Finite Group Databasebeta

← C23×#36700 #19184326

Subgroups — C23×#36700 #19184326

Order 5888.

1356 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2839C214yesmaximal, complemented
2C23×#2836C214yesmaximal
3C23×#2833C214yesmaximal, complemented
4C23×#2831C214yesmaximal
5C23×#2402C211yesmaximal, characteristic, complemented
6C23×#2402C214yesmaximal, complemented
7C23×#2401C214yesmaximal
8C23×#2399C211yesmaximal, characteristic, complemented
9C23×#2396C212yesmaximal
10C23×#2298C211yesmaximal, characteristic
11C23×#2295C212yesmaximal, complemented
12C2×C4×Q8×C23C2212yes—
13C2×C23×#117C2211yescharacteristic
14C23×#401C2214yes—
15C23×#400C2214yes—
16C23×#400C2214yes—
17C23×#400C2214yes—
18C23×#400C2214yes—
19C23×#400C2214yes—
20C23×#400C2214yes—
21C23×#400C2214yes—
22C23×#400C2214yes—
23C23×#400C2212yes—
24C23×#399C2212yes—
25C23×#398C2214yes—
26C23×#398C2214yes—
27C23×#398C2214yes—
28C23×#398C2214yes—
29C23×#398C2214yes—
30C23×#398C2214yes—
31C23×#398C2214yes—
32C23×#397C2214yes—
33C23×#397C2214yes—
34C23×#397C2214yes—
35C23×#396C2214yes—
36C23×#396C2214yes—
37C23×#395C2214yes—
38C23×#395C2212yes—
39C23×#393C2212yes—
40C23×#391C2214yes—
41C23×#389C2214yes—
42C23×#388C2214yes—
43C23×#387C2214yes—
44C23×#387C2214yes—
45C23×#387C2214yes—
46C4×C4⋊C4×C23C2214yes—
47C4×C4⋊C4×C23C2214yes—
48C4×C4⋊C4×C23C2214yes—
49C4×C4⋊C4×C23C2212yes—
50C4×C4⋊C4×C23C2214yes—
51C4×C4⋊C4×C23C2212yes—
52C4×C22⋊C4×C23C2212yes—
53C4×C22⋊C4×C23C2214yes—
54C23×#376C2211yescharacteristic
55C23×#376C2214yes—
56C43×C23C2211yescharacteristic
57C22×Q8×C23C2312yes—
58C23×C4×C23C2311yescharacteristic
59C4×Q8×C23—28no—
60C23×#117—24no—
61C2×C4⋊C4×C23C2314yes—
62C2×C4⋊C4×C23C2312yes—
63C2×C4⋊C4×C23C2312yes—
64C2×C4⋊C4×C23C2314yes—
65C2×C4⋊C4×C23C2314yes—
66C2×C4⋊C4×C23C2314yes—
67C2×C4⋊C4×C23C2314yes—
68C2×C4⋊C4×C23C2314yes—
69C2×C4⋊C4×C23C2314yes—
70C2×C4⋊C4×C23C2314yes—
71C2×C4⋊C4×C23C2314yes—
72C2×C4⋊C4×C23C2314yes—
73C2×C4⋊C4×C23C2314yes—
74C2×C22⋊C4×C23C2312yes—
75C2×C22⋊C4×C23C2314yes—
76C2×C22⋊C4×C23C2314yes—
77C2×C22⋊C4×C23C2314yes—
78C2×C42×C23C2312yes—
79C2×C42×C23C2312yes—
80C2×C42×C23C2314yes—
81C2×C42×C23C2314yes—
82C2×C42×C23C2311yescharacteristic
83C2×C42×C23C2314yes—
84C2×C42×C23C2311yescharacteristic
85C23×#95C2314yes—
86C23×#95C2314yes—
87C23×#95C2314yes—
88C23×#95C2314yes—
89C23×#95C2314yes—
90C23×#95C2314yes—
91C23×#95C2314yes—
92C23×#95C2314yes—
93C23×#95C2312yes—
94C23×#95C2312yes—
95C23×#95C2314yes—
96C23×#95C2314yes—
97C23×#95C2314yes—
98C23×#95C2314yes—
99C23×#95C2314yes—
100C23×#95C2314yes—
101C23×#95C2314yes—
102C23×#95C2314yes—
103C23×#95C2314yes—
104C24×C23C2411yescharacteristic
105C2×Q8×C23—24no—
106C2×Q8×C23—44no—
107C22×C4×C23—22no—
108C22×C4×C23—24no—
109C22×C4×C23C2414yes—
110C22×C4×C23C2412yes—
111C22×C4×C23C2414yes—
112C22×C4×C23C2414yes—
113C22×C4×C23C2414yes—
114C22×C4×C23C2414yes—
115C22×C4×C23C2414yes—
116C22×C4×C23C2411yescharacteristic
