Finite Group Databasebeta

← C23×#9210 #19156836

Subgroups — C23×#9210 #19156836

Order 5888.

510 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3035C211yesmaximal, characteristic
2C23×#2965C211yesmaximal, characteristic, complemented
3C23×#1806C211yesmaximal, characteristic, complemented
4C23×#1502C211yesmaximal, characteristic, complemented
5C23×#1492C211yesmaximal, characteristic, complemented
6C23×#1301C211yesmaximal, characteristic, complemented
7C23×#1279C211yesmaximal, characteristic, complemented
8C2×C23×#135—21no—
9C23×#532C2211yescharacteristic
10C4×#41×C23C2211yescharacteristic
11C23×#464—21no—
12C23×#463—21no—
13C23×#462C411yescharacteristic, complemented
14C23×#461C411yescharacteristic, complemented
15C23×#460C411yescharacteristic, complemented
16C23×#459C411yescharacteristic, complemented
17C23×#446—21no—
18C23×#436—21no—
19C23×#435—22no—
20C23×#431—21no—
21C23×#431C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C2×C23×C4⋊C8C2211yescharacteristic
24C2×C23×#104—21no—
25C23×#416—21no—
26C4×M16×C23—21no—
27C2×C4×C8×C23C2211yescharacteristic
28C4×C4⋊C4×C23C2211yescharacteristic
29C23×#339—21no—
30C23×#327—22no—
31C23×D4⋊C8—21no—
32C23×D4⋊C8—21no—
33C2×#41×C23—21no—
34C2×#41×C23D411yescharacteristic
35C23×#135—42no—
36C23×#135—42no—
37C2×Q16×C23—21no—
38C2×SD16×C23—21no—
39C2×SD16×C23—21no—
40C2×D8×C23—21no—
41C2×M16×C23—22no—
42C2×M16×C23—21no—
43C22×C8×C23—21no—
44C22×C8×C23—21no—
45C23×C4⋊Q8C2×C411yescharacteristic
46C23×#127C2×C411yescharacteristic
47C23×#124—21no—
48C23×#121—21no—
49C23×#121—21no—
50C4×Q8×C23C2×C411yescharacteristic
51C4×Q8×C23D411yescharacteristic
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C4×D4×C23—21no—
55C4×D4×C23—21no—
56C4×D4×C23C2×C411yescharacteristic
57C4×D4×C23D411yescharacteristic
58C23×#117—21no—
59C23×#117D411yescharacteristic
60C2×C4⋊C4×C23—21no—
61C2×C42×C23—21no—
62C2×C42×C23—21no—
63C2×C42×C23C2311yescharacteristic
64C23×C4⋊C8—22no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×C4⋊C8—21no—
68C23×C4⋊C8—21no—
69C23×C4⋊C8C2×C411yescharacteristic
70C23×C4⋊C8C2×C411yescharacteristic
71C23×#104—42no—
72C23×#103—21no—
73C23×#103—21no—
74C23×#102—21no—
75C23×#102—21no—
76C23×C22⋊C8—22no—
77C23×C22⋊C8D412yes—
78C23×#97—22no—
79C4×C8×C23—21no—
80C4×C8×C23—21no—
81C4×C8×C23D411yescharacteristic
82C4×C8×C23D411yescharacteristic
83C23×#95—21no—
84#41×C23—21no—
85#41×C23—21no—
86#41×C23—21no—
87#41×C23—21no—
88#41×C23—41no—
89#41×C23—41no—
90#41×C23—41no—
91#41×C23—41no—
92C2×Q8×C23—21no—
93C2×Q8×C23C22⋊C411yescharacteristic
94C2×D4×C23—21no—
95C2×D4×C23—21no—
96C2×D4×C23—21no—
97C2×D4×C23—21no—
98C2×D4×C23—41no—
99C2×D4×C23C22⋊C411yescharacteristic
100C22×C4×C23—21no—
101C22×C4×C23—21no—
102C22×C4×C23—21no—
103C22×C4×C23—41no—
104C22×C4×C23—41no—
105C22×C4×C23C2×D411yescharacteristic
106C22×C4×C23C2×D411yescharacteristic
107C22×C4×C23#4111yescharacteristic
108Q16×C23—42no—
109SD16×C23—42no—
110SD16×C23—42no—
111D8×C23—42no—
112M16×C23—44no—
113M16×C23—41no—
114M16×C23—41no—
115C2×C8×C23—21no—
116C2×C8×C23—21no—
117C2×C8×C23—21no—
118C2×C8×C23—21no—
119C2×C8×C23—22no—
120C2×C8×C23—42no—
121C2×C8×C23—41no—
122C2×C8×C23—41no—
123C2×C8×C23—41no—
124C4⋊C4×C23—21no—
125C4⋊C4×C23—21no—
126C4⋊C4×C23—22no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—41no—
130C4⋊C4×C23C22⋊C411yescharacteristic
131C4⋊C4×C23C22⋊C411yescharacteristic
132C22⋊C4×C23—21no—
133C22⋊C4×C23—21no—
134C22⋊C4×C23—21no—
135C22⋊C4×C23—41no—
