Finite Group Databasebeta

← C2×C23×#1318 #19159055

Subgroups — C2×C23×#1318 #19159055

Order 5888.

834 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×C23×#390C211yesmaximal, characteristic
2C2×C23×#324C212yesmaximal, complemented
3C23×#1318C214yesmaximal, direct product factor
4C23×D4×C23C411yescharacteristic
5C22×C4⋊C4×C23C2211yescharacteristic
6C22×C22⋊C4×C23—22no—
7C22×C42×C23C411yescharacteristic
8C2×C23×C22⋊C8—22no—
9C23×#390—22no—
10C23×#390C412yes—
11C23×#390C2212yes—
12C23×#324—22no—
13C23×#324C2214yescomplemented
14C25×C23—21no—
15C22×D4×C23—21no—
16C22×D4×C23—22no—
17C22×D4×C23—41no—
18C22×D4×C23—41no—
19C22×D4×C23—42no—
20C22×D4×C23C814yescomplemented
21C22×D4×C23C2×C412yes—
22C23×C4×C23—21no—
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23D412yes—
25C22×C8×C23—42no—
26C2×C4⋊C4×C23—41no—
27C2×C4⋊C4×C23—41no—
28C2×C4⋊C4×C23C2×C411yescharacteristic
29C2×C4⋊C4×C23D412yes—
30C2×C4⋊C4×C23C2311yescharacteristic
31C2×C22⋊C4×C23—24no—
32C2×C22⋊C4×C23—24no—
33C2×C22⋊C4×C23—48no—
34C2×C42×C23—21no—
35C2×C42×C23—42no—
36C2×C42×C23C2×C412yes—
37C23×C22⋊C8—24no—
38C23×C22⋊C8—42no—
39C24×C23—24no—
40C24×C23—22no—
41C24×C23—44no—
42C24×C23—44no—
43C24×C23—41no—
44C24×C23—41no—
45C24×C23—42no—
46C24×C23C22⋊C411yescharacteristic
47C2×D4×C23—24no—
48C2×D4×C23—22no—
49C2×D4×C23—44no—
50C2×D4×C23—44no—
51C2×D4×C23—42no—
52C2×D4×C23—42no—
53C2×D4×C23—44no—
54C2×D4×C23—44no—
55C2×D4×C23—42no—
56C2×D4×C23—42no—
57C2×D4×C23C2×C814yescomplemented
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—22no—
61C22×C4×C23—24no—
62C22×C4×C23—22no—
63C22×C4×C23—22no—
64C22×C4×C23—22no—
65C22×C4×C23—42no—
66C22×C4×C23—42no—
67C22×C4×C23—44no—
68C22×C4×C23C22⋊C411yescharacteristic
69C22×C4×C23C22⋊C412yes—
70C22×C4×C23C2×C812yes—
71C22×C4×C23M1612yes—
72C22×C4×C23C22×C411yescharacteristic
73C22×C4×C23C2×D412yes—
74C2×C8×C23—44no—
75C2×C8×C23—84no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—42no—
78C4⋊C4×C23—41no—
79C22⋊C4×C23—48no—
80C22⋊C4×C23—48no—
81C42×C23—42no—
82C42×C23—42no—
83C2×#1318C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—22no—
85C23×C23—22no—
86C23×C23—24no—
87C23×C23—44no—
88C23×C23—416no—
89C23×C23—44no—
90C23×C23—44no—
91C23×C23—44no—
92C23×C23—42no—
93C23×C23—42no—
94C23×C23—88no—
95C23×C23—82no—
96C23×C23—82no—
97C23×C23—82no—
98C23×C23—82no—
99C23×C23C22⋊C812yes—
100C23×C23#9911yescharacteristic
101C23×C23#9911yescharacteristic
102C23×C23#10012yes—
103C23×C23C2×C22⋊C411yescharacteristic
104D4×C23—44no—
105D4×C23—48no—
106D4×C23—44no—
107C2×C4×C23—22no—
108C2×C4×C23—22no—
109C2×C4×C23—22no—
110C2×C4×C23—22no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—24no—
116C2×C4×C23—22no—
117C2×C4×C23—22no—
118C2×C4×C23—48no—
119C2×C4×C23—42no—
120C2×C4×C23—44no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—82no—
125C2×C4×C23C22⋊C812yes—
126C2×C4×C23C22×C811yescharacteristic
127C2×C4×C23C2×M1611yescharacteristic
128C8×C23—84no—
129C2×#390C2×C2311yescharacteristic
130C2×#324C2×C2312yescomplemented
131#1318C2×C2314yesdirect product factor
132C22×C23—24no—
133C22×C23—24no—
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—21no—
138C22×C23—48no—
139C22×C23—44no—
140C22×C23—84no—
141C22×C23—88no—
142C22×C23—88no—
143C22×C23—82no—
144C22×C23—82no—
145C22×C23#32312yes—
146C22×C23#32312yes—
147C22×C23#35312yes—
148C22×C23#35512yes—
149C22×C23C2×C22⋊C811yescharacteristic
150C22×C23C2×#9911yescharacteristic
151C22×C23C2×#10011yescharacteristic
152C4×C23—22no—
153C4×C23—21no—
154C4×C23—21no—
155C4×C23—44no—
156C4×C23—42no—
157C4×C23—42no—
