Finite Group Databasebeta

← #13056369

Subgroups — #13056369

Order 1344.

396 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 79 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×D84C211yesmaximal, characteristic, complemented
2#10621413C211yesmaximal, characteristic, complemented
3C2×D168C212yesmaximal, complemented
4C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
5#10620660C212yesmaximal, complemented
6#89504—31nomaximal, Hall
7C23×D21—21no—
8C22×C3×C4×C7C2211yescharacteristic
9C2×#4268—21no—
10C2×D84—21no—
11C2×D84—24no—
12C2×D84C2212yes—
13C2×D84C2211yescharacteristic, complemented
14C2×C3×C7×C8C2212yescomplemented
15#66631—21no—
16C4⋊Dic21C2211yescharacteristic, complemented
17D168—24no—
18C22×D28—31no—
19#6493—31no—
20C2×D56—32no—
21C7×C22⋊C8D311yescharacteristic, complemented
22#6395—32no—
23#4732—71nomaximal, Hall
24C23×C3×C7D411yescharacteristic
25C22×D21—24no—
26C22×D21—22no—
27C22×D21—21no—
28C22×D21—42no—
29C22×D21—42no—
30C2×C3×C4×C7—22no—
31C2×C3×C4×C7D411yescharacteristic
32C2×C3×C4×C7C2311yescharacteristic
33#4268—42no—
34C2×Dic21—21no—
35D84—24no—
36D84—22no—
37D84—22no—
38D84D414yescomplemented
39C3×C7×C8—22no—
40C23×D7—61no—
41C22×C4×C7C2×D311yescharacteristic
42C2×C22⋊D7—61no—
43C2×D28—32no—
44C2×D28—31no—
45C2×D28—64no—
46C2×D28—61no—
47C2×C7×C8C2×D312yescomplemented
48(C2×D7)⋊C4—61no—
49C4⋊Dic7—31no—
50D56—64no—
51C22×D12—71no—
52#930—71no—
53C2×D24—72no—
54C3×C22⋊C8D711yescharacteristic, complemented
55#821—72no—
56C22×C3×C7—21no—
57C22×C3×C7—21no—
58C22×C3×C7D812yescomplemented
59C22×C3×C7C2×D411yescharacteristic
60C2×D21—24no—
61C2×D21—48no—
62C2×D21—42no—
63C2×D21—42no—
64C2×D21—41no—
65C2×D21—41no—
66C2×D21—41no—
67C3×C4×C7—21no—
68C3×C4×C7C2×D412yes—
69Dic21—41no—
70#447—211noSylow
71C23×C7D1211yescharacteristic
72C22×D7—64no—
73C22×D7—62no—
74C22×D7—61no—
75C22×D7—122no—
76C22×D7—122no—
77C2×C4×C7—22no—
78C2×C4×C7D1211yescharacteristic
79C2×C4×C7C22×D311yescharacteristic
80C22⋊D7—122no—
81C2×Dic7—61no—
82D28—34no—
83D28—64no—
84D28—62no—
85D28—62no—
86C7×C8—22no—
87C23×D3—141no—
88C22×C3×C4C2×D711yescharacteristic
89C2×#67—141no—
90C2×D12—72no—
91C2×D12—71no—
92C2×D12—144no—
93C2×D12—141no—
94C2×C3×C8C2×D712yescomplemented
95(C2×D3)⋊C4—141no—
96C4⋊Dic3—71no—
97D24—144no—
98C2×C3×C7—22no—
99C2×C3×C7#12011yescharacteristic
100C2×C3×C7C2×D811yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102D21—44no—
103D21—81no—
104C22×D4—211no—
105C2×D8—212no—
106#121—211no—
107#102—212no—
108C22⋊C8D2111yescharacteristic, complemented
109C22×C7—21no—
110C22×C7—21no—
111C22×C7D2412yescomplemented
112C22×C7C2×D1211yescharacteristic
113C2×D7—64no—
114C2×D7—128no—
115C2×D7—122no—
116C2×D7—122no—
117C2×D7—121no—
118C2×D7—121no—
119C2×D7—121no—
120C4×C7—21no—
121C4×C7D3×D412yes—
