Finite Group Databasebeta

← C23×#13353 #19160979

Subgroups — C23×#13353 #19160979

Order 5888.

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

normal subgroups only

This group has 720 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×#3199C211yesmaximal, characteristic, complemented
2C23×#3027C211yesmaximal, characteristic, complemented
3C23×#3021C211yesmaximal, characteristic, complemented
4C23×#2134C211yesmaximal, characteristic, complemented
5C23×#2122C211yesmaximal, characteristic, complemented
6C23×#1791C211yesmaximal, characteristic, complemented
7C23×#1395C211yesmaximal, characteristic, complemented
8C23×#585C2211yescharacteristic
9C2×C23×AGL(1,ℤ/8)—21no—
10C23×#535—21no—
11C23×#535C2211yescharacteristic
12C23×#535C2211yescharacteristic
13C23×#482—21no—
14C23×#482—21no—
15C23×#469—21no—
16C23×#466—21no—
17C23×B(4,𝔽2)—22no—
18C23×B(4,𝔽2)—22no—
19C23×#454—22no—
20C23×#453—22no—
21C23×#447—21no—
22C23×#447—21no—
23C23×#421—21no—
24C23×#413—21no—
25C23×#413C2211yescharacteristic
26C23×#413C2211yescharacteristic
27C23×#410—21no—
28C23×#410C2211yescharacteristic
29C23×#408—21no—
30C23×#408C2211yescharacteristic
31C23×#355—22no—
32C23×AΣL(1,𝔽16)—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×D4×C23D411yescharacteristic
45C22×D4×C23D411yescharacteristic, complemented
46C23×AGL(1,ℤ/8)—24no—
47C23×AGL(1,ℤ/8)—42no—
48C2×SD16×C23—41no—
49C2×D8×C23—21no—
50C2×D8×C23—21no—
51C2×D8×C23—41no—
52C2×M16×C23—21no—
53C2×M16×C23—21no—
54C2×M16×C23D411yescharacteristic, complemented
55C22×C8×C23D411yescharacteristic, complemented
56C23×#127—21no—
57C23×#127—21no—
58C23×#127—41no—
59C23×#124—21no—
60C23×#124—21no—
61C23×#124—41no—
62C23×#121—21no—
63C23×#121—21no—
64C23×#121—21no—
65C23×#121—21no—
66C23×#121—42no—
67C23×#120—24no—
68C23×#120—21no—
69C23×#120—21no—
70C23×#120—21no—
71C23×#120—21no—
72C23×#117—21no—
73C23×#117D411yescharacteristic
74C23×#117D411yescharacteristic, complemented
75C23×#107—21no—
76C23×#106—21no—
77C23×#104—24no—
78C23×#102—21no—
79C23×#102—21no—
80C23×#102—21no—
81C23×#102—21no—
82C23×#101—21no—
83C23×#101—41no—
84C23×#101D412yescomplemented
85C23×#100—21no—
86C23×#100—21no—
87C23×#100—41no—
88C23×#99—24no—
89C23×#99—21no—
90C23×#99D412yes—
91C23×C22⋊C8—21no—
92C23×C22⋊C8—21no—
93C23×C22⋊C8—42no—
94C24×C23—21no—
95C24×C23—21no—
96#41×C23—22no—
97#41×C23—22no—
98#41×C23—41no—
99#41×C23—41no—
100#41×C23—42no—
101#41×C23—41no—
102#41×C23—41no—
103#41×C23—41no—
104#41×C23—41no—
105#41×C23—41no—
106#41×C23—42no—
107#41×C23—42no—
108#41×C23—42no—
109#41×C23—41no—
110#41×C23—41no—
111#41×C23—41no—
112C2×Q8×C23—21no—
113C2×Q8×C23—21no—
114C2×Q8×C23—21no—
115C2×Q8×C23—42no—
116C2×Q8×C23C2×D411yescharacteristic
117C2×D4×C23—21no—
118C2×D4×C23—21no—
119C2×D4×C23—21no—
120C2×D4×C23—22no—
121C2×D4×C23—22no—
122C2×D4×C23—21no—
123C2×D4×C23—21no—
124C2×D4×C23—21no—
125C2×D4×C23—21no—
126C2×D4×C23—41no—
127C2×D4×C23—41no—
128C2×D4×C23—41no—
129C2×D4×C23—41no—
130C2×D4×C23—42no—
131C2×D4×C23—42no—
132C2×D4×C23—42no—
133C2×D4×C23—41no—
134C2×D4×C23—44no—
135C2×D4×C23—41no—
136C2×D4×C23—41no—
137C2×D4×C23—41no—
138C2×D4×C23—41no—
139C2×D4×C23C2×D411yescharacteristic
140C2×D4×C23C2×D411yescharacteristic
141C2×D4×C23#4111yescharacteristic
142C22×C4×C23—22no—
143C22×C4×C23—22no—
144C22×C4×C23—41no—
145C22×C4×C23C2×D411yescharacteristic
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D411yescharacteristic
148SD16×C23—44no—
149D8×C23—44no—
150D8×C23—81no—
151D8×C23—81no—
152M16×C23—22no—
153M16×C23—41no—
154M16×C23—41no—
155M16×C23—41no—
156M16×C23—41no—
157C2×C8×C23—21no—
