Finite Group Databasebeta

← C3×#11716776 #72259828

Subgroups — C3×#11716776 #72259828

Order 2400.

412 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C22×C3×#1044C211yesmaximal, characteristic
2C22×C3×#1043C211yesmaximal, characteristic
3C2×C3×#6036C211yesmaximal, characteristic
4#11716776C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5⋊D5C2211yescharacteristic
6C2×C3×#1044—21no—
7C2×C3×#1044—21no—
8C2×C3×#1044C412yescomplemented
9C2×C3×#1043—21no—
10C2×C3×#1043—21no—
11C2×C3×#1043C412yescomplemented
12C2×C3×C5×Dic5C412yescomplemented
13C3×#6036—21no—
14C3×#6036—21no—
15C3×(C2×AGL(1,𝔽5))⋊C4—52nomaximal
16C22×#1044C2×C311yescharacteristic
17C22×#1043C2×C311yescharacteristic
18C2×#6036C2×C311yescharacteristic
19C22×C3×C52C2×C411yescharacteristic
20C2×C3×C5⋊D5C2×C411yescharacteristic
21C2×C3×C5⋊D5C2×C411yescharacteristic
22C2×C3×C5⋊D5D411yescharacteristic
23C2×C3×C5⋊D5D411yescharacteristic
24C2×C3×C5⋊D5D411yescharacteristic
25C2×C3×C5⋊D5Q811yescharacteristic
26C3×#1044—22no—
27C3×#1043—22no—
28C3×C5×Dic5—22no—
29C22×C3×AGL(1,𝔽5)—52no—
30C22×C3×AGL(1,𝔽5)—52no—
31C22×C3×AGL(1,𝔽5)—102no—
32C2×C3×C4×D5—52no—
33C22×C5⋊D5C22×C311yescharacteristic
34C2×#1044—21no—
35C2×#1044—21no—
36C2×#1044C3×C412yescomplemented
37C2×#1043—21no—
38C2×#1043—21no—
39C2×#1043C3×C412yescomplemented
40C2×C5×Dic5C3×C412yescomplemented
41#6036—21no—
42#6036—21no—
43(C2×AGL(1,𝔽5))⋊C4—52no—
44C2×C3×C52C4211yescharacteristic
45C2×C3×C52C22⋊C411yescharacteristic
46C2×C3×C52C4⋊C411yescharacteristic
47C3×C5⋊D5C22⋊C411yescharacteristic
48C3×C5⋊D5C22⋊C411yescharacteristic
49C3×C5⋊D5C4⋊C411yescharacteristic
50C3×C5⋊D5C4⋊C411yescharacteristic
51C22×C3×D5—52no—
52C22×C3×D5—102no—
53C2×C3×AGL(1,𝔽5)—54no—
54C2×C3×AGL(1,𝔽5)—54no—
55C2×C3×AGL(1,𝔽5)—102no—
56C2×C3×AGL(1,𝔽5)—102no—
57C2×C3×AGL(1,𝔽5)—102no—
58C2×C3×AGL(1,𝔽5)—102no—
59C2×C3×AGL(1,𝔽5)—104no—
60C2×C3×AGL(1,𝔽5)—104no—
61C2×C3×AGL(1,𝔽5)—104no—
62C2×C3×C4×C5—52no—
63C2×C3×Dic5AGL(1,𝔽5)12yescomplemented
64C3×C4×D5—102no—
65C3×C4×D5—102no—
66C22×C52C2×C3×C411yescharacteristic
67C2×C5⋊D5C2×C3×C411yescharacteristic
68C2×C5⋊D5C2×C3×C411yescharacteristic
69C2×C5⋊D5C3×D411yescharacteristic
70C2×C5⋊D5C3×D411yescharacteristic
71C2×C5⋊D5C3×D411yescharacteristic
72C2×C5⋊D5C3×Q811yescharacteristic
73#1044—22no—
74#1043—22no—
75C5×Dic5—22no—
76C3×#95—251noHall
77C22×AGL(1,𝔽5)—52no—
78C22×AGL(1,𝔽5)—52no—
79C22×AGL(1,𝔽5)—102no—
80C2×C4×D5—52no—
81C3×C52#9511yescharacteristic, complemented, Hall
82C22×C3×C5—22no—
83C22×C3×C5C2×AGL(1,𝔽5)12yes—
84C2×C3×D5—52no—
85C2×C3×D5—52no—
86C2×C3×D5—52no—
87C2×C3×D5—52no—
88C2×C3×D5—52no—
89C2×C3×D5—52no—
90C2×C3×D5—102no—
91C2×C3×D5—102no—
92C2×C3×D5—102no—
93C2×C3×D5—102no—
94C2×C3×D5—102no—
95C2×C3×D5—102no—
96C3×AGL(1,𝔽5)—104no—
97C3×AGL(1,𝔽5)—104no—
98C3×AGL(1,𝔽5)—108no—
99C3×C4×C5—102no—
100C3×Dic5—22no—
101C2×C52C3×C4211yescharacteristic
102C2×C52C3×C22⋊C411yescharacteristic
103C2×C52C3×C4⋊C411yescharacteristic
104C5⋊D5C3×C22⋊C411yescharacteristic
105C5⋊D5C3×C22⋊C411yescharacteristic
106C5⋊D5C3×C4⋊C411yescharacteristic
107C5⋊D5C3×C4⋊C411yescharacteristic
108C22×C3×C4—251no—
109C22×C3×C4—251no—
110C22×C3×C4—251no—
111C22×D5—52no—
112C22×D5—102no—
113C2×AGL(1,𝔽5)—54no—
114C2×AGL(1,𝔽5)—54no—
