Finite Group Databasebeta

← C23×#13354 #19160980

Subgroups — C23×#13354 #19160980

Order 5888.

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

normal subgroups only

This group has 692 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×#3021C211yesmaximal, characteristic
4C23×#2135C211yesmaximal, characteristic, complemented
5C23×#2122C211yesmaximal, characteristic, complemented
6C23×#1791C211yesmaximal, characteristic, complemented
7C23×#1395C211yesmaximal, characteristic, complemented
8C23×#585C2211yescharacteristic
9C2×C23×AGL(1,ℤ/8)—21no—
10C23×#536—21no—
11C23×#536C2211yescharacteristic
12C23×#535C2211yescharacteristic
13C23×#485—21no—
14C23×#482—21no—
15C23×#468—21no—
16C23×#465—21no—
17C23×#458—22no—
18C23×#458—22no—
19C23×#454—22no—
20C23×#453—22no—
21C23×#450—21no—
22C23×#448—21no—
23C23×#421—21no—
24C23×#413—21no—
25C23×#413C2211yescharacteristic
26C23×#413C2211yescharacteristic
27C23×#410—21no—
28C23×#410C2211yescharacteristic
29C23×#408—21no—
30C23×#408C2211yescharacteristic
31C23×#356—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×Q8×C23D411yescharacteristic
45C22×D4×C23D411yescharacteristic
46C23×AGL(1,ℤ/8)—24no—
47C23×AGL(1,ℤ/8)—42no—
48C2×SD16×C23—21no—
49C2×SD16×C23—21no—
50C2×SD16×C23—41no—
51C2×D8×C23—41no—
52C2×M16×C23—21no—
53C2×M16×C23—21no—
54C2×M16×C23D411yescharacteristic
55C22×C8×C23D411yescharacteristic
56C23×C4⋊Q8—21no—
57C23×C4⋊Q8—41no—
58C23×#127—21no—
59C23×#124—21no—
60C23×#124—21no—
61C23×#124—41no—
62C23×#123—24no—
63C23×#123—21no—
64C23×#123—21no—
65C23×C22⋊Q8—21no—
66C23×C22⋊Q8—21no—
67C23×C22⋊Q8—42no—
68C23×#121—21no—
69C23×#121—21no—
70C23×#120—21no—
71C23×#120—21no—
72C23×#117—21no—
73C23×#117D411yescharacteristic
74C23×#117D411yescharacteristic
75C23×#107—21no—
76C23×#106—21no—
77C23×#104—24no—
78C23×#103—21no—
79C23×#103—21no—
80C23×#102—21no—
81C23×#102—21no—
82C23×#101—21no—
83C23×#101—41no—
84C23×#101D412yes—
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—
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×C23C2×D411yescharacteristic
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—41no—
134C2×D4×C23—41no—
135C2×D4×C23C2×D411yescharacteristic
136C2×D4×C23C2×D411yescharacteristic
137C2×D4×C23#4111yescharacteristic
138C22×C4×C23—22no—
139C22×C4×C23—22no—
140C22×C4×C23—21no—
141C22×C4×C23—41no—
142C22×C4×C23C2×D411yescharacteristic
143C22×C4×C23C2×D411yescharacteristic
144C22×C4×C23C2×D411yescharacteristic
145SD16×C23—44no—
146SD16×C23—81no—
147SD16×C23—81no—
148D8×C23—44no—
149M16×C23—22no—
150M16×C23—41no—
151M16×C23—41no—
152M16×C23—41no—
153M16×C23—41no—
154C2×C8×C23—21no—
155C2×C8×C23—21no—
156C2×C8×C23—41no—
157C2×C8×C23—41no—
158C2×C8×C23—41no—
159C4⋊C4×C23—21no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—41no—
162C4⋊C4×C23—44no—
163C4⋊C4×C23—41no—
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#13354C2311yesmaximal, characteristic, direct product factor, Sylow
177C23×C23—22no—
178C23×C23—22no—
179C23×C23—41no—
180C23×C23—41no—
181C23×C23—81no—
182C23×C23—161no—
183C23×C23#12011yescharacteristic
184C23×C23#12111yescharacteristic
185C23×C23#12111yescharacteristic
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—42no—
205D4×C23—41no—
206D4×C23—41no—
207D4×C23—41no—
208D4×C23—161no—
209D4×C23—161no—
210C2×C4×C23—21no—
211C2×C4×C23—21no—
212C2×C4×C23—21no—
213C2×C4×C23—21no—
214C2×C4×C23—21no—
215C2×C4×C23—21no—
216C2×C4×C23—22no—
217C2×C4×C23—22no—
218C2×C4×C23—42no—
219C2×C4×C23—42no—
220C2×C4×C23—42no—
221C2×C4×C23—41no—
222C2×C4×C23—42no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—41no—
226C2×C4×C23—41no—
227C2×C4×C23—41no—
228C2×C4×C23#12011yescharacteristic
229C2×C4×C23#12011yescharacteristic
230C2×C4×C23#12111yescharacteristic
231C2×C4×C23#12711yescharacteristic
232C8×C23—42no—
233C8×C23—41no—
234C8×C23—41no—
235C8×C23—81no—
236#3200C2×C2311yescharacteristic, complemented
237#3028C2×C2311yescharacteristic, complemented
238#3021C2×C2311yescharacteristic
239#2135C2×C2311yescharacteristic, complemented
240#2122C2×C2311yescharacteristic, complemented
241#1791C2×C2311yescharacteristic, complemented
242#1395C2×C2311yescharacteristic, complemented
243C22×C23—21no—
244C22×C23—21no—
245C22×C23—21no—
246C22×C23—41no—
247C22×C23—41no—
248C22×C23—41no—
249C22×C23—42no—
250C22×C23—81no—
251C22×C23—81no—
252C22×C23—82no—
253C22×C23—82no—
254C22×C23—81no—
255C22×C23—161no—
256C22×C23—162no—
257C22×C23#39211yescharacteristic
