Finite Group Databasebeta

← C23×#13065 #19160691

Subgroups — C23×#13065 #19160691

Order 5888.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×M32C211yesmaximal, characteristic, complemented
2C23×#3050C211yesmaximal, characteristic
3C2×C23×#360C212yesmaximal, complemented
4C23×#1807C211yesmaximal, characteristic, complemented
5C23×#1386C212yesmaximal, complemented
6C2×C23×#135C2211yescharacteristic
7C23×C8×C23C2211yescharacteristic
8C23×#525—21no—
9C2×C23×M32—21no—
10C2×C23×M32—24no—
11C2×C23×M32C2212yes—
12C22×C16×C23—21no—
13C23×#467C411yescharacteristic, complemented
14C23×#467C411yescharacteristic, complemented
15C23×#466C411yescharacteristic, complemented
16C23×#466C411yescharacteristic, complemented
17C23×#465—21no—
18C23×#465—21no—
19C23×#427C2211yescharacteristic
20C2×C23×#106C2212yes—
21C23×#416—21no—
22C23×#380—21no—
23C23×#360—24no—
24C23×#340—21no—
25C2×#41×C23—21no—
26C23×C4×C23D411yescharacteristic
27C23×#135—22no—
28C23×#135—22no—
29C2×Q16×C23C2×C411yescharacteristic
30C2×SD16×C23—21no—
31C2×D8×C23C2×C411yescharacteristic
32C22×C8×C23—22no—
33C22×C8×C23—22no—
34C22×C8×C23C2×C411yescharacteristic
35C22×C8×C23C2×C411yescharacteristic
36C22×C8×C23D411yescharacteristic
37C22×C8×C23D411yescharacteristic
38C22×C8×C23D411yescharacteristic
39C22×C8×C23C2311yescharacteristic
40C23×#123—41no—
41C23×C22⋊Q8—21no—
42C23×C22⋊Q8—21no—
43C23×#121—21no—
44C23×#121—21no—
45C23×#120—41no—
46C4×D4×C23—41no—
47C4×D4×C23—41no—
48C23×#117—21no—
49C2×C4⋊C4×C23—22no—
50C2×C22⋊C4×C23—41no—
51C23×M32—24no—
52C23×M32—22no—
53C23×M32D414yescomplemented
54C2×C16×C23—24no—
55C2×C16×C23—22no—
56C23×#107C2×C411yescharacteristic
57C23×#107C2×C411yescharacteristic
58C23×#106—21no—
59C23×#106—22no—
60C23×#106—22no—
61C23×#103—21no—
62C23×#103—21no—
63C23×#102—21no—
64C23×#102—21no—
65C23×#95—41no—
66C24×C23D811yescharacteristic
67#41×C23—42no—
68#41×C23—42no—
69C2×Q8×C23—21no—
70C2×D4×C23—21no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23—41no—
74C22×C4×C23—22no—
75C22×C4×C23—22no—
76C22×C4×C23—41no—
77C22×C4×C23—42no—
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23D811yescharacteristic
81C22×C4×C23SD1611yescharacteristic
82C22×C4×C23SD1611yescharacteristic
83C22×C4×C23C2×D411yescharacteristic
84Q16×C23—22no—
85SD16×C23—41no—
86SD16×C23—41no—
87D8×C23—22no—
88C2×C8×C23—22no—
89C2×C8×C23—22no—
90C2×C8×C23—22no—
91C2×C8×C23—21no—
92C2×C8×C23—21no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23C22⋊C411yescharacteristic
96C2×C8×C23C22⋊C411yescharacteristic
97C2×C8×C23C22×C411yescharacteristic
98C2×C8×C23C2×D412yes—
99C2×C8×C23C2×D411yescharacteristic
100C2×C8×C23#4112yes—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23—42no—
106C4⋊C4×C23—42no—
107C22⋊C4×C23—41no—
108C22⋊C4×C23—41no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—82no—
113C42×C23—41no—
114C16×C23—24no—
115#13065C2311yesmaximal, characteristic, direct product factor, Sylow
116C23×C23—22no—
117C23×C23—22no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23#10211yescharacteristic
121C23×C23#10211yescharacteristic
122C23×C23C2×D811yescharacteristic
123Q8×C23—41no—
124Q8×C23—41no—
125D4×C23—41no—
126D4×C23—41no—
127D4×C23—82no—
128D4×C23—82no—
129D4×C23—82no—
130C2×C4×C23—22no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—22no—
136C2×C4×C23—22no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—82no—
141C2×C4×C23—82no—
142C2×C4×C23—82no—
143C2×C4×C23—82no—
144C2×C4×C23#10211yescharacteristic
145C2×C4×C23#10211yescharacteristic
146C2×C4×C23C2×C22⋊C411yescharacteristic
147C2×C4×C23#12112yes—
148C2×C4×C23C2×SD1611yescharacteristic
149C2×C4×C23#13512yes—
150C8×C23—21no—
151C8×C23—21no—
152C8×C23C4×D412yes—
153C8×C23#12011yescharacteristic
154C8×C23#12311yescharacteristic
155C22×M32C2×C2311yescharacteristic, complemented
156#3050C2×C2311yescharacteristic
157C2×#360C2×C2312yescomplemented
158#1807C2×C2311yescharacteristic, complemented
159#1386C2×C2312yescomplemented
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—44no—
168C22×C23—81no—
169C22×C23—81no—
170C22×C23—81no—
171C22×C23#36012yescomplemented
172C22×C23C2×#10211yescharacteristic
173C22×C23#46612yes—
174C4×C23—21no—
175C4×C23—21no—
176C4×C23—81no—
177C4×C23—81no—
178C4×C23—81no—
179C4×C23—162no—
180C4×C23#38611yescharacteristic
181C4×C23#41611yescharacteristic
