8—21no—
94C24×C23—21no—
95#41×C23—22no—
96#41×C23—22no—
97#41×C23—41no—
98#41×C23—41no—
99#41×C23—42no—
100#41×C23—41no—
101#41×C23—41no—
102#41×C23—41no—
103#41×C23—41no—
104#41×C23—41no—
105#41×C23—42no—
106#41×C23—42no—
107#41×C23—42no—
108#41×C23—41no—
109#41×C23—41no—
110#41×C23—41no—
111C2×Q8×C23—21no—
112C2×Q8×C23—21no—
113C2×Q8×C23—21no—
114C2×Q8×C23—21no—
115C2×Q8×C23—21no—
116C2×Q8×C23—41no—
117C2×Q8×C23—41no—
118C2×Q8×C23—42no—
119C2×Q8×C23#4111yescharacteristic
120C2×D4×C23—21no—
121C2×D4×C23—21no—
122C2×D4×C23—21no—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—21no—
126C2×D4×C23—21no—
127C2×D4×C23—41no—
128C2×D4×C23—41no—
129C2×D4×C23—42no—
130C2×D4×C23—42no—
131C2×D4×C23—42no—
132C2×D4×C23—41no—
133C2×D4×C23—44no—
134C2×D4×C23—41no—
135C2×D4×C23—41no—
136C2×D4×C23C2×D411yescharacteristic
137C2×D4×C23C2×D411yescharacteristic
138C2×D4×C23C2×D411yescharacteristic
139C22×C4×C23—22no—
140C22×C4×C23—22no—
141C22×C4×C23—21no—
142C22×C4×C23—41no—
143C22×C4×C23C2×D411yescharacteristic
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23C2×D411yescharacteristic
146SD16×C23—81no—
147SD16×C23—81no—
148M16×C23—41no—
149C2×C8×C23—21no—
150C2×C8×C23—21no—
151C2×C8×C23—41no—
152C4⋊C4×C23—21no—
153C4⋊C4×C23—21no—
154C4⋊C4×C23—41no—
155C4⋊C4×C23—41no—
156C4⋊C4×C23—44no—
157C4⋊C4×C23—41no—
158C4⋊C4×C23—41no—
159C4⋊C4×C23—41no—
160C22⋊C4×C23—21no—
161C22⋊C4×C23—21no—
162C22⋊C4×C23—22no—
163C22⋊C4×C23—22no—
164C22⋊C4×C23—44no—
165C22⋊C4×C23—44no—
166C22⋊C4×C23—41no—
167C22⋊C4×C23—41no—
168C22⋊C4×C23—41no—
169C22⋊C4×C23—41no—
170C22⋊C4×C23—41no—
171C42×C23—21no—
172C42×C23—21no—
173C42×C23—41no—
174C42×C23—41no—
175C42×C23—41no—
176#13351C2311yesmaximal, characteristic, direct product factor, Sylow
177C23×C23—22no—
178C23×C23—22no—
179C23×C23—41no—
180C23×C23—42no—
181C23×C23—41no—
182C23×C23—161no—
183C23×C23#12011yescharacteristic
184C23×C23#12011yescharacteristic
185C23×C23#12711yescharacteristic
186Q8×C23—22no—
187Q8×C23—41no—
188Q8×C23—41no—
189Q8×C23—41no—
190Q8×C23—41no—
191Q8×C23—41no—
192Q8×C23—41no—
193Q8×C23—81no—
194D4×C23—22no—
195D4×C23—41no—
196D4×C23—41no—
197D4×C23—41no—
198D4×C23—41no—
199D4×C23—41no—
200D4×C23—42no—
201D4×C23—42no—
202D4×C23—42no—
203D4×C23—42no—
204D4×C23—41no—
205D4×C23—41no—
206D4×C23—41no—
207D4×C23—81no—
208D4×C23—161no—
209D4×C23—161no—
210D4×C23—164no—
211C2×C4×C23—21no—
212C2×C4×C23—21no—
213C2×C4×C23—21no—
214C2×C4×C23—21no—
215C2×C4×C23—21no—
216C2×C4×C23—21no—
217C2×C4×C23—22no—
218C2×C4×C23—22no—
219C2×C4×C23—42no—
220C2×C4×C23—42no—
221C2×C4×C23—42no—
222C2×C4×C23—41no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—41no—
230C2×C4×C23—81no—
231C2×C4×C23#12011yescharacteristic
232C2×C4×C23#12111yescharacteristic
233C2×C4×C23#12111yescharacteristic
234C2×C4×C23#12111yescharacteristic
235C8×C23—41no—
236C8×C23—41no—
237#3200C2×C2311yescharacteristic, complemented
238#3028C2×C2311yescharacteristic, complemented
239#3027C2×C2311yescharacteristic
240#2129C2×C2311yescharacteristic, complemented
241#2128C2×C2311yescharacteristic, complemented
242#1794C2×C2311yescharacteristic, complemented
243#1391C2×C2311yescharacteristic, complemented
244C22×C23—21no—
245C22×C23—21no—
246C22×C23—21no—
247C22×C23—41no—
248C22×C23—41no—
249C22×C23—41no—
250C22×C23—42no—
251C22×C23—81no—
252C22×C23—81no—
253C22×C23—82no—
254C22×C23—82no—
255C22×C23—84no—
256C22×C23—81no—
257C22×C23—161no—
258C22×C23#39211yescharacteristic
259C4×C23—21no—
260C4×C23—21no—
261C4×C23—21no—
262C4×C23—42no—
263C4×C23—81no—
264C4×C23—81no—
265C4×C23—82no—
266C4×C23—82no—
267C4×C23—81no—
268C4×C23—161no—
269C4×C23B(4,𝔽2)11yescharacteristic
270C4×C23B(4,𝔽2)11yescharacteristic
271#585C22×C2311yescharacteristic
272#536—21no—
273#536C22×C2311yescharacteristic
274#535—21no—
275#535C22×C2311yescharacteristic
276#485—21no—
277#482—21no—
278#468—21no—
279#465—21no—
280#458—22no—
281#458—22no—
282B(4,𝔽2)—22no—
283B(4,𝔽2)—22no—
284#450—21no—
285#448—21no—
286#421—21no—
287#416—21no—
288#410—21no—
289#410—21no—
290#410C22×C2311yescharacteristic
291#410C22×C2311yescharacteristic
292#410C22×C2311yescharacteristic
293#408C22×C2311yescharacteristic
