Finite Group Databasebeta

← C4×C23×#352 #19160246

Subgroups — C4×C23×#352 #19160246

Order 5888.

576 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C23×#123C211yesmaximal, characteristic
2C2×C23×#352C212yesmaximal
3C4×C23×#100C211yesmaximal, characteristic
4C4×C23×#99C211yesmaximal, characteristic
5C23×#1390C212yesmaximal
6C2×C23×#123C2211yescharacteristic
7C2×C4×D4×C23C2211yescharacteristic
8C22×C42×C23C411yescharacteristic, complemented
9C2×C23×#100C2212yes—
10C2×C23×#99C2212yes—
11C4×M16×C23—21no—
12C23×#388—21no—
13C23×#387—21no—
14C23×#386C2211yescharacteristic
15C23×#385C411yescharacteristic, complemented
16C23×#381C411yescharacteristic, complemented
17C4×C4⋊C4×C23—21no—
18C4×C22⋊C4×C23—21no—
19C4×C22⋊C4×C23—21no—
20C4×C22⋊C4×C23C411yescharacteristic, complemented
21C23×#352C418yesdirect product factor
22C23×#343—21no—
23C23×#342—21no—
24C22×D4×C23C2311yescharacteristic
25C23×C4×C23C2×C411yescharacteristic
26C23×C4×C23C2×C411yescharacteristic
27C23×C4×C23C2×C411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C2×M16×C23—22no—
30C23×#123—22no—
31C23×#123C2×C414yes—
32C4×D4×C23—41no—
33C4×D4×C23—41no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23D411yescharacteristic
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23C2×C411yescharacteristic
41C2×C22⋊C4×C23C2×C411yescharacteristic
42C2×C22⋊C4×C23C2×C411yescharacteristic
43C2×C22⋊C4×C23D411yescharacteristic
44C2×C42×C23—21no—
45C2×C42×C23—21no—
46C2×C42×C23—21no—
47C2×C42×C23—41no—
48C2×C42×C23—41no—
49C2×C42×C23—41no—
50C2×C42×C23D411yescharacteristic
51C23×#100—22no—
52C23×#100C2×C414yes—
53C23×#99—22no—
54C23×#99C2×C414yes—
55C23×#97—41no—
56C4×C8×C23—41no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—21no—
60C23×#95—21no—
61C23×#95—21no—
62C23×#95—41no—
63C24×C23C22×C411yescharacteristic
64C24×C23C2×D411yescharacteristic
65C2×D4×C23—41no—
66C2×D4×C23—41no—
67C2×D4×C23C22×C412yes—
68C2×D4×C23#4112yes—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—42no—
79C22×C4×C23—42no—
80C22×C4×C23—42no—
81C22×C4×C23—41no—
82C22×C4×C23—41no—
83C22×C4×C23—41no—
84C22×C4×C23—42no—
85C22×C4×C23C4212yescomplemented
86C22×C4×C23C22⋊C411yescharacteristic
87C22×C4×C23C22⋊C411yescharacteristic
88C22×C4×C23C22⋊C411yescharacteristic
89C22×C4×C23C22⋊C411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91M16×C23—24no—
92M16×C23—42no—
93C2×C8×C23—42no—
94C4⋊C4×C23—24no—
95C4⋊C4×C23—41no—
96C4⋊C4×C23—42no—
97C22⋊C4×C23—21no—
98C22⋊C4×C23—24no—
99C22⋊C4×C23—24no—
100C22⋊C4×C23—41no—
101C22⋊C4×C23—41no—
102C22⋊C4×C23—42no—
103C22⋊C4×C23—42no—
104C22⋊C4×C23—41no—
105C22⋊C4×C23C4212yescomplemented
106C42×C23—21no—
107C42×C23—21no—
108C42×C23—42no—
109C42×C23—42no—
110C42×C23—41no—
111C42×C23—41no—
112C42×C23—82no—
113C4×#352C2311yesmaximal, characteristic, direct product factor, Sylow
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23C2×C4211yescharacteristic
123C23×C23C2×C22⋊C411yescharacteristic
124C23×C23#11711yescharacteristic
125C23×C23C4×D412yes—
126D4×C23—42no—
127D4×C23—42no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—22no—
131C2×C4×C23—22no—
132C2×C4×C23—22no—
133C2×C4×C23—22no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—42no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—42no—
142C2×C4×C23—42no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—42no—
147C2×C4×C23—42no—
148C2×C4×C23—41no—
149C2×C4×C23—42no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—42no—
153C2×C4×C23—41no—
154C2×C4×C23—44no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—81no—
158C2×C4×C23—82no—
159C2×C4×C23—82no—
160C2×C4×C23#9911yescharacteristic
161C2×C4×C23#9911yescharacteristic
162C2×C4×C23C4×D412yes—
163C8×C23—44no—
164C4×#123C2×C2311yescharacteristic
165C2×#352C2×C2312yes—
166C4×#100C2×C2311yescharacteristic
167C4×#99C2×C2311yescharacteristic
168#1390C2×C2312yes—
169C22×C23—21no—
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—22no—
174C22×C23—41no—
175C22×C23—41no—
176C22×C23—42no—
177C22×C23—42no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—81no—
183C22×C23C4×C22⋊C411yescharacteristic
