Finite Group Databasebeta

← C23×#12726 #19160352

Subgroups — C23×#12726 #19160352

Order 5888.

708 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3491C211yesmaximal, characteristic
2C23×#2121C212yesmaximal
3C2×C23×#352C212yesmaximal, complemented
4C23×#1829C211yesmaximal, characteristic, complemented
5C23×#1828C211yesmaximal, characteristic, complemented
6C23×#552—22no—
7C23×#552—22no—
8C23×#551—22no—
9C23×#551—22no—
10C23×#543C411yescharacteristic, complemented
11C23×#539—22no—
12C23×#534C2211yescharacteristic
13C23×#529—22no—
14C2×C23×#123C411yescharacteristic, complemented
15C2×C23×#123C2211yescharacteristic
16C23×#518C411yescharacteristic, complemented
17C23×#518C2211yescharacteristic
18C2×C23×#117C411yescharacteristic, complemented
19C2×C23×#100C2212yes—
20C2×C23×#99C2212yes—
21C23×#352—22no—
22C23×#352—22no—
23C23×#352—22no—
24C23×AΣL(1,𝔽16)—22no—
25C23×#344—22no—
26C23×#342—22no—
27C22×D4×C23C2×C411yescharacteristic
28C22×D4×C23C2311yescharacteristic
29C23×C4×C23C2×C411yescharacteristic
30C2×M16×C23—22no—
31C23×#126—22no—
32C23×#126—22no—
33C23×#126—22no—
34C23×#126—22no—
35C23×#125—22no—
36C23×#125—22no—
37C23×#125—22no—
38C23×#124—22no—
39C23×#123—22no—
40C23×#123—22no—
41C23×#123—22no—
42C23×#123—21no—
43C23×#123—21no—
44C23×#123—21no—
45C23×#123—22no—
46C23×#123D412yes—
47C23×#123D412yes—
48C23×#123D412yes—
49C23×C22⋊Q8—22no—
50C23×C22⋊Q8—22no—
51C23×C22⋊Q8—22no—
52C23×C22⋊Q8—22no—
53C23×C22⋊Q8—22no—
54C23×#121—22no—
55C23×#120—21no—
56C23×#120D412yes—
57C4×Q8×C23—22no—
58C4×D4×C23—22no—
59C4×D4×C23—22no—
60C4×D4×C23—22no—
61C4×D4×C23—22no—
62C4×D4×C23—22no—
63C23×#117—22no—
64C23×#117—22no—
65C23×#117—21no—
66C23×#117—22no—
67C23×#117—22no—
68C23×#117D412yes—
69C2×C4⋊C4×C23—21no—
70C2×C4⋊C4×C23—21no—
71C2×C4⋊C4×C23—21no—
72C2×C22⋊C4×C23—21no—
73C2×C22⋊C4×C23—21no—
74C2×C22⋊C4×C23—21no—
75C2×C22⋊C4×C23—22no—
76C2×C22⋊C4×C23C2×C411yescharacteristic
77C2×C22⋊C4×C23C2×C411yescharacteristic
78C2×C22⋊C4×C23C2×C411yescharacteristic
79C2×C22⋊C4×C23C2×C411yescharacteristic
80C2×C22⋊C4×C23D412yes—
81C2×C42×C23—21no—
82C23×#100—22no—
83C23×#100—22no—
84C23×#99—22no—
85C23×#99—22no—
86C24×C23C22⋊C411yescharacteristic
87C24×C23C22×C411yescharacteristic
88C24×C23C2×D411yescharacteristic
89C2×Q8×C23—22no—
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—22no—
93C2×D4×C23—42no—
94C2×D4×C23—42no—
95C2×D4×C23C2×D412yes—
96C2×D4×C23C2×D412yes—
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—22no—
100C22×C4×C23—21no—
101C22×C4×C23—21no—
102C22×C4×C23—21no—
103C22×C4×C23—21no—
104C22×C4×C23—22no—
105C22×C4×C23—42no—
106C22×C4×C23—42no—
107C22×C4×C23C22⋊C411yescharacteristic
108C22×C4×C23C22⋊C411yescharacteristic
109C22×C4×C23C22⋊C411yescharacteristic
110C22×C4×C23C2×D411yescharacteristic
111M16×C23—42no—
112M16×C23—42no—
113C2×C8×C23—42no—
114C4⋊C4×C23—22no—
115C4⋊C4×C23—22no—
116C4⋊C4×C23—22no—
117C4⋊C4×C23—22no—
118C4⋊C4×C23—22no—
