Finite Group Databasebeta

← C23×#10267 #19157893

Subgroups — C23×#10267 #19157893

Order 5888.

646 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#486C211yesmaximal, characteristic, complemented
2C2×C23×#476C211yesmaximal, characteristic, complemented
3C2×C23×#459C211yesmaximal, characteristic
4C23×#1954C211yesmaximal, characteristic, complemented
5C2×C23×#333C211yesmaximal, characteristic, complemented
6C2×C23×#331C211yesmaximal, characteristic, complemented
7C23×#1277C211yesmaximal, characteristic, complemented
8C22×D8×C23—21no—
9C2×C23×C4⋊Q8C2211yescharacteristic
10C2×C23×#127C2211yescharacteristic
11C2×C4×D4×C23C2211yescharacteristic
12C23×#486—22no—
13C23×#486—22no—
14C23×#476—22no—
15C23×#476—22no—
16C23×#459—22no—
17C23×#459—22no—
18C2×C23×#106—21no—
19C2×C23×C4⋊C8C2211yescharacteristic
20C2×C23×C4⋊C8C2211yescharacteristic
21C2×C23×C4⋊C8C2211yescharacteristic
22C2×C23×#102—21no—
23C2×C23×#102—22no—
24C2×C23×#102—21no—
25C2×C23×C22⋊C8—22no—
26C2×C4×C8×C23C2211yescharacteristic
27C23×#333—22no—
28C23×#331—22no—
29C22×Q8×C23—21no—
30C22×D4×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23D411yescharacteristic
34C23×C4×C23—21no—
35C2×D8×C23—24no—
36C2×D8×C23—42no—
37C2×D8×C23—42no—
38C22×C8×C23—21no—
39C22×C8×C23—21no—
40C22×C8×C23—21no—
41C22×C8×C23D412yes—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8—21no—
44C23×C4⋊Q8—21no—
45C23×C4⋊Q8D412yes—
46C23×#127—21no—
47C23×#127—21no—
48C23×#127—21no—
49C23×#127D412yes—
50C4×D4×C23—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23D411yescharacteristic
57C2×C22⋊C4×C23—21no—
58C2×C42×C23C2311yescharacteristic
59C23×#106—24no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8D412yes—
65C23×C4⋊C8D412yes—
66C23×#102—24no—
67C23×#102—24no—
68C23×#102—28no—
69C23×C22⋊C8—44no—
70C4×C8×C23—21no—
71C4×C8×C23—21no—
72C24×C23—21no—
73C24×C23—41no—
74C2×Q8×C23—22no—
75C2×Q8×C23—41no—
76C2×Q8×C23—82no—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—22no—
80C2×D4×C23—22no—
81C2×D4×C23—22no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—21no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—41no—
88C2×D4×C23—41no—
89C2×D4×C23—41no—
90C2×D4×C23—42no—
91C2×D4×C23—41no—
92C2×D4×C23—42no—
93C2×D4×C23—41no—
94C2×D4×C23—82no—
95C2×D4×C23SD1612yes—
96C22×C4×C23—21no—
97C22×C4×C23—41no—
98C22×C4×C23—41no—
99C22×C4×C23—41no—
100C22×C4×C23—42no—
101C22×C4×C23—42no—
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104C22×C4×C23#4111yescharacteristic
105D8×C23—48no—
106C2×C8×C23—22no—
107C2×C8×C23—22no—
108C2×C8×C23—22no—
109C2×C8×C23—22no—
110C2×C8×C23—22no—
111C2×C8×C23—41no—
112C2×C8×C23—41no—
113C2×C8×C23—41no—
114C2×C8×C23—41no—
115C2×C8×C23—41no—
116C2×C8×C23—42no—
117C4⋊C4×C23—22no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—22no—
120C4⋊C4×C23—21no—
121C4⋊C4×C23—41no—
122C4⋊C4×C23—41no—
123C22⋊C4×C23—41no—
124C22⋊C4×C23—41no—
125C42×C23C2×D411yescharacteristic
126C42×C23C2×D411yescharacteristic
127C42×C23C2×D411yescharacteristic
128C42×C23C2×D411yescharacteristic
129#10267C2311yesmaximal, characteristic, direct product factor, Sylow
