Finite Group Databasebeta

← #13093155

Subgroups — #13093155

Order 1472.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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×#121C211yesmaximal, characteristic, complemented
2#10625111C211yesmaximal, characteristic, complemented
3#10625110C211yesmaximal, characteristic, complemented
4#10625110C211yesmaximal, characteristic, complemented
5#10625109C211yesmaximal, characteristic, complemented
6#10625108C211yesmaximal, characteristic, complemented
7#10625108C211yesmaximal, characteristic, complemented
8#10625101C211yesmaximal, characteristic, complemented
9#10625100C211yesmaximal, characteristic, complemented
10#10625069C211yesmaximal, characteristic, complemented
11#10625065C211yesmaximal, characteristic, complemented
12#10625057C211yesmaximal, characteristic, complemented
13#10625055C211yesmaximal, characteristic, complemented
14#10625053C211yesmaximal, characteristic, complemented
15#10625050C211yesmaximal, characteristic, complemented
16C23×D23C2211yescharacteristic
17C2×D4×C23C2211yescharacteristic
18C2×D4×C23C2211yescharacteristic
19C2×D4×C23C2211yescharacteristic, complemented
20C22×C4×C23C2211yescharacteristic, complemented
21C2×C22⋊D23C2211yescharacteristic
22C2×C22⋊D23C2211yescharacteristic
23C2×C22⋊D23C2211yescharacteristic
24C2×C22⋊D23C2211yescharacteristic
25C2×C22⋊D23C2211yescharacteristic
26C2×C22⋊D23C2211yescharacteristic, complemented
27C2×C22⋊D23C2211yescharacteristic, complemented
28C22×Dic23C2211yescharacteristic, complemented
29C2×D92C2211yescharacteristic
30C2×D92C2211yescharacteristic
31C2×C4×D23C2211yescharacteristic, complemented
32C4⋊C4×C23C2211yescharacteristic, complemented
33C22⋊C4×C23C2211yescharacteristic
34C22⋊C4×C23C2211yescharacteristic, complemented
35C22⋊Dic23C2211yescharacteristic
36C22⋊Dic23C2211yescharacteristic
37C22⋊Dic23C2211yescharacteristic, complemented
38(C2×D23)⋊C4C2211yescharacteristic
39(C2×D23)⋊C4C2211yescharacteristic
40(C2×D23)⋊C4C2211yescharacteristic
41(C2×D23)⋊C4C2211yescharacteristic
42(C2×D23)⋊C4C2211yescharacteristic
43(C2×D23)⋊C4C2211yescharacteristic
44(C2×D23)⋊C4C2211yescharacteristic, complemented
45C4⋊Dic23C2211yescharacteristic, complemented
46C4⋊Dic23C2211yescharacteristic, complemented
47Dic23⋊C4C2211yescharacteristic, complemented
48Dic23⋊C4C2211yescharacteristic, complemented
49Dic23⋊C4C2211yescharacteristic, complemented
50C4×Dic23C2211yescharacteristic, complemented
51C23×C23C2311yescharacteristic
52C23×C23C2311yescharacteristic
53C23×C23C2311yescharacteristic
54C22×D23—41no—
55C22×D23—41no—
56C22×D23—41no—
57C22×D23C2311yescharacteristic
58C22×D23C2311yescharacteristic
59C22×D23C2311yescharacteristic
60D4×C23—41no—
61D4×C23—41no—
62D4×C23—41no—
63C2×C4×C23—41no—
64C2×C4×C23C2311yescharacteristic
65C2×C4×C23C2311yescharacteristic
66C2×C4×C23C2311yescharacteristic
67C2×C4×C23C2311yescharacteristic
68C22⋊D23—41no—
69C22⋊D23—41no—
70C22⋊D23—41no—
71C22⋊D23—41no—
72C22⋊D23—41no—
73C22⋊D23—41no—
74C22⋊D23—41no—
75C2×Dic23—41no—
76C2×Dic23C2311yescharacteristic
77C2×Dic23C2311yescharacteristic
78C2×Dic23C2311yescharacteristic
79C2×Dic23C2311yescharacteristic
80C2×Dic23C2311yescharacteristic
81D92—41no—
82D92—41no—
83C4×D23—41no—
84C22×C23—21no—
85C22×C23—21no—
86C22×C23—21no—
87C22×C23—21no—
88C22×C23—21no—
89C22×C23—21no—
90C22×C23—21no—
91C22×C23—21no—
92C22×C23—21no—
93C22×C23C2411yescharacteristic
94C2×D23—21no—
95C2×D23—21no—
96C2×D23—21no—
97C2×D23—21no—
98C2×D23—21no—
99C2×D23—21no—
100C2×D23—21no—
101C2×D23—21no—
102C2×D23—21no—
103C2×D23—44no—
104C4×C23—21no—
105C4×C23—21no—
106C4×C23—21no—
107C4×C23—21no—
108Dic23—21no—
109Dic23—21no—
110Dic23—21no—
111Dic23—21no—
112Dic23—21no—