119C4⋊C4×C23—22no—
120C4⋊C4×C23—22no—
121C4⋊C4×C23—22no—
122C4⋊C4×C23—22no—
123C4⋊C4×C23—22no—
124C4⋊C4×C23—22no—
125C4⋊C4×C23—22no—
126C22⋊C4×C23—22no—
127C22⋊C4×C23—22no—
128C22⋊C4×C23—22no—
129C22⋊C4×C23—21no—
130C22⋊C4×C23—21no—
131C22⋊C4×C23—21no—
132C22⋊C4×C23—21no—
133C22⋊C4×C23—22no—
134C22⋊C4×C23—22no—
135C22⋊C4×C23—22no—
136C22⋊C4×C23—22no—
137C22⋊C4×C23—22no—
138C22⋊C4×C23—42no—
139C22⋊C4×C23—42no—
140C22⋊C4×C23C22⋊C412yes—
141C22⋊C4×C23C22⋊C412yes—
142C22⋊C4×C23C22⋊C412yes—
143C22⋊C4×C23C22⋊C412yes—
144C42×C23—22no—
145C42×C23—22no—
146C42×C23—22no—
147C42×C23—22no—
148#12726C2311yesmaximal, characteristic, direct product factor, Sylow
149C23×C23—22no—
150C23×C23—22no—
151C23×C23—21no—
152C23×C23—41no—
153C23×C23—81no—
154C23×C23—81no—
155C23×C23—81no—
156C23×C23C2×C22⋊C411yescharacteristic
157C23×C23C2×C22⋊C411yescharacteristic
158C23×C23C2×C22⋊C411yescharacteristic
159C23×C23#12012yes—
160Q8×C23—82no—
161D4×C23—42no—
162D4×C23—42no—
163D4×C23—42no—
164D4×C23—42no—
165D4×C23—82no—
166C2×C4×C23—22no—
167C2×C4×C23—21no—
168C2×C4×C23—22no—
169C2×C4×C23—22no—
170C2×C4×C23—21no—
171C2×C4×C23—22no—
172C2×C4×C23—22no—
173C2×C4×C23—21no—
174C2×C4×C23—42no—
175C2×C4×C23—42no—
176C2×C4×C23—41no—
177C2×C4×C23—41no—
178C2×C4×C23—41no—
179C2×C4×C23—81no—
180C2×C4×C23—82no—
181C2×C4×C23—82no—
182C2×C4×C23—81no—
183C2×C4×C23—81no—
184C2×C4×C23—81no—
185C2×C4×C23—82no—
186C2×C4×C23—162no—
187C2×C4×C23#12012yes—
188C8×C23—82no—
189#3491C2×C2311yescharacteristic
190#2121C2×C2312yes—
191C2×#352C2×C2312yescomplemented
192#1829C2×C2311yescharacteristic, complemented
193#1828C2×C2311yescharacteristic, complemented
194C22×C23—21no—
195C22×C23—21no—
196C22×C23—22no—
197C22×C23—41no—
198C22×C23—41no—
199C22×C23—42no—
200C22×C23—41no—
201C22×C23—42no—
202C22×C23—41no—
203C22×C23—82no—
204C22×C23—82no—
205C22×C23—161no—
206C22×C23#37911yescharacteristic
207C22×C23#41012yes—
208C4×C23—22no—
209C4×C23—42no—
210C4×C23—42no—
211C4×C23—42no—
212C4×C23—42no—
213C4×C23—41no—
214C4×C23—41no—
215C4×C23—41no—
216C4×C23—162no—
217#552—22no—
218#552—22no—
219#551—22no—
220#551—22no—
221#543C4×C2311yescharacteristic, complemented
222#539—22no—
223#534C22×C2311yescharacteristic
224#529—22no—
225C2×#123C4×C2311yescharacteristic, complemented
226C2×#123C22×C2311yescharacteristic
227#518C4×C2311yescharacteristic, complemented
228#518C22×C2311yescharacteristic
229C2×#117C4×C2311yescharacteristic, complemented
230C2×#100C22×C2312yes—
231C2×#99C22×C2312yes—
232#352—22no—
233#352—22no—
234#352—22no—
235AΣL(1,𝔽16)—22no—
236#344—22no—
237#342—22no—
238C2×C23—21no—
239C2×C23—22no—
240C2×C23—42no—
241C2×C23—41no—
242C2×C23—41no—
243C2×C23—81no—
244C2×C23#178211yescharacteristic
245C22×D4C2×C4×C2311yescharacteristic
246C22×D4C23×C2311yescharacteristic
247C23×C4C2×C4×C2311yescharacteristic
248C2×M16—22no—
249#126—22no—
250#126—22no—
251#126—22no—
252#126—22no—
253#125—22no—
254#125—22no—
255#125—22no—