158C23×D4C4×C2311yescharacteristic
159C22×C4⋊C4C22×C2311yescharacteristic
160C22×C22⋊C4—22no—
161C22×C42C4×C2311yescharacteristic
162C2×C22⋊C8—22no—
163#390—22no—
164#390C4×C2312yes—
165#390C22×C2312yes—
166#324—22no—
167#324C22×C2314yescomplemented
168C2×C23—21no—
169C2×C23—21no—
170C2×C23—22no—
171C2×C23—84no—
172C2×C23#131814yesdirect product factor
173C2×C23C2×#32311yescharacteristic
174C2×C23C2×#35311yescharacteristic
175C2×C23C2×#35511yescharacteristic
176C25—21no—
177C22×D4—21no—
178C22×D4—22no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4C8×C2314yescomplemented
183C22×D4C2×C4×C2312yes—
184C23×C4—21no—
185C23×C4C2×C4×C2311yescharacteristic
186C23×C4D4×C2312yes—
187C22×C8—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×C4×C2311yescharacteristic
191C2×C4⋊C4D4×C2312yes—
192C2×C4⋊C4C23×C2311yescharacteristic
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—48no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42C2×C4×C2312yes—
199C22⋊C8—24no—
200C22⋊C8—42no—
201C23C2×#131811yescharacteristic, direct product factor, Sylow
202C24—22no—
203C24—24no—
204C24—44no—
205C24—44no—
206C24—41no—
207C24—41no—
208C24—42no—
209C24C22⋊C4×C2311yescharacteristic
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4C2×C8×C2314yescomplemented
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4C22⋊C4×C2311yescharacteristic
232C22×C4C22⋊C4×C2312yes—
233C22×C4C2×C8×C2312yes—
234C22×C4M16×C2312yes—
235C22×C4C22×C4×C2311yescharacteristic
236C22×C4C2×D4×C2312yes—
237C2×C8—44no—
238C2×C8—84no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C42—42no—
245C42—42no—
246C23—22no—
247C23—22no—
248C23—24no—
249C23—44no—
250C23—416no—
251C23—44no—
252C23—44no—
253C23—44no—
254C23—42no—
255C23—42no—
256C23—88no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23—82no—
261C23C23×C22⋊C812yes—
262C23C23×#9911yescharacteristic
263C23C23×#9911yescharacteristic
264C23C23×#10012yes—
265C23C2×C22⋊C4×C2311yescharacteristic
266D4—44no—
267D4—44no—
268D4—48no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—48no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4C23×C22⋊C812yes—
288C2×C4C22×C8×C2311yescharacteristic
289C2×C4C2×M16×C2311yescharacteristic
290C8—84no—
291C22—24no—
292C22—24no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—48no—
298C22—44no—
299C22—84no—
300C22—88no—
301C22—88no—
302C22—82no—
303C22—82no—
304C22C23×#32312yes—
305C22C23×#32312yes—
306C22C23×#35312yes—
307C22C23×#35512yes—
308C22C2×C23×C22⋊C811yescharacteristic
309C22C2×C23×#9911yescharacteristic
310C22C2×C23×#10011yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—44no—
315C4—42no—
316C4—42no—
317C2—21no—
318C2—21no—
319C2—22no—
320C2—84no—
321C2C23×#131814yesdirect product factor
322C2C2×C23×#32311yescharacteristic
323C2C2×C23×#35311yescharacteristic
324C2C2×C23×#35511yescharacteristic
subgroupG/H
center, socleC23×C23#99
commutatorC2×C4C22×C8×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×D4×C23C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C23 ◁ C22×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#324C2semidirect
3. C23×#1318C2direct (N × H)
13. C23×#324C22semidirect
20. C22×D4×C23C8semidirect
57. C2×D4×C23C2×C8semidirect
83. C2×#1318C23direct (N × H)
130. C2×#324C2×C23semidirect
131. #1318C2×C23direct (N × H)
167. #324C22×C23semidirect
172. C2×C23#1318direct (N × H)
182. C22×D4C8×C23semidirect
201. C23C2×#1318direct (N × H)
220. C2×D4C2×C8×C23semidirect
321. C2C23×#1318direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3902
C2×C23×#3242
C23×#13182
C2×#131823