Finite Group Databasebeta

← C23×#36696 #19184322

Subgroups — C23×#36696 #19184322

Order 5888.

1908 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2843C211yesmaximal, characteristic, complemented
2C23×#2830C213yesmaximal, complemented
3C23×#2829C216yesmaximal, complemented
4C23×#2828C213yesmaximal, complemented
5C23×#2823C211yesmaximal, characteristic, complemented
6C23×#2818C212yesmaximal, complemented
7C23×#2395C213yesmaximal, complemented
8C23×#2392C216yesmaximal, complemented
9C23×#2388C213yesmaximal, complemented
10C4×C23×#127C211yesmaximal, characteristic, complemented
11C4×C23×#126C212yesmaximal, complemented
12C22×#41×C23C2211yescharacteristic
13C2×C23×#127C2211yescharacteristic
14C2×C23×#126C2212yes—
15C2×C4×D4×C23C2213yes—
16C2×C4×D4×C23C2213yes—
17C2×C4×D4×C23C2216yes—
18C2×C23×#117C2213yes—
19C23×#400C2213yes—
20C23×#399C2213yes—
21C23×#398C2213yes—
22C23×#398C2216yes—
23C23×#397C2216yes—
24C23×#397C2213yes—
25C23×#397C2216yes—
26C23×#396C2216yes—
27C23×#396C2216yes—
28C23×#395C2213yes—
29C23×#394C2216yes—
30C23×#394C2216yes—
31C23×#394C2213yes—
32C23×#394C2216yes—
33C23×#394C2216yes—
34C23×#394C2216yes—
35C23×#393C2213yes—
36C23×#392C2216yes—
37C23×#392C2216yes—
38C23×#392C2213yes—
39C23×#390C2213yes—
40C23×#390C2213yes—
41C23×#388C2216yes—
42C23×#387C2216yes—
43C23×#384C2211yescharacteristic
44C23×#383C2212yes—
45C4×C4⋊C4×C23C2216yes—
46C4×C22⋊C4×C23C2216yes—
47C4×C22⋊C4×C23C2213yes—
48C4×C22⋊C4×C23C2213yes—
49C43×C23C2211yescharacteristic
50C2×#41×C23—46no—
51C22×Q8×C23C2311yescharacteristic
52C22×D4×C23C2313yes—
53C22×D4×C23C2313yes—
54C22×D4×C23C2312yes—
55C22×D4×C23C2311yescharacteristic
56C22×D4×C23C2316yes—
57C23×C4×C23C2312yes—
58C23×C4×C23C2311yescharacteristic
59C23×C4×C23C2313yes—
60C23×#127—24no—
61C23×#126—28no—
62C4×D4×C23—412no—
63C4×D4×C23—46no—
64C4×D4×C23—46no—
65C23×#117—46no—
66C2×C4⋊C4×C23C2316yes—
67C2×C4⋊C4×C23C2316yes—
68C2×C4⋊C4×C23C2313yes—
69C2×C4⋊C4×C23C2313yes—
70C2×C4⋊C4×C23C2313yes—
71C2×C4⋊C4×C23C2316yes—
72C2×C4⋊C4×C23C2313yes—
73C2×C22⋊C4×C23C2316yes—
74C2×C22⋊C4×C23C2316yes—
75C2×C22⋊C4×C23C2316yes—
76C2×C22⋊C4×C23C2316yes—
77C2×C22⋊C4×C23C2313yes—
78C2×C22⋊C4×C23C2316yes—
79C2×C22⋊C4×C23C2313yes—
80C2×C22⋊C4×C23C2316yes—
81C2×C22⋊C4×C23C2316yes—
82C2×C22⋊C4×C23C2316yes—
83C2×C42×C23C2313yes—
84C2×C42×C23C2316yes—
85C2×C42×C23C2313yes—
86C2×C42×C23C2311yescharacteristic
87C23×#95C2316yes—
88C23×#95C2316yes—
89C23×#95C2313yes—
90C23×#95C2313yes—
91C23×#95C2313yes—
92C23×#95C2313yes—
93C23×#95C2316yes—
