Finite Group Databasebeta

← C23×#32708 #19180334

Subgroups — C23×#32708 #19180334

Order 5888.

1062 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 226 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×#2848C211yesmaximal, characteristic, complemented
2C23×#2847C211yesmaximal, characteristic
3C23×#2077C213yesmaximal, complemented
4C23×#2035C213yesmaximal, complemented
5C23×#2034C213yesmaximal, complemented
6C23×#2000C213yesmaximal, complemented
7C23×#1779C211yesmaximal, characteristic, complemented
8C24×C4×C23C2211yescharacteristic
9C22×SD16×C23C2211yescharacteristic, complemented
10C2×C23×C22⋊Q8C2213yes—
11C2×C23×#121C2213yes—
12C2×C23×#120—21no—
13C2×C23×#106C2213yescomplemented
14C2×C23×#103C2213yescomplemented
15C2×C23×#102C2213yescomplemented
16C2×C23×C22⋊C8C2213yes—
17C2×C23×C22⋊C8C2213yes—
18C23×#386—23no—
19C23×#385—23no—
20C23×#380C2213yes—
21C23×#380C2213yes—
22C23×#379—21no—
23C23×#340C2213yescomplemented
24C23×#340C2213yescomplemented
25C25×C23D411yescharacteristic
26C22×Q8×C23C2311yescharacteristic
27C22×D4×C23—23no—
28C22×D4×C23C2311yescharacteristic
29C23×C4×C23—43no—
30C23×C4×C23—43no—
31C23×C4×C23D413yes—
32C23×C4×C23C2313yes—
33C2×SD16×C23—43no—
34C22×C8×C23C2311yescharacteristic
35C22×C8×C23C2313yes—
36C23×C22⋊Q8—43no—
37C23×C22⋊Q8—43no—
38C23×#121—43no—
39C23×#121—43no—
40C23×#120—44no—
41C2×C4⋊C4×C23—23no—
42C2×C4⋊C4×C23C2313yes—
43C2×C4⋊C4×C23C2313yes—
44C2×C22⋊C4×C23—23no—
45C2×C22⋊C4×C23—23no—
46C2×C22⋊C4×C23—23no—
47C2×C22⋊C4×C23—23no—
48C23×#106—43no—
49C23×#103—43no—
50C23×#102—43no—
51C23×C22⋊C8—43no—
52C23×C22⋊C8—43no—
53C23×#95—23no—
54C23×#95—23no—
55C24×C23—21no—
56C24×C23—43no—
57C24×C23—43no—
58C24×C23SD1614yescomplemented
59C24×C23C2×D413yes—
60C2×Q8×C23—23no—
61C2×Q8×C23—23no—
62C2×D4×C23—23no—
63C2×D4×C23—23no—
64C2×D4×C23—46no—
65C2×D4×C23—46no—
66C2×D4×C23—46no—
67C22×C4×C23—23no—
68C22×C4×C23—23no—
69C22×C4×C23—23no—
70C22×C4×C23—23no—
71C22×C4×C23—23no—
72C22×C4×C23—23no—
73C22×C4×C23—21no—
74C22×C4×C23—23no—
75C22×C4×C23—23no—
76C22×C4×C23—43no—
77C22×C4×C23—43no—
78C22×C4×C23—43no—
79C22×C4×C23—412no—
80C22×C4×C23—44no—
81C22×C4×C23C2×D413yes—
82C22×C4×C23C2411yescharacteristic
83SD16×C23—44no—
84C2×C8×C23—23no—
85C2×C8×C23—23no—
86C2×C8×C23—23no—
87C2×C8×C23—23no—
88C4⋊C4×C23—23no—
89C4⋊C4×C23—23no—
90C4⋊C4×C23—23no—
91C4⋊C4×C23—23no—
92C4⋊C4×C23—46no—
93C22⋊C4×C23—46no—
94C22⋊C4×C23—46no—
95C22⋊C4×C23—46no—
96C22⋊C4×C23—46no—
97#32708C2311yesmaximal, characteristic, direct product factor, Sylow
98C23×C23—23no—
99C23×C23—23no—
100C23×C23—23no—
101C23×C23—23no—
102C23×C23—23no—
103C23×C23—23no—