136C22⋊C4×C23—41no—
137C42×C23—21no—
138C42×C23—21no—
139C42×C23—21no—
140C42×C23—41no—
141C42×C23—41no—
142C42×C23C22×C411yescharacteristic
143C42×C23C2×D411yescharacteristic
144C42×C23C2×D411yescharacteristic
145C42×C23#4111yescharacteristic
146#9210C2311yesmaximal, characteristic, direct product factor, Sylow
147C23×C23—21no—
148C23×C23—41no—
149C23×C23#12111yescharacteristic
150Q8×C23—21no—
151Q8×C23—21no—
152Q8×C23—41no—
153Q8×C23—41no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23—41no—
157D4×C23—41no—
158D4×C23—41no—
159D4×C23—41no—
160D4×C23—41no—
161D4×C23—41no—
162D4×C23—81no—
163D4×C23—81no—
164D4×C23—81no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—21no—
168C2×C4×C23—21no—
169C2×C4×C23—21no—
170C2×C4×C23—41no—
171C2×C4×C23—42no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—41no—
176C2×C4×C23—42no—
177C2×C4×C23—41no—
178C2×C4×C23—81no—
179C2×C4×C23—81no—
180C2×C4×C23—81no—
181C2×C4×C23—81no—
182C2×C4×C23C2×C22⋊C411yescharacteristic
183C2×C4×C23C4×D411yescharacteristic
184C2×C4×C23C4×D411yescharacteristic
185C2×C4×C23#12011yescharacteristic
186C2×C4×C23#12111yescharacteristic
187C2×C4×C23#12311yescharacteristic
188C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C23#13711yescharacteristic
192C8×C23—22no—
193C8×C23—41no—
194C8×C23—42no—
195C8×C23—41no—
196C8×C23—41no—
197C8×C23—81no—
198#3035C2×C2311yescharacteristic
199#2965C2×C2311yescharacteristic, complemented
200#1806C2×C2311yescharacteristic, complemented
201#1502C2×C2311yescharacteristic, complemented
202#1492C2×C2311yescharacteristic, complemented
203#1301C2×C2311yescharacteristic, complemented
204#1279C2×C2311yescharacteristic, complemented
205C22×C23—21no—
206C22×C23—21no—
207C22×C23—41no—
208C22×C23—41no—
209C22×C23—41no—
210C22×C23—81no—
211C22×C23—81no—
212C22×C23—81no—
213C22×C23#38611yescharacteristic
214C22×C23#46811yescharacteristic
215C22×C23#46911yescharacteristic
216C4×C23—21no—
217C4×C23—21no—
218C4×C23—21no—
219C4×C23—41no—
220C4×C23—41no—
221C4×C23—41no—
222C4×C23—82no—
223C4×C23—81no—
224C4×C23#41711yescharacteristic
225C4×C23#41811yescharacteristic
226C4×C23#44311yescharacteristic
227C4×C23#44411yescharacteristic
228C4×C23#46811yescharacteristic
229C4×C23#46911yescharacteristic
230C2×#135—21no—
231#532C22×C2311yescharacteristic
232C4×#41C22×C2311yescharacteristic
233#464—21no—
234#463—21no—
235#462C4×C2311yescharacteristic, complemented
236#461C4×C2311yescharacteristic, complemented
237#460C4×C2311yescharacteristic, complemented
238#459C4×C2311yescharacteristic, complemented
239#446—21no—
240#436—21no—
241#435—22no—
242#431—21no—
243#431C22×C2311yescharacteristic
244#423C22×C2311yescharacteristic
245C2×C4⋊C8C22×C2311yescharacteristic
246C2×#104—21no—
247#416—21no—
248C4×M16—21no—
249C2×C4×C8C22×C2311yescharacteristic
250C4×C4⋊C4C22×C2311yescharacteristic
251#339—21no—
252#327—22no—
253D4⋊C8—21no—
254D4⋊C8—21no—
255C2×C23—21no—
256C2×C23—21no—
257C2×C23—81no—
258C2×C23—161no—
259C2×C23#187611yescharacteristic
260C2×C23#189511yescharacteristic
261C2×C23#190611yescharacteristic
262C2×#41—21no—
263C2×#41D4×C2311yescharacteristic
264#135—42no—
265#135—42no—
266C2×Q16—21no—
267C2×SD16—21no—
268C2×SD16—21no—
269C2×D8—21no—
270C2×M16—22no—
271C2×M16—21no—
272C22×C8—21no—
273C22×C8—21no—
274C4⋊Q8C2×C4×C2311yescharacteristic
275#127C2×C4×C2311yescharacteristic
276#124—21no—
277#121—21no—
278#121—21no—
279C4×Q8C2×C4×C2311yescharacteristic
280C4×Q8D4×C2311yescharacteristic