94C23×#95C2316yes—
95C24×C23C2412yes—
96C24×C23C2413yes—
97C24×C23C2411yescharacteristic
98#41×C23—416no—
99C2×Q8×C23—43no—
100C2×D4×C23—26no—
101C2×D4×C23—26no—
102C2×D4×C23—43no—
103C2×D4×C23—43no—
104C2×D4×C23—46no—
105C2×D4×C23—43no—
106C2×D4×C23—43no—
107C2×D4×C23—46no—
108C2×D4×C23—46no—
109C2×D4×C23—46no—
110C2×D4×C23—43no—
111C22×C4×C23—26no—
112C22×C4×C23—212no—
113C22×C4×C23—26no—
114C22×C4×C23—212no—
115C22×C4×C23C2413yes—
116C22×C4×C23C2416yes—
117C22×C4×C23C2416yes—
118C22×C4×C23C2413yes—
119C22×C4×C23C2411yescharacteristic
120C22×C4×C23C2412yes—
121C22×C4×C23C2413yes—
122C22×C4×C23C2411yescharacteristic
123C4⋊C4×C23—212no—
124C4⋊C4×C23—46no—
125C4⋊C4×C23—43no—
126C4⋊C4×C23—43no—
127C4⋊C4×C23—43no—
128C4⋊C4×C23—43no—
129C4⋊C4×C23—46no—
130C22⋊C4×C23—212no—
131C22⋊C4×C23—46no—
132C22⋊C4×C23—46no—
133C22⋊C4×C23—46no—
134C22⋊C4×C23—46no—
135C22⋊C4×C23—46no—
136C22⋊C4×C23—43no—
137C22⋊C4×C23—43no—
138C22⋊C4×C23—46no—
139C22⋊C4×C23—46no—
140C42×C23—212no—
141C42×C23—26no—
142C42×C23—43no—
143C42×C23#4114yescomplemented
144#36696C2311yesmaximal, characteristic, direct product factor, Sylow
145C23×C23—22no—
146C23×C23—23no—
147C23×C23—21no—
148C23×C23—23no—
149C23×C23—26no—
150C23×C23—23no—
151C23×C23—26no—
152C23×C23—23no—
153C23×C23—26no—
154C23×C23—26no—
155C23×C23—23no—
156C23×C23C2511yescharacteristic
157Q8×C23—44no—
158D4×C23—412no—
159D4×C23—412no—
160D4×C23—424no—
161D4×C23—48no—
162D4×C23—44no—
163C2×C4×C23—23no—
164C2×C4×C23—23no—
165C2×C4×C23—26no—
166C2×C4×C23—23no—
167C2×C4×C23—26no—
168C2×C4×C23—26no—
169C2×C4×C23—26no—
170C2×C4×C23—23no—
171C2×C4×C23—23no—
172C2×C4×C23—26no—
173C2×C4×C23—26no—
174C2×C4×C23—26no—
175C2×C4×C23—26no—
176C2×C4×C23—26no—
177C2×C4×C23—412no—
178C2×C4×C23—44no—
179C2×C4×C23—48no—
180C2×C4×C23C2×#4116yes—
181C2×C4×C23H(5,𝔽2)16yes—
182#2843C2×C2311yescharacteristic, complemented
183#2830C2×C2313yescomplemented
184#2829C2×C2316yescomplemented
185#2828C2×C2313yescomplemented
186#2823C2×C2311yescharacteristic, complemented
187#2818C2×C2312yescomplemented
188#2395C2×C2313yescomplemented
189#2392C2×C2316yescomplemented
190#2388C2×C2313yescomplemented
191C4×#127C2×C2311yescharacteristic, complemented
192C4×#126C2×C2312yescomplemented
193C22×C23—43no—
194C22×C23—46no—
195C22×C23—43no—
196C22×C23—46no—
197C22×C23—41no—
198C22×C23—42no—
199C22×C23—43no—
200C22×C23—46no—
201C22×C23—46no—
202C22×C23—43no—
203C22×C23—43no—
204C22×C23C22×#4111yescharacteristic
205C22×C23C2×H(5,𝔽2)13yes—
206C22×C23#58513yes—
207C4×C23—26no—
