Finite Group Databasebeta

← C3×#10604352 #14284490

Subgroups — C3×#10604352 #14284490

Order 1920.

388 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C3×#65851C211yesmaximal, characteristic
2C3×#65759C212yesmaximal
3C3×#64870C214yesmaximal, complemented
4#10604352C311yesmaximal, characteristic, direct product factor, Hall
5C3×C5×#117C411yescharacteristic
6C3×#4072C411yescharacteristic
7C2×C3×C4×Dic5C2211yescharacteristic
8C3×#4036C2212yes—
9C3×#4034C412yes—
10C3×#4024—22no—
11C3×C4×Dic10—22no—
12C3×C22⋊#170—22no—
13C3×#4002C2214yes—
14C3×C4⋊#170—22no—
15C3×#1537—51nomaximal, Hall
16#65851C2×C311yescharacteristic
17#65759C2×C312yes—
18#64870C2×C314yescomplemented
19C22×C3×C4×C5C2×C411yescharacteristic
20C22×C3×Dic5D412yes—
21C2×C3×Dic10—22no—
22C2×C3×#170—24no—
23C3×C5×C4⋊C4C2×C412yes—
24C3×C5×C22⋊C4—21no—
25C3×C42×C5—21no—
26C3×C22⋊Dic5—21no—
27C3×C4⋊Dic5—22no—
28C3×C4⋊Dic5C2×C412yes—
29C3×#683—22no—
30C3×#683—22no—
31C3×C4×Dic5—21no—
32C3×C4×Dic5C2×C411yescharacteristic
33C3×C4×Dic5D412yes—
34C3×C4×Dic5C2311yescharacteristic
35C3×#533—51no—
36C3×#432—52no—
37C3×#330—54no—
38C5×#117C3×C411yescharacteristic
39#4072C3×C411yescharacteristic
40C2×C4×Dic5C22×C311yescharacteristic
41#4036C22×C312yes—
42#4034C3×C412yes—
43#4024—22no—
44C4×Dic10—22no—
45C22⋊#170—22no—
46#4002C22×C314yes—
47C4⋊#170—22no—
48#1537—51noSylow
49C23×C3×C5C22⋊C411yescharacteristic
50C2×C3×C4×C5—22no—
51C2×C3×C4×C5—22no—
52C2×C3×C4×C5C22⋊C411yescharacteristic
53C2×C3×C4×C5C22×C411yescharacteristic
54C2×C3×Dic5—22no—
55C2×C3×Dic5—42no—
56C2×C3×Dic5C22⋊C412yes—
57C2×C3×Dic5C2×D412yes—
58C3×Dic10—44no—
59C3×#170—44no—
60C3×C4⋊Q8—52no—
61C3×#125—52no—
62C3×C22⋊Q8—102no—
63C3×C4×Q8—102no—
64C3×#117—51no—
65C3×#117AGL(1,𝔽5)11yescharacteristic
66C2×C3×C42—51no—
67C3×C4⋊C8—102no—
68C3×C22⋊C8—102no—
69C3×#97—54no—
70C22×C4×C5C2×C3×C411yescharacteristic
71C22×Dic5C3×D412yes—
72C2×Dic10—22no—
73C2×#170—24no—
74C5×C4⋊C4C2×C3×C412yes—
75C5×C22⋊C4—21no—
76C42×C5—21no—
77C22⋊Dic5—21no—
78C4⋊Dic5—22no—
79C4⋊Dic5C2×C3×C412yes—
80#683—22no—
81#683—22no—
82C4×Dic5—21no—
83C4×Dic5C2×C3×C411yescharacteristic
84C4×Dic5C3×D412yes—
85C4×Dic5C23×C311yescharacteristic
86#533—51no—
87#432—52no—
88#330—54no—
89C22×C3×C5—21no—
90C22×C3×C5—22no—
91C22×C3×C5C2×C22⋊C411yescharacteristic
92C3×C4×C5—21no—
93C3×C4×C5—42no—
94C3×C4×C5#10112yes—
95C3×Dic5—22no—
96C3×Dic5—22no—
97C3×Dic5—42no—
98C2×C3×Q8—102no—
99C22×C3×C4—52no—
100C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
101C2×C3×C8—104no—
102C3×C4⋊C4—52no—
103C3×C4⋊C4—102no—
104C3×C4⋊C4—102no—
105C3×C4⋊C4—102no—
106C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
107C3×C22⋊C4—21no—
108C3×C22⋊C4—101no—
109C3×C42—21no—
110C3×C42—52no—
111C3×C42—51no—
112C3×C42—51no—
113C3×C42—101no—
114C23×C5C3×C22⋊C411yescharacteristic
