Finite Group Databasebeta

← C23×#19074 #19166700

Subgroups — C23×#19074 #19166700

Order 5888.

1072 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3549C211yesmaximal, characteristic
2C23×#3152C211yesmaximal, characteristic, complemented
3C23×#3141C212yesmaximal, complemented
4C23×#1659C218yesmaximal, complemented
5C23×#1525C211yesmaximal, characteristic, complemented
6C23×#1525C212yesmaximal, complemented
7C2×C23×#137—22no—
8C23×H(3,𝔽4)C2214yes—
9C23×#555—22no—
10C23×#555—21no—
11D42×C23—22no—
12D42×C23—21no—
13C23×#535—22no—
14C23×#535—24no—
15C23×#532C2212yes—
16C23×#532C2211yescharacteristic
17C2×C23×C4⋊Q8C2211yescharacteristic, complemented
18C23×#497C2214yescomplemented
19C23×#494—21no—
20C23×#492—21no—
21C23×#489C2218yescomplemented
22C23×#487—22no—
23C23×#486—22no—
24C23×#417—24no—
25C4×M16×C23C2212yescomplemented
26C4×M16×C23C2211yescharacteristic, complemented
27C23×#329C2218yescomplemented
28C23×#329C2214yescomplemented
29C2×#41×C23D412yes—
30C2×#41×C23D414yes—
31C22×Q8×C23—21no—
32C22×D4×C23—22no—
33C22×D4×C23—24no—
34C23×#137—48no—
35C2×Q16×C23—24no—
36C2×SD16×C23—24no—
37C2×M16×C23—21no—
38C2×M16×C23—22no—
39C23×C4⋊Q8—22no—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8D412yes—
42C23×C4⋊Q8D411yescharacteristic
43C23×C4⋊Q8C2312yes—
44C23×#127—24no—
45C23×#127D412yes—
46C23×#127D411yescharacteristic
47C23×#124—28no—
48C23×#124—24no—
49C23×#124C2314yes—
50C23×#124C2312yes—
51C23×#121—24no—
52C23×#121—24no—
53C23×#121—24no—
54C23×#121—22no—
55C23×#121—22no—
56C23×#121—22no—
57C23×#120—24no—
58C23×#120—28no—
59C23×#120—24no—
60C4×D4×C23—22no—
61C4×D4×C23—22no—
62C4×D4×C23—22no—
63C23×#117—22no—
64C23×#117—24no—
65C2×C4⋊C4×C23—22no—
66C2×C42×C23C2311yescharacteristic
67C23×#103—28no—
68C23×#102—28no—
69C23×#97C2314yes—
70C23×#97C2312yes—
71C4×C8×C23—22no—
72C4×C8×C23—21no—
73C24×C23—24no—
74#41×C23—44no—
75#41×C23—48no—
76C2×Q8×C23—22no—
77C2×Q8×C23—44no—
78C2×Q8×C23—41no—
79C2×Q8×C23C2×D414yes—
80C2×Q8×C23C2×D412yes—
81C2×D4×C23—24no—
82C2×D4×C23—28no—
83C2×D4×C23—22no—
84C2×D4×C23—24no—
85C2×D4×C23—22no—
86C2×D4×C23—22no—
87C2×D4×C23—22no—
88C2×D4×C23—84no—
89C2×D4×C23—88no—
90C2×D4×C23C2×D414yes—
91C2×D4×C23C2×D412yes—
92C22×C4×C23—22no—
93C22×C4×C23—24no—
94C22×C4×C23—41no—
95C22×C4×C23C2×D412yes—
96C22×C4×C23C2×D411yescharacteristic
97Q16×C23—48no—
98SD16×C23—48no—
99M16×C23—24no—
100M16×C23—44no—
101C2×C8×C23—22no—
102C2×C8×C23—24no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—22no—
105C4⋊C4×C23—22no—
106C4⋊C4×C23—24no—
107C4⋊C4×C23—42no—
108C22⋊C4×C23—28no—
109C22⋊C4×C23—24no—
110C22⋊C4×C23—24no—
111C22⋊C4×C23—24no—
112C22⋊C4×C23—24no—
113C42×C23—24no—
114C42×C23C2×D412yes—
115C42×C23C2×D411yescharacteristic
116C42×C23C2411yescharacteristic
117#19074C2311yesmaximal, characteristic, direct product factor, Sylow
118C23×C23—22no—
119C23×C23—24no—
120C23×C23—84no—
121C23×C23—88no—
122C23×C23#12011yescharacteristic
123Q8×C23—42no—
124Q8×C23—44no—
125Q8×C23—44no—
126Q8×C23—82no—
127D4×C23—42no—
128D4×C23—44no—
129D4×C23—44no—
130D4×C23—44no—
131D4×C23—44no—
132D4×C23—82no—
133D4×C23—84no—
134D4×C23—84no—
135D4×C23—88no—
136C2×C4×C23—22no—
137C2×C4×C23—24no—
138C2×C4×C23—22no—
139C2×C4×C23—24no—
140C2×C4×C23—44no—
141C2×C4×C23—48no—
142C2×C4×C23—42no—
143C2×C4×C23—82no—
144C2×C4×C23#12011yescharacteristic
145C2×C4×C23#12012yes—
146C2×C4×C23C22×D412yes—
147C2×C4×C23C22×D411yescharacteristic
148C8×C23—24no—
149C8×C23—44no—
150#3549C2×C2311yescharacteristic
151#3152C2×C2311yescharacteristic, complemented
152#3141C2×C2312yescomplemented
153#1659C2×C2318yescomplemented
154#1525C2×C2312yescomplemented
155#1525C2×C2311yescharacteristic, complemented
156C22×C23—21no—
157C22×C23—22no—
158C22×C23—42no—
159C22×C23—44no—
160C22×C23—44no—
161C22×C23—44no—
162C22×C23—44no—
163C22×C23—168no—
164C22×C23C2×#12011yescharacteristic
165C4×C23—21no—
166C4×C23—22no—
167C4×C23—42no—
