Finite Group Databasebeta

← C3×#89504 #13055165

Subgroups — C3×#89504 #13055165

Order 1344.

396 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 94 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×D28C211yesmaximal, characteristic, complemented
2C3×#6493C211yesmaximal, characteristic, complemented
3C2×C3×D56C212yesmaximal, complemented
4C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
5C3×#6395C212yesmaximal, complemented
6#89504C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×D7—21no—
8C22×C3×C4×C7C2211yescharacteristic
9C2×C3×C22⋊D7—21no—
10C2×C3×D28—21no—
11C2×C3×D28—24no—
12C2×C3×D28C2212yes—
13C2×C3×D28C2211yescharacteristic, complemented
14C2×C3×C7×C8C2212yescomplemented
15C3×(C2×D7)⋊C4—21no—
16C3×C4⋊Dic7C2211yescharacteristic, complemented
17C3×D56—24no—
18C22×D28C2×C311yescharacteristic, complemented
19#6493C2×C311yescharacteristic, complemented
20C2×D56C2×C312yescomplemented
21C7×C22⋊C8C2×C311yescharacteristic, complemented
22#6395C2×C312yescomplemented
23C3×#447—71nomaximal, Hall
24C23×C3×C7D411yescharacteristic
25C22×C3×D7—24no—
26C22×C3×D7—22no—
27C22×C3×D7—21no—
28C22×C3×D7—42no—
29C22×C3×D7—42no—
30C2×C3×C4×C7—22no—
31C2×C3×C4×C7D411yescharacteristic
32C2×C3×C4×C7C2311yescharacteristic
33C3×C22⋊D7—42no—
34C2×C3×Dic7—21no—
35C3×D28—24no—
36C3×D28—22no—
37C3×D28—22no—
38C3×D28D414yescomplemented
39C3×C7×C8—22no—
40C23×D7—21no—
41C22×C4×C7C22×C311yescharacteristic
42C2×C22⋊D7—21no—
43C2×D28—21no—
44C2×D28—24no—
45C2×D28C22×C312yes—
46C2×D28C22×C311yescharacteristic, complemented
47C2×C7×C8C22×C312yescomplemented
48(C2×D7)⋊C4—21no—
49C4⋊Dic7C22×C311yescharacteristic, complemented
50D56—24no—
51C22×C3×D4—71no—
52C2×C3×D8—72no—
53C3×#121—71no—
54C3×#102—72no—
55C3×C22⋊C8D711yescharacteristic, complemented
56C22×C3×C7—21no—
57C22×C3×C7—21no—
58C22×C3×C7D812yescomplemented
59C22×C3×C7C2×D411yescharacteristic
60C2×C3×D7—24no—
61C2×C3×D7—48no—
62C2×C3×D7—42no—
63C2×C3×D7—42no—
64C2×C3×D7—41no—
65C2×C3×D7—41no—
66C2×C3×D7—41no—
67C3×C4×C7—21no—
68C3×C4×C7C2×D412yes—
69C3×Dic7—41no—
70#447—71noSylow
71C23×C7C3×D411yescharacteristic
72C22×D7—24no—
73C22×D7—22no—
74C22×D7—21no—
75C22×D7—42no—
76C22×D7—42no—
77C2×C4×C7—22no—
78C2×C4×C7C3×D411yescharacteristic
79C2×C4×C7C23×C311yescharacteristic
80C22⋊D7—42no—
81C2×Dic7—21no—
82D28—24no—
83D28—22no—
84D28—22no—
85D28C3×D414yescomplemented
86C7×C8—22no—
87C24×C3—141no—
88C2×C3×D4—72no—
89C2×C3×D4—71no—
90C2×C3×D4—144no—
91C2×C3×D4—141no—
92C2×C3×D4—141no—
93C22×C3×C4C2×D711yescharacteristic
94C3×D8—144no—
95C2×C3×C8C2×D712yescomplemented
96C3×C4⋊C4—71no—
97C3×C22⋊C4—141no—
98C2×C3×C7—22no—
99C2×C3×C7#12011yescharacteristic
100C2×C3×C7C2×D811yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C3×D7—44no—
103C3×D7—81no—
104C22×D4—71no—
105C2×D8—72no—
106#121—71no—
107#102—72no—
108C22⋊C8C3×D711yescharacteristic, complemented
109C22×C7—21no—
110C22×C7—21no—
111C22×C7C3×D812yescomplemented
112C22×C7C2×C3×D411yescharacteristic
113C2×D7—24no—
114C2×D7—48no—
115C2×D7—42no—
116C2×D7—42no—
117C2×D7—41no—
118C2×D7—41no—
119C2×D7—41no—