208C4×C23—46no—
209C4×C23—43no—
210C4×C23—41no—
211C4×C23—42no—
212C4×C23—43no—
213C4×C23—46no—
214C4×C23#56014yes—
215C22×#41C22×C2311yescharacteristic
216C2×#127C22×C2311yescharacteristic
217C2×#126C22×C2312yes—
218C2×C4×D4C22×C2313yes—
219C2×C4×D4C22×C2313yes—
220C2×C4×D4C22×C2316yes—
221C2×#117C22×C2313yes—
222#400C22×C2313yes—
223#399C22×C2313yes—
224#398C22×C2313yes—
225#398C22×C2316yes—
226#397C22×C2316yes—
227#397C22×C2313yes—
228#397C22×C2316yes—
229#396C22×C2316yes—
230#396C22×C2316yes—
231#395C22×C2313yes—
232#394C22×C2316yes—
233#394C22×C2316yes—
234#394C22×C2313yes—
235#394C22×C2316yes—
236#394C22×C2316yes—
237#394C22×C2316yes—
238#393C22×C2313yes—
239#392C22×C2316yes—
240#392C22×C2316yes—
241#392C22×C2313yes—
242#390C22×C2313yes—
243#390C22×C2313yes—
244#388C22×C2316yes—
245#387C22×C2316yes—
246#384C22×C2311yescharacteristic
247#383C22×C2312yes—
248C4×C4⋊C4C22×C2316yes—
249C4×C22⋊C4C22×C2316yes—
250C4×C22⋊C4C22×C2313yes—
251C4×C22⋊C4C22×C2313yes—
252C43C22×C2311yescharacteristic
253C2×C23—81no—
254C2×C23—83no—
255C2×C23—82no—
256C2×C23#346013yes—
257C2×C23C2×#56011yescharacteristic
258C2×C23#352913yes—
259C2×#41—46no—
260C22×Q8C23×C2311yescharacteristic
261C22×D4C23×C2313yes—
262C22×D4C23×C2313yes—
263C22×D4C23×C2316yes—
264C22×D4C23×C2312yes—
265C22×D4C23×C2311yescharacteristic
266C23×C4C23×C2313yes—
267C23×C4C23×C2311yescharacteristic
268C23×C4C23×C2312yes—
269#127—24no—
270#126—28no—
271C4×D4—412no—
272C4×D4—46no—
273C4×D4—46no—
274#117—46no—
275C2×C4⋊C4C23×C2316yes—
276C2×C4⋊C4C23×C2313yes—
277C2×C4⋊C4C23×C2313yes—
278C2×C4⋊C4C23×C2313yes—
279C2×C4⋊C4C23×C2313yes—
280C2×C4⋊C4C23×C2316yes—
281C2×C4⋊C4C23×C2316yes—
282C2×C22⋊C4C23×C2316yes—
283C2×C22⋊C4C23×C2316yes—
284C2×C22⋊C4C23×C2313yes—
285C2×C22⋊C4C23×C2316yes—
286C2×C22⋊C4C23×C2313yes—
287C2×C22⋊C4C23×C2316yes—
288C2×C22⋊C4C23×C2316yes—
289C2×C22⋊C4C23×C2316yes—
290C2×C22⋊C4C23×C2316yes—
291C2×C22⋊C4C23×C2316yes—
292C2×C42C23×C2316yes—
293C2×C42C23×C2313yes—
294C2×C42C23×C2313yes—
295C2×C42C23×C2311yescharacteristic
296#95C23×C2316yes—
297#95C23×C2316yes—
298#95C23×C2313yes—
299#95C23×C2313yes—
300#95C23×C2313yes—
301#95C23×C2313yes—
302#95C23×C2316yes—
303#95C23×C2316yes—
304C23#3669611yescharacteristic, direct product factor, Sylow
305C24C24×C2312yes—
306C24C24×C2313yes—
307C24C24×C2311yescharacteristic
308#41—416no—
309C2×Q8—43no—
310C2×D4—26no—
311C2×D4—26no—
312C2×D4—43no—
313C2×D4—43no—
314C2×D4—46no—
315C2×D4—46no—
316C2×D4—43no—
317C2×D4—43no—
318C2×D4—43no—
319C2×D4—46no—
320C2×D4—46no—