168C4×C23—44no—
169C4×C23—82no—
170C4×C23#45312yes—
171C4×C23#45514yes—
172C2×#137—22no—
173H(3,𝔽4)C22×C2314yes—
174#555—22no—
175#555—21no—
176D42—22no—
177D42—21no—
178#535—22no—
179#535—24no—
180#532C22×C2312yes—
181#532C22×C2311yescharacteristic
182C2×C4⋊Q8C22×C2311yescharacteristic, complemented
183#497C22×C2314yescomplemented
184#494—21no—
185#492—21no—
186#489C22×C2318yescomplemented
187#487—22no—
188#486—22no—
189#417—24no—
190C4×M16C22×C2312yescomplemented
191C4×M16C22×C2311yescharacteristic, complemented
192#329C22×C2314yescomplemented
193#329C22×C2318yescomplemented
194C2×C23—41no—
195C2×C23—82no—
196C2×C23—84no—
197C2×C23C2×#45311yescharacteristic
198C2×C23C2×#45512yes—
199C2×#41D4×C2312yes—
200C2×#41D4×C2314yes—
201C22×Q8—21no—
202C22×D4—22no—
203C22×D4—24no—
204#137—48no—
205C2×Q16—24no—
206C2×SD16—24no—
207C2×M16—21no—
208C2×M16—22no—
209C4⋊Q8—22no—
210C4⋊Q8—21no—
211C4⋊Q8D4×C2312yes—
212C4⋊Q8D4×C2311yescharacteristic
213C4⋊Q8C23×C2312yes—
214#127—24no—
215#127D4×C2312yes—
216#127D4×C2311yescharacteristic
217#124—28no—
218#124—24no—
219#124C23×C2314yes—
220#124C23×C2312yes—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—22no—
225#121—22no—
226#121—22no—
227#120—24no—
228#120—28no—
229#120—24no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233#117—22no—
234#117—24no—
235C2×C4⋊C4—22no—
236C2×C42C23×C2311yescharacteristic
237#103—28no—
238#102—28no—
239#97C23×C2314yes—
240#97C23×C2312yes—
241C4×C8—22no—
242C4×C8—21no—
243C23#1907411yescharacteristic, direct product factor, Sylow
244C24—24no—
245#41—44no—
246#41—48no—
247C2×Q8—22no—
248C2×Q8—44no—
249C2×Q8—41no—
250C2×Q8C2×D4×C2314yes—
251C2×Q8C2×D4×C2312yes—
252C2×D4—24no—
253C2×D4—28no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—84no—
260C2×D4—88no—
261C2×D4C2×D4×C2314yes—
262C2×D4C2×D4×C2312yes—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—41no—
266C22×C4C2×D4×C2312yes—
267C22×C4C2×D4×C2311yescharacteristic
268Q16—48no—
269SD16—48no—
270M16—24no—
271M16—44no—
272C2×C8—22no—
273C2×C8—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C22⋊C4—28no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C42—24no—
285C42C2×D4×C2312yes—
286C42C2×D4×C2311yescharacteristic
287C42C24×C2311yescharacteristic
288C23—22no—
289C23—24no—
290C23—84no—
291C23—88no—
292C23C23×#12011yescharacteristic
293Q8—42no—
294Q8—44no—
295Q8—44no—
296Q8—82no—
297D4—42no—
298D4—44no—
299D4—44no—
300D4—44no—
301D4—44no—
302D4—82no—
303D4—84no—
304D4—84no—
305D4—88no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4C23×#12011yescharacteristic
315C2×C4C23×#12012yes—
316C2×C4C22×D4×C2312yes—
317C2×C4C22×D4×C2311yescharacteristic
318C8—24no—
319C8—44no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—168no—
328C22C2×C23×#12011yescharacteristic
329C4—21no—
330C4—22no—
331C4—42no—
332C4—44no—
333C4—82no—
334C4C23×#45312yes—
335C4C23×#45514yes—
336C2—41no—
337C2—82no—
338C2—84no—
339C2C2×C23×#45311yescharacteristic
340C2C2×C23×#45512yes—
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3549C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3152C2semidirect
3. C23×#3141C2semidirect
4. C23×#1659C2semidirect
5. C23×#1525C2semidirect
6. C23×#1525C2semidirect
17. C2×C23×C4⋊Q8C22semidirect
18. C23×#497C22semidirect
21. C23×#489C22semidirect
25. C4×M16×C23C22semidirect
26. C4×M16×C23C22semidirect
27. C23×#329C22semidirect
28. C23×#329C22semidirect
117. #19074C23direct (N × H)
151. #3152C2×C23semidirect
152. #3141C2×C23semidirect
153. #1659C2×C23semidirect
154. #1525C2×C23semidirect
155. #1525C2×C23semidirect
182. C2×C4⋊Q8C22×C23semidirect
183. #497C22×C23semidirect
186. #489C22×C23semidirect
190. C4×M16C22×C23semidirect
191. C4×M16C22×C23semidirect
192. #329C22×C23semidirect
193. #329C22×C23semidirect
243. C23#19074direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35492
C23×#31522
C23×#31412
C23×#16592
C23×#15252
#1907423