Finite Group Databasebeta

← C23×#11530 #19159156

Subgroups — C23×#11530 #19159156

Order 5888.

444 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C4×C23×#124C211yesmaximal, characteristic
2C23×#1467C212yesmaximal, complemented
3C23×#1324C212yesmaximal, complemented
4C23×#1318C212yesmaximal
5C2×C23×#124C411yescharacteristic
6C2×C4×Q8×C23C2211yescharacteristic
7C2×C4×D4×C23C411yescharacteristic
8C23×#432—22no—
9C23×#391C411yescharacteristic
10C23×#390C2211yescharacteristic
11C23×#388—22no—
12C23×#382C2211yescharacteristic
13C4×C22⋊C4×C23—22no—
14C43×C23C411yescharacteristic
15C23×#324—22no—
16C23×#324C2214yes—
17C22×Q8×C23C2×C411yescharacteristic
18C22×D4×C23C2×C411yescharacteristic
19C23×C4×C23—21no—
20C23×#124—22no—
21C23×#124—22no—
22C4×Q8×C23—41no—
23C4×Q8×C23—41no—
24C4×D4×C23—41no—
25C4×D4×C23—41no—
26C2×C4⋊C4×C23C2×C411yescharacteristic
27C2×C4⋊C4×C23D412yes—
28C2×C4⋊C4×C23C2311yescharacteristic
29C2×C22⋊C4×C23—22no—
30C2×C22⋊C4×C23—21no—
31C2×C22⋊C4×C23—22no—
32C2×C42×C23—21no—
33C2×C42×C23—21no—
34C2×C42×C23C2×C411yescharacteristic
35C2×C42×C23C2×C411yescharacteristic
36C2×C42×C23C2×C411yescharacteristic
37C2×C42×C23D412yes—
38C23×C4⋊C8—42no—
39C23×C22⋊C8—24no—
40C23×#97—42no—
41C23×#95—22no—
42C23×#95—22no—
43C24×C23—21no—
44C2×Q8×C23—21no—
45C2×Q8×C23—21no—
46C2×Q8×C23—41no—
47C2×Q8×C23—41no—
48C2×D4×C23—21no—
49C2×D4×C23—41no—
50C2×D4×C23—41no—
51C2×D4×C23M1612yes—
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—44no—
56C22×C4×C23—41no—
57C22×C4×C23—41no—
58C22×C4×C23C22⋊C412yes—
59C22×C4×C23C22⋊C411yescharacteristic
60C22×C4×C23C22⋊C411yescharacteristic
61C22×C4×C23C22×C411yescharacteristic
62C22×C4×C23C2×D412yes—
63C2×C8×C23—44no—
64C4⋊C4×C23—42no—
65C4⋊C4×C23—41no—
66C4⋊C4×C23—41no—
67C22⋊C4×C23—28no—
68C22⋊C4×C23—44no—
69C22⋊C4×C23—42no—
70C42×C23—22no—
71C42×C23—21no—
72C42×C23—21no—
73C42×C23—22no—
74C42×C23—21no—
75C42×C23—41no—
76C42×C23—42no—
77C42×C23—42no—
78C42×C23M1612yes—
79#11530C2311yesmaximal, characteristic, direct product factor, Sylow
80C23×C23—22no—
81C23×C23—42no—
82C23×C23—42no—
83C23×C23—41no—
84C23×C23—41no—
85C23×C23C2×C22⋊C411yescharacteristic
86Q8×C23—42no—
87Q8×C23—42no—
88D4×C23—42no—
89D4×C23—42no—
90C2×C4×C23—21no—
91C2×C4×C23—21no—
92C2×C4×C23—22no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—21no—
97C2×C4×C23—22no—
98C2×C4×C23—22no—
99C2×C4×C23—22no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—41no—
104C2×C4×C23—41no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—41no—
108C2×C4×C23—82no—
109C2×C4×C23—82no—
110C2×C4×C23#9911yescharacteristic
111C2×C4×C23#9911yescharacteristic
112C2×C4×C23C2×M1611yescharacteristic
113C2×C4×C23C2×M1611yescharacteristic
114C8×C23—84no—
115C4×#124C2×C2311yescharacteristic
116#1467C2×C2312yescomplemented
117#1324C2×C2312yescomplemented
118#1318C2×C2312yes—
119C22×C23—21no—
120C22×C23—21no—
121C22×C23—44no—
122C22×C23—42no—
123C22×C23—82no—
124C22×C23—81no—
125C22×C23—81no—
126C22×C23#40711yescharacteristic
127C22×C23C2×#9911yescharacteristic
128C22×C23#41311yescharacteristic
129C4×C23—21no—
130C4×C23—21no—
131C4×C23—21no—
132C4×C23—42no—
133C4×C23—42no—
134C4×C23—42no—
135C4×C23—41no—
136C4×C23—42no—
137C4×C23—81no—
138C4×C23#35512yes—
139C2×#124C4×C2311yescharacteristic
140C2×C4×Q8C22×C2311yescharacteristic
141C2×C4×D4C4×C2311yescharacteristic
142#432—22no—
143#391C4×C2311yescharacteristic
144#390C22×C2311yescharacteristic
145#388—22no—
146#382C22×C2311yescharacteristic
147C4×C22⋊C4—22no—
148C43C4×C2311yescharacteristic
149#324—22no—
150#324C22×C2314yes—
151C2×C23—21no—
152C2×C23—21no—
153C2×C23—82no—
154C2×C23#146511yescharacteristic
155C2×C23#212611yescharacteristic
156C2×C23C2×#35511yescharacteristic
157C22×Q8C2×C4×C2311yescharacteristic
158C22×D4C2×C4×C2311yescharacteristic
159C23×C4—21no—
160#124—22no—
161#124—22no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×D4—41no—
165C4×D4—41no—
166C2×C4⋊C4C2×C4×C2311yescharacteristic
167C2×C4⋊C4D4×C2312yes—
168C2×C4⋊C4C23×C2311yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C2×C4×C2311yescharacteristic
175C2×C42C2×C4×C2311yescharacteristic
176C2×C42C2×C4×C2311yescharacteristic
177C2×C42D4×C2312yes—
178C4⋊C8—42no—
179C22⋊C8—24no—
180#97—42no—
181#95—22no—
182#95—22no—
183C23#1153011yescharacteristic, direct product factor, Sylow
184C24—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4M16×C2312yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C22⋊C4×C2312yes—
200C22×C4C22⋊C4×C2311yescharacteristic
201C22×C4C22⋊C4×C2311yescharacteristic
202C22×C4C22×C4×C2311yescharacteristic
203C22×C4C2×D4×C2312yes—
204C2×C8—44no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—44no—
211C42—22no—
212C42—22no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42M16×C2312yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊C4×C2311yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4C23×#9911yescharacteristic
251C2×C4C23×#9911yescharacteristic
252C2×C4C2×M16×C2311yescharacteristic
253C2×C4C2×M16×C2311yescharacteristic
254C8—84no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22C23×#40711yescharacteristic
263C22C2×C23×#9911yescharacteristic
264C22C23×#41311yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—42no—
272C4—41no—
273C4—81no—
274C4C23×#35512yes—
275C2—21no—
276C2—21no—
277C2—82no—
278C2C23×#146511yescharacteristic
279C2C23×#212611yescharacteristic
280C2C2×C23×#35511yescharacteristic
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×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1467C2semidirect
3. C23×#1324C2semidirect
79. #11530C23direct (N × H)
116. #1467C2×C23semidirect
117. #1324C2×C23semidirect
183. C23#11530direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C23×#1242
C23×#14672
C23×#13242
C23×#13182
#1153023