321C22×C4—26no—
322C22×C4—212no—
323C22×C4—26no—
324C22×C4—212no—
325C22×C4C24×C2316yes—
326C22×C4C24×C2312yes—
327C22×C4C24×C2311yescharacteristic
328C22×C4C24×C2313yes—
329C22×C4C24×C2316yes—
330C22×C4C24×C2313yes—
331C22×C4C24×C2313yes—
332C22×C4C24×C2311yescharacteristic
333C4⋊C4—212no—
334C4⋊C4—43no—
335C4⋊C4—43no—
336C4⋊C4—43no—
337C4⋊C4—43no—
338C4⋊C4—46no—
339C4⋊C4—46no—
340C22⋊C4—212no—
341C22⋊C4—46no—
342C22⋊C4—46no—
343C22⋊C4—43no—
344C22⋊C4—43no—
345C22⋊C4—46no—
346C22⋊C4—46no—
347C22⋊C4—46no—
348C22⋊C4—46no—
349C22⋊C4—46no—
350C42—212no—
351C42—26no—
352C42—43no—
353C42#41×C2314yescomplemented
354C23—22no—
355C23—23no—
356C23—21no—
357C23—23no—
358C23—26no—
359C23—23no—
360C23—26no—
361C23—23no—
362C23—26no—
363C23—26no—
364C23—23no—
365C23C25×C2311yescharacteristic
366Q8—44no—
367D4—412no—
368D4—412no—
369D4—48no—
370D4—44no—
371D4—424no—
372C2×C4—23no—
373C2×C4—23no—
374C2×C4—26no—
375C2×C4—23no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—23no—
380C2×C4—23no—
381C2×C4—26no—
382C2×C4—26no—
383C2×C4—26no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—412no—
389C2×C4C2×#41×C2316yes—
390C2×C4C23×H(5,𝔽2)16yes—
391C22—43no—
392C22—46no—
393C22—43no—
394C22—46no—
395C22—41no—
396C22—42no—
397C22—43no—
398C22—46no—
399C22—46no—
400C22—43no—
401C22—43no—
402C22C22×#41×C2311yescharacteristic
403C22C2×C23×H(5,𝔽2)13yes—
404C22C23×#58513yes—
405C4—26no—
406C4—46no—
407C4—41no—
408C4—42no—
409C4—43no—
410C4—46no—
411C4—43no—
412C4C23×#56014yes—
413C2—81no—
414C2—83no—
415C2—82no—
416C2C23×#346013yes—
417C2C2×C23×#56011yescharacteristic
418C2C23×#352913yes—
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2843C2semidirect
2. C23×#2830C2semidirect
3. C23×#2829C2semidirect
4. C23×#2828C2semidirect
5. C23×#2823C2semidirect
6. C23×#2818C2semidirect
7. C23×#2395C2semidirect
8. C23×#2392C2semidirect
9. C23×#2388C2semidirect
10. C4×C23×#127C2semidirect
11. C4×C23×#126C2semidirect
143. C42×C23#41semidirect
144. #36696C23direct (N × H)
182. #2843C2×C23semidirect
183. #2830C2×C23semidirect
184. #2829C2×C23semidirect
185. #2828C2×C23semidirect
186. #2823C2×C23semidirect
187. #2818C2×C23semidirect
188. #2395C2×C23semidirect
189. #2392C2×C23semidirect
190. #2388C2×C23semidirect
191. C4×#127C2×C23semidirect
192. C4×#126C2×C23semidirect
304. C23#36696direct (N × H)
353. C42#41×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28432
C23×#28302
C23×#28292
C23×#28282
C23×#28232
C23×#28182
C23×#23952
C23×#23922
C23×#23882
C4×C23×#1272
C4×C23×#1262
#3669623