258C4×C23—21no—
259C4×C23—21no—
260C4×C23—21no—
261C4×C23—42no—
262C4×C23—81no—
263C4×C23—81no—
264C4×C23—82no—
265C4×C23—81no—
266C4×C23B(4,𝔽2)11yescharacteristic
267C4×C23B(4,𝔽2)11yescharacteristic
268#585C22×C2311yescharacteristic
269C2×AGL(1,ℤ/8)—21no—
270#536—21no—
271#536C22×C2311yescharacteristic
272#535C22×C2311yescharacteristic
273#485—21no—
274#482—21no—
275#468—21no—
276#465—21no—
277#458—22no—
278#458—22no—
279#454—22no—
280#453—22no—
281#450—21no—
282#448—21no—
283#421—21no—
284#413—21no—
285#413C22×C2311yescharacteristic
286#413C22×C2311yescharacteristic
287#410—21no—
288#410C22×C2311yescharacteristic
289#408—21no—
290#408C22×C2311yescharacteristic
291#356—22no—
292AΣL(1,𝔽16)—22no—
293C2×C23—21no—
294C2×C23—41no—
295C2×C23—41no—
296C2×C23—41no—
297C2×C23—82no—
298C2×C23—161no—
299C2×C23#202111yescharacteristic
3002−1+4—21no—
3012−1+4—41no—
302H(5,𝔽2)—21no—
303H(5,𝔽2)—21no—
304H(5,𝔽2)—41no—
305H(5,𝔽2)D4×C2312yes—
306C2×#41—21no—
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—42no—
310C2×#41C23×C2311yescharacteristic
311C22×Q8D4×C2311yescharacteristic
312C22×D4D4×C2311yescharacteristic
313AGL(1,ℤ/8)—24no—
314AGL(1,ℤ/8)—42no—
315C2×SD16—21no—
316C2×SD16—21no—
317C2×SD16—41no—
318C2×D8—41no—
319C2×M16—21no—
320C2×M16—21no—
321C2×M16D4×C2311yescharacteristic
322C22×C8D4×C2311yescharacteristic
323C4⋊Q8—21no—
324C4⋊Q8—41no—
325#127—21no—
326#124—21no—
327#124—21no—
328#124—41no—
329#123—24no—
330#123—21no—
331#123—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—42no—
335#121—21no—
336#121—21no—
337#120—21no—
338#120—21no—
339#117—21no—
340#117D4×C2311yescharacteristic
341#117D4×C2311yescharacteristic
342#107—21no—
343#106—21no—
344#104—24no—
345#103—21no—
346#103—21no—
347#102—21no—
348#102—21no—
349#101—21no—
350#101—41no—
351#101D4×C2312yes—
352#100—21no—
353#100—21no—
354#100—41no—
355#99—24no—
356#99—21no—
357#99D4×C2312yes—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—42no—
361C23#1335411yescharacteristic, direct product factor, Sylow
362C24—21no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367#41—42no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—42no—
374#41—42no—
375#41—42no—
376#41—41no—
377#41—41no—
378#41—41no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—42no—
387C2×Q8C2×D4×C2311yescharacteristic
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4C2×D4×C2311yescharacteristic
404C2×D4C2×D4×C2311yescharacteristic
405C2×D4#41×C2311yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4C2×D4×C2311yescharacteristic
411C22×C4C2×D4×C2311yescharacteristic
412C22×C4C2×D4×C2311yescharacteristic
413SD16—44no—
414SD16—81no—
415SD16—81no—
416D8—44no—
417M16—22no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—44no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—21no—
434C22⋊C4—22no—
435C22⋊C4—21no—
436C22⋊C4—44no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C42—21no—
442C42—21no—
443C42—41no—
444C23—22no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—81no—
449C23—161no—
450C23C23×#12011yescharacteristic
451C23C23×#12111yescharacteristic
452C23C23×#12111yescharacteristic
453Q8—22no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—81no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—161no—
476D4—161no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4C23×#12011yescharacteristic
496C2×C4C23×#12011yescharacteristic
497C2×C4C23×#12111yescharacteristic
498C2×C4C23×#12711yescharacteristic
499C8—42no—
500C8—41no—
501C8—41no—
502C8—81no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—161no—
516C22—162no—
517C22C23×#39211yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—82no—
525C4—81no—
526C4C23×B(4,𝔽2)11yescharacteristic
527C4C23×B(4,𝔽2)11yescharacteristic
528C2—21no—
529C2—41no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2C23×#202111yescharacteristic
subgroupG/H
center, socleC2×C23#2021
commutator, Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3021C2
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×#2135C2semidirect
5. C23×#2122C2semidirect
6. C23×#1791C2semidirect
7. C23×#1395C2semidirect
176. #13354C23direct (N × H)
236. #3200C2×C23semidirect
237. #3028C2×C23semidirect
239. #2135C2×C23semidirect
240. #2122C2×C23semidirect
241. #1791C2×C23semidirect
242. #1395C2×C23semidirect
361. C23#13354direct (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×#30212
C23×#21352
C23×#21222
C23×#17912
C23×#13952
#1335423