Finite Group Databasebeta

← C23×#9480 #19157106

Subgroups — C23×#9480 #19157106

Order 5888.

726 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2406C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic, complemented
3C23×#1789C212yesmaximal
4C23×#1463C211yesmaximal, characteristic, complemented
5C23×#1284C212yesmaximal, complemented
6C22×#41×C23C2211yescharacteristic
7C2×C23×#123—21no—
8C22×C4⋊C4×C23C2211yescharacteristic
9C2×C23×#104—24no—
10C23×#407C2212yes—
11C23×#395C412yescomplemented
12C23×#394C412yescomplemented
13C23×#387—24no—
14C23×#386—21no—
15C23×#385—22no—
16C23×#381—21no—
17C23×#381C2211yescharacteristic
18C23×#380—21no—
19C4×C22⋊C4×C23C2212yes—
20C23×#324—22no—
21C2×#41×C23—22no—
22C2×#41×C23—22no—
23C2×#41×C23—44no—
24C22×Q8×C23C2×C411yescharacteristic
25C22×D4×C23—21no—
26C22×D4×C23C2×C411yescharacteristic
27C23×C4×C23—21no—
28C23×C4×C23D411yescharacteristic
29C23×C4×C23D411yescharacteristic
30C23×C4×C23C2311yescharacteristic
31C2×M16×C23—22no—
32C23×#123—44no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—22no—
36C2×C4⋊C4×C23—22no—
37C2×C4⋊C4×C23C2×C412yes—
38C2×C4⋊C4×C23D412yes—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—22no—
43C2×C42×C23D414yes—
44C23×#104—48no—
45C23×C22⋊C8—22no—
46C23×C22⋊C8—22no—
47C23×#95—22no—
48C23×#95—24no—
49C23×#95—22no—
50C23×#95—22no—
51C23×#95—21no—
52C23×#95C2×C412yes—
53C24×C23—21no—
54C24×C23C2×D411yescharacteristic
55#41×C23—28no—
56#41×C23—44no—
57#41×C23—44no—
58#41×C23—44no—
59C2×Q8×C23—22no—
60C2×Q8×C23—42no—
61C2×D4×C23—24no—
62C2×D4×C23—22no—
63C2×D4×C23—42no—
64C2×D4×C23—42no—
65C2×D4×C23—41no—
66C2×D4×C23—41no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—22no—
73C22×C4×C23—24no—
74C22×C4×C23—42no—
75C22×C4×C23—42no—
76C22×C4×C23—42no—
77C22×C4×C23—42no—
78C22×C4×C23—42no—
79C22×C4×C23—42no—
80C22×C4×C23C22⋊C412yes—
81C22×C4×C23C22⋊C412yes—
82C22×C4×C23C22×C411yescharacteristic
83C22×C4×C23C2×D412yes—
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23#4112yes—
86M16×C23—44no—
87C2×C8×C23—42no—
88C4⋊C4×C23—24no—
89C4⋊C4×C23—24no—
90C4⋊C4×C23—44no—
91C4⋊C4×C23—82no—
92C22⋊C4×C23—44no—
93C22⋊C4×C23—44no—
94C22⋊C4×C23—44no—
95C22⋊C4×C23—42no—
96C42×C23—44no—
97#9480C2311yesmaximal, characteristic, direct product factor, Sylow
98C23×C23—22no—
99C23×C23—22no—
100C23×C23—22no—
101C23×C23—41no—
102C23×C23—41no—
103C23×C23—42no—
104C23×C23—41no—
105C23×C23—42no—
106C23×C23C2×C22⋊C411yescharacteristic
107C23×C23#12011yescharacteristic
108C23×C23#12311yescharacteristic
109Q8×C23—24no—
110Q8×C23—42no—
111D4×C23—24no—
112D4×C23—42no—
113D4×C23—44no—
114D4×C23—44no—
115C2×C4×C23—24no—
116C2×C4×C23—22no—
117C2×C4×C23—22no—
118C2×C4×C23—22no—
119C2×C4×C23—22no—
120C2×C4×C23—22no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—42no—
124C2×C4×C23—48no—
125C2×C4×C23—42no—
126C2×C4×C23—42no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
131C2×C4×C23—42no—
132C2×C4×C23—44no—
133C2×C4×C23—44no—
134C2×C4×C23—42no—
135C2×C4×C23—42no—
136C2×C4×C23—44no—
137C2×C4×C23—81no—
138C2×C4×C23C4×D412yes—
139C2×C4×C23#12112yes—
140C8×C23—82no—
141#2406C2×C2311yescharacteristic
142#2313C2×C2311yescharacteristic, complemented
143#1789C2×C2312yes—
144#1463C2×C2311yescharacteristic, complemented
145#1284C2×C2312yescomplemented
146C22×C23—21no—
147C22×C23—21no—
148C22×C23—21no—
149C22×C23—21no—
