Finite Group Databasebeta

← C3×#89806 #13055467

Subgroups — C3×#89806 #13055467

Order 1344.

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

normal subgroups only

This group has 228 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
1C3×C7×#121C211yesmaximal, characteristic
2C2×C3×C4⋊Dic7C211yesmaximal, characteristic, complemented
3C3×#6406C212yesmaximal, complemented
4C3×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
5C3×#6382C212yesmaximal, complemented
6#89806C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C7×D4—21no—
8C2×C3×C7×D4C2211yescharacteristic
9C22×C3×C4×C7C2211yescharacteristic
10C22×C3×Dic7—21no—
11C3×C7×C4⋊C4C2211yescharacteristic
12C3×C7×C22⋊C4—21no—
13C3×C4⋊Dic7—21no—
14C3×C4⋊Dic7C2212yes—
15C2×C3×C7⋊C8C2212yes—
16C7×#121C2×C311yescharacteristic
17C2×C4⋊Dic7C2×C311yescharacteristic, complemented
18#6406C2×C312yescomplemented
19C22⋊(C7⋊C8)C2×C311yescharacteristic, complemented
20#6382C2×C312yescomplemented
21C3×#481—71nomaximal, Hall
22C23×C3×C7—21no—
23C23×C3×C7D411yescharacteristic
24C3×C7×D4—22no—
25C3×C7×D4—42no—
26C2×C3×C4×C7—22no—
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7D411yescharacteristic
29C2×C3×C4×C7C2311yescharacteristic
30C2×C3×Dic7—22no—
31C2×C3×Dic7—42no—
32C3×C7⋊C8—22no—
33C2×C7×D4—21no—
34C2×C7×D4C22×C311yescharacteristic
35C22×C4×C7C22×C311yescharacteristic
36C22×Dic7—21no—
37C7×C4⋊C4C22×C311yescharacteristic
38C7×C22⋊C4—21no—
39C4⋊Dic7—21no—
40C4⋊Dic7C22×C312yes—
41C2×C7⋊C8C22×C312yes—
42C3×#121D711yescharacteristic
43C2×C3×C4⋊C4—71no—
44C3×#106—72no—
45C3×#102—72no—
46C3×C22⋊C8—71no—
47C22×C3×C7—21no—
48C22×C3×C7—21no—
49C22×C3×C7—41no—
50C22×C3×C7—41no—
51C22×C3×C7—41no—
52C22×C3×C7SD1612yes—
53C22×C3×C7C2×D411yescharacteristic
54C3×C4×C7—21no—
55C3×C4×C7—41no—
56C3×C4×C7#4112yes—
57C3×Dic7—42no—
58#481—71noSylow
59C23×C7—21no—
60C23×C7C3×D411yescharacteristic
61C7×D4—22no—
62C7×D4—42no—
63C2×C4×C7—22no—
64C2×C4×C7—21no—
65C2×C4×C7C3×D411yescharacteristic
66C2×C4×C7C23×C311yescharacteristic
67C2×Dic7—22no—
68C2×Dic7—42no—
69C7⋊C8—22no—
70C2×C3×D4—21no—
71C2×C3×D4C2×D711yescharacteristic
72C22×C3×C4—141no—
73C22×C3×C4C2×D711yescharacteristic
74C2×C3×C8—72no—
75C3×C4⋊C4—72no—
76C3×C4⋊C4—141no—
77C3×C4⋊C4C2×D711yescharacteristic
78C3×C22⋊C4—21no—
79C2×C3×C7—22no—
80C2×C3×C7—81no—
81C2×C3×C7#12311yescharacteristic
82C2×C3×C7C2×SD1611yescharacteristic
83C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
84#121C3×D711yescharacteristic
85C2×C4⋊C4—71no—
86#106—72no—
87#102—72no—
88C22⋊C8—71no—
89C22×C7—21no—
90C22×C7—21no—
91C22×C7—41no—
92C22×C7—41no—
93C22×C7—41no—
94C22×C7C3×SD1612yes—
95C22×C7C2×C3×D411yescharacteristic
96C4×C7—21no—
97C4×C7—41no—
98C4×C7C3×#4112yes—
99Dic7—42no—
100C23×C3—21no—
101C23×C3C22⋊D711yescharacteristic
102C3×D4—22no—
103C3×D4—42no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—142no—
107C2×C3×C4—282no—
108C2×C3×C4C22⋊D711yescharacteristic
109C2×C3×C4C22×D711yescharacteristic
110C3×C8—142no—
111C3×C7#48111yescharacteristic, complemented, Hall
112C2×D4—21no—
113C2×D4C2×C3×D711yescharacteristic
114C22×C4—141no—
115C22×C4C2×C3×D711yescharacteristic
116C2×C8—72no—
117C4⋊C4—72no—
118C4⋊C4—141no—
119C4⋊C4C2×C3×D711yescharacteristic
120C22⋊C4—21no—
121C2×C7—22no—
122C2×C7—81no—
123C2×C7C3×#12311yescharacteristic
124C2×C7C2×C3×SD1611yescharacteristic
125C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
126C22×C3—21no—
127C22×C3—21no—
128C22×C3—41no—
129C22×C3—41no—
130C22×C3—41no—
131C22×C3#114212yes—
132C22×C3C2×C22⋊D711yescharacteristic
133C3×C4—21no—
134C3×C4—41no—
135C3×C4—282no—
136C3×C4#115912yes—
137C23—21no—
138C23C3×C22⋊D711yescharacteristic
139D4—22no—
140D4—42no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—142no—
144C2×C4—282no—
145C2×C4C3×C22⋊D711yescharacteristic
146C2×C4C22×C3×D711yescharacteristic
147C8—142no—
148C7C3×#48111yescharacteristic, complemented, Sylow
149C2×C3—22no—
150C2×C3—81no—
151C2×C3C2×#114211yescharacteristic
152C2×C3#649811yescharacteristic
153C2×C3#651211yescharacteristic
154C22—21no—
155C22—21no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22C3×#114212yes—
160C22C2×C3×C22⋊D711yescharacteristic
161C4—21no—
162C4—41no—
163C4—282no—
164C4C3×#115912yes—
165C3#8980611yescharacteristic, direct product factor, Sylow
166C2—22no—
167C2—81no—
168C2C2×C3×#114211yescharacteristic
169C2C3×#649811yescharacteristic
170C2C3×#651211yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
Fitting, prime radicalC3×C7×#121C2
socleC22×C3×C7C2×D4
hypercenterC3×#121D7
hypocenterC7C3×#481
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊Dic7C2semidirect
3. C3×#6406C2semidirect
4. C3×C22⋊(C7⋊C8)C2semidirect
5. C3×#6382C2semidirect
6. #89806C3direct (N × H)
17. C2×C4⋊Dic7C2×C3semidirect
18. #6406C2×C3semidirect
19. C22⋊(C7⋊C8)C2×C3semidirect
20. #6382C2×C3semidirect
111. C3×C7#481semidirect
148. C7C3×#481semidirect
165. C3#89806direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C7×#1212
C2×C3×C4⋊Dic72
C3×#64062
C3×C22⋊(C7⋊C8)2
C3×#63822
#898063
C3×#4817