184C22×C23C2×#9911yescharacteristic
185C22×C23#41011yescharacteristic
186C4×C23—21no—
187C4×C23—22no—
188C4×C23—42no—
189C4×C23—41no—
190C4×C23—41no—
191C4×C23—41no—
192C4×C23—42no—
193C4×C23—42no—
194C4×C23—81no—
195C4×C23—84no—
196C4×C23#35212yesdirect product factor
197C2×#123C22×C2311yescharacteristic
198C2×C4×D4C22×C2311yescharacteristic
199C22×C42C4×C2311yescharacteristic, complemented
200C2×#100C22×C2312yes—
201C2×#99C22×C2312yes—
202C4×M16—21no—
203#388—21no—
204#387—21no—
205#386C22×C2311yescharacteristic
206#385C4×C2311yescharacteristic, complemented
207#381C4×C2311yescharacteristic, complemented
208C4×C4⋊C4—21no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4C4×C2311yescharacteristic, complemented
212#352C4×C2318yesdirect product factor
213#343—21no—
214#342—21no—
215C2×C23—21no—
216C2×C23—21no—
217C2×C23—42no—
218C2×C23—41no—
219C2×C23—41no—
220C2×C23C4×#9911yescharacteristic
221C2×C23C2×#35211yescharacteristic
222C2×C23#212011yescharacteristic
223C22×D4C23×C2311yescharacteristic
224C23×C4C2×C4×C2311yescharacteristic
225C23×C4C2×C4×C2311yescharacteristic
226C23×C4C2×C4×C2311yescharacteristic
227C23×C4D4×C2311yescharacteristic
228C2×M16—22no—
229#123—22no—
230#123C2×C4×C2314yes—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4D4×C2311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4C2×C4×C2311yescharacteristic
240C2×C22⋊C4C2×C4×C2311yescharacteristic
241C2×C22⋊C4C2×C4×C2311yescharacteristic
242C2×C22⋊C4D4×C2311yescharacteristic
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42D4×C2311yescharacteristic
250#100—22no—
251#100C2×C4×C2314yes—
252#99—22no—
253#99C2×C4×C2314yes—
254#97—41no—
255C4×C8—41no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262C23C4×#35211yescharacteristic, direct product factor, Sylow
263C24C22×C4×C2311yescharacteristic
264C24C2×D4×C2311yescharacteristic
265C2×D4—41no—
266C2×D4—41no—
267C2×D4C22×C4×C2312yes—
268C2×D4#41×C2312yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4C42×C2312yescomplemented
286C22×C4C22⋊C4×C2311yescharacteristic
287C22×C4C22⋊C4×C2311yescharacteristic
288C22×C4C22⋊C4×C2311yescharacteristic
289C22×C4C22⋊C4×C2311yescharacteristic
290C22×C4C2×D4×C2311yescharacteristic
291M16—24no—
292M16—42no—
293C2×C8—42no—
294C4⋊C4—24no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C22⋊C4—21no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4C42×C2312yescomplemented
306C42—21no—
307C42—21no—
308C42—42no—
309C42—42no—
310C42—41no—
311C42—41no—
312C42—82no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C2×C42×C2311yescharacteristic
322C23C2×C22⋊C4×C2311yescharacteristic
323C23C23×#11711yescharacteristic
324C23C4×D4×C2312yes—
325D4—42no—
326D4—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4C23×#9911yescharacteristic
360C2×C4C23×#9911yescharacteristic
361C2×C4C4×D4×C2312yes—
362C8—44no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22C4×C22⋊C4×C2311yescharacteristic
378C22C2×C23×#9911yescharacteristic
379C22C23×#41011yescharacteristic
380C4—22no—
381C4—21no—
382C4—42no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—81no—
389C4—84no—
390C4C23×#35212yesdirect product factor
391C2—21no—
392C2—21no—
393C2—42no—
394C2—41no—
395C2—41no—
396C2C4×C23×#9911yescharacteristic
397C2C2×C23×#35211yescharacteristic
398C2C23×#212011yescharacteristic
subgroupG/H
centerC2×C4×C23#99
commutatorC23C2×C42×C23
Frattini, cosocleC22×D4C23×C23
socleC22×C23C2×#99
prime radicalC22×D4×C23C23
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
8. C22×C42×C23C4semidirect
15. C23×#385C4semidirect
16. C23×#381C4semidirect
20. C4×C22⋊C4×C23C4semidirect
21. C23×#352C4direct (N × H)
85. C22×C4×C23C42semidirect
105. C22⋊C4×C23C42semidirect
113. C4×#352C23direct (N × H)
196. C4×C23#352direct (N × H)
199. C22×C42C4×C23semidirect
206. #385C4×C23semidirect
207. #381C4×C23semidirect
211. C4×C22⋊C4C4×C23semidirect
212. #352C4×C23direct (N × H)
262. C23C4×#352direct (N × H)
285. C22×C4C42×C23semidirect
305. C22⋊C4C42×C23semidirect
390. C4C23×#352direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C23×#1232
C2×C23×#3522
C4×C23×#1002
C4×C23×#992
C23×#13902
C4×#35223