150C22×C23—21no—
151C22×C23—21no—
152C22×C23—21no—
153C22×C23—44no—
154C22×C23—44no—
155C22×C23—42no—
156C22×C23—82no—
157C22×C23—81no—
158C22×C23—81no—
159C22×C23#38611yescharacteristic
160C22×C23#42112yes—
161C22×C23B(4,𝔽2)12yes—
162C4×C23—22no—
163C4×C23—22no—
164C4×C23—42no—
165C4×C23—42no—
166C4×C23—42no—
167C4×C23—84no—
168C4×C23—82no—
169C22×#41C22×C2311yescharacteristic
170C2×#123—21no—
171C22×C4⋊C4C22×C2311yescharacteristic
172C2×#104—24no—
173#407C22×C2312yes—
174#395C4×C2312yescomplemented
175#394C4×C2312yescomplemented
176#387—24no—
177#386—21no—
178#385—22no—
179#381—21no—
180#381C22×C2311yescharacteristic
181#380—21no—
182C4×C22⋊C4C22×C2312yes—
183#324—22no—
184C2×C23—22no—
185C2×C23—22no—
186C2×C23—41no—
187C2×C23—82no—
188C2×C23#188911yescharacteristic
189C2×C23#189711yescharacteristic
190C2×C23#191211yescharacteristic
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—44no—
194C22×Q8C2×C4×C2311yescharacteristic
195C22×D4—21no—
196C22×D4C2×C4×C2311yescharacteristic
197C23×C4—21no—
198C23×C4D4×C2311yescharacteristic
199C23×C4D4×C2311yescharacteristic
200C23×C4C23×C2311yescharacteristic
201C2×M16—22no—
202#123—44no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C2×C4×C2312yes—
208C2×C4⋊C4D4×C2312yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42D4×C2314yes—
214#104—48no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217#95—22no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95C2×C4×C2312yes—
223C23#948011yescharacteristic, direct product factor, Sylow
224C24—21no—
225C24C2×D4×C2311yescharacteristic
226#41—28no—
227#41—44no—
228#41—44no—
229#41—44no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊C4×C2312yes—
252C22×C4C22⋊C4×C2312yes—
253C22×C4C22×C4×C2311yescharacteristic
254C22×C4C2×D4×C2312yes—
255C22×C4C2×D4×C2311yescharacteristic
256C22×C4#41×C2312yes—
257M16—44no—
258C2×C8—42no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—82no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—44no—
267C42—44no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—41no—
275C23—42no—
276C23C2×C22⋊C4×C2311yescharacteristic
277C23C23×#12011yescharacteristic
278C23C23×#12311yescharacteristic
279Q8—24no—
280Q8—42no—
281D4—24no—
282D4—42no—
283D4—44no—
284D4—44no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—48no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—81no—
308C2×C4C4×D4×C2312yes—
309C2×C4C23×#12112yes—
310C8—82no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—44no—
320C22—42no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22C23×#38611yescharacteristic
325C22C23×#42112yes—
326C22C23×B(4,𝔽2)12yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—84no—
333C4—82no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—82no—
338C2C23×#188911yescharacteristic
339C2C23×#189711yescharacteristic
340C2C23×#191211yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC22×#41×C23C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2313C2semidirect
4. C23×#1463C2semidirect
5. C23×#1284C2semidirect
11. C23×#395C4semidirect
12. C23×#394C4semidirect
97. #9480C23direct (N × H)
142. #2313C2×C23semidirect
144. #1463C2×C23semidirect
145. #1284C2×C23semidirect
174. #395C4×C23semidirect
175. #394C4×C23semidirect
223. C23#9480direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24062
C23×#23132
C23×#17892
C23×#14632
C23×#12842
#948023