Finite Group Databasebeta

← C3×#10590053 #13154903

Subgroups — C3×#10590053 #13154903

Order 1728.

372 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 70 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⋊#157C211yesmaximal, characteristic
2C3×#63818C212yesmaximal
3#10590053C311yesmaximal, characteristic, direct product factor
4C22×C3×#157C412yes—
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C3×C4⋊#157—22no—
7C32×#208—22no—
8C3×#3712C414yescomplemented
9C3×#3710—22no—
10C2×C4⋊#157C2×C311yescharacteristic
11C3×D3×M16—62no—
12#63818C2×C312yes—
13C22×C3×C3⋊D3C2×C411yescharacteristic
14C2×C3×#157—22no—
15C2×C3×#157—42no—
16C2×C33×C4D411yescharacteristic
17C2×C3×C3⋊Dic3Q811yescharacteristic
18C3×C4×C3⋊D3C2×C412yes—
19C3×C4×C3⋊D3D412yes—
20C32×C3⋊C8—24no—
21C3×#343—91nomaximal
22C22×#157C3×C412yes—
23C2×C4×C3⋊D3C22×C311yescharacteristic
24C2×C3×C4×D3—62no—
25C4⋊#157—22no—
26C32×M16—62no—
27C3×#208—22no—
28C3×#208—22no—
29C3×#208—42no—
30C3×ΓL(1,𝔽25)—64no—
31C3×D3×C8—64no—
32#3712C3×C414yescomplemented
33#3710—22no—
34C22×C33C4⋊C411yescharacteristic
35C2×C3×C3⋊D3—21no—
36C2×C3×C3⋊D3—21no—
37C2×C3×C3⋊D3C4211yescharacteristic
38C2×C3×C3⋊D3C22⋊C411yescharacteristic
39C3×#157—42no—
40C33×C4C22⋊C412yes—
41C3×C3⋊Dic3C4⋊C412yes—
42C2×C3×M16—92no—
43C2×C3×C4⋊C4—91no—
44D3×M16—62no—
45C22×C3⋊D3C2×C3×C411yescharacteristic
46C22×C3×D3—62no—
47C2×#157—22no—
48C2×#157—42no—
49C2×C32×C4—22no—
50C2×C32×C4—42no—
51C2×C32×C4C3×D411yescharacteristic
52C2×C3⋊Dic3C3×Q811yescharacteristic
53C4×C3⋊D3C2×C3×C412yes—
54C4×C3⋊D3C3×D412yes—
55C2×C3×Dic3—62no—
56C3×C4×D3—62no—
57C3×C4×D3—62no—
58C3×C4×D3—62no—
59C3×C4×D3—62no—
60C32×C8—64no—
61C3×C3⋊C8—24no—
62C3×C3⋊C8—24no—
63C3×C3⋊C8—44no—
64#343—91noSylow
65C2×C33—21no—
66C2×C33#9511yescharacteristic
67C3×C3⋊D3—21no—
68C3×C3⋊D3#10011yescharacteristic
69C3×C3⋊D3#10111yescharacteristic
70C22×C3×C4—91no—
71C22×C3×C4—92no—
72C2×C4×D3—62no—
73C3×M16—62no—
74C3×M16—62no—
75C3×M16—94no—
76C3×M16—122no—
77C2×C3×C8—182no—
78C3×C4⋊C4—182no—
79#208—22no—
80ΓL(1,𝔽25)—64no—
81D3×C8—64no—
82C22×C32—22no—
83C22×C32—42no—
84C22×C32C3×C4⋊C411yescharacteristic
85C2×C3⋊D3—21no—
86C2×C3⋊D3—21no—
87C2×C3⋊D3C3×C4211yescharacteristic
88C2×C3⋊D3C3×C22⋊C411yescharacteristic
89C2×C3×D3—62no—
90C2×C3×D3—62no—
91C2×C3×D3—122no—
92C2×C3×D3—122no—
93#157—42no—
94C32×C4—22no—
95C32×C4—22no—
96C32×C4—42no—
97C32×C4—42no—
98C32×C4C3×C22⋊C412yes—
99C3⋊Dic3C3×C4⋊C412yes—
100C3×Dic3—62no—
101C3×Dic3—62no—
102C2×M16—92no—
103C2×C4⋊C4—91no—
104C33#34311yescharacteristic, complemented, Sylow
105C23×C3—91no—
106C22×D3—62no—
107C2×C3×C4—22no—
108C2×C3×C4—42no—
109C2×C3×C4—91no—
110C2×C3×C4—92no—
111C2×C3×C4—92no—
112C2×C3×C4—182no—
113C2×C3×C4—362no—
114C2×C3×C4GO+(4,𝔽2)11yescharacteristic
115C2×Dic3—62no—
116C4×D3—62no—
117C4×D3—62no—
118C4×D3—62no—
119C4×D3—62no—
120C3×C8—64no—
121C3×C8—64no—
122C3×C8—124no—
123C3⋊C8—24no—
124C2×C32—22no—
125C2×C32—21no—
126C2×C32—42no—
127C2×C32—42no—
128C2×C32—82no—
129C2×C32C3×#9511yescharacteristic
130C3⋊D3—21no—
131C3⋊D3C3×#10011yescharacteristic
132C3⋊D3C3×#10111yescharacteristic
133C3×D3—62no—
134C3×D3—62no—
135C3×D3—122no—
136C22×C4—91no—
137C22×C4—92no—
138M16—62no—
139M16—94no—
140C2×C8—182no—
141C4⋊C4—182no—
142C22×C3—22no—
143C22×C3—42no—
144C22×C3—91no—
145C22×C3—91no—
146C22×C3—181no—
147C22×C3—181no—
148C22×C3#377311yescharacteristic
149C2×D3—62no—
150C2×D3—62no—
151C2×D3—122no—
152C2×D3—122no—
153C3×C4—22no—
154C3×C4—22no—
155C3×C4—42no—
156C3×C4—42no—
157C3×C4—92no—
158C3×C4—362no—
159C3×C4D32⋊C412yes—
160Dic3—62no—
161Dic3—62no—
162C32—22no—
163C32—42no—
164C32C3×#34311yescharacteristic, complemented
165C23—91no—
166C2×C4—91no—
167C2×C4—92no—
168C2×C4—92no—
169C2×C4—182no—
170C2×C4—362no—
171C2×C4C3×GO+(4,𝔽2)11yescharacteristic
172C8—64no—
173C2×C3—21no—
174C2×C3—22no—
175C2×C3—42no—
176C2×C3—42no—
177C2×C3—82no—
178C2×C3—91no—
179C2×C3—91no—
180C2×C3—181no—
181C2×C3#6373711yescharacteristic
182D3—62no—
183D3—62no—
184D3—122no—
185C22—91no—
186C22—91no—
187C22—181no—
188C22—181no—
189C22C3×#377311yescharacteristic
190C4—92no—
191C4—362no—
192C4C3×D32⋊C412yes—
193C3—22no—
194C3—42no—
195C3#1059005311yescharacteristic, direct product factor
196C2—21no—
197C2—91no—
198C2—91no—
199C2—181no—
200C2C3×#6373711yescharacteristic
subgroupG/H
centerC2×C3#63737
commutatorC2×C3⋊D3C3×C42
FrattiniC2×C4C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#343
LCSG ▷ C2×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. #10590053C3direct (N × H)
8. C3×#3712C4semidirect
32. #3712C3×C4semidirect
104. C33#343semidirect
164. C32C3×#343semidirect
195. C3#10590053direct (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⋊#1572
C3×#638182
#105900533
C3×#3439