Finite Group Databasebeta

← #13061752

Subgroups — #13061752

Order 1344.

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

normal subgroups only

This group has 496 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#10621420C211yesmaximal, characteristic
3#10621419C211yesmaximal, characteristic, complemented
4C4×#4268C212yesmaximal, complemented
5#10621379C212yesmaximal, complemented
6#10621375C212yesmaximal, complemented
7#10621366C212yesmaximal, complemented
8C22⋊C4×D21C212yesmaximal, complemented
9#10621362C212yesmaximal, complemented
10#90350—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C22×C3×C4×C7C2211yescharacteristic, complemented
14C2×#4268C2212yes—
15C2×C4×D21C2212yes—
16C2×C4×D21C2212yes—
17C2×Dic42C2211yescharacteristic, complemented
18C3×C7×C4⋊C4C2212yescomplemented
19C3×C7×C22⋊C4C2212yes—
20C3×C7×C22⋊C4C2211yescharacteristic, complemented
21#66636C2211yescharacteristic
22#66636C2212yes—
23#66636C2212yescomplemented
24#66631C2212yes—
25#66631C2212yes—
26#66631C2212yes—
27C4⋊Dic21C2212yescomplemented
28#66629C2212yescomplemented
29#66629C2212yescomplemented
30C4×Dic21C2212yescomplemented
31C4×Dic21C2211yescharacteristic, complemented
32C7×#123D311yescharacteristic, complemented
33#6500—31no—
34#6499—31no—
35C4×C22⋊D7—32no—
36#6459—32no—
37#6455—32no—
38#6446—32no—
39D7×C22⋊C4—32no—
40#6442—32no—
41#5659—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C23×C3×C7C2311yescharacteristic
44C22×D21—24no—
45C22×D21—41no—
46C22×D21C2312yes—
47C3×C7×D4—41no—
48C2×C3×C4×C7—41no—
49C2×C3×C4×C7C2312yes—
50C2×C3×C4×C7C2311yescharacteristic
51C2×C3×C4×C7C2312yes—
52#4268—24no—
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55C2×Dic21C2312yes—
56C4×D21—24no—
57C4×D21—42no—
58Dic42—41no—
59C23×D7—31no—
60C2×C7×D4C2×D311yescharacteristic
61C22×C4×C7C2×D311yescharacteristic, complemented
62C2×C22⋊D7—32no—
63C2×C4×D7—32no—
64C2×C4×D7—32no—
65C2×Dic14—31no—
66C7×C4⋊C4C2×D312yescomplemented
67C7×C22⋊C4C2×D312yes—
68C7×C22⋊C4C2×D311yescharacteristic, complemented
69C22⋊Dic7—31no—
70C22⋊Dic7—32no—
71C22⋊Dic7—32no—
72(C2×D7)⋊C4—32no—
73(C2×D7)⋊C4—32no—
74(C2×D7)⋊C4—32no—
75C4⋊Dic7—32no—
76Dic7⋊C4—32no—
77Dic7⋊C4—32no—
78C4×Dic7—32no—
79C4×Dic7—31no—
80C3×#123D711yescharacteristic, complemented
81#937—71no—
82#936—71no—
83C4×#67—72no—
84#896—72no—
85#892—72no—
86#883—72no—
87D3×C22⋊C4—72no—
88#879—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—44no—
97C2×D21#4114yes—
98C3×C4×C7—22no—
99C3×C4×C7—21no—
100C3×C4×C7—22no—
101Dic21—22no—
102Dic21—22no—
103Dic21—22no—
104#551—211noSylow
105C23×C7C22×D311yescharacteristic
106C23×C7C22×D311yescharacteristic
107C22×D7—32no—
108C22×D7—64no—
109C22×D7—121no—
110C7×D4—41no—
111C2×C4×C7—41no—
112C2×C4×C7C22×D312yes—
113C2×C4×C7C22×D311yescharacteristic
114C2×C4×C7C22×D312yes—
115C22⋊D7—64no—
116C2×Dic7—32no—
117C2×Dic7—32no—
118C2×Dic7—32no—
119C4×D7—64no—
120C4×D7—122no—
121Dic14—121no—
122C23×D3—71no—
123C2×C3×D4C2×D711yescharacteristic
124C22×C3×C4C2×D711yescharacteristic, complemented
125C2×#67—72no—
126C2×C4×D3—72no—
127C2×C4×D3—72no—
128C2×Dic6—71no—
129C3×C4⋊C4C2×D712yescomplemented
130C3×C22⋊C4C2×D712yes—
131C3×C22⋊C4C2×D711yescharacteristic, complemented
132#217—71no—
133#217—72no—
134#217—72no—
135(C2×D3)⋊C4—72no—
136(C2×D3)⋊C4—72no—
137(C2×D3)⋊C4—72no—
138C4⋊Dic3—72no—
139Dic3⋊C4—72no—
140Dic3⋊C4—72no—
141C4×Dic3—72no—
142C4×Dic3—71no—
143C2×C3×C7—41no—
144C2×C3×C7—41no—
145C2×C3×C7C2×#4112yes—
146C2×C3×C7H(5,𝔽2)11yescharacteristic
147D21—24no—
148#126—212no—
149#124—211no—
150#123—212no—