294#354—22no—
295#353—22no—
296C2×C23—21no—
297C2×C23—41no—
298C2×C23—41no—
299C2×C23—41no—
300C2×C23—82no—
301C2×C23—161no—
302C2×C23#202111yescharacteristic
3032−1+4—21no—
3042−1+4—41no—
305H(5,𝔽2)—21no—
306H(5,𝔽2)—21no—
307H(5,𝔽2)—41no—
308H(5,𝔽2)D4×C2312yes—
309C2×#41—21no—
310C2×#41—21no—
311C2×#41—21no—
312C2×#41—42no—
313C2×#41C23×C2311yescharacteristic
314C22×Q8D4×C2311yescharacteristic, complemented
315C22×D4D4×C2311yescharacteristic
316C2×SD16—21no—
317C2×SD16—21no—
318C2×M16D4×C2311yescharacteristic, complemented
319C22×C8D4×C2311yescharacteristic, complemented
320C4⋊Q8—21no—
321C4⋊Q8—41no—
322#127—21no—
323#127—41no—
324#124—21no—
325#124—21no—
326#124—41no—
327#124—41no—
328#123—24no—
329#123—21no—
330#123—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—42no—
334#121—21no—
335#121—21no—
336#121—42no—
337#120—24no—
338#120—21no—
339#120—21no—
340#117—21no—
341#117—21no—
342#117—21no—
343#117D4×C2311yescharacteristic
344#117D4×C2311yescharacteristic, complemented
345#107—21no—
346#106—21no—
347#104—42no—
348#103—21no—
349#103—21no—
350#103—41no—
351#102—21no—
352#102—21no—
353#102—41no—
354#99—24no—
355#99—24no—
356#99—21no—
357#99—21no—
358#99—21no—
359#99—21no—
360#99D4×C2312yes—
361#99D4×C2312yescomplemented
362C22⋊C8—21no—
363C22⋊C8—21no—
364C23#1335111yescharacteristic, direct product factor, Sylow
365C24—21no—
366#41—22no—
367#41—22no—
368#41—41no—
369#41—41no—
370#41—42no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—41no—
380#41—41no—
381#41—41no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—42no—
390C2×Q8#41×C2311yescharacteristic
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—44no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4C2×D4×C2311yescharacteristic
408C2×D4C2×D4×C2311yescharacteristic
409C2×D4C2×D4×C2311yescharacteristic
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—41no—
414C22×C4C2×D4×C2311yescharacteristic
415C22×C4C2×D4×C2311yescharacteristic
416C22×C4C2×D4×C2311yescharacteristic
417SD16—81no—
418SD16—81no—
419M16—41no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—41no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—44no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—41no—
446C42—41no—
447C23—22no—
448C23—22no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—161no—
453C23C23×#12011yescharacteristic
454C23C23×#12011yescharacteristic
455C23C23×#12711yescharacteristic
456Q8—22no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—81no—
464D4—22no—
465D4—41no—
466D4—41no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—81no—
478D4—161no—
479D4—161no—
480D4—164no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—81no—
501C2×C4C23×#12011yescharacteristic
502C2×C4C23×#12111yescharacteristic
503C2×C4C23×#12111yescharacteristic
504C2×C4C23×#12111yescharacteristic
505C8—41no—
506C8—41no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—81no—
520C22—161no—
521C22C23×#39211yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—82no—
529C4—82no—
530C4—81no—
531C4—161no—
532C4C23×B(4,𝔽2)11yescharacteristic
533C4C23×B(4,𝔽2)11yescharacteristic
534C2—21no—
535C2—41no—
536C2—41no—
537C2—41no—
538C2—82no—
539C2—161no—
540C2C23×#202111yescharacteristic
subgroupG/H
center, socleC2×C23#2021
commutator, Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3027C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3200C2semidirect
2. C23×#3028C2semidirect
4. C23×#2129C2semidirect
5. C23×#2128C2semidirect
6. C23×#1794C2semidirect
7. C23×#1391C2semidirect
44. C22×Q8×C23D4semidirect
48. C2×M16×C23D4semidirect
49. C22×C8×C23D4semidirect
74. C23×#117D4semidirect
91. C23×#99D4semidirect
176. #13351C23direct (N × H)
237. #3200C2×C23semidirect
238. #3028C2×C23semidirect
240. #2129C2×C23semidirect
241. #2128C2×C23semidirect
242. #1794C2×C23semidirect
243. #1391C2×C23semidirect
314. C22×Q8D4×C23semidirect
318. C2×M16D4×C23semidirect
319. C22×C8D4×C23semidirect
344. #117D4×C23semidirect
361. #99D4×C23semidirect
364. C23#13351direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32002
C23×#30282
C23×#30272
C23×#21292
C23×#21282
C23×#17942
C23×#13912
#1335123