158C2×C8×C23—21no—
159C2×C8×C23—41no—
160C2×C8×C23—41no—
161C2×C8×C23—41no—
162C4⋊C4×C23—21no—
163C4⋊C4×C23—21no—
164C4⋊C4×C23—41no—
165C22⋊C4×C23—21no—
166C22⋊C4×C23—22no—
167C22⋊C4×C23—21no—
168C22⋊C4×C23—44no—
169C22⋊C4×C23—41no—
170C22⋊C4×C23—41no—
171C22⋊C4×C23—41no—
172C22⋊C4×C23—41no—
173C42×C23—21no—
174C42×C23—21no—
175C42×C23—41no—
176#13353C2311yesmaximal, characteristic, direct product factor, Sylow
177C23×C23—22no—
178C23×C23—22no—
179C23×C23—41no—
180C23×C23—42no—
181C23×C23—41no—
182C23×C23—41no—
183C23×C23—81no—
184C23×C23—161no—
185C23×C23—161no—
186C23×C23#12011yescharacteristic
187C23×C23#12111yescharacteristic
188C23×C23#12111yescharacteristic
189Q8×C23—22no—
190Q8×C23—41no—
191Q8×C23—41no—
192Q8×C23—41no—
193Q8×C23—41no—
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—42no—
205D4×C23—41no—
206D4×C23—41no—
207D4×C23—41no—
208D4×C23—41no—
209D4×C23—41no—
210D4×C23—81no—
211D4×C23—161no—
212D4×C23—161no—
213D4×C23—161no—
214D4×C23—161no—
215D4×C23—164no—
216C2×C4×C23—21no—
217C2×C4×C23—21no—
218C2×C4×C23—21no—
219C2×C4×C23—21no—
220C2×C4×C23—21no—
221C2×C4×C23—21no—
222C2×C4×C23—22no—
223C2×C4×C23—22no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—41no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—41no—
231C2×C4×C23—41no—
232C2×C4×C23#12011yescharacteristic
233C2×C4×C23#12011yescharacteristic
234C2×C4×C23#12111yescharacteristic
235C2×C4×C23#12711yescharacteristic
236C8×C23—42no—
237C8×C23—41no—
238C8×C23—41no—
239C8×C23—81no—
240#3199C2×C2311yescharacteristic, complemented
241#3027C2×C2311yescharacteristic, complemented
242#3021C2×C2311yescharacteristic, complemented
243#2134C2×C2311yescharacteristic, complemented
244#2122C2×C2311yescharacteristic, complemented
245#1791C2×C2311yescharacteristic, complemented
246#1395C2×C2311yescharacteristic, complemented
247C22×C23—21no—
248C22×C23—21no—
249C22×C23—21no—
250C22×C23—41no—
251C22×C23—41no—
252C22×C23—41no—
253C22×C23—42no—
254C22×C23—81no—
255C22×C23—81no—
256C22×C23—81no—
257C22×C23—81no—
258C22×C23—82no—
259C22×C23—82no—
260C22×C23—84no—
261C22×C23—81no—
262C22×C23—81no—
263C22×C23—161no—
264C22×C23—162no—
265C22×C23#39211yescharacteristic
266C4×C23—21no—
267C4×C23—21no—
268C4×C23—21no—
269C4×C23—42no—
270C4×C23—81no—
271C4×C23—81no—
272C4×C23—82no—
273C4×C23B(4,𝔽2)11yescharacteristic
274C4×C23B(4,𝔽2)11yescharacteristic
275#585C22×C2311yescharacteristic
276C2×AGL(1,ℤ/8)—21no—
277#535—21no—
278#535C22×C2311yescharacteristic
279#535C22×C2311yescharacteristic
280#482—21no—
281#482—21no—
282#469—21no—
283#466—21no—
284B(4,𝔽2)—22no—
285B(4,𝔽2)—22no—
286#454—22no—
287#453—22no—
288#447—21no—
289#447—21no—
290#421—21no—
291#413—21no—
292#413C22×C2311yescharacteristic
293#413C22×C2311yescharacteristic
294#410—21no—
295#410C22×C2311yescharacteristic
296#408—21no—
297#408C22×C2311yescharacteristic
298#355—22no—
299AΣL(1,𝔽16)—22no—
300C2×C23—21no—
301C2×C23—41no—
302C2×C23—41no—
303C2×C23—41no—
304C2×C23—82no—
305C2×C23—161no—
306C2×C23—161no—
307C2×C23#202111yescharacteristic
3082−1+4—21no—
3092−1+4—41no—
310H(5,𝔽2)—21no—
311H(5,𝔽2)—21no—
312H(5,𝔽2)—41no—
313H(5,𝔽2)D4×C2312yes—
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—42no—
318C2×#41C23×C2311yescharacteristic
319C22×D4D4×C2311yescharacteristic
320C22×D4D4×C2311yescharacteristic, complemented
321AGL(1,ℤ/8)—24no—
322AGL(1,ℤ/8)—42no—
323C2×SD16—41no—
324C2×D8—21no—
325C2×D8—21no—
326C2×D8—41no—
327C2×M16—21no—
328C2×M16—21no—
329C2×M16D4×C2311yescharacteristic, complemented
330C22×C8D4×C2311yescharacteristic, complemented
331#127—21no—
332#127—21no—
333#127—41no—