151#123D2111yescharacteristic, complemented
152C22⋊Q8—212no—
153#120—211no—
154C4×D4—212no—
155#117—212no—
156C2×C22⋊C4—212no—
157C22×C7—21no—
158C22×C7—21no—
159C22×C7—22no—
160C22×C7—22no—
161C22×C7C23×D311yescharacteristic
162C2×D7—34no—
163C2×D7—62no—
164C2×D7—62no—
165C2×D7—124no—
166C4×C7—21no—
167C4×C7—22no—
168C4×C7—22no—
169Dic7—62no—
170Dic7—62no—
171Dic7—62no—
172C23×C3C22×D711yescharacteristic
173C23×C3C22×D711yescharacteristic
174C22×D3—72no—
175C22×D3—144no—
176C22×D3—281no—
177C3×D4—41no—
178C2×C3×C4—41no—
179C2×C3×C4C22×D712yes—
180C2×C3×C4C22×D711yescharacteristic
181C2×C3×C4C22×D712yes—
182#67—144no—
183C2×Dic3—72no—
184C2×Dic3—72no—
185C2×Dic3—72no—
186C4×D3—144no—
187C4×D3—282no—
188Dic6—281no—
189C3×C7#55111yescharacteristic, complemented, Hall
190C24—211no—
191C2×Q8—211no—
192C2×D4—212no—
193C2×D4C2×D2111yescharacteristic
194C22×C4—212no—
195C22×C4—212no—
196C22×C4C2×D2111yescharacteristic, complemented
197C4⋊C4—212no—
198C4⋊C4—212no—
199C4⋊C4—212no—
200C4⋊C4C2×D2112yescomplemented
201C22⋊C4—212no—
202C22⋊C4—212no—
203C22⋊C4—211no—
204C22⋊C4—212no—
205C22⋊C4—212no—
206C22⋊C4—212no—
207C22⋊C4C2×D2112yes—
208C22⋊C4C2×D2111yescharacteristic, complemented
209C42—211no—
210C42—212no—
211C2×C7—41no—
212C2×C7—41no—
213C2×C7#100411yescharacteristic
214C2×C7D3×#4112yes—
215D7—64no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—22no—
219C22×C3—22no—
220C22×C3C23×D711yescharacteristic
221C2×D3—74no—
222C2×D3—142no—
223C2×D3—142no—
224C2×D3—284no—
225C3×C4—21no—
226C3×C4—22no—
227C3×C4—22no—
228Dic3—142no—
229Dic3—142no—
230Dic3—142no—
231C23—212no—
232C23—424no—
233C23—841no—
234C23C22×D2111yescharacteristic
235C23C22×D2111yescharacteristic
236Q8—841no—
237D4—41no—
238D4—424no—
239C2×C4—41no—
240C2×C4—212no—
241C2×C4—212no—
242C2×C4—212no—
243C2×C4—424no—
244C2×C4—842no—
245C2×C4C22×D2111yescharacteristic
246C2×C4C22×D2112yes—
247C2×C4C22×D2112yes—
248C7#565911yescharacteristic, complemented, Sylow
249C2×C3—41no—
250C2×C3—41no—
251C2×C3#654811yescharacteristic
252C2×C3D7×#4112yes—
253D3—144no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—214no—
259C22—422no—
260C22—422no—
261C22—844no—
262C22C23×D2111yescharacteristic
263C4—22no—
264C4—21no—
265C4—22no—
266C4—422no—
267C4—422no—
268C4—422no—
269C3#9035011yescharacteristic, complemented, Sylow
270C2—41no—
271C2—41no—
272C2—424no—
273C2#1062165711yescharacteristic
274C2#41×D2112yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#123C2
prime radical#10621420C2
hypercenter#123D21
hypocenterC3×C7#551
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. #10621419C2semidirect
4. C4×#4268C2semidirect
5. #10621379C2semidirect
6. #10621375C2semidirect
7. #10621366C2semidirect
8. C22⋊C4×D21C2semidirect
9. #10621362C2semidirect
13. C22×C3×C4×C7C22semidirect
17. C2×Dic42C22semidirect
18. C3×C7×C4⋊C4C22semidirect
20. C3×C7×C22⋊C4C22semidirect
23. #66636C22semidirect
27. C4⋊Dic21C22semidirect
28. #66629C22semidirect
29. #66629C22semidirect
30. C4×Dic21C22semidirect
31. C4×Dic21C22semidirect
32. C7×#123D3semidirect
61. C22×C4×C7C2×D3semidirect
66. C7×C4⋊C4C2×D3semidirect
68. C7×C22⋊C4C2×D3semidirect
80. C3×#123D7semidirect
124. C22×C3×C4C2×D7semidirect
129. C3×C4⋊C4C2×D7semidirect
131. C3×C22⋊C4C2×D7semidirect
151. #123D21semidirect
189. C3×C7#551semidirect
196. C22×C4C2×D21semidirect
200. C4⋊C4C2×D21semidirect
208. C22⋊C4C2×D21semidirect
248. C7#5659semidirect
269. C3#90350semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1232
#106214202
#106214192
C4×#42682
#106213792
#106213752
#106213662
C22⋊C4×D212
#106213622
#903503
#56597