Finite Group Databasebeta

← #13061744

Subgroups — #13061744

Order 1344.

472 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 135 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×#123C211yesmaximal, characteristic, complemented
2#10621423C211yesmaximal, characteristic
3#10621418C211yesmaximal, characteristic, complemented
4C4×#4268C212yesmaximal, complemented
5#10621377C212yesmaximal, complemented
6#10621370C212yesmaximal, complemented
7#10621368C212yesmaximal, complemented
8#10621368C212yesmaximal, complemented
9#10621364C212yesmaximal, complemented
10#90355—31nomaximal, Hall
11C2×C3×C7×D4C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C2×#4268C2212yes—
14C2×#4268C2212yes—
15C22×Dic21C2211yescharacteristic
16C2×D84C2211yescharacteristic
17C2×C4×D21C2212yes—
18C2×Dic42C2212yes—
19C3×C7×C4⋊C4C2212yes—
20C3×C7×C22⋊C4C2212yes—
21C3×C7×C22⋊C4C2211yescharacteristic
22#66636C2211yescharacteristic
23#66636C2212yes—
24#66631C2212yes—
25#66631C2212yes—
26#66631C2212yes—
27#66629C2212yes—
28#66629C2212yes—
29C4×Dic21C2212yes—
30C4×Dic21C2212yes—
31C4×Dic21C2211yescharacteristic
32C7×#123D311yescharacteristic, complemented
33#6503—31no—
34#6498—31no—
35C4×C22⋊D7—32no—
36#6457—32no—
37#6450—32no—
38#6448—32no—
39#6448—32no—
40(C22⋊D7)⋊C4—32no—
41#5664—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C23×C3×C7C2311yescharacteristic
44C22×D21C2312yes—
45C3×C7×D4—41no—
46C2×C3×C4×C7—41no—
47C2×C3×C4×C7C2311yescharacteristic
48C2×C3×C4×C7C2312yes—
49C2×C3×C4×C7C2312yes—
50#4268—24no—
51#4268—24no—
52C2×Dic21—41no—
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55C2×Dic21C2312yes—
56D84—41no—
57C4×D21—42no—
58Dic42—24no—
59C2×C7×D4C2×D311yescharacteristic
60C22×C4×C7C2×D311yescharacteristic
61C2×C22⋊D7—32no—
62C2×C22⋊D7—32no—
63C22×Dic7—31no—
64C2×D28—31no—
65C2×C4×D7—32no—
66C2×Dic14—32no—
67C7×C4⋊C4C2×D312yes—
68C7×C22⋊C4C2×D312yes—
69C7×C22⋊C4C2×D311yescharacteristic
70C22⋊Dic7—31no—
71C22⋊Dic7—32no—
72(C2×D7)⋊C4—32no—
73(C2×D7)⋊C4—32no—
74(C2×D7)⋊C4—32no—
75Dic7⋊C4—32no—
76Dic7⋊C4—32no—
77C4×Dic7—32no—
78C4×Dic7—32no—
79C4×Dic7—31no—
80C3×#123D711yescharacteristic, complemented
81#940—71no—
82#935—71no—
83C4×#67—72no—
84#894—72no—
85#887—72no—
86#885—72no—
87#885—72no—
88#67⋊C4—72no—
89C22×C3×C7—22no—
90C22×C3×C7—22no—
91C22×C3×C7—21no—
92C22×C3×C7—21no—
93C22×C3×C7C2411yescharacteristic
94C2×D21—22no—
95C2×D21—22no—
96C2×D21—22no—
97C3×C4×C7—22no—
98C3×C4×C7—22no—
99C3×C4×C7—21no—
100Dic21—22no—
101Dic21—22no—
102Dic21#4114yes—
103#559—211noSylow
104C23×C7C22×D311yescharacteristic
105C23×C7C22×D311yescharacteristic
106C22×D7—32no—
107C7×D4—41no—
108C2×C4×C7—41no—
109C2×C4×C7C22×D311yescharacteristic
110C2×C4×C7C22×D312yes—
111C2×C4×C7C22×D312yes—
112C22⋊D7—64no—
113C22⋊D7—64no—
114C2×Dic7—32no—
115C2×Dic7—32no—
116C2×Dic7—32no—
117C2×Dic7—121no—
118D28—121no—
119C4×D7—122no—
120Dic14—64no—
121C2×C3×D4C2×D711yescharacteristic
122C22×C3×C4C2×D711yescharacteristic
123C2×#67—72no—
124C2×#67—72no—
125C22×Dic3—71no—
126C2×D12—71no—
127C2×C4×D3—72no—
128C2×Dic6—72no—
129C3×C4⋊C4C2×D712yes—
130C3×C22⋊C4C2×D712yes—
131C3×C22⋊C4C2×D711yescharacteristic
132#217—71no—
133#217—72no—
134(C2×D3)⋊C4—72no—
135(C2×D3)⋊C4—72no—
