Finite Group Databasebeta

← C23×#23897 #19171523

Subgroups — C23×#23897 #19171523

Order 5888.

1952 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 282 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
1C23×#3484C212yesmaximal, complemented
2C23×#3019C214yesmaximal, complemented
3C23×#2010C214yesmaximal, complemented
4C23×#1911C214yesmaximal, complemented
5C23×#1783C211yesmaximal, characteristic, complemented
6C2×C23×H(5,𝔽2)C2212yescomplemented
7C23×D4×C23C2211yescharacteristic
8C23×#575C2214yescomplemented
9C23×#546—28no—
10D42×C23—28no—
11C23×#535C2214yes—
12C23×#534—24no—
13C23×#525C2214yescomplemented
14C2×C23×#120—24no—
15C23×#518C2212yescomplemented
16C23×#455—28no—
17C23×#453—28no—
18C23×#421C2218yescomplemented
19C2×C23×#100C2214yescomplemented
20C23×#407C2212yescomplemented
21C23×H(3,ℤ/4)C2214yescomplemented
22C25×C23D412yes—
23C23×H(5,𝔽2)—28no—
24C23×H(5,𝔽2)—44no—
25C2×#41×C23—24no—
26C2×#41×C23C2314yes—
27C22×D4×C23—24no—
28C22×D4×C23—28no—
29C22×D4×C23—28no—
30C22×D4×C23—24no—
31C22×D4×C23—42no—
32C22×D4×C23D412yes—
33C22×D4×C23C2312yes—
34C23×C4×C23C2311yescharacteristic
35C23×#137—28no—
36C23×AGL(1,ℤ/8)—28no—
37C23×#135—44no—
38C2×M16×C23C2314yes—
39C23×#127D418yescomplemented
40C23×#124D418yescomplemented
41C23×#123—28no—
42C23×#123—24no—
43C23×C22⋊Q8—24no—
44C23×#121—216no—
45C23×#121—28no—
46C23×#121—24no—
47C23×#120—216no—
48C23×#120—28no—
49C23×#120—24no—
50C23×#120—28no—
51C4×D4×C23—28no—
52C4×D4×C23—28no—
53C23×#117C2314yes—
54C2×C22⋊C4×C23—28no—
55C2×C22⋊C4×C23—24no—
56C23×#104—216no—
57C23×#100—28no—
58C23×C22⋊C8—42no—
59C24×C23—24no—
60C24×C23—28no—
61C24×C23—42no—
62C24×C23—84no—
63C24×C23C2×D411yescharacteristic
64C24×C23C2×D414yes—
65#41×C23—28no—
66#41×C23—28no—
67#41×C23—416no—
68C2×Q8×C23C2×D414yescomplemented
69C2×D4×C23—28no—
70C2×D4×C23—28no—
71C2×D4×C23—28no—
72C2×D4×C23—24no—
73C2×D4×C23—216no—
74C2×D4×C23—24no—
75C2×D4×C23—24no—
76C2×D4×C23—42no—
77C2×D4×C23—416no—
78C2×D4×C23—416no—
79C2×D4×C23—48no—
80C2×D4×C23—48no—
81C2×D4×C23—816no—
82C2×D4×C23C2×D418yes—
83C2×D4×C23C2×D414yescomplemented
84C22×C4×C23—24no—
85C22×C4×C23—28no—
86C22×C4×C23—24no—
87C22×C4×C23—24no—
88C22×C4×C23—22no—
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2411yescharacteristic
91Q16×C23—44no—
92SD16×C23—48no—
93D8×C23—44no—
94M16×C23—28no—
95C2×C8×C23—24no—
96C4⋊C4×C23—28no—
97C4⋊C4×C23—24no—
98C22⋊C4×C23—28no—
99C22⋊C4×C23—28no—
100C22⋊C4×C23—28no—
101C22⋊C4×C23—216no—
102C22⋊C4×C23—24no—
103C42×C23C2×D418yescomplemented
104#23897C2311yesmaximal, characteristic, direct product factor, Sylow
105C23×C23—22no—
106C23×C23—24no—
107C23×C23—21no—
108C23×C23—24no—
