Finite Group Databasebeta

← #13061760

Subgroups — #13061760

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×#124C211yesmaximal, characteristic, complemented
2#10621430C211yesmaximal, characteristic, complemented
3#10621427C211yesmaximal, characteristic, complemented
4#10621422C212yesmaximal, complemented
5#10621368C212yesmaximal, complemented
6#10621367C212yesmaximal, complemented
7#10621364C212yesmaximal, complemented
8#10621360C212yesmaximal, complemented
9#10621355C212yesmaximal, complemented
10#90361—31nomaximal, Hall
11C2×C3×C7×Q8C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2212yes—
14C2×#4268C2212yes—
15C22×Dic21C2212yes—
16C2×D84C2211yescharacteristic
17C2×C4×D21C2212yes—
18C2×Dic42C2211yescharacteristic
19C3×C7×C22⋊C4C2212yes—
20C3×C7×C22⋊C4C2212yes—
21C3×C42×C7C2211yescharacteristic
22#66636C2212yes—
23#66631C2212yes—
24#66631C2212yes—
25#66631C2212yes—
26#66629C2212yes—
27#66629C2212yes—
28#66629C2212yes—
29C4×Dic21C2212yes—
30C4×Dic21C2211yescharacteristic
31C4×Dic21C2211yescharacteristic
32C7×#124D311yescharacteristic, complemented
33#6510—31no—
34#6507—31no—
35#6502—32no—
36#6448—32no—
37#6447—32no—
38(C22⋊D7)⋊C4—32no—
39#6440—32no—
40#6435—32no—
41#5670—71nomaximal, Hall
42C23×C3×C7C2312yes—
43C22×D21C2312yes—
44C3×C7×Q8—41no—
45C3×C7×D4—41no—
46C2×C3×C4×C7C2311yescharacteristic
47C2×C3×C4×C7C2312yes—
48C2×C3×C4×C7C2311yescharacteristic
49C2×C3×C4×C7C2311yescharacteristic
50#4268—24no—
51#4268—42no—
52C2×Dic21—24no—
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55C2×Dic21C2312yes—
56D84—41no—
57C4×D21—24no—
58Dic42—41no—
59C2×C7×Q8C2×D311yescharacteristic
60C2×C7×D4C2×D311yescharacteristic
61C2×C22⋊D7—32no—
62C2×C22⋊D7—32no—
63C22×Dic7—32no—
64C2×D28—31no—
65C2×C4×D7—32no—
66C2×Dic14—31no—
67C7×C22⋊C4C2×D312yes—
68C7×C22⋊C4C2×D312yes—
69C42×C7C2×D311yescharacteristic
70C22⋊Dic7—32no—
71(C2×D7)⋊C4—32no—
72(C2×D7)⋊C4—32no—
73(C2×D7)⋊C4—32no—
74Dic7⋊C4—32no—
75Dic7⋊C4—32no—
76Dic7⋊C4—32no—
77C4×Dic7—32no—
78C4×Dic7—31no—
79C4×Dic7—31no—
80C3×#124D711yescharacteristic, complemented
81#947—71no—
82#944—71no—
83#939—72no—
84#885—72no—
85#884—72no—
86#67⋊C4—72no—
87#877—72no—
88#872—72no—
89C22×C3×C7—22no—
90C22×C3×C7—22no—
91C22×C3×C7—22no—
92C22×C3×C7C2411yescharacteristic
93C2×D21—22no—
94C2×D21—22no—
95C2×D21—22no—
96C3×C4×C7—22no—
97C3×C4×C7—21no—
98C3×C4×C7—21no—
99C3×C4×C7—21no—
100Dic21—22no—
101Dic21—22no—
102Dic21#4114yes—
103#555—211noSylow
104C23×C7C22×D312yes—
105C22×D7—32no—
106C7×Q8—41no—
107C7×D4—41no—
108C2×C4×C7C22×D311yescharacteristic
109C2×C4×C7C22×D312yes—
110C2×C4×C7C22×D311yescharacteristic
111C2×C4×C7C22×D311yescharacteristic
112C22⋊D7—64no—
113C22⋊D7—122no—
114C2×Dic7—32no—
115C2×Dic7—32no—
116C2×Dic7—32no—
117C2×Dic7—64no—
118D28—121no—
119C4×D7—64no—
120Dic14—121no—
121C2×C3×Q8C2×D711yescharacteristic
122C2×C3×D4C2×D711yescharacteristic
123C2×#67—72no—
124C2×#67—72no—
125C22×Dic3—72no—
126C2×D12—71no—
127C2×C4×D3—72no—
128C2×Dic6—71no—
129C3×C22⋊C4C2×D712yes—
130C3×C22⋊C4C2×D712yes—
131C3×C42C2×D711yescharacteristic
132#217—72no—
133(C2×D3)⋊C4—72no—
134(C2×D3)⋊C4—72no—
135(C2×D3)⋊C4—72no—
136Dic3⋊C4—72no—
