Finite Group Databasebeta

← C3×#65712 #11752245

Subgroups — C3×#65712 #11752245

Order 960.

340 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 128 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×C4⋊AGL(1,𝔽5)C212yesmaximal
2C2×C3×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C2×C3×C4×AGL(1,𝔽5)C211yesmaximal, characteristic
4C3×D5×C4⋊C4C211yesmaximal, characteristic
5C3×(C2×AGL(1,𝔽5))⋊C4C212yesmaximal
6#65712C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×AGL(1,𝔽5)C2212yes—
8C22×C3×AGL(1,𝔽5)C2211yescharacteristic
9C22×C3×AGL(1,𝔽5)C2211yescharacteristic
10C2×C3×C4×D5C2212yes—
11C2×C3×C4×D5C2211yescharacteristic
12C3×C4⋊AGL(1,𝔽5)—24no—
13C3×C4⋊AGL(1,𝔽5)—22no—
14C3×C4×AGL(1,𝔽5)—22no—
15C3×C5×C4⋊C4C411yescharacteristic, complemented
16C3×C4⋊Dic5C411yescharacteristic, complemented
17C3×#683C412yescomplemented
18C3×#389—51nomaximal, Hall
19C2×C4⋊AGL(1,𝔽5)C2×C312yes—
20C2×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic
21C2×C4×AGL(1,𝔽5)C2×C311yescharacteristic
22D5×C4⋊C4C2×C311yescharacteristic
23(C2×AGL(1,𝔽5))⋊C4C2×C312yes—
24C22×C3×D5C2311yescharacteristic
25C2×C3×AGL(1,𝔽5)—21no—
26C2×C3×AGL(1,𝔽5)—21no—
27C2×C3×AGL(1,𝔽5)—21no—
28C2×C3×AGL(1,𝔽5)—21no—
29C2×C3×AGL(1,𝔽5)—22no—
30C2×C3×AGL(1,𝔽5)—22no—
31C2×C3×AGL(1,𝔽5)—22no—
32C2×C3×AGL(1,𝔽5)D412yes—
33C2×C3×AGL(1,𝔽5)Q812yes—
34C2×C3×C4×C5C2×C412yes—
35C2×C3×C4×C5C2×C411yescharacteristic
36C2×C3×Dic5C2×C412yes—
37C2×C3×Dic5C2×C411yescharacteristic
38C3×C4×D5—22no—
39C3×C4×D5—22no—
40C3×C4×D5D412yes—
41C3×C4×D5Q812yes—
42C2×C3×C4⋊C4—51no—
43C2×C3×C4⋊C4—51no—
44C2×C3×C4⋊C4—52no—
45C2×C3×C42—51no—
46C3×#95—52no—
47C22×AGL(1,𝔽5)C22×C312yes—
48C22×AGL(1,𝔽5)C22×C311yescharacteristic
49C22×AGL(1,𝔽5)C22×C311yescharacteristic
50C2×C4×D5C22×C312yes—
51C2×C4×D5C22×C311yescharacteristic
52C4⋊AGL(1,𝔽5)—24no—
53C4⋊AGL(1,𝔽5)—22no—
54C4×AGL(1,𝔽5)—22no—
55C5×C4⋊C4C3×C411yescharacteristic, complemented
56C4⋊Dic5C3×C411yescharacteristic, complemented
57#683C3×C412yescomplemented
58#389—51noSylow
59C22×C3×C5C22×C411yescharacteristic
60C2×C3×D5C2×D411yescharacteristic
61C2×C3×D5C2×D411yescharacteristic
62C2×C3×D5C2×Q811yescharacteristic
63C2×C3×D5C2×Q811yescharacteristic
64C2×C3×D5#4111yescharacteristic
65C2×C3×D5#4111yescharacteristic
66C3×AGL(1,𝔽5)—22no—
67C3×AGL(1,𝔽5)—22no—
68C3×AGL(1,𝔽5)—42no—
69C3×C4×C5—22no—
70C3×C4×C5C4⋊C412yescomplemented
71C3×Dic5—22no—
72C3×Dic5C4⋊C412yescomplemented
73C22×C3×C4—51no—
74C22×C3×C4—51no—
75C22×C3×C4—51no—
76C22×C3×C4—52no—
77C22×C3×C4—52no—
78C3×C4⋊C4—52no—
79C3×C4⋊C4—51no—
80C3×C4⋊C4—102no—
81C3×C4⋊C4—104no—
82C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
83C3×C42—102no—
84C22×D5C23×C311yescharacteristic
85C2×AGL(1,𝔽5)—21no—
86C2×AGL(1,𝔽5)—21no—
87C2×AGL(1,𝔽5)—21no—
88C2×AGL(1,𝔽5)—21no—
89C2×AGL(1,𝔽5)—22no—
90C2×AGL(1,𝔽5)—22no—
91C2×AGL(1,𝔽5)—22no—
92C2×AGL(1,𝔽5)C3×D412yes—
93C2×AGL(1,𝔽5)C3×Q812yes—
94C2×C4×C5C2×C3×C412yes—
95C2×C4×C5C2×C3×C411yescharacteristic
96C2×Dic5C2×C3×C412yes—
97C2×Dic5C2×C3×C411yescharacteristic
98C4×D5—22no—
99C4×D5—22no—
100C4×D5C3×D412yes—
101C4×D5C3×Q812yes—
102C2×C4⋊C4—51no—
103C2×C4⋊C4—51no—
104C2×C4⋊C4—52no—
105C2×C42—51no—
106#95—52no—
107C2×C3×C5C2×C4⋊C411yescharacteristic
108C2×C3×C5C4×D411yescharacteristic
109C2×C3×C5C4×Q811yescharacteristic
110C3×D5#12111yescharacteristic
111C3×D5C22⋊Q811yescharacteristic
112C3×D5#12511yescharacteristic
113C3×D5C4⋊Q811yescharacteristic
114C23×C3—51no—
115C2×C3×C4—51no—
116C2×C3×C4—52no—
117C2×C3×C4—52no—
118C2×C3×C4—52no—
119C2×C3×C4—52no—
