Finite Group Databasebeta

← C23×#11513 #19159139

Subgroups — C23×#11513 #19159139

Order 5888.

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

normal subgroups only

This group has 446 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
1C23×#2285C211yesmaximal, characteristic
2C23×#1464C212yesmaximal
3C23×#1325C212yesmaximal, complemented
4C23×#1324C212yesmaximal, complemented
5C2×C23×#117C411yescharacteristic
6C2×C23×#117C2211yescharacteristic
7C22×C42×C23C411yescharacteristic
8C23×#432—22no—
9C23×#384C411yescharacteristic
10C23×#384C2211yescharacteristic
11C23×#382C411yescharacteristic
12C23×#382C2211yescharacteristic
13C23×#380—22no—
14C4×C4⋊C4×C23—22no—
15C23×#324C2214yes—
16C23×#323—22no—
17C23×C4×C23—21no—
18C23×C4×C23C2×C411yescharacteristic
19C23×#117—41no—
20C23×#117—41no—
21C23×#117—41no—
22C23×#117—41no—
23C2×C4⋊C4×C23—22no—
24C2×C4⋊C4×C23—22no—
25C2×C4⋊C4×C23C2×C411yescharacteristic
26C2×C4⋊C4×C23C2×C411yescharacteristic
27C2×C4⋊C4×C23C2×C411yescharacteristic
28C2×C4⋊C4×C23C2311yescharacteristic
29C2×C22⋊C4×C23—21no—
30C2×C22⋊C4×C23D412yes—
31C2×C42×C23—21no—
32C2×C42×C23—22no—
33C2×C42×C23—22no—
34C2×C42×C23—21no—
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×C23C22⋊C411yescharacteristic
44C22×C4×C23—22no—
45C22×C4×C23—21no—
46C22×C4×C23—21no—
47C22×C4×C23—21no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—24no—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—42no—
55C22×C4×C23—42no—
56C22×C4×C23C22⋊C412yes—
57C22×C4×C23C22⋊C411yescharacteristic
58C22×C4×C23C22×C411yescharacteristic
59C22×C4×C23C2×D412yes—
60C2×C8×C23—44no—
61C4⋊C4×C23—41no—
62C4⋊C4×C23—41no—
63C4⋊C4×C23—41no—
64C4⋊C4×C23—41no—
65C4⋊C4×C23—44no—
66C4⋊C4×C23—44no—
67C22⋊C4×C23—42no—
68C22⋊C4×C23—42no—
69C42×C23—24no—
70C42×C23—22no—
71C42×C23—22no—
72C42×C23—21no—
73C42×C23—21no—
74C42×C23—41no—
75C42×C23—41no—
76C42×C23M1612yes—
77C42×C23M1612yes—
78#11513C2311yesmaximal, characteristic, direct product factor, Sylow
79C23×C23—21no—
80C23×C23—21no—
81C23×C23—41no—
82C23×C23—41no—
83C23×C23#10112yes—
84C23×C23C2×C22⋊C411yescharacteristic
85C2×C4×C23—22no—
86C2×C4×C23—22no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—21no—
92C2×C4×C23—22no—
93C2×C4×C23—21no—
94C2×C4×C23—21no—
95C2×C4×C23—21no—
96C2×C4×C23—22no—
97C2×C4×C23—22no—
98C2×C4×C23—22no—
99C2×C4×C23—48no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—41no—
107C2×C4×C23—41no—
108C2×C4×C23#9911yescharacteristic
109C2×C4×C23#9911yescharacteristic
110C2×C4×C23C2×M1611yescharacteristic
111C2×C4×C23C2×M1611yescharacteristic
112C8×C23—84no—
113#2285C2×C2311yescharacteristic
114#1464C2×C2312yes—
115#1325C2×C2312yescomplemented
116#1324C2×C2312yescomplemented
117C22×C23—21no—
118C22×C23—21no—
119C22×C23—22no—
120C22×C23—42no—
121C22×C23—42no—
122C22×C23—41no—
123C22×C23—41no—
124C22×C23#40711yescharacteristic
125C22×C23C2×#9911yescharacteristic
126C22×C23C2×#10111yescharacteristic
127C4×C23—24no—
128C4×C23—21no—
129C4×C23—21no—
130C4×C23—21no—
131C4×C23—21no—
132C4×C23—42no—
133C4×C23—42no—
134C4×C23—42no—
135C4×C23—81no—
136C4×C23—81no—
137C2×#117C4×C2311yescharacteristic
138C2×#117C22×C2311yescharacteristic
139C22×C42C4×C2311yescharacteristic
140#432—22no—
141#384C4×C2311yescharacteristic
142#384C22×C2311yescharacteristic
143#382C4×C2311yescharacteristic
144#382C22×C2311yescharacteristic
145#380—22no—
146C4×C4⋊C4—22no—
147#324C22×C2314yes—
148#323—22no—
149C2×C23—21no—
150C2×C23—21no—
151C2×C23—42no—
152C2×C23#146311yescharacteristic
153C2×C23#212611yescharacteristic
154C2×C23#213211yescharacteristic
155C23×C4—21no—
156C23×C4C2×C4×C2311yescharacteristic
157#117—41no—
158#117—41no—
159#117—41no—
160#117—41no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C2×C4×C2311yescharacteristic
164C2×C4⋊C4C2×C4×C2311yescharacteristic
165C2×C4⋊C4C2×C4×C2311yescharacteristic
166C2×C4⋊C4C23×C2311yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4D4×C2312yes—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42C2×C4×C2311yescharacteristic
174C2×C42C2×C4×C2311yescharacteristic
175C2×C42D4×C2312yes—
176C4⋊C8—42no—
177C22⋊C8—24no—
178#97—42no—
179#95—22no—
180#95—22no—
181C23#1151311yescharacteristic, direct product factor, Sylow
182C24C22⋊C4×C2311yescharacteristic
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C22⋊C4×C2311yescharacteristic
196C22×C4C22⋊C4×C2312yes—
197C22×C4C22×C4×C2311yescharacteristic
198C22×C4C2×D4×C2312yes—
199C2×C8—44no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—24no—
209C42—22no—
210C42—22no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C42M16×C2312yes—
216C42M16×C2312yes—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23C23×#10112yes—
222C23C2×C22⋊C4×C2311yescharacteristic
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4C23×#9911yescharacteristic
247C2×C4C23×#9911yescharacteristic
248C2×C4C2×M16×C2311yescharacteristic
249C2×C4C2×M16×C2311yescharacteristic
250C8—84no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22C23×#40711yescharacteristic
259C22C2×C23×#9911yescharacteristic
260C22C2×C23×#10111yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—81no—
270C4—81no—
271C2—21no—
272C2—21no—
273C2—42no—
274C2C23×#146311yescharacteristic
275C2C23×#212611yescharacteristic
276C2C23×#213211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1325C2semidirect
4. C23×#1324C2semidirect
78. #11513C23direct (N × H)
115. #1325C2×C23semidirect
116. #1324C2×C23semidirect
181. C23#11513direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#22852
C23×#14642
C23×#13252
C23×#13242
#1151323