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5C3×C22⋊C411yescharacteristic
118C2×C4×C5C22×C3×C411yescharacteristic
119C2×Dic5—22no—
120C2×Dic5—42no—
121C2×Dic5C3×C22⋊C412yes—
122C2×Dic5C2×C3×D412yes—
123Dic10—44no—
124#170—44no—
125C4⋊Q8—52no—
126#125—52no—
127C22⋊Q8—102no—
128C4×Q8—102no—
129#117—51no—
130#117C3×AGL(1,𝔽5)11yescharacteristic
131C2×C42—51no—
132C4⋊C8—102no—
133C22⋊C8—102no—
134#97—54no—
135C2×C3×C5—41no—
136C2×C3×C5C2×#10111yescharacteristic
137C2×C3×C5#42112yes—
138C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
139C3×Q8—204no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—52no—
143C2×C3×C4—52no—
144C2×C3×C4—102no—
145C2×C3×C4—202no—
146C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
147C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
148C3×C8—204no—
149C22×C5—21no—
150C22×C5—22no—
151C22×C5C2×C3×C22⋊C411yescharacteristic
152C4×C5—21no—
153C4×C5—42no—
154C4×C5C3×#10112yes—
155Dic5—22no—
156Dic5—22no—
157Dic5—42no—
158C2×Q8—102no—
159C22×C4—52no—
160C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
161C2×C8—104no—
162C4⋊C4—52no—
163C4⋊C4—102no—
164C4⋊C4—102no—
165C4⋊C4—102no—
166C4⋊C4C2×C3×AGL(1,𝔽5)12yes—
167C22⋊C4—21no—
168C22⋊C4—101no—
169C42—21no—
170C42—52no—
171C42—51no—
172C42—51no—
173C42—101no—
174C3×C5#153711yescharacteristic, complemented, Hall
175C22×C3—22no—
176C22×C3—21no—
177C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
178C3×C4—21no—
179C3×C4—42no—
180C3×C4—102no—
181C3×C4—102no—
182C3×C4—202no—
183C3×C4#400512yes—
184C2×C5—41no—
185C2×C5C2×C3×#10111yescharacteristic
186C2×C5C3×#42112yes—
187C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
188Q8—204no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—52no—
192C2×C4—52no—
193C2×C4—102no—
194C2×C4—202no—
195C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
196C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
197C8—204no—
198C2×C3—41no—
199C2×C3C2×#400511yescharacteristic
200C2×C3#6579512yes—
201C5C3×#153711yescharacteristic, complemented, Sylow
202C22—21no—
203C22—22no—
204C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
205C4—21no—
206C4—42no—
207C4—102no—
208C4—102no—
209C4—202no—
210C4C3×#400512yes—
211C3#1060435211yescharacteristic, direct product factor, Sylow
212C2—41no—
213C2C2×C3×#400511yescharacteristic
214C2C3×#6579512yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×C5C22⋊C4
cosocleC4×Dic5C23×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#1537
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#64870C2semidirect
4. #10604352C3direct (N × H)
18. #64870C2×C3semidirect
174. C3×C5#1537semidirect
201. C5C3×#1537semidirect
211. C3#10604352direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658512
C3×#657592
C3×#648702
#106043523
C3×#15375