Finite Group Databasebeta

← C23×#61900 #19209526

Subgroups — C23×#61900 #19209526

Order 5888.

1692 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3523C211yesmaximal, characteristic
2C2×C23×#557C211yesmaximal, characteristic, complemented
3C23×#3398C216yesmaximal, complemented
4C23×#3395C212yesmaximal, complemented
5C23×#3374C216yesmaximal, complemented
6C23×#3359C212yesmaximal, complemented
7C23×#3218C213yesmaximal
8C23×#3209C213yesmaximal, complemented
9C23×#3113C213yesmaximal, complemented
10C23×#3108C213yesmaximal
11C23×#2984C211yesmaximal, characteristic, complemented
12C23×#557—21no—
13C23×#557—23no—
14C23×#557C2212yes—
15C23×#557C2216yes—
16C23×#557C2211yescharacteristic
17C23×#557C2213yes—
18C23×#556C2213yes—
19C23×#552—23no—
20D4×Q8×C23—21no—
21D4×Q8×C23C2211yescharacteristic
22C23×#541—23no—
23C23×#533C2213yes—
24C2×C23×C4⋊Q8C2213yes—
25C23×#529C2213yes—
26C2×C23×#125—23no—
27C23×#519—21no—
28C2×C4×Q8×C23—21no—
29C2×C4×Q8×C23C2213yes—
30C2×C4×Q8×C23C2211yescharacteristic
31C23×#497C2216yes—
32C23×#494C2216yes—
33C23×#491C2216yes—
34C23×#487C2216yes—
35C23×#483C2213yes—
36C23×#480C2211yescharacteristic
37C23×#479C2216yes—
38C23×#475C2216yes—
39C23×#464C2216yes—
40C23×#462C22112yes—
41C23×#462C2216yes—
42C23×#462C2216yes—
43C23×#461C2216yes—
44C23×#451C2213yes—
45C23×#451C2216yes—
46C23×#448C2213yes—
47C23×#446C2216yes—
48C8×Q8×C23C2212yes—
49C23×#441C22112yes—
50C4×Q16×C23C2216yes—
51C4×SD16×C23C2216yes—
52C23×#432C2213yes—
53C23×#431C2213yes—
54C22×Q8×C23—21no—
55C22×Q8×C23D411yescharacteristic
56C22×Q8×C23C2313yes—
57C2×Q16×C23C2316yes—
58C2×Q16×C23C23112yes—
59C2×SD16×C23C2316yes—
60C23×C4⋊Q8—23no—
61C23×C4⋊Q8—23no—
62C23×C4⋊Q8—23no—
63C23×C4⋊Q8C2313yes—
64C23×C4⋊Q8C2316yes—
65C23×C4⋊Q8C2316yes—
66C23×C4⋊Q8C2313yes—
67C23×#126—26no—
68C23×#126—23no—
69C23×#125—26no—
70C23×#125—23no—
71C23×#125—26no—
72C23×#125—26no—
73C23×#125—212no—
74C23×#125—23no—
75C23×#125C2313yes—
76C23×#124C2313yes—
77C23×#123—23no—
78C23×C22⋊Q8—23no—
79C23×C22⋊Q8—23no—
80C23×C22⋊Q8—23no—
81C23×C22⋊Q8C2313yes—
82C23×#121C2311yescharacteristic
83C4×Q8×C23—21no—
84C4×Q8×C23—26no—
85C4×Q8×C23—22no—
86C4×Q8×C23—23no—
87C4×Q8×C23—23no—
88C4×Q8×C23—23no—
89C4×Q8×C23—23no—
90C4×Q8×C23D411yescharacteristic
91C4×Q8×C23D413yes—
92C4×Q8×C23C2316yes—
93C4×Q8×C23C2316yes—
94C4×Q8×C23C2313yes—
95C4×Q8×C23C2316yes—
96C4×Q8×C23C2311yescharacteristic
97C4×Q8×C23C2313yes—
98C4×D4×C23—23no—
99C4×D4×C23C2313yes—
100C23×#117—23no—
101C23×#117—23no—
102C23×#117—23no—
103C2×C4⋊C4×C23—23no—
104C2×C4⋊C4×C23—23no—
105C2×C4⋊C4×C23—23no—
106C2×C4⋊C4×C23D413yes—
107C2×C4⋊C4×C23C2311yescharacteristic
108C2×C42×C23—23no—
109C2×C42×C23C2313yes—
110C23×#107C2312yes—
111C23×#106C2316yes—
112C23×C4⋊C8C2316yes—