256#124—22no—
257#123—22no—
258#123—22no—
259#123—22no—
260#123—21no—
261#123—21no—
262#123—21no—
263#123—22no—
264#123D4×C2312yes—
265#123D4×C2312yes—
266#123D4×C2312yes—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272#121—22no—
273#120—21no—
274#120D4×C2312yes—
275C4×Q8—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281#117—22no—
282#117—22no—
283#117—21no—
284#117—22no—
285#117—22no—
286#117D4×C2312yes—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4C2×C4×C2311yescharacteristic
295C2×C22⋊C4C2×C4×C2311yescharacteristic
296C2×C22⋊C4C2×C4×C2311yescharacteristic
297C2×C22⋊C4C2×C4×C2311yescharacteristic
298C2×C22⋊C4D4×C2312yes—
299C2×C42—21no—
300#100—22no—
301#100—22no—
302#99—22no—
303#99—22no—
304C23#1272611yescharacteristic, direct product factor, Sylow
305C24C22⋊C4×C2311yescharacteristic
306C24C22×C4×C2311yescharacteristic
307C24C2×D4×C2311yescharacteristic
308C2×Q8—22no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—22no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4C2×D4×C2312yes—
315C2×D4C2×D4×C2312yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4C22⋊C4×C2311yescharacteristic
327C22×C4C22⋊C4×C2311yescharacteristic
328C22×C4C22⋊C4×C2311yescharacteristic
329C22×C4C2×D4×C2311yescharacteristic
330M16—42no—
331M16—42no—
332C2×C8—42no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4C22⋊C4×C2312yes—
360C22⋊C4C22⋊C4×C2312yes—
361C22⋊C4C22⋊C4×C2312yes—
362C22⋊C4C22⋊C4×C2312yes—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C23—22no—
368C23—22no—
369C23—21no—
370C23—41no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C2×C22⋊C4×C2311yescharacteristic
375C23C2×C22⋊C4×C2311yescharacteristic
376C23C2×C22⋊C4×C2311yescharacteristic
377C23C23×#12012yes—
378Q8—82no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—82no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—162no—
405C2×C4C23×#12012yes—
406C8—82no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22C23×#37911yescharacteristic
420C22C23×#41012yes—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—162no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—41no—
434C2—41no—
435C2—81no—
436C2C23×#178211yescharacteristic
subgroupG/H
center, socleC2×C23#1782
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#534C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C23×#518 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#352C2semidirect
4. C23×#1829C2semidirect
5. C23×#1828C2semidirect
10. C23×#543C4semidirect
14. C2×C23×#123C4semidirect
16. C23×#518C4semidirect
18. C2×C23×#117C4semidirect
148. #12726C23direct (N × H)
191. C2×#352C2×C23semidirect
192. #1829C2×C23semidirect
193. #1828C2×C23semidirect
221. #543C4×C23semidirect
225. C2×#123C4×C23semidirect
227. #518C4×C23semidirect
229. C2×#117C4×C23semidirect
304. C23#12726direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34912
C23×#21212
C2×C23×#3522
C23×#18292
C23×#18282
#1272623