117C22×C4×C23C2412yes—
118C22×C4×C23C2411yescharacteristic
119C4⋊C4×C23—24no—
120C4⋊C4×C23—24no—
121C4⋊C4×C23—28no—
122C4⋊C4×C23—44no—
123C4⋊C4×C23—44no—
124C4⋊C4×C23—44no—
125C4⋊C4×C23—44no—
126C4⋊C4×C23—44no—
127C4⋊C4×C23—44no—
128C4⋊C4×C23—44no—
129C4⋊C4×C23—44no—
130C4⋊C4×C23—44no—
131C4⋊C4×C23—44no—
132C22⋊C4×C23—24no—
133C22⋊C4×C23—44no—
134C22⋊C4×C23—44no—
135C22⋊C4×C23—44no—
136C42×C23—22no—
137C42×C23—28no—
138C42×C23—24no—
139C42×C23—24no—
140C42×C23—28no—
141C42×C23—44no—
142C42×C23#4114yes—
143#36700C2311yesmaximal, characteristic, direct product factor, Sylow
144C23×C23—21no—
145C23×C23—22no—
146C23×C23—24no—
147C23×C23C2511yescharacteristic
148Q8×C23—48no—
149C2×C4×C23—24no—
150C2×C4×C23—22no—
151C2×C4×C23—24no—
152C2×C4×C23—24no—
153C2×C4×C23—21no—
154C2×C4×C23—24no—
155C2×C4×C23—24no—
156C2×C4×C23—24no—
157C2×C4×C23—24no—
158C2×C4×C23—22no—
159C2×C4×C23—24no—
160C2×C4×C23—24no—
161C2×C4×C23—24no—
162C2×C4×C23—24no—
163C2×C4×C23—24no—
164C2×C4×C23—24no—
165C2×C4×C23—24no—
166C2×C4×C23—24no—
167C2×C4×C23—24no—
168C2×C4×C23—24no—
169C2×C4×C23—24no—
170C2×C4×C23—24no—
171C2×C4×C23—24no—
172C2×C4×C23—44no—
173C2×C4×C23C2×#4114yes—
174C2×C4×C23C2×#4112yes—
175C2×C4×C232−1+414yes—
176#2839C2×C2314yescomplemented
177#2836C2×C2314yes—
178#2833C2×C2314yescomplemented
179#2831C2×C2314yes—
180#2402C2×C2311yescharacteristic, complemented
181#2402C2×C2314yescomplemented
182#2401C2×C2314yes—
183#2399C2×C2311yescharacteristic, complemented
184#2396C2×C2312yes—
185#2298C2×C2311yescharacteristic
186#2295C2×C2312yescomplemented
187C22×C23—41no—
188C22×C23—42no—
189C22×C23—44no—
190C22×C23C22×#4111yescharacteristic
191C22×C23C2×2−1+412yes—
192C22×C23#58514yes—
193C4×C23—24no—
194C4×C23—41no—
195C4×C23—42no—
196C4×C23—44no—
197C4×C23—44no—
198C4×C23—44no—
199C4×C23—44no—
200C4×C23—44no—
201C4×C23—44no—
202C4×C23#54114yes—
203C2×C4×Q8C22×C2312yes—
204C2×#117C22×C2311yescharacteristic
205#401C22×C2314yes—
206#400C22×C2314yes—
207#400C22×C2314yes—
208#400C22×C2314yes—
209#400C22×C2314yes—
210#400C22×C2314yes—
211#400C22×C2314yes—
212#400C22×C2314yes—
213#400C22×C2312yes—
214#400C22×C2314yes—
215#399C22×C2312yes—
216#398C22×C2314yes—
217#398C22×C2314yes—
218#398C22×C2314yes—
219#398C22×C2314yes—
220#398C22×C2314yes—
221#398C22×C2314yes—
222#398C22×C2314yes—
223#397C22×C2314yes—
224#397C22×C2314yes—
225#397C22×C2314yes—
226#396C22×C2314yes—
227#396C22×C2314yes—
228#395C22×C2314yes—
229#395C22×C2312yes—
230#393C22×C2312yes—
231#391C22×C2314yes—
232#389C22×C2314yes—
233#388C22×C2314yes—
234#387C22×C2314yes—
235#387C22×C2314yes—
236#387C22×C2314yes—
237C4×C4⋊C4C22×C2314yes—
238C4×C4⋊C4C22×C2312yes—
239C4×C4⋊C4C22×C2314yes—
240C4×C4⋊C4C22×C2314yes—
241C4×C4⋊C4C22×C2314yes—
242C4×C4⋊C4C22×C2312yes—
243C4×C22⋊C4C22×C2314yes—
244C4×C22⋊C4C22×C2312yes—
245#376C22×C2311yescharacteristic
246#376C22×C2314yes—
247C43C22×C2311yescharacteristic
248C2×C23—81no—
249C2×C23C2×#54111yescharacteristic
