Finite Group Databasebeta

← #13087031

Subgroups — #13087031

Order 1440.

564 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 118 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×#67184C211yesmaximal, characteristic, complemented
2C2×C32×C22⋊D5C211yesmaximal, characteristic, complemented
3C2×#67139C211yesmaximal, characteristic, complemented
4C2×#67137C211yesmaximal, characteristic, complemented
5C5×#3757C211yesmaximal, characteristic, complemented
6#10623773C211yesmaximal, characteristic, complemented
7#10623770C211yesmaximal, characteristic, complemented
8#91533—34nomaximal
9C23×C32×C5C2211yescharacteristic
10C22×C3⋊D15C2211yescharacteristic
11C22×C32×D5C2211yescharacteristic
12#67184—22no—
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C5×C3⋊Dic3C2211yescharacteristic
15C2×C5×C3⋊Dic3C2211yescharacteristic
16C32×C22⋊D5—22no—
17C2×C32×Dic5C2211yescharacteristic
18#67139—22no—
19#67137—22no—
20#64147—51nomaximal, Hall
21C2×#1220—34no—
22C2×C3×C22⋊D5D314yescomplemented
23C2×#1202—34no—
24C2×#1200—34no—
25C5×#217—34no—
26#6668—34no—
27#6665—34no—
28C22×C32×C5—21no—
29C22×C32×C5—21no—
30C22×C32×C5—21no—
31C22×C32×C5C2311yescharacteristic
32C2×C3⋊D15—21no—
33C2×C3⋊D15—21no—
34C2×C3⋊D15D412yescomplemented
35C2×C32×D5—21no—
36C2×C32×D5—21no—
37C2×C32×D5—21no—
38C3⋊Dic15—21no—
39C5×C3⋊Dic3—21no—
40C5×C3⋊Dic3—21no—
41C32×Dic5D412yescomplemented
42#4030—91noHall
43C2×C32×D4—51no—
44C2×#638—51no—
45C2×#638—51no—
46C2×C4×C3⋊D3—51no—
47#3757—51no—
48#3757D511yescharacteristic, complemented
49C4⋊(C3⋊Dic3)—51no—
50C23×C3×C5C2×D314yes—
51C22×D15—34no—
52C22×C3×D5C2×D314yes—
53#1220—68no—
54C2×Dic15—34no—
55C2×C5×Dic3—34no—
56C2×C5×Dic3—34no—
57C3×C22⋊D5—28no—
58C2×C3×Dic5C2×D314yes—
59#1202—68no—
60#1200—68no—
61#938—154no—
62C2×C32×C5—41no—
63C2×C32×C5C2×D411yescharacteristic
64C2×C32×C5C2×D411yescharacteristic
65C2×C32×C5#4111yescharacteristic
66C3⋊D15—22no—
67C32×D5—41no—
68C2×C22⋊D5—91no—
69C2×C22⋊D5C3⋊D311yescharacteristic, complemented
70C2×D20—91no—
71C2×C4×D5—91no—
72C5×C22⋊C4—91no—
73#685—91no—
74#683—91no—
75C23×C32—51no—
76C23×C32C2×D511yescharacteristic
77C22×C3⋊D3—51no—
78C32×D4—102no—
79C2×C32×C4—51no—
80#638—102no—
81#638—102no—
82C2×C3⋊Dic3—51no—
83C2×C3⋊Dic3C2×D511yescharacteristic
84C2×C3⋊Dic3C2×D511yescharacteristic
85C4×C3⋊D3—102no—
86C22×C3×C5—24no—
87C22×C3×C5—24no—
88C22×C3×C5—24no—
89C22×C3×C5C22×D314yes—
90C2×D15—38no—
91C2×D15—64no—
92C2×D15—64no—
93C2×C3×D5—24no—
94C2×C3×D5—24no—
95C2×C3×D5—24no—
96Dic15—64no—
97C3×Dic5#6718yescomplemented
98C5×Dic3—64no—
99C5×Dic3—64no—
100C2×C3×D4—54no—
101C2×#67—154no—
102C2×#67—154no—
103C2×C4×D3—154no—
104#217—34no—
105#217—154no—
106C4⋊Dic3—154no—
107C32×C5#12111yescharacteristic, complemented, Hall
108C23×C5C2×C3⋊D311yescharacteristic
109C22×D5—91no—
110C22×D5C2×C3⋊D311yescharacteristic
111C2×C4×C5—91no—
112C2×C4×C5—91no—
113C22⋊D5—22no—
114C22⋊D5—182no—
115C2×Dic5—91no—
116C2×Dic5C2×C3⋊D311yescharacteristic
117D20—182no—
118C4×D5—182no—
119C22×C32—21no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—101no—
123C22×C32—101no—