182C4×C23#46512yes—
183C2×#135C22×C2311yescharacteristic
184C23×C8C22×C2311yescharacteristic
185#525—21no—
186C2×M32—21no—
187C2×M32—24no—
188C2×M32C22×C2312yes—
189C22×C16—21no—
190#467C4×C2311yescharacteristic, complemented
191#467C4×C2311yescharacteristic, complemented
192#466C4×C2311yescharacteristic, complemented
193#466C4×C2311yescharacteristic, complemented
194#465—21no—
195#465—21no—
196#427C22×C2311yescharacteristic
197C2×#106C22×C2312yes—
198#416—21no—
199#380—21no—
200#360—24no—
201#340—21no—
202C2×C23—22no—
203C2×C23—22no—
204C2×C23—41no—
205C2×C23—161no—
206C2×C23#189311yescharacteristic
207C2×C23C2×#36011yescharacteristic
208C2×C23#214511yescharacteristic
209C2×#41—21no—
210C23×C4D4×C2311yescharacteristic
211#135—22no—
212#135—22no—
213C2×Q16C2×C4×C2311yescharacteristic
214C2×SD16—21no—
215C2×D8C2×C4×C2311yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8C2×C4×C2311yescharacteristic
219C22×C8C2×C4×C2311yescharacteristic
220C22×C8D4×C2311yescharacteristic
221C22×C8D4×C2311yescharacteristic
222C22×C8D4×C2311yescharacteristic
223C22×C8C23×C2311yescharacteristic
224#123—41no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227#121—21no—
228#121—21no—
229#120—41no—
230C4×D4—41no—
231C4×D4—41no—
232#117—21no—
233C2×C4⋊C4—22no—
234C2×C22⋊C4—41no—
235M32—24no—
236M32—22no—
237M32D4×C2314yescomplemented
238C2×C16—24no—
239C2×C16—22no—
240#107C2×C4×C2311yescharacteristic
241#107C2×C4×C2311yescharacteristic
242#106—21no—
243#106—22no—
244#106—22no—
245#103—21no—
246#103—21no—
247#102—21no—
248#102—21no—
249#95—41no—
250C23#1306511yescharacteristic, direct product factor, Sylow
251C24D8×C2311yescharacteristic
252#41—42no—
253#41—42no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4C22⋊C4×C2311yescharacteristic
264C22×C4C22⋊C4×C2311yescharacteristic
265C22×C4D8×C2311yescharacteristic
266C22×C4SD16×C2311yescharacteristic
267C22×C4SD16×C2311yescharacteristic
268C22×C4C2×D4×C2311yescharacteristic
269Q16—22no—
270SD16—41no—
271SD16—41no—
272D8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8C22⋊C4×C2311yescharacteristic
281C2×C8C22⋊C4×C2311yescharacteristic
282C2×C8C22×C4×C2311yescharacteristic
283C2×C8C2×D4×C2312yes—
284C2×C8C2×D4×C2311yescharacteristic
285C2×C8#41×C2312yes—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—82no—
298C42—41no—
299C16—24no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23C23×#10211yescharacteristic
305C23C23×#10211yescharacteristic
306C23C2×D8×C2311yescharacteristic
307Q8—41no—
308Q8—41no—
309D4—41no—
310D4—41no—
311D4—82no—
312D4—82no—
313D4—82no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4C23×#10211yescharacteristic
329C2×C4C23×#10211yescharacteristic
330C2×C4C2×C22⋊C4×C2311yescharacteristic
331C2×C4C23×#12112yes—
332C2×C4C2×SD16×C2311yescharacteristic
333C2×C4C23×#13512yes—
334C8—21no—
335C8—21no—
336C8C4×D4×C2312yes—
337C8C23×#12011yescharacteristic
338C8C23×#12311yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C23×#36012yescomplemented
351C22C2×C23×#10211yescharacteristic
352C22C23×#46612yes—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—162no—
359C4C23×#38611yescharacteristic
360C4C23×#41611yescharacteristic
361C4C23×#46512yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—161no—
366C2C23×#189311yescharacteristic
367C2C2×C23×#36011yescharacteristic
368C2C23×#214511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#102
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
prime radicalC23×#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ C23×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C23×M32C2semidirect
3. C2×C23×#360C2semidirect
4. C23×#1807C2semidirect
5. C23×#1386C2semidirect
13. C23×#467C4semidirect
14. C23×#467C4semidirect
15. C23×#466C4semidirect
16. C23×#466C4semidirect
53. C23×M32D4semidirect
115. #13065C23direct (N × H)
155. C22×M32C2×C23semidirect
157. C2×#360C2×C23semidirect
158. #1807C2×C23semidirect
159. #1386C2×C23semidirect
171. C22×C23#360semidirect
190. #467C4×C23semidirect
191. #467C4×C23semidirect
192. #466C4×C23semidirect
193. #466C4×C23semidirect
237. M32D4×C23semidirect
250. C23#13065direct (N × H)
350. C22C23×#360semidirect

Maximal subgroups

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

subgroupindex
C22×C23×M322
C23×#30502
C2×C23×#3602
C23×#18072
C23×#13862
#1306523