Finite Group Databasebeta

← C2×#5720⋊C3 #11923235

Subgroups — C2×#5720⋊C3 #11923235

Order 1152.

623 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 67 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×C3×#127⋊C3C211yesmaximal, characteristic, complemented
2#5720⋊C3C214yesmaximal, direct product factor
3C2×C3⋊(#126⋊C3)C212yesmaximal, complemented
4C2×#5720C311yesmaximal, characteristic, complemented
5C2×#560⋊C3—31nomaximal
6C22×D3×A4—41nomaximal
7C3×#127⋊C3C2212yescomplemented
8C2×C3×C42⋊C3C2211yescharacteristic, complemented
9C3⋊(#126⋊C3)C2214yescomplemented
10C2×C3×#127C2×C311yescharacteristic, complemented
11C2×#939—32no—
12C2×#937—31no—
13C2×#935—31no—
14#5720—34no—
15#5720C2×C314yescomplemented
16C2×#876C2×C312yescomplemented
17#560⋊C3—34no—
18C2×#127⋊C3—21no—
19C2×#127⋊C3D311yescharacteristic, complemented
20C2×#126⋊C3—32no—
21C22×C3×A4—41no—
22C2×D3×A4—42no—
23C2×D3×A4—44no—
24C3×C42⋊C3C2311yescharacteristic, complemented
25C2×#560—31noSylow
26C24×D3A411yescharacteristic
27C23×A4—121no—
28C22×C3×D4—31no—
29C22×C3×D4—31no—
30C22×#67—32no—
31C23×Dic3—31no—
32C3×#127—32no—
33C3×#127C22×C312yescomplemented
34C2×C3×C42C22×C311yescharacteristic, complemented
35C2×#217—31no—
36C2×#217—31no—
37#939—38no—
38#939—38no—
39#937—34no—
40#937—34no—
41#935—34no—
42#935—34no—
43C2×(C2×D3)⋊C4—32no—
44C2×Dic3⋊C4—32no—
45#876—34no—
46#876C22×C314yescomplemented
47#127⋊C3—22no—
48#127⋊C3C2×D312yescomplemented
49#126⋊C3—34no—
50C2×C42⋊C3—21no—
51C2×C42⋊C3C2×D311yescharacteristic, complemented
52C22×C3×D3—161no—
53C2×C3×A4—41no—
54C2×C3×A4—42no—
55D3×A4—44no—
56#560—34no—
57#560—94no—
58C2×#127C3×D311yescharacteristic, complemented
59C2×#126—32no—
60C2×#123—91no—
61C2×#121—92no—
62C2×#120—91no—
63C24×C3—31no—
64C24×C3C2×A411yescharacteristic
65C23×D3—121no—
66C23×D3—121no—
67C23×D3C2×A412yes—
68C23×D3C2×A414yes—
69C22×A4—41no—
70C22×A4—81no—
71C22×A4—122no—
72C22×A4—124no—
73C2×C3×D4—32no—
74C2×C3×D4—32no—
75C2×C3×D4—32no—
76C2×C3×D4—32no—
77C2×C3×D4—121no—
78C2×C3×D4—121no—
79C2×C3×D4—121no—
80C2×C3×D4—121no—
81C22×C3×C4—31no—
82C2×#67—38no—
83C2×#67—122no—
84C2×#67—122no—
85C22×Dic3—32no—
86C22×Dic3—34no—
87C22×Dic3—121no—
88C22×Dic3—121no—
89C3×C42—31no—
90C3×C42C23×C311yescharacteristic, complemented
91#217—34no—
92#217—34no—
93(C2×D3)⋊C4—34no—
94(C2×D3)⋊C4—34no—
95Dic3⋊C4—34no—
96Dic3⋊C4—34no—
97C42⋊C3—21no—
98C42⋊C3C22×D311yescharacteristic, complemented
99C22×C32—161no—
100C2×C3×D3—164no—
101C2×C3×D3—162no—
102C3×A4—41no—
103C25—31no—
104C22×D4—31no—
105C22×D4—31no—
106C22×D4—92no—
107C23×C4—91no—
108#127—32no—
109#127C2×C3×D312yescomplemented
110#126—34no—
111#126—94no—
112#123—94no—
113#123—94no—
114#121—98no—
115#121—98no—
116#120—94no—
117#120—94no—
118C2×C4⋊C4—92no—
119C2×C22⋊C4—92no—
120C2×C22⋊C4—91no—
121C2×C22⋊C4—91no—
122C2×C42C2×C3×D311yescharacteristic, complemented
123C23×C3—32no—
124C23×C3—61no—
125C23×C3—61no—
126C23×C3—61no—
127C23×C3—61no—
128C23×C3—62no—
129C23×C3—62no—
130C23×C3—481no—
131C23×C3C22×A411yescharacteristic
132C23×C3C22×A412yes—
133C22×D3—41no—
134C22×D3—62no—
135C22×D3—62no—
136C22×D3—121no—
137C22×D3—124no—
138C22×D3—124no—
139C22×D3C22×A414yes—
140C2×A4—41no—
141C2×A4—42no—
142C2×A4—81no—
143C2×A4—82no—
144C2×A4—124no—
145C3×D4—122no—
146C3×D4—122no—
147C3×D4—122no—