113C23×C4⋊C8C2316yes—
114C23×#103C2316yes—
115C23×#103C2316yes—
116C23×#103C2316yes—
117C23×#103C23112yes—
118C23×#102C2312yes—
119C23×C22⋊C8C2311yescharacteristic
120C23×C22⋊C8C2313yes—
121C23×#97C2316yes—
122C4×C8×C23C2316yes—
123C2×Q8×C23—24no—
124C2×Q8×C23—23no—
125C2×Q8×C23—23no—
126C2×Q8×C23—46no—
127C2×Q8×C23—44no—
128C2×Q8×C23C2×D413yes—
129C2×Q8×C23C2×D411yescharacteristic
130C2×Q8×C23C2416yes—
131C2×Q8×C23C2413yes—
132C2×D4×C23—21no—
133C2×D4×C23C2411yescharacteristic
134C22×C4×C23—21no—
135C22×C4×C23—23no—
136C22×C4×C23—23no—
137C22×C4×C23C2×D413yes—
138C22×C4×C23C2×D413yes—
139C22×C4×C23C2411yescharacteristic
140Q16×C23—412no—
141Q16×C23—46no—
142SD16×C23—46no—
143C2×C8×C23C2412yes—
144C2×C8×C23C2416yes—
145C4⋊C4×C23—26no—
146C4⋊C4×C23—26no—
147C4⋊C4×C23—26no—
148C4⋊C4×C23—26no—
149C4⋊C4×C23—26no—
150C4⋊C4×C23—26no—
151C4⋊C4×C23—21no—
152C4⋊C4×C23—23no—
153C4⋊C4×C23—23no—
154C4⋊C4×C23—23no—
155C4⋊C4×C23—23no—
156C4⋊C4×C23—23no—
157C4⋊C4×C23—23no—
158C4⋊C4×C23—26no—
159C4⋊C4×C23—23no—
160C4⋊C4×C23C2×D413yes—
161C4⋊C4×C23C2×D413yes—
162C4⋊C4×C23C2×D416yes—
163C4⋊C4×C23C2412yes—
164C4⋊C4×C23C2413yes—
165C4⋊C4×C23C2411yescharacteristic
166C22⋊C4×C23—23no—
167C22⋊C4×C23—21no—
168C22⋊C4×C23—23no—
169C22⋊C4×C23—23no—
170C22⋊C4×C23—23no—
171C42×C23—26no—
172C42×C23—26no—
173C42×C23—26no—
174C42×C23—23no—
175C42×C23C2×D413yes—
176C42×C23C2×D413yes—
177C42×C23C2413yes—
178C42×C23C2413yes—
179#61900C2311yesmaximal, characteristic, direct product factor, Sylow
180C23×C23—21no—
181C23×C23C22×D411yescharacteristic
182Q8×C23—26no—
183Q8×C23—26no—
184Q8×C23—26no—
185Q8×C23—26no—
186Q8×C23—46no—
187Q8×C23—46no—
188Q8×C23#13714yes—
189D4×C23—22no—
190D4×C23—42no—
191C2×C4×C23—26no—
192C2×C4×C23—26no—
193C2×C4×C23—22no—
194C2×C4×C23—22no—
195C2×C4×C23—23no—
196C2×C4×C23—21no—
197C2×C4×C23—23no—
198C2×C4×C23—26no—
199C2×C4×C23—46no—
200C2×C4×C23—42no—
201C2×C4×C23—46no—
202C2×C4×C23C22×D413yes—
203C2×C4×C23C22×D413yes—
204C2×C4×C23C22×D411yescharacteristic
205C2×C4×C23C22×D413yes—
206C2×C4×C23C22×D413yes—
207C2×C4×C23C2511yescharacteristic
208C8×C23—26no—
209C8×C23—22no—
210#3523C2×C2311yescharacteristic
211C2×#557C2×C2311yescharacteristic, complemented
212#3398C2×C2316yescomplemented
213#3395C2×C2312yescomplemented
214#3374C2×C2316yescomplemented
215#3359C2×C2312yescomplemented
216#3218C2×C2313yes—
217#3209C2×C2313yescomplemented
218#3113C2×C2313yescomplemented
219#3108C2×C2313yes—
220#2984C2×C2311yescharacteristic, complemented
221C22×C23—21no—
222C22×C23—21no—
223C22×C23—41no—
224C22×C23—41no—
225C22×C23—41no—
226C22×C23#57812yes—
227C22×C23C23×D411yescharacteristic
228C4×C23—21no—
229C4×C23—23no—
230C4×C23—23no—
231C4×C23—23no—
232C4×C23—46no—
233C4×C23—43no—