250C2×C23#345512yes—
251C2×C23#353114yes—
252C22×Q8C23×C2312yes—
253C23×C4C23×C2311yescharacteristic
254C4×Q8—28no—
255#117—24no—
256C2×C4⋊C4C23×C2312yes—
257C2×C4⋊C4C23×C2314yes—
258C2×C4⋊C4C23×C2312yes—
259C2×C4⋊C4C23×C2314yes—
260C2×C4⋊C4C23×C2314yes—
261C2×C4⋊C4C23×C2314yes—
262C2×C4⋊C4C23×C2314yes—
263C2×C4⋊C4C23×C2314yes—
264C2×C4⋊C4C23×C2314yes—
265C2×C4⋊C4C23×C2314yes—
266C2×C4⋊C4C23×C2314yes—
267C2×C4⋊C4C23×C2314yes—
268C2×C4⋊C4C23×C2314yes—
269C2×C22⋊C4C23×C2312yes—
270C2×C22⋊C4C23×C2314yes—
271C2×C22⋊C4C23×C2314yes—
272C2×C22⋊C4C23×C2314yes—
273C2×C42C23×C2311yescharacteristic
274C2×C42C23×C2314yes—
275C2×C42C23×C2312yes—
276C2×C42C23×C2312yes—
277C2×C42C23×C2314yes—
278C2×C42C23×C2314yes—
279C2×C42C23×C2311yescharacteristic
280#95C23×C2314yes—
281#95C23×C2314yes—
282#95C23×C2314yes—
283#95C23×C2314yes—
284#95C23×C2314yes—
285#95C23×C2314yes—
286#95C23×C2314yes—
287#95C23×C2314yes—
288#95C23×C2312yes—
289#95C23×C2312yes—
290#95C23×C2314yes—
291#95C23×C2314yes—
292#95C23×C2314yes—
293#95C23×C2314yes—
294#95C23×C2314yes—
295#95C23×C2314yes—
296#95C23×C2314yes—
297#95C23×C2314yes—
298#95C23×C2314yes—
299C23#3670011yescharacteristic, direct product factor, Sylow
300C24C24×C2311yescharacteristic
301C2×Q8—24no—
302C2×Q8—44no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4C24×C2311yescharacteristic
306C22×C4C24×C2314yes—
307C22×C4C24×C2314yes—
308C22×C4C24×C2314yes—
309C22×C4C24×C2314yes—
310C22×C4C24×C2314yes—
311C22×C4C24×C2312yes—
312C22×C4C24×C2314yes—
313C22×C4C24×C2312yes—
314C22×C4C24×C2311yescharacteristic
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C42—22no—
333C42—28no—
334C42—24no—
335C42—24no—
336C42—28no—
337C42—44no—
338C42#41×C2314yes—
339C23—21no—
340C23—22no—
341C23—24no—
342C23C25×C2311yescharacteristic
343Q8—48no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—44no—
368C2×C4C2×#41×C2314yes—
369C2×C4C2×#41×C2312yes—
370C2×C4C23×2−1+414yes—
371C22—41no—
372C22—42no—
373C22—44no—
374C22C22×#41×C2311yescharacteristic
375C22C2×C23×2−1+412yes—
376C22C23×#58514yes—
377C4—24no—
378C4—41no—
379C4—44no—
380C4—42no—
381C4—44no—
382C4—44no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4C23×#54114yes—
387C2—81no—
388C2C2×C23×#54111yescharacteristic
389C2C23×#345512yes—
390C2C23×#353114yes—
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC24×C23C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2839C2semidirect
3. C23×#2833C2semidirect
5. C23×#2402C2semidirect
6. C23×#2402C2semidirect
8. C23×#2399C2semidirect
11. C23×#2295C2semidirect
143. #36700C23direct (N × H)
176. #2839C2×C23semidirect
178. #2833C2×C23semidirect
180. #2402C2×C23semidirect
181. #2402C2×C23semidirect
183. #2399C2×C23semidirect
186. #2295C2×C23semidirect
299. C23#36700direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28392
C23×#28362
C23×#28332
C23×#28312
C23×#24022
C23×#24012
C23×#23992
C23×#23962
C23×#22982
C23×#22952
#3670023