334#124—21no—
335#124—21no—
336#124—41no—
337#121—21no—
338#121—21no—
339#121—21no—
340#121—21no—
341#121—42no—
342#120—24no—
343#120—21no—
344#120—21no—
345#120—21no—
346#120—21no—
347#117—21no—
348#117D4×C2311yescharacteristic
349#117D4×C2311yescharacteristic, complemented
350#107—21no—
351#106—21no—
352#104—24no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#101—21no—
358#101—41no—
359#101D4×C2312yescomplemented
360#100—21no—
361#100—21no—
362#100—41no—
363#99—24no—
364#99—21no—
365#99D4×C2312yes—
366C22⋊C8—21no—
367C22⋊C8—21no—
368C22⋊C8—42no—
369C23#1335311yescharacteristic, direct product factor, Sylow
370C24—21no—
371C24—21no—
372#41—22no—
373#41—22no—
374#41—41no—
375#41—41no—
376#41—42no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—42no—
383#41—42no—
384#41—42no—
385#41—41no—
386#41—41no—
387#41—41no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8C2×D4×C2311yescharacteristic
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—44no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C2×D4×C2311yescharacteristic
416C2×D4C2×D4×C2311yescharacteristic
417C2×D4#41×C2311yescharacteristic
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—41no—
421C22×C4C2×D4×C2311yescharacteristic
422C22×C4C2×D4×C2311yescharacteristic
423C22×C4C2×D4×C2311yescharacteristic
424SD16—44no—
425D8—44no—
426D8—81no—
427D8—81no—
428M16—22no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—41no—
441C22⋊C4—21no—
442C22⋊C4—22no—
443C22⋊C4—21no—
444C22⋊C4—44no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C42—21no—
450C42—21no—
451C42—41no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—81no—
459C23—161no—
460C23—161no—
461C23C23×#12011yescharacteristic
462C23C23×#12111yescharacteristic
463C23C23×#12111yescharacteristic
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469D4—22no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—81no—
486D4—161no—
487D4—161no—
488D4—161no—
489D4—161no—
490D4—164no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4C23×#12011yescharacteristic
508C2×C4C23×#12011yescharacteristic
509C2×C4C23×#12111yescharacteristic
510C2×C4C23×#12711yescharacteristic
511C8—42no—
512C8—41no—
513C8—41no—
514C8—81no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—82no—
527C22—82no—
528C22—84no—
529C22—81no—
530C22—81no—
531C22—161no—
532C22—162no—
533C22C23×#39211yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—42no—
538C4—81no—
539C4—81no—
540C4—82no—
541C4C23×B(4,𝔽2)11yescharacteristic
542C4C23×B(4,𝔽2)11yescharacteristic
543C2—21no—
544C2—41no—
545C2—41no—
546C2—41no—
547C2—82no—
548C2—161no—
549C2—161no—
550C2C23×#202111yescharacteristic
subgroupG/H
center, socleC2×C23#2021
commutator, Frattini, cosocleC2×#41C23×C23
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×#3199C2semidirect
2. C23×#3027C2semidirect
3. C23×#3021C2semidirect
4. C23×#2134C2semidirect
5. C23×#2122C2semidirect
6. C23×#1791C2semidirect
7. C23×#1395C2semidirect
45. C22×D4×C23D4semidirect
54. C2×M16×C23D4semidirect
55. C22×C8×C23D4semidirect
74. C23×#117D4semidirect
84. C23×#101D4semidirect
176. #13353C23direct (N × H)
240. #3199C2×C23semidirect
241. #3027C2×C23semidirect
242. #3021C2×C23semidirect
243. #2134C2×C23semidirect
244. #2122C2×C23semidirect
245. #1791C2×C23semidirect
246. #1395C2×C23semidirect
320. C22×D4D4×C23semidirect
329. C2×M16D4×C23semidirect
330. C22×C8D4×C23semidirect
349. #117D4×C23semidirect
359. #101D4×C23semidirect
369. C23#13353direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31992
C23×#30272
C23×#30212
C23×#21342
C23×#21222
C23×#17912
C23×#13952
#1335323