234C4×C23—43no—
235C4×C23—42no—
236C4×C23—41no—
237C4×C23C2×#13716yes—
238C4×C23#58512yes—
239#557—21no—
240#557—23no—
241#557C22×C2312yes—
242#557C22×C2316yes—
243#557C22×C2311yescharacteristic
244#557C22×C2313yes—
245#556C22×C2313yes—
246#552—23no—
247D4×Q8—21no—
248D4×Q8C22×C2311yescharacteristic
249#541—23no—
250#533C22×C2313yes—
251C2×C4⋊Q8C22×C2313yes—
252#529C22×C2313yes—
253C2×#125—23no—
254#519—21no—
255C2×C4×Q8—21no—
256C2×C4×Q8C22×C2313yes—
257C2×C4×Q8C22×C2311yescharacteristic
258#497C22×C2316yes—
259#494C22×C2316yes—
260#491C22×C2316yes—
261#487C22×C2316yes—
262#483C22×C2313yes—
263#480C22×C2311yescharacteristic
264#479C22×C2316yes—
265#475C22×C2316yes—
266#464C22×C2316yes—
267#462C22×C2316yes—
268#462C22×C2316yes—
269#462C22×C23112yes—
270#461C22×C2316yes—
271#451C22×C2313yes—
272#451C22×C2316yes—
273#448C22×C2313yes—
274#446C22×C2316yes—
275C8×Q8C22×C2312yes—
276#441C22×C23112yes—
277C4×Q16C22×C2316yes—
278C4×SD16C22×C2316yes—
279#432C22×C2313yes—
280#431C22×C2313yes—
281C2×C23—22no—
282C2×C23—81no—
283C2×C23#348711yescharacteristic
284C2×C23C22×#13711yescharacteristic
285C2×C23C2×#57811yescharacteristic
286C22×Q8—21no—
287C22×Q8D4×C2311yescharacteristic
288C22×Q8C23×C2313yes—
289C2×Q16C23×C2316yes—
290C2×Q16C23×C23112yes—
291C2×SD16C23×C2316yes—
292C4⋊Q8—23no—
293C4⋊Q8—23no—
294C4⋊Q8—23no—
295C4⋊Q8C23×C2316yes—
296C4⋊Q8C23×C2316yes—
297C4⋊Q8C23×C2313yes—
298C4⋊Q8C23×C2313yes—
299#126—26no—
300#126—23no—
301#125—26no—
302#125—26no—
303#125—212no—
304#125—26no—
305#125—23no—
306#125—23no—
307#125C23×C2313yes—
308#124C23×C2313yes—
309#123—23no—
310C22⋊Q8—23no—
311C22⋊Q8—23no—
312C22⋊Q8—23no—
313C22⋊Q8C23×C2313yes—
314#121C23×C2311yescharacteristic
315C4×Q8—26no—
316C4×Q8—22no—
317C4×Q8—21no—
318C4×Q8—23no—
319C4×Q8—23no—
320C4×Q8—23no—
321C4×Q8—23no—
322C4×Q8D4×C2311yescharacteristic
323C4×Q8D4×C2313yes—
324C4×Q8C23×C2316yes—
325C4×Q8C23×C2316yes—
326C4×Q8C23×C2313yes—
327C4×Q8C23×C2316yes—
328C4×Q8C23×C2311yescharacteristic
329C4×Q8C23×C2313yes—
330C4×D4—23no—
331C4×D4C23×C2313yes—
332#117—23no—
333#117—23no—
334#117—23no—
335C2×C4⋊C4—23no—
336C2×C4⋊C4—23no—
337C2×C4⋊C4—23no—
338C2×C4⋊C4D4×C2313yes—
339C2×C4⋊C4C23×C2311yescharacteristic
340C2×C42—23no—
341C2×C42C23×C2313yes—
342#107C23×C2312yes—
343#106C23×C2316yes—
344C4⋊C8C23×C2316yes—
345C4⋊C8C23×C2316yes—
346#103C23×C2316yes—
347#103C23×C2316yes—
348#103C23×C2316yes—
349#103C23×C23112yes—
350#102C23×C2312yes—
351C22⋊C8C23×C2311yescharacteristic
352C22⋊C8C23×C2313yes—
353#97C23×C2316yes—
354C4×C8C23×C2316yes—
355C23#6190011yescharacteristic, direct product factor, Sylow
356C2×Q8—24no—
357C2×Q8—23no—
358C2×Q8—23no—
359C2×Q8—46no—
360C2×Q8—44no—
361C2×Q8C2×D4×C2313yes—
362C2×Q8C2×D4×C2311yescharacteristic