148C3×D4—122no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—61no—
152C2×C3×C4—61no—
153#67—128no—
154C2×Dic3—34no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—124no—
158C2×C32—162no—
159C2×C32—161no—
160C3×D3—164no—
161C24—31no—
162C24—32no—
163C24—34no—
164C24—361no—
165C24—361no—
166C24D3×A411yescharacteristic
167C2×D4—32no—
168C2×D4—32no—
169C2×D4—32no—
170C2×D4—32no—
171C2×D4—98no—
172C2×D4—121no—
173C2×D4—121no—
174C2×D4—121no—
175C2×D4—121no—
176C2×D4—362no—
177C2×D4—362no—
178C22×C4—31no—
179C22×C4—92no—
180C22×C4—94no—
181C22×C4—361no—
182C22×C4—361no—
183C4⋊C4—94no—
184C4⋊C4—94no—
185C22⋊C4—94no—
186C22⋊C4—94no—
187C22⋊C4—94no—
188C22⋊C4—94no—
189C42—31no—
190C42C22×C3×D311yescharacteristic, complemented
191C22×C3—31no—
192C22×C3—31no—
193C22×C3—41no—
194C22×C3—62no—
195C22×C3—62no—
196C22×C3—64no—
197C22×C3—121no—
198C22×C3—122no—
199C22×C3—121no—
200C22×C3—121no—
201C22×C3—161no—
202C22×C3—321no—
203C22×C3—484no—
204C22×C3—482no—
205C22×C3C23×A411yescharacteristic
206C2×D3—44no—
207C2×D3—42no—
208C2×D3—64no—
209C2×D3—122no—
210C2×D3—124no—
211A4—41no—
212A4—81no—
213C3×C4—61no—
214C3×C4—61no—
215Dic3—64no—
216C32—161noSylow
217C23—32no—
218C23—34no—
219C23—61no—
220C23—61no—
221C23—61no—
222C23—61no—
223C23—62no—
224C23—62no—
225C23—121no—
226C23—182no—
227C23—182no—
228C23—361no—
229C23—364no—
230C23—364no—
231C23C2×D3×A411yescharacteristic
232C23C2×D3×A412yes—
233D4—122no—
234D4—122no—
235D4—122no—
236D4—122no—
237D4—368no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—94no—
243C2×C4—182no—
244C2×C4—182no—
245C2×C4—364no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—42no—
249C2×C3—122no—
250C2×C3—162no—
251C2×C3—161no—
252C2×C3—322no—
253C2×C3—321no—
254C2×C3—484no—
255C2×C3#560⋊C311yescharacteristic, complemented
256D3—44no—
257C22—31no—
258C22—31no—
259C22—41no—
260C22—62no—
261C22—62no—
262C22—64no—
263C22—121no—
264C22—122no—
265C22—121no—
266C22—124no—
267C22—121no—
268C22—122no—
269C22—184no—
270C22—362no—
271C22—364no—
272C22C22×D3×A411yescharacteristic
273C4—61no—
274C4—61no—
275C4—184no—
276C3—161no—
277C3—321no—
278C3C2×#560⋊C311yescharacteristic, complemented
279C2—31no—
280C2—31no—
281C2—42no—
282C2—122no—
283C2—124no—
284C2#5720⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#5720⋊C3
commutator, cosocle, hypocenterC3×C42C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×#127C2×C3
socleC23×C3C22×A4
LCSG ▷ C3×C42
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×#127⋊C3C2semidirect
2. #5720⋊C3C2direct (N × H)
3. C2×C3⋊(#126⋊C3)C2semidirect
4. C2×#5720C3semidirect
7. C3×#127⋊C3C22semidirect
8. C2×C3×C42⋊C3C22semidirect
9. C3⋊(#126⋊C3)C22semidirect
10. C2×C3×#127C2×C3semidirect
15. #5720C2×C3semidirect
16. C2×#876C2×C3semidirect
19. C2×#127⋊C3D3semidirect
24. C3×C42⋊C3C23semidirect
33. C3×#127C22×C3semidirect
34. C2×C3×C42C22×C3semidirect
46. #876C22×C3semidirect
48. #127⋊C3C2×D3semidirect
51. C2×C42⋊C3C2×D3semidirect
58. C2×#127C3×D3semidirect
90. C3×C42C23×C3semidirect
98. C42⋊C3C22×D3semidirect
109. #127C2×C3×D3semidirect
122. C2×C42C2×C3×D3semidirect
190. C42C22×C3×D3semidirect
255. C2×C3#560⋊C3semidirect
278. C3C2×#560⋊C3semidirect
284. C2#5720⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#127⋊C32
#5720⋊C32
C2×C3⋊(#126⋊C3)2
C2×#57203
C2×#560⋊C33
C22×D3×A44