Finite Group Databasebeta

← C23×#32878 #19180504

Subgroups — C23×#32878 #19180504

Order 5888.

6020 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 6020 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×D4×C23C211yesmaximal, characteristic, complemented
2C2×C23×#447C216yesmaximal, complemented
3C23×#2847C211yesmaximal, characteristic, complemented
4C23×#1786C216yesmaximal, complemented
5C23×#1779C211yesmaximal, characteristic, complemented
6C26×C23—21no—
7C23×D4×C23—224no—
8C23×D4×C23—23no—
9C23×D4×C23C2216yes—
10C24×C4×C23C2211yescharacteristic
11C22×D8×C23C2214yescomplemented
12C2×C23×#121C2213yescomplemented
13C2×C23×#120—21no—
14C23×#447—224no—
15C2×C23×#102C22112yescomplemented
16C2×C23×C22⋊C8C2216yescomplemented
17C23×#386—23no—
18C23×#380C2213yescomplemented
19C25×C23—224no—
20C25×C23—26no—
21C25×C23—412no—
22C25×C23—44no—
23C25×C23D411yescharacteristic
24C22×D4×C23—2128no—
25C22×D4×C23—224no—
26C22×D4×C23—248no—
27C22×D4×C23—248no—
28C22×D4×C23—23no—
29C22×D4×C23—26no—
30C22×D4×C23—212no—
31C22×D4×C23D4124yescomplemented
32C22×D4×C23C2314yes—
33C22×D4×C23C2311yescharacteristic, complemented
34C23×C4×C23—212no—
35C23×C4×C23D413yes—
36C23×C4×C23C2313yes—
37C2×D8×C23—224no—
38C22×C8×C23C2314yescomplemented
39C23×#121—212no—
40C23×#120—44no—
41C2×C4⋊C4×C23C2313yescomplemented
42C2×C22⋊C4×C23—23no—
43C2×C22⋊C4×C23—23no—
44C23×#102—224no—
45C23×C22⋊C8—212no—
46C23×#95—23no—
47C24×C23—26no—
48C24×C23—26no—
49C24×C23—264no—
50C24×C23—248no—
51C24×C23—224no—
52C24×C23—24no—
53C24×C23—21no—
54C24×C23—496no—
55C24×C23—424no—
56C24×C23—496no—
57C24×C23—412no—
58C24×C23—412no—
59C24×C23D814yescomplemented
60C24×C23C2×D413yes—
61C2×D4×C23—2192no—
62C2×D4×C23—248no—
63C2×D4×C23—296no—
64C2×D4×C23—296no—
65C2×D4×C23—224no—
66C2×D4×C23—212no—
67C2×D4×C23—26no—
68C2×D4×C23—46no—
69C2×D4×C23—46no—
70C2×D4×C23—46no—
71C2×D4×C23C2×D4124yes—
72C22×C4×C23—232no—
73C22×C4×C23—26no—
74C22×C4×C23—212no—
75C22×C4×C23—212no—
76C22×C4×C23—23no—
77C22×C4×C23C2×D413yes—
78C22×C4×C23C2×D4112yes—
79C22×C4×C23C2411yescharacteristic
80D8×C23—416no—
81C2×C8×C23—212no—
82C4⋊C4×C23—26no—
83C22⋊C4×C23—46no—
84C22⋊C4×C23—46no—
85#32878C2311yesmaximal, characteristic, direct product factor, Sylow
86C23×C23—28no—
87C23×C23—224no—
88C23×C23—26no—
89C23×C23—26no—
90C23×C23—224no—
91C23×C23—296no—
92C23×C23—224no—
93C23×C23—4256no—
94C23×C23—496no—
95C23×C23—448no—
96C23×C23—496no—
97C23×C23—44no—
98C23×C23—412no—
99C23×C23—448no—
100C23×C23—43no—
101C23×C23—41no—
102C23×C23—43no—
103C23×C23#12016yes—
104C23×C23C2×D816yes—
105C23×C23AGL(1,ℤ/8)16yes—
106C23×C23C22×D411yescharacteristic
107D4×C23—296no—
108D4×C23—224no—
109D4×C23—44no—
110D4×C23—812no—
111D4×C23#120116yescomplemented
112C2×C4×C23—224no—
113C2×C4×C23—26no—
114C2×C4×C23—224no—
115C2×C4×C23—43no—
116C2×C4×C23—43no—
117C2×C4×C23—43no—
118C2×C4×C23#12016yes—
119C2×C4×C23C22×D416yes—
120C8×C23—44no—
121C24×D4C2×C2311yescharacteristic, complemented
122C2×#447C2×C2316yescomplemented
123#2847C2×C2311yescharacteristic, complemented
124#1786C2×C2316yescomplemented
125#1779C2×C2311yescharacteristic, complemented
126C22×C23—232no—
127C22×C23—26no—
128C22×C23—212no—
129C22×C23—26no—
130C22×C23—212no—
131C22×C23—216no—
132C22×C23—4192no—
133C22×C23—424no—
134C22×C23—424no—
135C22×C23—81no—
136C22×C23—83no—
137C22×C23—83no—
138C22×C23#447112yescomplemented
139C22×C23C2×#12013yes—
140C22×C23C22×D811yescharacteristic
141C22×C23C2×AGL(1,ℤ/8)13yes—
142C4×C23—26no—
143C4×C23—83no—
144C4×C23C2×#12014yes—
145C26—21no—
146C23×D4—224no—
147C23×D4—23no—
148C23×D4C22×C2316yes—
149C24×C4C22×C2311yescharacteristic
150C22×D8C22×C2314yescomplemented
151C2×#121C22×C2313yescomplemented
152C2×#120—21no—
153#447—224no—
154C2×#102C22×C23112yescomplemented
155C2×C22⋊C8C22×C2316yescomplemented