104C23×C23—412no—
105C23×C23—44no—
106C23×C23—412no—
107C23×C23—43no—
108C23×C23—41no—
109C23×C23—43no—
110C23×C23C2×SD1616yes—
111C23×C23C22×D411yescharacteristic
112Q8×C23—43no—
113Q8×C23—41no—
114D4×C23—43no—
115D4×C23—41no—
116D4×C23—812no—
117C2×C4×C23—23no—
118C2×C4×C23—23no—
119C2×C4×C23—23no—
120C2×C4×C23—412no—
121C2×C4×C23—43no—
122C2×C4×C23—43no—
123C2×C4×C23—43no—
124C2×C4×C23—43no—
125C2×C4×C23—43no—
126C2×C4×C23—43no—
127C2×C4×C23—43no—
128C2×C4×C23—43no—
129C2×C4×C23—46no—
130C2×C4×C23—43no—
131C2×C4×C23H(5,𝔽2)13yes—
132C2×C4×C23H(5,𝔽2)13yes—
133C8×C23—43no—
134C8×C23—41no—
135#2848C2×C2311yescharacteristic, complemented
136#2847C2×C2311yescharacteristic
137#2077C2×C2313yescomplemented
138#2035C2×C2313yescomplemented
139#2034C2×C2313yescomplemented
140#2000C2×C2313yescomplemented
141#1779C2×C2311yescharacteristic, complemented
142C22×C23—26no—
143C22×C23—26no—
144C22×C23—26no—
145C22×C23—416no—
146C22×C23—43no—
147C22×C23—43no—
148C22×C23—43no—
149C22×C23—43no—
150C22×C23—83no—
151C22×C23—83no—
152C22×C23—81no—
153C22×C23#53413yes—
154C22×C23C22×SD1611yescharacteristic
155C22×C23#57513yes—
156C4×C23—43no—
157C4×C23—83no—
158C4×C23—83no—
159C4×C23—81no—
160C4×C23#56013yes—
161C4×C23H(3,𝔽4)11yescharacteristic
162C24×C4C22×C2311yescharacteristic
163C22×SD16C22×C2311yescharacteristic, complemented
164C2×C22⋊Q8C22×C2313yes—
165C2×#121C22×C2313yes—
166C2×#120—21no—
167C2×#106C22×C2313yescomplemented
168C2×#103C22×C2313yescomplemented
169C2×#102C22×C2313yescomplemented
170C2×C22⋊C8C22×C2313yes—
171C2×C22⋊C8C22×C2313yes—
172#386—23no—
173#385—23no—
174#380C22×C2313yes—
175#380C22×C2313yes—
176#379—21no—
177#340C22×C2313yescomplemented
178#340C22×C2313yescomplemented
179C2×C23—46no—
180C2×C23—161no—
181C2×C23#281211yescharacteristic
182C2×C23#318713yes—
183C2×C23#319713yes—
184C25D4×C2311yescharacteristic
185C22×Q8C23×C2311yescharacteristic
186C22×D4—23no—
187C22×D4C23×C2311yescharacteristic
188C23×C4—43no—
189C23×C4—43no—
190C23×C4D4×C2313yes—
191C23×C4C23×C2313yes—
192C2×SD16—43no—
193C22×C8C23×C2311yescharacteristic
194C22×C8C23×C2313yes—
195C22⋊Q8—43no—
196C22⋊Q8—43no—
197#121—43no—
198#121—43no—
199#120—44no—
200C2×C4⋊C4—23no—
201C2×C4⋊C4C23×C2313yes—
202C2×C4⋊C4C23×C2313yes—
203C2×C22⋊C4—23no—
204C2×C22⋊C4—23no—
205C2×C22⋊C4—23no—
206C2×C22⋊C4—23no—
207#106—43no—
208#103—43no—
209#102—43no—
210C22⋊C8—43no—
211C22⋊C8—43no—
212#95—23no—
213#95—23no—
214C23#3270811yescharacteristic, direct product factor, Sylow
215C24—21no—
216C24—43no—
217C24—43no—
218C24SD16×C2314yescomplemented
219C24C2×D4×C2313yes—
220C2×Q8—23no—
221C2×Q8—23no—