137Dic3⋊C4—72no—
138Dic3⋊C4—72no—
139C4×Dic3—72no—
140C4×Dic3—71no—
141C4×Dic3—71no—
142C2×C3×C7—42no—
143C2×C3×C7C2×#4112yes—
144C2×C3×C7H(5,𝔽2)11yescharacteristic
145D21—42no—
146C4⋊Q8—211no—
147#124—211no—
148#124—212no—
149#124D2111yescharacteristic, complemented
150#121—212no—
151#121—212no—
152C4×D4—212no—
153#117—212no—
154#117—212no—
155C22×C7—22no—
156C22×C7—22no—
157C22×C7—22no—
158C22×C7C23×D311yescharacteristic
159C2×D7—62no—
160C2×D7—62no—
161C2×D7—62no—
162C4×C7—21no—
163C4×C7—21no—
164C4×C7—21no—
165C4×C7—22no—
166Dic7—34no—
167Dic7—62no—
168Dic7—62no—
169C23×C3C22×D712yes—
170C22×D3—72no—
171C3×Q8—41no—
172C3×D4—41no—
173C2×C3×C4C22×D711yescharacteristic
174C2×C3×C4C22×D712yes—
175C2×C3×C4C22×D711yescharacteristic
176C2×C3×C4C22×D711yescharacteristic
177#67—144no—
178#67—282no—
179C2×Dic3—72no—
180C2×Dic3—72no—
181C2×Dic3—72no—
182C2×Dic3—144no—
183D12—281no—
184C4×D3—144no—
185Dic6—281no—
186C3×C7#55511yescharacteristic, complemented, Hall
187C2×Q8—211no—
188C2×Q8C2×D2111yescharacteristic
189C2×D4—212no—
190C2×D4—212no—
191C2×D4—211no—
192C2×D4C2×D2111yescharacteristic
193C22×C4—212no—
194C22×C4—212no—
195C4⋊C4—212no—
196C4⋊C4—212no—
197C4⋊C4—212no—
198C22⋊C4—212no—
199C22⋊C4—212no—
200C22⋊C4—212no—
201C22⋊C4—212no—
202C22⋊C4C2×D2112yes—
203C22⋊C4C2×D2112yes—
204C42—211no—
205C42—212no—
206C42—211no—
207C42C2×D2111yescharacteristic
208C2×C7—42no—
209C2×C7#100411yescharacteristic
210C2×C7D3×#4112yes—
211D7—122no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3C23×D711yescharacteristic
216C2×D3—142no—
217C2×D3—142no—
218C2×D3—142no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—22no—
223Dic3—74no—
224Dic3—142no—
225Dic3—142no—
226C23—212no—
227C23C22×D2112yes—
228Q8—41no—
229Q8—841no—
230D4—41no—
231D4—424no—
232D4—841no—
233D4—842no—
234C2×C4—212no—
235C2×C4—212no—
236C2×C4—212no—
237C2×C4—424no—
238C2×C4—424no—
239C2×C4C22×D2112yes—
240C2×C4C22×D2111yescharacteristic
241C2×C4C22×D2111yescharacteristic
242C2×C4C22×D2111yescharacteristic
243C7#567011yescharacteristic, complemented, Sylow
244C2×C3—42no—
245C2×C3#654811yescharacteristic
246C2×C3D7×#4112yes—
247D3—282no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—422no—
252C22—422no—
253C22—422no—
254C22C23×D2111yescharacteristic
255C4—21no—
256C4—21no—
257C4—21no—
258C4—22no—
259C4—214no—
260C4—422no—
261C4—422no—
262C3#9036111yescharacteristic, complemented, Sylow
263C2—42no—
264C2—842no—
265C2#1062165711yescharacteristic
266C2#41×D2112yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#124C2
hypercenter#124D21
hypocenterC3×C7#555
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×C7×#124C2semidirect
2. #10621430C2semidirect
3. #10621427C2semidirect
4. #10621422C2semidirect
5. #10621368C2semidirect
6. #10621367C2semidirect
7. #10621364C2semidirect
8. #10621360C2semidirect
9. #10621355C2semidirect
32. C7×#124D3semidirect
80. C3×#124D7semidirect
149. #124D21semidirect
186. C3×C7#555semidirect
243. C7#5670semidirect
262. C3#90361semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1242
#106214302
#106214272
#106214222
#106213682
#106213672
#106213642
#106213602
#106213552
#903613
#56707