120C2×C3×C4—52no—
121C2×C3×C4—102no—
122C2×C3×C4—102no—
123C2×C3×C4—101no—
124C2×C3×C4—101no—
125C2×C3×C4—101no—
126C2×C3×C4—101no—
127C2×C3×C4—102no—
128C2×C3×C4—102no—
129C2×C3×C4—102no—
130C2×C3×C4C2×AGL(1,𝔽5)12yes—
131C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
132C22×C5C22×C3×C411yescharacteristic
133C2×D5C2×C3×D411yescharacteristic
134C2×D5C2×C3×D411yescharacteristic
135C2×D5C2×C3×Q811yescharacteristic
136C2×D5C2×C3×Q811yescharacteristic
137C2×D5C3×#4111yescharacteristic
138C2×D5C3×#4111yescharacteristic
139AGL(1,𝔽5)—22no—
140AGL(1,𝔽5)—22no—
141AGL(1,𝔽5)—42no—
142C4×C5—22no—
143C4×C5C3×C4⋊C412yescomplemented
144Dic5—22no—
145Dic5C3×C4⋊C412yescomplemented
146C22×C4—51no—
147C22×C4—51no—
148C22×C4—51no—
149C22×C4—52no—
150C22×C4—52no—
151C4⋊C4—52no—
152C4⋊C4—51no—
153C4⋊C4—102no—
154C4⋊C4—104no—
155C4⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
156C42—102no—
157C3×C5#38911yescharacteristic, complemented, Hall
158C22×C3—51no—
159C22×C3—51no—
160C22×C3—51no—
161C22×C3—51no—
162C22×C3—51no—
163C22×C3—51no—
164C22×C3C22×AGL(1,𝔽5)11yescharacteristic
165C3×C4—22no—
166C3×C4—52no—
167C3×C4—102no—
168C3×C4—102no—
169C3×C4—102no—
170C3×C4—202no—
171C3×C4C4⋊AGL(1,𝔽5)12yescomplemented
172C2×C5C2×C3×C4⋊C411yescharacteristic
173C2×C5C3×C4×D411yescharacteristic
174C2×C5C3×C4×Q811yescharacteristic
175D5C3×#12111yescharacteristic
176D5C3×C22⋊Q811yescharacteristic
177D5C3×#12511yescharacteristic
178D5C3×C4⋊Q811yescharacteristic
179C23—51no—
180C2×C4—51no—
181C2×C4—52no—
182C2×C4—52no—
183C2×C4—52no—
184C2×C4—52no—
185C2×C4—52no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—101no—
189C2×C4—101no—
190C2×C4—101no—
191C2×C4—101no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4C2×C3×AGL(1,𝔽5)12yes—
196C2×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
197C2×C3—51no—
198C2×C3—51no—
199C2×C3—51no—
200C2×C3—51no—
201C2×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
202C2×C3D4×AGL(1,𝔽5)11yescharacteristic
203C2×C3Q8×AGL(1,𝔽5)11yescharacteristic
204C5C3×#38911yescharacteristic, complemented, Sylow
205C22—51no—
206C22—51no—
207C22—51no—
208C22—51no—
209C22—51no—
210C22—51no—
211C22C22×C3×AGL(1,𝔽5)11yescharacteristic
212C4—22no—
213C4—52no—
214C4—102no—
215C4—102no—
216C4—102no—
217C4—202no—
218C4C3×C4⋊AGL(1,𝔽5)12yescomplemented
219C3#6571211yescharacteristic, direct product factor, Sylow
220C2—51no—
221C2—51no—
222C2—51no—
223C2—51no—
224C2C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
225C2C3×D4×AGL(1,𝔽5)11yescharacteristic
226C2C3×Q8×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C22×AGL(1,𝔽5)
commutatorC22×C5C22×C3×C4
FrattiniC22C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C4
socleC22×C3×C5C22×C4
prime radicalC22×C3×D5C23
cosocleC22×D5C23×C3
hypercenterC3×C4⋊C4AGL(1,𝔽5)
hypocenterC5C3×#389
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #65712C3direct (N × H)
15. C3×C5×C4⋊C4C4semidirect
16. C3×C4⋊Dic5C4semidirect
17. C3×#683C4semidirect
55. C5×C4⋊C4C3×C4semidirect
56. C4⋊Dic5C3×C4semidirect
57. #683C3×C4semidirect
70. C3×C4×C5C4⋊C4semidirect
72. C3×Dic5C4⋊C4semidirect
82. C3×C4⋊C4AGL(1,𝔽5)semidirect
143. C4×C5C3×C4⋊C4semidirect
145. Dic5C3×C4⋊C4semidirect
155. C4⋊C4C3×AGL(1,𝔽5)semidirect
157. C3×C5#389semidirect
171. C3×C4C4⋊AGL(1,𝔽5)semidirect
204. C5C3×#389semidirect
218. C4C3×C4⋊AGL(1,𝔽5)semidirect
219. C3#65712direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊AGL(1,𝔽5)2
C2×C3×C4×AGL(1,𝔽5)2
C3×D5×C4⋊C42
C3×(C2×AGL(1,𝔽5))⋊C42
#657123
C3×#3895