120C4×C7—21no—
121C4×C7C2×C3×D412yes—
122Dic7—41no—
123C23×C3—144no—
124C23×C3—142no—
125C23×C3—141no—
126C23×C3—282no—
127C23×C3—282no—
128C23×C3D2811yescharacteristic
129C3×D4—74no—
130C3×D4—144no—
131C3×D4—142no—
132C3×D4—142no—
133C3×D4—282no—
134C2×C3×C4—22no—
135C2×C3×C4—141no—
136C2×C3×C4D2811yescharacteristic
137C2×C3×C4C22×D711yescharacteristic
138C3×C8—22no—
139C3×C7#44711yescharacteristic, complemented, Hall
140C24—141no—
141C2×D4—72no—
142C2×D4—71no—
143C2×D4—144no—
144C2×D4—141no—
145C2×D4—141no—
146C22×C4C2×C3×D711yescharacteristic
147D8—144no—
148C2×C8C2×C3×D712yescomplemented
149C4⋊C4—71no—
150C22⋊C4—141no—
151C2×C7—22no—
152C2×C7C3×#12011yescharacteristic
153C2×C7C2×C3×D811yescharacteristic
154C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
155D7—44no—
156D7—81no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3—144no—
160C22×C3—288no—
161C22×C3—282no—
162C22×C3—282no—
163C22×C3—281no—
164C22×C3—281no—
165C22×C3—281no—
166C22×C3D5612yescomplemented
167C22×C3C2×D2811yescharacteristic
168C3×C4—21no—
169C3×C4—281no—
170C3×C4D4×D712yes—
171C23—144no—
172C23—142no—
173C23—141no—
174C23—282no—
175C23—282no—
176C23C3×D2811yescharacteristic
177D4—74no—
178D4—144no—
179D4—142no—
180D4—142no—
181D4—282no—
182C2×C4—22no—
183C2×C4—141no—
184C2×C4C3×D2811yescharacteristic
185C2×C4C22×C3×D711yescharacteristic
186C8—22no—
187C7C3×#44711yescharacteristic, complemented, Sylow
188C2×C3—22no—
189C2×C3—284no—
190C2×C3—561no—
191C2×C3#644511yescharacteristic
192C2×C3C2×D5611yescharacteristic
193C2×C3#647111yescharacteristic
194C22—21no—
195C22—21no—
196C22—144no—
197C22—288no—
198C22—282no—
199C22—282no—
200C22—281no—
201C22—281no—
202C22—281no—
203C22C3×D5612yescomplemented
204C22C2×C3×D2811yescharacteristic
205C4—21no—
206C4—281no—
207C4C3×D4×D712yes—
208C3#8950411yescharacteristic, direct product factor, Sylow
209C2—22no—
210C2—284no—
211C2—561no—
212C2C3×#644511yescharacteristic
213C2C2×C3×D5611yescharacteristic
214C2C3×#647111yescharacteristic
subgroupG/H
centerC22×C3C2×D28
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C2×D4
hypercenterC3×C22⋊C8D7
hypocenterC7C3×#447
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×D28C2semidirect
2. C3×#6493C2semidirect
3. C2×C3×D56C2semidirect
4. C3×C7×C22⋊C8C2semidirect
5. C3×#6395C2semidirect
6. #89504C3direct (N × H)
13. C2×C3×D28C22semidirect
14. C2×C3×C7×C8C22semidirect
16. C3×C4⋊Dic7C22semidirect
18. C22×D28C2×C3semidirect
19. #6493C2×C3semidirect
20. C2×D56C2×C3semidirect
21. C7×C22⋊C8C2×C3semidirect
22. #6395C2×C3semidirect
38. C3×D28D4semidirect
46. C2×D28C22×C3semidirect
47. C2×C7×C8C22×C3semidirect
49. C4⋊Dic7C22×C3semidirect
55. C3×C22⋊C8D7semidirect
58. C22×C3×C7D8semidirect
85. D28C3×D4semidirect
95. C2×C3×C8C2×D7semidirect
108. C22⋊C8C3×D7semidirect
111. C22×C7C3×D8semidirect
139. C3×C7#447semidirect
148. C2×C8C2×C3×D7semidirect
166. C22×C3D56semidirect
187. C7C3×#447semidirect
203. C22C3×D56semidirect
208. C3#89504direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D282
C3×#64932
C2×C3×D562
C3×C7×C22⋊C82
C3×#63952
#895043
C3×#4477