109C23×C23—24no—
110C23×C23—42no—
111C23×C23—48no—
112C23×C23—416no—
113C23×C23—48no—
114C23×C23—84no—
115C23×C23—84no—
116C23×C23—88no—
117C23×C23—88no—
118C23×C23—84no—
119C23×C23#12018yes—
120C23×C23C22×D411yescharacteristic
121Q8×C23—28no—
122D4×C23—216no—
123D4×C23—28no—
124D4×C23—48no—
125D4×C23—416no—
126D4×C23—416no—
127D4×C23—44no—
128D4×C23—44no—
129D4×C23—816no—
130C2×C4×C23—28no—
131C2×C4×C23—24no—
132C2×C4×C23—22no—
133C2×C4×C23—21no—
134C2×C4×C23—48no—
135C2×C4×C23—44no—
136C2×C4×C23—48no—
137C2×C4×C23—88no—
138C2×C4×C23C22×D414yes—
139C2×C4×C23H(5,𝔽2)12yes—
140C8×C23—44no—
141#3484C2×C2312yescomplemented
142#3019C2×C2314yescomplemented
143#2010C2×C2314yescomplemented
144#1911C2×C2314yescomplemented
145#1783C2×C2311yescharacteristic, complemented
146C22×C23—24no—
147C22×C23—22no—
148C22×C23—28no—
149C22×C23—41no—
150C22×C23—48no—
151C22×C23—48no—
152C22×C23—42no—
153C22×C23—41no—
154C22×C23—48no—
155C22×C23—41no—
156C22×C23—44no—
157C22×C23—84no—
158C22×C23—816no—
159C22×C23—168no—
160C22×C23C2×#12012yes—
161C22×C23#53411yescharacteristic
162C4×C23—48no—
163C4×C23—41no—
164C4×C23—44no—
165C4×C23D4214yes—
166C2×H(5,𝔽2)C22×C2312yescomplemented
167C23×D4C22×C2311yescharacteristic
168#575C22×C2314yescomplemented
169#546—28no—
170D42—28no—
171#535C22×C2314yes—
172#534—24no—
173#525C22×C2314yescomplemented
174C2×#120—24no—
175#518C22×C2312yescomplemented
176#455—28no—
177#453—28no—
178#421C22×C2318yescomplemented
179C2×#100C22×C2314yescomplemented
180#407C22×C2312yescomplemented
181H(3,ℤ/4)C22×C2314yescomplemented
182C2×C23—22no—
183C2×C23—21no—
184C2×C23—42no—
185C2×C23—48no—
186C2×C23—84no—
187C2×C23#240311yescharacteristic
188C25D4×C2312yes—
189H(5,𝔽2)—28no—
190H(5,𝔽2)—44no—
191C2×#41—24no—
192C2×#41C23×C2314yes—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4D4×C2312yes—
199C22×D4C23×C2312yes—
200C23×C4C23×C2311yescharacteristic
201#137—28no—
202AGL(1,ℤ/8)—28no—
203#135—44no—
204C2×M16C23×C2314yes—
205#127D4×C2318yescomplemented
206#124D4×C2318yescomplemented
207#123—28no—
208#123—24no—
209C22⋊Q8—24no—
210#121—216no—
211#121—28no—
212#121—24no—
213#120—216no—
214#120—28no—
215#120—24no—
216#120—28no—
217C4×D4—28no—
218C4×D4—28no—
219#117C23×C2314yes—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222#104—216no—
223#100—28no—
224C22⋊C8—42no—
225C23#2389711yescharacteristic, direct product factor, Sylow
226C24—24no—
227C24—28no—
228C24—42no—
229C24—84no—
230C24C2×D4×C2311yescharacteristic
231C24C2×D4×C2314yes—
232#41—28no—
233#41—28no—
234#41—416no—
235C2×Q8C2×D4×C2314yescomplemented
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—24no—