113#560—231nomaximal, Sylow
114C2×C23—41no—
115C2×C23—41no—
116C2×C23—41no—
117C2×C23H(5,𝔽2)11yescharacteristic
118C2×C23H(5,𝔽2)11yescharacteristic
119C2×C23H(5,𝔽2)11yescharacteristic
120D23—41no—
121D23—41no—
122D23—41no—
123#127—231no—
124#126—231no—
125#126—231no—
126#123—231no—
127#123—231no—
128#123—231no—
129#121—231no—
130#121—231no—
131#121—231no—
132#121—231no—
133#121—231no—
134#121D2311yescharacteristic, complemented
135#120—231no—
136#120—231no—
137#120—231no—
138C23#56011yescharacteristic, complemented, Sylow
139C24—231no—
140C2×D4—231no—
141C2×D4—231no—
142C2×D4—231no—
143C2×D4—231no—
144C2×D4—231no—
145C2×D4—231no—
146C2×D4—231no—
147C2×D4—231no—
148C2×D4—231no—
149C2×D4C2×D2311yescharacteristic
150C2×D4C2×D2311yescharacteristic
151C2×D4C2×D2311yescharacteristic, complemented
152C22×C4—231no—
153C22×C4—231no—
154C22×C4C2×D2311yescharacteristic, complemented
155C4⋊C4—231no—
156C4⋊C4—231no—
157C4⋊C4—231no—
158C4⋊C4—231no—
159C4⋊C4—231no—
160C4⋊C4C2×D2311yescharacteristic, complemented
161C22⋊C4—231no—
162C22⋊C4—231no—
163C22⋊C4—231no—
164C22⋊C4—231no—
165C22⋊C4—231no—
166C22⋊C4—231no—
167C22⋊C4—231no—
168C22⋊C4—231no—
169C22⋊C4—231no—
170C22⋊C4—231no—
171C22⋊C4C2×D2311yescharacteristic
172C22⋊C4C2×D2311yescharacteristic, complemented
173C42—231no—
174C23—231no—
175C23—231no—
176C23—231no—
177C23—921no—
178C23—921no—
179C23—921no—
180C23C22×D2311yescharacteristic
181C23C22×D2311yescharacteristic
182C23C22×D2311yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—921no—
187D4—921no—
188D4—921no—
189D4—921no—
190D4—921no—
191D4—921no—
192D4—921no—
193D4—921no—
194D4—921no—
195C2×C4—41no—
196C2×C4—231no—
197C2×C4—231no—
198C2×C4—231no—
199C2×C4—231no—
200C2×C4—231no—
201C2×C4—921no—
202C2×C4—921no—
203C2×C4C22×D2311yescharacteristic
204C2×C4C22×D2311yescharacteristic
205C2×C4C22×D2311yescharacteristic
206C2×C4C22×D2311yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—461no—
217C22—461no—
218C22—461no—
219C22—461no—
220C22—461no—
221C22—461no—
222C22—461no—
223C22—461no—
224C22—461no—
225C22—924no—
226C22C23×D2311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—461no—
232C4—461no—
233C4—461no—
234C4—461no—
235C4—461no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—921no—
240C2—921no—
241C2—921no—
242C2#1062515511yescharacteristic
243C2#1062515511yescharacteristic
244C2#1062516011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×#121C2
hypercenter#121D23
hypocenterC23#560
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C23×#121C2semidirect
2. #10625111C2semidirect
3. #10625110C2semidirect
4. #10625110C2semidirect
5. #10625109C2semidirect
6. #10625108C2semidirect
7. #10625108C2semidirect
8. #10625101C2semidirect
9. #10625100C2semidirect
10. #10625069C2semidirect
11. #10625065C2semidirect
12. #10625057C2semidirect
13. #10625055C2semidirect
14. #10625053C2semidirect
15. #10625050C2semidirect
19. C2×D4×C23C22semidirect
20. C22×C4×C23C22semidirect
26. C2×C22⋊D23C22semidirect
27. C2×C22⋊D23C22semidirect
28. C22×Dic23C22semidirect
31. C2×C4×D23C22semidirect
32. C4⋊C4×C23C22semidirect
34. C22⋊C4×C23C22semidirect
37. C22⋊Dic23C22semidirect
44. (C2×D23)⋊C4C22semidirect
45. C4⋊Dic23C22semidirect
46. C4⋊Dic23C22semidirect
47. Dic23⋊C4C22semidirect
48. Dic23⋊C4C22semidirect
49. Dic23⋊C4C22semidirect
50. C4×Dic23C22semidirect
134. #121D23semidirect
138. C23#560semidirect
151. C2×D4C2×D23semidirect
154. C22×C4C2×D23semidirect
160. C4⋊C4C2×D23semidirect
172. C22⋊C4C2×D23semidirect

Maximal subgroups

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

subgroupindex
C23×#1212
#106251112
#106251102
#106251092
#106251082
#106251012
#106251002
#106250692
#106250652
#106250572
#106250552
#106250532
#106250502
#56023