222C2×D4—23no—
223C2×D4—23no—
224C2×D4—46no—
225C2×D4—46no—
226C2×D4—46no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—23no—
232C22×C4—23no—
233C22×C4—23no—
234C22×C4—23no—
235C22×C4—21no—
236C22×C4—43no—
237C22×C4—43no—
238C22×C4—43no—
239C22×C4—412no—
240C22×C4—44no—
241C22×C4C2×D4×C2313yes—
242C22×C4C24×C2311yescharacteristic
243SD16—44no—
244C2×C8—23no—
245C2×C8—23no—
246C2×C8—23no—
247C2×C8—23no—
248C4⋊C4—23no—
249C4⋊C4—23no—
250C4⋊C4—23no—
251C4⋊C4—23no—
252C4⋊C4—46no—
253C22⋊C4—46no—
254C22⋊C4—46no—
255C22⋊C4—46no—
256C22⋊C4—46no—
257C23—23no—
258C23—23no—
259C23—23no—
260C23—23no—
261C23—23no—
262C23—23no—
263C23—412no—
264C23—44no—
265C23—412no—
266C23—41no—
267C23—43no—
268C23—43no—
269C23C2×SD16×C2316yes—
270C23C22×D4×C2311yescharacteristic
271Q8—43no—
272Q8—41no—
273D4—43no—
274D4—41no—
275D4—812no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—23no—
279C2×C4—412no—
280C2×C4—43no—
281C2×C4—43no—
282C2×C4—43no—
283C2×C4—43no—
284C2×C4—43no—
285C2×C4—46no—
286C2×C4—43no—
287C2×C4—43no—
288C2×C4—43no—
289C2×C4—43no—
290C2×C4C23×H(5,𝔽2)13yes—
291C2×C4C23×H(5,𝔽2)13yes—
292C8—43no—
293C8—41no—
294C22—26no—
295C22—26no—
296C22—26no—
297C22—416no—
298C22—43no—
299C22—43no—
300C22—43no—
301C22—43no—
302C22—83no—
303C22—83no—
304C22—81no—
305C22C23×#53413yes—
306C22C22×SD16×C2311yescharacteristic
307C22C23×#57513yes—
308C4—43no—
309C4—83no—
310C4—83no—
311C4—81no—
312C4C23×#56013yes—
313C4C23×H(3,𝔽4)11yescharacteristic
314C2—46no—
315C2—161no—
316C2C23×#281211yescharacteristic
317C2C23×#318713yes—
318C2C23×#319713yes—
subgroupG/H
center, socleC23×C23C22×D4
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#2847C2
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. C23×#2848C2semidirect
3. C23×#2077C2semidirect
4. C23×#2035C2semidirect
5. C23×#2034C2semidirect
6. C23×#2000C2semidirect
7. C23×#1779C2semidirect
9. C22×SD16×C23C22semidirect
13. C2×C23×#106C22semidirect
14. C2×C23×#103C22semidirect
15. C2×C23×#102C22semidirect
23. C23×#340C22semidirect
24. C23×#340C22semidirect
58. C24×C23SD16semidirect
97. #32708C23direct (N × H)
135. #2848C2×C23semidirect
137. #2077C2×C23semidirect
138. #2035C2×C23semidirect
139. #2034C2×C23semidirect
140. #2000C2×C23semidirect
141. #1779C2×C23semidirect
163. C22×SD16C22×C23semidirect
167. C2×#106C22×C23semidirect
168. C2×#103C22×C23semidirect
169. C2×#102C22×C23semidirect
177. #340C22×C23semidirect
178. #340C22×C23semidirect
214. C23#32708direct (N × H)
218. C24SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28482
C23×#28472
C23×#20772
C23×#20352
C23×#20342
C23×#20002
C23×#17792
#3270823