363C2×Q8C24×C2316yes—
364C2×Q8C24×C2313yes—
365C2×D4—21no—
366C2×D4C24×C2311yescharacteristic
367C22×C4—21no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4C2×D4×C2313yes—
371C22×C4C2×D4×C2313yes—
372C22×C4C24×C2311yescharacteristic
373Q16—412no—
374Q16—46no—
375SD16—46no—
376C2×C8C24×C2312yes—
377C2×C8C24×C2316yes—
378C4⋊C4—26no—
379C4⋊C4—26no—
380C4⋊C4—26no—
381C4⋊C4—26no—
382C4⋊C4—26no—
383C4⋊C4—26no—
384C4⋊C4—21no—
385C4⋊C4—23no—
386C4⋊C4—23no—
387C4⋊C4—23no—
388C4⋊C4—23no—
389C4⋊C4—23no—
390C4⋊C4—23no—
391C4⋊C4—26no—
392C4⋊C4—23no—
393C4⋊C4C2×D4×C2313yes—
394C4⋊C4C2×D4×C2313yes—
395C4⋊C4C2×D4×C2316yes—
396C4⋊C4C24×C2312yes—
397C4⋊C4C24×C2313yes—
398C4⋊C4C24×C2311yescharacteristic
399C22⋊C4—23no—
400C22⋊C4—21no—
401C22⋊C4—23no—
402C22⋊C4—23no—
403C22⋊C4—23no—
404C42—26no—
405C42—26no—
406C42—26no—
407C42—23no—
408C42C2×D4×C2313yes—
409C42C2×D4×C2313yes—
410C42C24×C2313yes—
411C42C24×C2313yes—
412C23—21no—
413C23C22×D4×C2311yescharacteristic
414Q8—26no—
415Q8—26no—
416Q8—26no—
417Q8—26no—
418Q8—46no—
419Q8—46no—
420Q8C23×#13714yes—
421D4—22no—
422D4—42no—
423C2×C4—26no—
424C2×C4—26no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—23no—
428C2×C4—21no—
429C2×C4—23no—
430C2×C4—26no—
431C2×C4—46no—
432C2×C4—42no—
433C2×C4—46no—
434C2×C4C22×D4×C2313yes—
435C2×C4C22×D4×C2313yes—
436C2×C4C22×D4×C2311yescharacteristic
437C2×C4C22×D4×C2313yes—
438C2×C4C22×D4×C2313yes—
439C2×C4C25×C2311yescharacteristic
440C8—26no—
441C8—22no—
442C22—21no—
443C22—21no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22C23×#57812yes—
448C22C23×D4×C2311yescharacteristic
449C4—21no—
450C4—23no—
451C4—23no—
452C4—23no—
453C4—46no—
454C4—43no—
455C4—43no—
456C4—42no—
457C4—41no—
458C4C2×C23×#13716yes—
459C4C23×#58512yes—
460C2—22no—
461C2—81no—
462C2C23×#348711yescharacteristic
463C2C22×C23×#13711yescharacteristic
464C2C2×C23×#57811yescharacteristic
subgroupG/H
center, socleC22×C23C23×D4
commutator, Frattini, cosocleC2×C4C25×C23
prime radicalC23×#121C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C4×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#557C2semidirect
3. C23×#3398C2semidirect
4. C23×#3395C2semidirect
5. C23×#3374C2semidirect
6. C23×#3359C2semidirect
8. C23×#3209C2semidirect
9. C23×#3113C2semidirect
11. C23×#2984C2semidirect
179. #61900C23direct (N × H)
211. C2×#557C2×C23semidirect
212. #3398C2×C23semidirect
213. #3395C2×C23semidirect
214. #3374C2×C23semidirect
215. #3359C2×C23semidirect
217. #3209C2×C23semidirect
218. #3113C2×C23semidirect
220. #2984C2×C23semidirect
355. C23#61900direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35232
C2×C23×#5572
C23×#33982
C23×#33952
C23×#33742
C23×#33592
C23×#32182
C23×#32092
C23×#31132
C23×#31082
C23×#29842
#6190023