136(C2×D3)⋊C4—72no—
137Dic3⋊C4—72no—
138Dic3⋊C4—72no—
139C4×Dic3—72no—
140C4×Dic3—72no—
141C4×Dic3—71no—
142C2×C3×C7—41no—
143C2×C3×C7—41no—
144C2×C3×C7C2×#4112yes—
145C2×C3×C7H(5,𝔽2)11yescharacteristic
146D21—42no—
147#127—211no—
148#124—212no—
149#124—212no—
150#123—211no—
151#123—212no—
152#123D2111yescharacteristic, complemented
153C4×Q8—212no—
154C4×D4—212no—
155C4×D4—212no—
156C22×C7—21no—
157C22×C7—21no—
158C22×C7—22no—
159C22×C7—22no—
160C22×C7C23×D311yescharacteristic
161C2×D7—62no—
162C2×D7—62no—
163C2×D7—62no—
164C4×C7—22no—
165C4×C7—21no—
166C4×C7—22no—
167Dic7—34no—
168Dic7—62no—
169Dic7—62no—
170C23×C3C22×D711yescharacteristic
171C23×C3C22×D711yescharacteristic
172C22×D3—72no—
173C3×D4—41no—
174C2×C3×C4—41no—
175C2×C3×C4C22×D711yescharacteristic
176C2×C3×C4C22×D712yes—
177C2×C3×C4C22×D712yes—
178#67—144no—
179#67—144no—
180C2×Dic3—72no—
181C2×Dic3—72no—
182C2×Dic3—72no—
183C2×Dic3—281no—
184D12—281no—
185C4×D3—282no—
186Dic6—144no—
187C3×C7#55911yescharacteristic, complemented, Hall
188C2×Q8—212no—
189C2×D4—212no—
190C2×D4—212no—
191C2×D4—211no—
192C2×D4C2×D2111yescharacteristic
193C22×C4—211no—
194C22×C4—212no—
195C22×C4C2×D2111yescharacteristic
196C4⋊C4—212no—
197C4⋊C4—212no—
198C4⋊C4C2×D2112yes—
199C22⋊C4—212no—
200C22⋊C4—211no—
201C22⋊C4—212no—
202C22⋊C4—212no—
203C22⋊C4—212no—
204C22⋊C4C2×D2112yes—
205C22⋊C4C2×D2111yescharacteristic
206C42—212no—
207C42—212no—
208C42—211no—
209C2×C7—41no—
210C2×C7—41no—
211C2×C7#100411yescharacteristic
212C2×C7D3×#4112yes—
213D7—122no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3C23×D711yescharacteristic
219C2×D3—142no—
220C2×D3—142no—
221C2×D3—142no—
222C3×C4—22no—
223C3×C4—21no—
224C3×C4—22no—
225Dic3—74no—
226Dic3—142no—
227Dic3—142no—
228C23—212no—
229C23C22×D2111yescharacteristic
230C23C22×D2111yescharacteristic
231Q8—424no—
232D4—41no—
233D4—424no—
234D4—424no—
235D4—841no—
236C2×C4—41no—
237C2×C4—212no—
238C2×C4—212no—
239C2×C4—212no—
240C2×C4—842no—
241C2×C4—841no—
242C2×C4C22×D2112yes—
243C2×C4C22×D2112yes—
244C2×C4C22×D2111yescharacteristic
245C7#566411yescharacteristic, complemented, Sylow
246C2×C3—41no—
247C2×C3—41no—
248C2×C3#654811yescharacteristic
249C2×C3D7×#4112yes—
250D3—282no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—422no—
256C22—422no—
257C22—422no—
258C22C23×D2111yescharacteristic
259C4—22no—
260C4—21no—
261C4—22no—
262C4—214no—
263C4—422no—
264C4—422no—
265C3#9035511yescharacteristic, complemented, Sylow
266C2—41no—
267C2—41no—
268C2—842no—
269C2#1062165711yescharacteristic
270C2#41×D2112yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#123C2
prime radical#10621423C2
hypercenter#123D21
hypocenterC3×C7#559
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C3×C7×#123C2semidirect
3. #10621418C2semidirect
4. C4×#4268C2semidirect
5. #10621377C2semidirect
6. #10621370C2semidirect
7. #10621368C2semidirect
8. #10621368C2semidirect
9. #10621364C2semidirect
32. C7×#123D3semidirect
80. C3×#123D7semidirect
152. #123D21semidirect
187. C3×C7#559semidirect
245. C7#5664semidirect
265. C3#90355semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1232
#106214232
#106214182
C4×#42682
#106213772
#106213702
#106213682
#106213642
#903553
#56647