124C22×C32—101no—
125C22×C32C22×D511yescharacteristic
126C2×C3⋊D3—52no—
127C2×C3⋊D3—101no—
128C2×C3⋊D3—101no—
129C32×C4—52no—
130C3⋊Dic3—21no—
131C3⋊Dic3—21no—
132C3⋊Dic3—101no—
133#121—451noSylow
134C2×C3×C5—44no—
135C2×C3×C5D3×D414yes—
136C2×C3×C5#23714yes—
137C2×C3×C5C2×#6714yes—
138D15—68no—
139C3×D5—44no—
140C23×C3—54no—
141C23×C3D3×D514yes—
142C22×D3—154no—
143C3×D4—108no—
144C2×C3×C4—54no—
145#67—308no—
146#67—308no—
147C2×Dic3—34no—
148C2×Dic3—34no—
149C2×Dic3—154no—
150C4×D3—308no—
151C22×C5—21no—
152C22×C5—21no—
153C22×C5—21no—
154C22×C5C22×C3⋊D311yescharacteristic
155C2×D5—21no—
156C2×D5—21no—
157C2×D5—21no—
158C2×D5—92no—
159C2×D5—181no—
160C2×D5—181no—
161C4×C5—181no—
162C4×C5—181no—
163Dic5—181no—
164Dic5#63812yescomplemented
165C2×C32—41no—
166C2×C32—201no—
167C2×C32#70911yescharacteristic
168C2×C32D4×D511yescharacteristic
169C2×C32D4×D511yescharacteristic
170C3⋊D3—102no—
171C2×D4—51no—
172C2×D4—451no—
173C2×D4—451no—
174C22×C4—451no—
175C4⋊C4—451no—
176C22⋊C4—91no—
177C22⋊C4—451no—
178C3×C5#93814yescomplemented
179C22×C3—24no—
180C22×C3—24no—
181C22×C3—24no—
182C22×C3—104no—
183C22×C3—104no—
184C22×C3—104no—
185C22×C3C2×D3×D514yes—
186C2×D3—158no—
187C2×D3—304no—
188C2×D3—304no—
189C3×C4—58no—
190Dic3—64no—
191Dic3—64no—
192Dic3—304no—
193C2×C5—41no—
194C2×C5D4×C3⋊D311yescharacteristic
195C2×C5#383011yescharacteristic
196C2×C5C2×#63811yescharacteristic
197D5—41no—
198D5—182no—
199C32#403011yescharacteristic, complemented, Sylow
200C23—51no—
201C23—451no—
202C23D5×C3⋊D311yescharacteristic
203D4—102no—
204D4—902no—
205D4—902no—
206C2×C4—51no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—451no—
210C2×C4—902no—
211C2×C3—44no—
212C2×C3—204no—
213C2×C3#677914yes—
214C2×C3D5×#6714yes—
215C2×C3#679014yes—
216D3—308no—
217C5#6414711yescharacteristic, complemented, Sylow
218C22—21no—
219C22—21no—
220C22—21no—
221C22—101no—
222C22—101no—
223C22—101no—
224C22—452no—
225C22—901no—
226C22—901no—
227C22C2×D5×C3⋊D311yescharacteristic
228C4—52no—
229C4—181no—
230C4—181no—
231C4—901no—
232C3#9153314yescomplemented
233C2—41no—
234C2—201no—
235C2—902no—
236C2#1062416211yescharacteristic
237C2D5×#63811yescharacteristic
238C2#1062417311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×C3⋊D3
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
hypercenterC23D5×C3⋊D3
hypocenterC32×C5#121
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#67184C2semidirect
2. C2×C32×C22⋊D5C2semidirect
3. C2×#67139C2semidirect
4. C2×#67137C2semidirect
5. C5×#3757C2semidirect
6. #10623773C2semidirect
7. #10623770C2semidirect
22. C2×C3×C22⋊D5D3semidirect
34. C2×C3⋊D15D4semidirect
41. C32×Dic5D4semidirect
48. #3757D5semidirect
69. C2×C22⋊D5C3⋊D3semidirect
97. C3×Dic5#67semidirect
107. C32×C5#121semidirect
164. Dic5#638semidirect
178. C3×C5#938semidirect
199. C32#4030semidirect
217. C5#64147semidirect
232. C3#91533semidirect

Maximal subgroups

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

subgroupindex
C2×#671842
C2×C32×C22⋊D52
C2×#671392
C2×#671372
C5×#37572
#106237732
#106237702
#915333
#641475