130C23×C23—22no—
131C23×C23—41no—
132C23×C23—42no—
133C23×C23—41no—
134C23×C23—42no—
135C23×C23—42no—
136C23×C23—82no—
137C23×C23—81no—
138C23×C23—81no—
139C23×C23—82no—
140C23×C23#12111yescharacteristic
141Q8×C23—82no—
142Q8×C23—82no—
143D4×C23—22no—
144D4×C23—22no—
145D4×C23—41no—
146D4×C23—44no—
147D4×C23—42no—
148D4×C23—41no—
149D4×C23—42no—
150D4×C23—82no—
151D4×C23—82no—
152D4×C23—81no—
153D4×C23—81no—
154D4×C23—81no—
155D4×C23—81no—
156C2×C4×C23—22no—
157C2×C4×C23—41no—
158C2×C4×C23—42no—
159C2×C4×C23—42no—
160C2×C4×C23—82no—
161C2×C4×C23—82no—
162C2×C4×C23—81no—
163C2×C4×C23—81no—
164C2×C4×C23#12011yescharacteristic
165C2×C4×C23#12011yescharacteristic
166C2×C4×C23#12011yescharacteristic
167C2×C4×C23#12111yescharacteristic
168C2×C4×C23#12111yescharacteristic
169C2×C4×C23#12711yescharacteristic
170C2×C4×C23C2×SD1611yescharacteristic
171C2×C4×C23#13511yescharacteristic
172C2×C4×C23#13512yes—
173C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C23AGL(1,ℤ/8)12yes—
175C2×C4×C23#13711yescharacteristic
176C8×C23—42no—
177C8×C23—42no—
178C8×C23—42no—
179C8×C23—81no—
180C2×#486C2×C2311yescharacteristic, complemented
181C2×#476C2×C2311yescharacteristic, complemented
182C2×#459C2×C2311yescharacteristic
183#1954C2×C2311yescharacteristic, complemented
184C2×#333C2×C2311yescharacteristic, complemented
185C2×#331C2×C2311yescharacteristic, complemented
186#1277C2×C2311yescharacteristic, complemented
187C22×C23—44no—
188C22×C23—42no—
189C22×C23—82no—
190C22×C23—81no—
191C22×C23—81no—
192C22×C23—82no—
193C22×C23—82no—
194C22×C23—82no—
195C22×C23—162no—
196C22×C23—162no—
197C22×C23#39211yescharacteristic
198C22×C23#46312yes—
199C22×C23#46511yescharacteristic
200C22×C23#46811yescharacteristic
201C22×C23#47212yes—
202C4×C23—21no—
203C4×C23—21no—
204C4×C23—81no—
205C4×C23—82no—
206C4×C23#44811yescharacteristic
207C4×C23#44911yescharacteristic
208C4×C23#44912yes—
209C4×C23#45011yescharacteristic
210C4×C23#45211yescharacteristic
211C4×C23#49511yescharacteristic
212C4×C23#49611yescharacteristic
213C22×D8—21no—
214C2×C4⋊Q8C22×C2311yescharacteristic
215C2×#127C22×C2311yescharacteristic
216C2×C4×D4C22×C2311yescharacteristic
217#486—22no—
218#486—22no—
219#476—22no—
220#476—22no—
221#459—22no—
222#459—22no—
223C2×#106—21no—
224C2×C4⋊C8C22×C2311yescharacteristic
225C2×C4⋊C8C22×C2311yescharacteristic
226C2×C4⋊C8C22×C2311yescharacteristic
227C2×#102—21no—
228C2×#102—22no—
229C2×#102—21no—
230C2×C22⋊C8—22no—
231C2×C4×C8C22×C2311yescharacteristic
232#333—22no—
233#331—22no—
234C2×C23—82no—
235C2×C23—162no—
236C2×C23#167712yes—
237C2×C23#169712yes—
238C2×C23#200011yescharacteristic
239C2×C23#201211yescharacteristic
240C2×C23#201811yescharacteristic
241C22×Q8—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4D4×C2311yescharacteristic
246C23×C4—21no—
247C2×D8—24no—
248C2×D8—42no—
249C2×D8—42no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8D4×C2312yes—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8D4×C2312yes—
258#127—21no—
259#127—21no—
260#127—21no—