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4C2×C4×C2311yescharacteristic
286C4×D4D4×C2311yescharacteristic
287#117—21no—
288#117D4×C2311yescharacteristic
289C2×C4⋊C4—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42C23×C2311yescharacteristic
293C4⋊C8—22no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8C2×C4×C2311yescharacteristic
299C4⋊C8C2×C4×C2311yescharacteristic
300#104—42no—
301#103—21no—
302#103—21no—
303#102—21no—
304#102—21no—
305C22⋊C8—22no—
306C22⋊C8D4×C2312yes—
307#97—22no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8D4×C2311yescharacteristic
311C4×C8D4×C2311yescharacteristic
312#95—21no—
313C23#921011yescharacteristic, direct product factor, Sylow
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8C22⋊C4×C2311yescharacteristic
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4C22⋊C4×C2311yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C2×D4×C2311yescharacteristic
336C22×C4C2×D4×C2311yescharacteristic
337C22×C4#41×C2311yescharacteristic
338Q16—42no—
339SD16—42no—
340SD16—42no—
341D8—42no—
342M16—44no—
343M16—41no—
344M16—41no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—41no—
360C4⋊C4C22⋊C4×C2311yescharacteristic
361C4⋊C4C22⋊C4×C2311yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42C22×C4×C2311yescharacteristic
373C42C2×D4×C2311yescharacteristic
374C42C2×D4×C2311yescharacteristic
375C42#41×C2311yescharacteristic
376C23—21no—
377C23—41no—
378C23C23×#12111yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—41no—
382Q8—41no—
383D4—21no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—81no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4C2×C22⋊C4×C2311yescharacteristic
412C2×C4C4×D4×C2311yescharacteristic
413C2×C4C4×D4×C2311yescharacteristic
414C2×C4C23×#12011yescharacteristic
415C2×C4C23×#12111yescharacteristic
416C2×C4C23×#12311yescharacteristic
417C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
418C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
419C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
420C2×C4C23×#13711yescharacteristic
421C8—22no—
422C8—41no—
423C8—42no—
424C8—41no—
425C8—41no—
426C8—81no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22C23×#38611yescharacteristic
436C22C23×#46811yescharacteristic
437C22C23×#46911yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—81no—
446C4C23×#41711yescharacteristic
447C4C23×#41811yescharacteristic
448C4C23×#44311yescharacteristic
449C4C23×#44411yescharacteristic
450C4C23×#46811yescharacteristic
451C4C23×#46911yescharacteristic
452C2—21no—
453C2—21no—
454C2—81no—
455C2—161no—
456C2C23×#187611yescharacteristic
457C2C23×#189511yescharacteristic
458C2C23×#190611yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC23×#3035C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2965C2semidirect
3. C23×#1806C2semidirect
4. C23×#1502C2semidirect
5. C23×#1492C2semidirect
6. C23×#1301C2semidirect
7. C23×#1279C2semidirect
13. C23×#462C4semidirect
14. C23×#461C4semidirect
15. C23×#460C4semidirect
16. C23×#459C4semidirect
146. #9210C23direct (N × H)
199. #2965C2×C23semidirect
200. #1806C2×C23semidirect
201. #1502C2×C23semidirect
202. #1492C2×C23semidirect
203. #1301C2×C23semidirect
204. #1279C2×C23semidirect
235. #462C4×C23semidirect
236. #461C4×C23semidirect
237. #460C4×C23semidirect
238. #459C4×C23semidirect
313. C23#9210direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30352
C23×#29652
C23×#18062
C23×#15022
C23×#14922
C23×#13012
C23×#12792
#921023