122Dic7—121no—
123C23×C3D2811yescharacteristic
124C22×D3—144no—
125C22×D3—142no—
126C22×D3—141no—
127C22×D3—282no—
128C22×D3—282no—
129C2×C3×C4—22no—
130C2×C3×C4D2811yescharacteristic
131C2×C3×C4C22×D711yescharacteristic
132#67—282no—
133C2×Dic3—141no—
134D12—74no—
135D12—144no—
136D12—142no—
137D12—142no—
138C3×C8—22no—
139C3×C7#44711yescharacteristic, complemented, Hall
140C24—421no—
141C2×D4—212no—
142C2×D4—211no—
143C2×D4—424no—
144C2×D4—421no—
145C2×D4—421no—
146C22×C4C2×D2111yescharacteristic
147D8—424no—
148C2×C8C2×D2112yescomplemented
149C4⋊C4—211no—
150C22⋊C4—421no—
151C2×C7—22no—
152C2×C7#88211yescharacteristic
153C2×C7C2×D2411yescharacteristic
154C2×C7#90811yescharacteristic
155D7—124no—
156D7—241no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3D5612yescomplemented
160C22×C3C2×D2811yescharacteristic
161C2×D3—144no—
162C2×D3—288no—
163C2×D3—282no—
164C2×D3—282no—
165C2×D3—281no—
166C2×D3—281no—
167C2×D3—281no—
168C3×C4—21no—
169C3×C4D4×D712yes—
170Dic3—281no—
171C23—424no—
172C23—422no—
173C23—421no—
174C23—842no—
175C23—842no—
176C23D8411yescharacteristic
177D4—214no—
178D4—424no—
179D4—422no—
180D4—422no—
181D4—842no—
182C2×C4—22no—
183C2×C4—421no—
184C2×C4D8411yescharacteristic
185C2×C4C22×D2111yescharacteristic
186C8—22no—
187C7#473211yescharacteristic, complemented, Sylow
188C2×C3—22no—
189C2×C3#644511yescharacteristic
190C2×C3C2×D5611yescharacteristic
191C2×C3#647111yescharacteristic
192D3—284no—
193D3—561no—
194C22—21no—
195C22—21no—
196C22—424no—
197C22—848no—
198C22—842no—
199C22—842no—
200C22—841no—
201C22—841no—
202C22—841no—
203C22D16812yescomplemented
204C22C2×D8411yescharacteristic
205C4—21no—
206C4—841no—
207C4D4×D2112yes—
208C3#8950411yescharacteristic, complemented, Sylow
209C2—22no—
210C2—844no—
211C2—1681no—
212C2#1062136511yescharacteristic
213C2C2×D16811yescharacteristic
214C2#1062139111yescharacteristic
subgroupG/H
centerC22C2×D84
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C2×D4
hypercenterC22⋊C8D21
hypocenterC3×C7#447
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×D84C2semidirect
2. #10621413C2semidirect
3. C2×D168C2semidirect
4. C3×C7×C22⋊C8C2semidirect
5. #10620660C2semidirect
13. C2×D84C22semidirect
14. C2×C3×C7×C8C22semidirect
16. C4⋊Dic21C22semidirect
21. C7×C22⋊C8D3semidirect
38. D84D4semidirect
47. C2×C7×C8C2×D3semidirect
54. C3×C22⋊C8D7semidirect
58. C22×C3×C7D8semidirect
94. C2×C3×C8C2×D7semidirect
108. C22⋊C8D21semidirect
111. C22×C7D24semidirect
139. C3×C7#447semidirect
148. C2×C8C2×D21semidirect
159. C22×C3D56semidirect
187. C7#4732semidirect
203. C22D168semidirect
208. C3#89504semidirect

Maximal subgroups

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

subgroupindex
C22×D842
#106214132
C2×D1682
C3×C7×C22⋊C82
#106206602
#895043
#47327