156#386—23no—
157#380C22×C2313yescomplemented
158C2×C23—212no—
159C2×C23—416no—
160C2×C23—161no—
161C2×C23#284611yescharacteristic
162C2×C23C2×#44713yes—
163C2×C23#300313yes—
164C25—224no—
165C25—26no—
166C25—412no—
167C25—44no—
168C25D4×C2311yescharacteristic
169C22×D4—2128no—
170C22×D4—224no—
171C22×D4—248no—
172C22×D4—248no—
173C22×D4—23no—
174C22×D4—26no—
175C22×D4—212no—
176C22×D4D4×C23124yescomplemented
177C22×D4C23×C2314yes—
178C22×D4C23×C2311yescharacteristic, complemented
179C23×C4—212no—
180C23×C4D4×C2313yes—
181C23×C4C23×C2313yes—
182C2×D8—224no—
183C22×C8C23×C2314yescomplemented
184#121—212no—
185#120—44no—
186C2×C4⋊C4C23×C2313yescomplemented
187C2×C22⋊C4—23no—
188C2×C22⋊C4—23no—
189#102—224no—
190C22⋊C8—212no—
191#95—23no—
192C23#3287811yescharacteristic, direct product factor, Sylow
193C24—26no—
194C24—26no—
195C24—264no—
196C24—248no—
197C24—224no—
198C24—24no—
199C24—21no—
200C24—496no—
201C24—424no—
202C24—496no—
203C24—412no—
204C24—412no—
205C24D8×C2314yescomplemented
206C24C2×D4×C2313yes—
207C2×D4—2192no—
208C2×D4—248no—
209C2×D4—296no—
210C2×D4—296no—
211C2×D4—224no—
212C2×D4—212no—
213C2×D4—26no—
214C2×D4—46no—
215C2×D4—46no—
216C2×D4—46no—
217C2×D4C2×D4×C23124yes—
218C22×C4—232no—
219C22×C4—26no—
220C22×C4—212no—
221C22×C4—212no—
222C22×C4—23no—
223C22×C4C2×D4×C2313yes—
224C22×C4C2×D4×C23112yes—
225C22×C4C24×C2311yescharacteristic
226D8—416no—
227C2×C8—212no—
228C4⋊C4—26no—
229C22⋊C4—46no—
230C22⋊C4—46no—
231C23—28no—
232C23—224no—
233C23—26no—
234C23—224no—
235C23—26no—
236C23—296no—
237C23—224no—
238C23—4256no—
239C23—496no—
240C23—448no—
241C23—496no—
242C23—44no—
243C23—412no—
244C23—448no—
245C23—41no—
246C23—43no—
247C23—43no—
248C23C23×#12016yes—
249C23C2×D8×C2316yes—
250C23C23×AGL(1,ℤ/8)16yes—
251C23C22×D4×C2311yescharacteristic
252D4—296no—
253D4—224no—
254D4—44no—
255D4—812no—
256D4C23×#120116yescomplemented
257C2×C4—224no—
258C2×C4—26no—
259C2×C4—224no—
260C2×C4—43no—
261C2×C4—43no—
262C2×C4—43no—
263C2×C4C23×#12016yes—
264C2×C4C22×D4×C2316yes—
265C8—44no—
266C22—232no—
267C22—212no—
268C22—26no—
269C22—212no—
270C22—26no—
271C22—216no—
272C22—4192no—
273C22—424no—
274C22—424no—
275C22—83no—
276C22—83no—
277C22—81no—
278C22C23×#447112yescomplemented
279C22C2×C23×#12013yes—
280C22C22×D8×C2311yescharacteristic
281C22C2×C23×AGL(1,ℤ/8)13yes—
282C4—26no—
283C4—83no—
284C4C2×C23×#12014yes—
285C2—212no—
286C2—416no—
287C2—161no—
288C2C23×#284611yescharacteristic
289C2C2×C23×#44713yes—
290C2C23×#300313yes—
subgroupG/H
center, socleC23×C23C22×D4
commutator, Frattini, cosocleC22×C4C24×C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C24×D4×C23C2semidirect
2. C2×C23×#447C2semidirect
3. C23×#2847C2semidirect
4. C23×#1786C2semidirect
5. C23×#1779C2semidirect
11. C22×D8×C23C22semidirect
12. C2×C23×#121C22semidirect
15. C2×C23×#102C22semidirect
16. C2×C23×C22⋊C8C22semidirect
18. C23×#380C22semidirect
31. C22×D4×C23D4semidirect
33. C22×D4×C23C23semidirect
38. C22×C8×C23C23semidirect
41. C2×C4⋊C4×C23C23semidirect
59. C24×C23D8semidirect
85. #32878C23direct (N × H)
111. D4×C23#120semidirect
121. C24×D4C2×C23semidirect
122. C2×#447C2×C23semidirect
123. #2847C2×C23semidirect
124. #1786C2×C23semidirect
125. #1779C2×C23semidirect
138. C22×C23#447semidirect
150. C22×D8C22×C23semidirect
151. C2×#121C22×C23semidirect
154. C2×#102C22×C23semidirect
155. C2×C22⋊C8C22×C23semidirect
157. #380C22×C23semidirect
176. C22×D4D4×C23semidirect
178. C22×D4C23×C23semidirect
183. C22×C8C23×C23semidirect
186. C2×C4⋊C4C23×C23semidirect
192. C23#32878direct (N × H)
205. C24D8×C23semidirect
256. D4C23×#120semidirect
278. C22C23×#447semidirect

Maximal subgroups

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

subgroupindex
C24×D4×C232
C2×C23×#4472
C23×#28472
C23×#17862
C23×#17792
#3287823