115C2×AGL(1,𝔽5)—104no—
116C2×AGL(1,𝔽5)—102no—
117C2×AGL(1,𝔽5)—102no—
118C2×AGL(1,𝔽5)—102no—
119C2×AGL(1,𝔽5)—102no—
120C2×AGL(1,𝔽5)—104no—
121C2×AGL(1,𝔽5)—104no—
122C2×C4×C5—52no—
123C2×Dic5C3×AGL(1,𝔽5)12yescomplemented
124C4×D5—102no—
125C4×D5—102no—
126#95—251noSylow
127C2×C3×C5—22no—
128C2×C3×C5—22no—
129C2×C3×C5—22no—
130C2×C3×C5C4×AGL(1,𝔽5)12yes—
131C2×C3×C5C4⋊AGL(1,𝔽5)12yes—
132C2×C3×C5C22⋊AGL(1,𝔽5)12yes—
133C3×D5—52no—
134C3×D5—52no—
135C3×D5—52no—
136C3×D5—52no—
137C3×D5—102no—
138C3×D5—102no—
139C3×D5—102no—
140C3×D5—102no—
141C52C3×#9511yescharacteristic, complemented, Sylow
142C23×C3—251no—
143C2×C3×C4—52no—
144C2×C3×C4—252no—
145C2×C3×C4—252no—
146C2×C3×C4—501no—
147C2×C3×C4—501no—
148C2×C3×C4—501no—
149C2×C3×C4—501no—
150C2×C3×C4—501no—
151C2×C3×C4—501no—
152C22×C5—22no—
153C22×C5C2×C3×AGL(1,𝔽5)12yes—
154C2×D5—52no—
155C2×D5—52no—
156C2×D5—52no—
157C2×D5—52no—
158C2×D5—52no—
159C2×D5—52no—
160C2×D5—102no—
161C2×D5—102no—
162C2×D5—102no—
163C2×D5—102no—
164C2×D5—102no—
165C2×D5—102no—
166AGL(1,𝔽5)—104no—
167AGL(1,𝔽5)—104no—
168AGL(1,𝔽5)—108no—
169C4×C5—102no—
170Dic5—22no—
171C22×C4—251no—
172C22×C4—251no—
173C22×C4—251no—
174C3×C5—22no—
175C3×C5(C2×AGL(1,𝔽5))⋊C412yescomplemented
176C22×C3—251no—
177C22×C3—251no—
178C22×C3—251no—
179C22×C3—251no—
180C22×C3—251no—
181C22×C3—251no—
182C22×C3D5⋊AGL(1,𝔽5)11yescharacteristic
183C3×C4—102no—
184C3×C4—502no—
185C3×C4—502no—
186C2×C5—22no—
187C2×C5—22no—
188C2×C5—22no—
189C2×C5C3×C4×AGL(1,𝔽5)12yes—
190C2×C5C3×C4⋊AGL(1,𝔽5)12yes—
191C2×C5C3×C22⋊AGL(1,𝔽5)12yes—
192D5—52no—
193D5—52no—
194D5—52no—
195D5—52no—
196D5—102no—
197D5—102no—
198D5—102no—
199D5—102no—
200C23—251no—
201C2×C4—52no—
202C2×C4—252no—
203C2×C4—252no—
204C2×C4—501no—
205C2×C4—501no—
206C2×C4—501no—
207C2×C4—501no—
208C2×C4—501no—
209C2×C4—501no—
210C2×C3—251no—
211C2×C3—251no—
212C2×C3—251no—
213C2×C3—251no—
214C2×C3#8783011yescharacteristic
215C2×C3#8783111yescharacteristic
216C2×C3#8783211yescharacteristic
217C5—22no—
218C5C3×(C2×AGL(1,𝔽5))⋊C412yescomplemented
219C22—251no—
220C22—251no—
221C22—251no—
222C22—251no—
223C22—251no—
224C22—251no—
225C22C3×D5⋊AGL(1,𝔽5)11yescharacteristic
226C4—102no—
227C4—502no—
228C4—502no—
229C3#1171677611yescharacteristic, direct product factor, Sylow
230C2—251no—
231C2—251no—
232C2—251no—
233C2—251no—
234C2C3×#8783011yescharacteristic
235C2C3×#8783111yescharacteristic
236C2C3×#8783211yescharacteristic
subgroupG/H
center, hypercenterC22×C3D5⋊AGL(1,𝔽5)
commutatorC2×C52C3×C42
FrattiniC22C3×D5⋊AGL(1,𝔽5)
Fitting, socleC22×C3×C52C2×C4
prime radicalC22×C3×C5⋊D5C22
cosocleC22×C5⋊D5C22×C3
hypocenterC52C3×#95
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
4. #11716776C3direct (N × H)
8. C2×C3×#1044C4semidirect
11. C2×C3×#1043C4semidirect
12. C2×C3×C5×Dic5C4semidirect
36. C2×#1044C3×C4semidirect
39. C2×#1043C3×C4semidirect
40. C2×C5×Dic5C3×C4semidirect
63. C2×C3×Dic5AGL(1,𝔽5)semidirect
81. C3×C52#95semidirect
123. C2×Dic5C3×AGL(1,𝔽5)semidirect
141. C52C3×#95semidirect
175. C3×C5(C2×AGL(1,𝔽5))⋊C4semidirect
218. C5C3×(C2×AGL(1,𝔽5))⋊C4semidirect
229. C3#11716776direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#10442
C22×C3×#10432
C2×C3×#60362
#117167763
C3×(C2×AGL(1,𝔽5))⋊C45