Finite Group Databasebeta

← C23×#11569 #19159195

Subgroups — C23×#11569 #19159195

Order 5888.

424 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2563C211yesmaximal, characteristic
2C23×#1465C211yesmaximal, characteristic
3C23×#1459C211yesmaximal, characteristic
4C23×#1324C214yesmaximal, complemented
5C2×C23×C4⋊Q8C411yescharacteristic
6C2×C4×D4×C23C2211yescharacteristic
7C23×#407—21no—
8C2×C23×C22⋊C8—21no—
9C23×#393—21no—
10C23×#393—21no—
11C23×#390C412yes—
12C23×#382C2212yes—
13C23×#380—21no—
14C23×#380—21no—
15C4×C4⋊C4×C23C411yescharacteristic
16C23×#324C2212yes—
17C23×#324C2212yes—
18C23×#323—21no—
19C23×#323—21no—
20C22×Q8×C23—21no—
21C22×D4×C23C2×C411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C2×M16×C23—41no—
25C22×C8×C23—41no—
26C23×C4⋊Q8—41no—
27C23×C4⋊Q8—41no—
28C4×D4×C23—41no—
29C4×D4×C23—41no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23C2×C412yes—
33C2×C4⋊C4×C23C2311yescharacteristic
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23D411yescharacteristic
37C2×C22⋊C4×C23D411yescharacteristic
38C2×C42×C23C2×C412yes—
39C2×C42×C23C2×C411yescharacteristic
40C23×C22⋊C8—22no—
41C23×C22⋊C8—22no—
42C23×C22⋊C8—41no—
43C23×C22⋊C8—41no—
44C23×#95—21no—
45C23×#95—22no—
46C23×#95—22no—
47C24×C23C22⋊C411yescharacteristic
48C24×C23C22⋊C411yescharacteristic
49C2×Q8×C23—42no—
50C2×Q8×C23—82no—
51C2×D4×C23—22no—
52C2×D4×C23—22no—
53C2×D4×C23—41no—
54C22×C4×C23—21no—
55C22×C4×C23—21no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—22no—
60C22×C4×C23—42no—
61C22×C4×C23—42no—
62C22×C4×C23C22⋊C411yescharacteristic
63C22×C4×C23C22⋊C411yescharacteristic
64C22×C4×C23C22×C411yescharacteristic
65C22×C4×C23C2×D411yescharacteristic
66C22×C4×C23C2×D411yescharacteristic
67M16×C23—82no—
68C2×C8×C23—42no—
69C2×C8×C23—42no—
70C2×C8×C23—82no—
71C4⋊C4×C23—41no—
72C4⋊C4×C23—42no—
73C4⋊C4×C23—42no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C22⋊C4×C23—41no—
77C22⋊C4×C23—41no—
78C22⋊C4×C23—82no—
79C22⋊C4×C23—82no—
80C42×C23—21no—
81C42×C23—21no—
82C42×C23—24no—
83#11569C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—21no—
85C23×C23—21no—
86C23×C23—21no—
87C23×C23—21no—
88C23×C23—21no—
89C23×C23—41no—
90C23×C23—41no—
91C23×C23—41no—
92C23×C23—41no—
93C23×C23#10112yes—
94C23×C23C2×C22⋊C411yescharacteristic
95Q8×C23—84no—
96D4×C23—44no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—22no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—41no—
106C2×C4×C23—41no—
107C2×C4×C23—41no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—81no—
115C2×C4×C23—81no—
116C2×C4×C23#9911yescharacteristic
117C2×C4×C23#9911yescharacteristic
118C2×C4×C23#13112yes—
119C8×C23—82no—
120C8×C23—82no—
121#2563C2×C2311yescharacteristic
122#1465C2×C2311yescharacteristic
123#1459C2×C2311yescharacteristic
124#1324C2×C2314yescomplemented
125C22×C23—21no—
126C22×C23—21no—
127C22×C23—22no—
128C22×C23—42no—
129C22×C23—42no—
130C22×C23—41no—
131C22×C23—41no—
132C22×C23—41no—
133C22×C23—41no—
134C22×C23—41no—
135C22×C23—81no—
136C22×C23—81no—
137C22×C23#40811yescharacteristic
138C22×C23C2×#9911yescharacteristic
139C22×C23C2×#10111yescharacteristic
140C4×C23—21no—
141C4×C23—22no—
142C4×C23—41no—
143C4×C23—41no—
144C4×C23—41no—
145C4×C23—82no—
146C4×C23—82no—
147C4×C23#35412yes—
148C2×C4⋊Q8C4×C2311yescharacteristic
149C2×C4×D4C22×C2311yescharacteristic
150#407—21no—
151C2×C22⋊C8—21no—
152#393—21no—
153#393—21no—
154#390C4×C2312yes—
155#382C22×C2312yes—
156#380—21no—
157#380—21no—
158C4×C4⋊C4C4×C2311yescharacteristic
159#324C22×C2312yes—
160#324C22×C2312yes—
161#323—21no—
162#323—21no—
163C2×C23—21no—
164C2×C23—21no—
165C2×C23—42no—
166C2×C23—81no—
167C2×C23#147211yescharacteristic
168C2×C23C2×#35411yescharacteristic
169C2×C23#213411yescharacteristic
170C22×Q8—21no—
171C22×D4C2×C4×C2311yescharacteristic
172C23×C4D4×C2311yescharacteristic
173C23×C4D4×C2311yescharacteristic
174C2×M16—41no—
175C22×C8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×C4×C2312yes—
183C2×C4⋊C4C23×C2311yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4D4×C2311yescharacteristic
187C2×C22⋊C4D4×C2311yescharacteristic
188C2×C42C2×C4×C2312yes—
189C2×C42C2×C4×C2311yescharacteristic
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#95—21no—
195#95—22no—
196#95—22no—
197C23#1156911yescharacteristic, direct product factor, Sylow
198C24C22⋊C4×C2311yescharacteristic
199C24C22⋊C4×C2311yescharacteristic
200C2×Q8—42no—
201C2×Q8—82no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C22⋊C4×C2311yescharacteristic
214C22×C4C22⋊C4×C2311yescharacteristic
215C22×C4C22×C4×C2311yescharacteristic
216C22×C4C2×D4×C2311yescharacteristic
217C22×C4C2×D4×C2311yescharacteristic
218M16—82no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—82no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C42—24no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C23×#10112yes—
244C23C2×C22⋊C4×C2311yescharacteristic
245Q8—84no—
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4C23×#9911yescharacteristic
267C2×C4C23×#9911yescharacteristic
268C2×C4C23×#13112yes—
269C8—82no—
270C8—82no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22C23×#40811yescharacteristic
284C22C2×C23×#9911yescharacteristic
285C22C2×C23×#10111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4C23×#35412yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2C23×#147211yescharacteristic
299C2C2×C23×#35411yescharacteristic
300C2C23×#213411yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1324C2semidirect
83. #11569C23direct (N × H)
124. #1324C2×C23semidirect
197. C23#11569direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25632
C23×#14652
C23×#14592
C23×#13242
#1156923