240C2×D4—216no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—42no—
244C2×D4—416no—
245C2×D4—416no—
246C2×D4—48no—
247C2×D4—48no—
248C2×D4—816no—
249C2×D4C2×D4×C2318yes—
250C2×D4C2×D4×C2314yescomplemented
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4C2×D4×C2311yescharacteristic
257C22×C4C24×C2311yescharacteristic
258Q16—44no—
259SD16—48no—
260D8—44no—
261M16—28no—
262C2×C8—24no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—28no—
268C22⋊C4—216no—
269C22⋊C4—24no—
270C42C2×D4×C2318yescomplemented
271C23—22no—
272C23—24no—
273C23—21no—
274C23—24no—
275C23—24no—
276C23—42no—
277C23—48no—
278C23—416no—
279C23—48no—
280C23—84no—
281C23—84no—
282C23—88no—
283C23—88no—
284C23—84no—
285C23C23×#12018yes—
286C23C22×D4×C2311yescharacteristic
287Q8—28no—
288D4—216no—
289D4—28no—
290D4—48no—
291D4—416no—
292D4—416no—
293D4—44no—
294D4—44no—
295D4—816no—
296C2×C4—28no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—88no—
304C2×C4C22×D4×C2314yes—
305C2×C4C23×H(5,𝔽2)12yes—
306C8—44no—
307C22—24no—
308C22—22no—
309C22—28no—
310C22—41no—
311C22—48no—
312C22—48no—
313C22—42no—
314C22—41no—
315C22—48no—
316C22—41no—
317C22—44no—
318C22—84no—
319C22—816no—
320C22—168no—
321C22C2×C23×#12012yes—
322C22C23×#53411yescharacteristic
323C4—48no—
324C4—41no—
325C4—44no—
326C4D42×C2314yes—
327C2—22no—
328C2—21no—
329C2—42no—
330C2—48no—
331C2—84no—
332C2C23×#240311yescharacteristic
subgroupG/H
center, socleC2×C23#2403
commutator, Frattini, cosocleC22×C4C24×C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3484C2semidirect
2. C23×#3019C2semidirect
3. C23×#2010C2semidirect
4. C23×#1911C2semidirect
5. C23×#1783C2semidirect
6. C2×C23×H(5,𝔽2)C22semidirect
8. C23×#575C22semidirect
13. C23×#525C22semidirect
15. C23×#518C22semidirect
18. C23×#421C22semidirect
19. C2×C23×#100C22semidirect
20. C23×#407C22semidirect
21. C23×H(3,ℤ/4)C22semidirect
39. C23×#127D4semidirect
40. C23×#124D4semidirect
68. C2×Q8×C23C2×D4semidirect
83. C2×D4×C23C2×D4semidirect
103. C42×C23C2×D4semidirect
104. #23897C23direct (N × H)
141. #3484C2×C23semidirect
142. #3019C2×C23semidirect
143. #2010C2×C23semidirect
144. #1911C2×C23semidirect
145. #1783C2×C23semidirect
166. C2×H(5,𝔽2)C22×C23semidirect
168. #575C22×C23semidirect
173. #525C22×C23semidirect
175. #518C22×C23semidirect
178. #421C22×C23semidirect
179. C2×#100C22×C23semidirect
180. #407C22×C23semidirect
181. H(3,ℤ/4)C22×C23semidirect
205. #127D4×C23semidirect
206. #124D4×C23semidirect
225. C23#23897direct (N × H)
235. C2×Q8C2×D4×C23semidirect
250. C2×D4C2×D4×C23semidirect
270. C42C2×D4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#34842
C23×#30192
C23×#20102
C23×#19112
C23×#17832
#2389723