261#127D4×C2312yes—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4D4×C2311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C23×C2311yescharacteristic
271#106—24no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8D4×C2312yes—
277C4⋊C8D4×C2312yes—
278#102—24no—
279#102—24no—
280#102—28no—
281C22⋊C8—44no—
282C4×C8—21no—
283C4×C8—21no—
284C23#1026711yescharacteristic, direct product factor, Sylow
285C24—21no—
286C24—41no—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—82no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—82no—
308C2×D4SD16×C2312yes—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4C2×D4×C2311yescharacteristic
316C22×C4C2×D4×C2311yescharacteristic
317C22×C4#41×C2311yescharacteristic
318D8—48no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42C2×D4×C2311yescharacteristic
339C42C2×D4×C2311yescharacteristic
340C42C2×D4×C2311yescharacteristic
341C42C2×D4×C2311yescharacteristic
342C23—22no—
343C23—41no—
344C23—42no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—82no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23C23×#12111yescharacteristic
353Q8—82no—
354Q8—82no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—44no—
359D4—42no—
360D4—41no—
361D4—42no—
362D4—82no—
363D4—82no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4C23×#12011yescharacteristic
377C2×C4C23×#12011yescharacteristic
378C2×C4C23×#12011yescharacteristic
379C2×C4C23×#12111yescharacteristic
380C2×C4C23×#12111yescharacteristic
381C2×C4C23×#12711yescharacteristic
382C2×C4C2×SD16×C2311yescharacteristic
383C2×C4C23×#13511yescharacteristic
384C2×C4C23×#13512yes—
385C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
386C2×C4C23×AGL(1,ℤ/8)12yes—
387C2×C4C23×#13711yescharacteristic
388C8—42no—
389C8—42no—
390C8—42no—
391C8—81no—
392C22—44no—
393C22—42no—
394C22—82no—
395C22—81no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—162no—
401C22—162no—
402C22C23×#39211yescharacteristic
403C22C23×#46312yes—
404C22C23×#46511yescharacteristic
405C22C23×#46811yescharacteristic
406C22C23×#47212yes—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—82no—
411C4C23×#44811yescharacteristic
412C4C23×#44911yescharacteristic
413C4C23×#44912yes—
414C4C23×#45011yescharacteristic
415C4C23×#45211yescharacteristic
416C4C23×#49511yescharacteristic
417C4C23×#49611yescharacteristic
418C2—82no—
419C2—162no—
420C2C23×#167712yes—
421C2C23×#169712yes—
422C2C23×#200011yescharacteristic
423C2C23×#201211yescharacteristic
424C2C23×#201811yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#459C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×#486C2semidirect
2. C2×C23×#476C2semidirect
4. C23×#1954C2semidirect
5. C2×C23×#333C2semidirect
6. C2×C23×#331C2semidirect
7. C23×#1277C2semidirect
129. #10267C23direct (N × H)
180. C2×#486C2×C23semidirect
181. C2×#476C2×C23semidirect
183. #1954C2×C23semidirect
184. C2×#333C2×C23semidirect
185. C2×#331C2×C23semidirect
186. #1277C2×C23semidirect
284. C23#10267direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4862
C2×C23×#4762
C2×C23×#4592
C23×#19542
C2×C23×#3332
C2×C23×#3312
C23×#12772
#1026723