Finite Group Databasebeta

← #13060043

Subgroups — #13060043

Order 1344.

496 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 124 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
1#10621420C211yesmaximal, characteristic
2C7×#936C211yesmaximal, characteristic, complemented
3C3×#6498C211yesmaximal, characteristic, complemented
4#10621124C212yesmaximal, complemented
5#10621123C212yesmaximal, complemented
6#10621114C212yesmaximal, complemented
7#10621093C212yesmaximal, complemented
8#10620986C212yesmaximal, complemented
9#10620965C212yesmaximal, complemented
10#90370—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2212yes—
14C2×C7×Dic6C2211yescharacteristic, complemented
15C22×C3×Dic7C2211yescharacteristic, complemented
16C2×D21⋊C4C2212yes—
17C2×D21⋊C4C2212yes—
18#66636C2211yescharacteristic
19#66631C2212yes—
20C7×#217C2212yes—
21C7×#217C2212yescomplemented
22C4×C7×Dic3C2211yescharacteristic, complemented
23C3×C22⋊Dic7C2212yes—
24C3×C22⋊Dic7C2211yescharacteristic, complemented
25C3×Dic7⋊C4C2212yescomplemented
26Dic3⋊Dic7C2212yescomplemented
27#66577C2212yescomplemented
28Dic7⋊Dic3C2212yescomplemented
29#66575C2212yes—
30#66575C2212yes—
31Dic3×Dic7C2212yescomplemented
32C7×#124—31no—
33#6509—32no—
34#6500—31no—
35#6498D311yescharacteristic, complemented
36#6446—32no—
37(C22⋊D7)⋊C4—32no—
38D7×C22⋊C4—32no—
39#6442—32no—
40#6435—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×C7C2311yescharacteristic
49#4268—24no—
50C2×Dic21C2312yes—
51C2×C7×Dic3C2312yes—
52C2×C7×Dic3C2312yes—
53C7×Dic6—41no—
54C2×C3×Dic7—41no—
55C2×C3×Dic7C2312yes—
56C2×C3×Dic7C2312yes—
57D21⋊C4—24no—
58D21⋊C4—42no—
59C23×D7—31no—
60C2×C7×Q8—31no—
61C2×C7×D4C2×D311yescharacteristic
62C2×C22⋊D7—32no—
63C22×Dic7C2×D311yescharacteristic, complemented
64C2×C4×D7—32no—
65C2×C4×D7—32no—
66C7×C22⋊C4—32no—
67C7×C22⋊C4—32no—
68C42×C7—31no—
69C22⋊Dic7—31no—
70C22⋊Dic7C2×D312yes—
71C22⋊Dic7C2×D311yescharacteristic, complemented
72(C2×D7)⋊C4—32no—
73(C2×D7)⋊C4—32no—
74(C2×D7)⋊C4—32no—
75C4⋊Dic7—32no—
76Dic7⋊C4—32no—
77Dic7⋊C4—32no—
78Dic7⋊C4C2×D312yescomplemented
79C4×Dic7—32no—
80C3×#123—71no—
81#937—71no—
82#936D711yescharacteristic, complemented
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—21no—
99Dic21—22no—
100C3×Dic7—22no—
101C3×Dic7—22no—
102C7×Dic3—22no—
103C7×Dic3—22no—
104#551—211noSylow
105C23×C7C22×D311yescharacteristic
106C23×C7C22×D311yescharacteristic
107C22×D7—32no—
108C22×D7—64no—
109C22×D7—121no—
110C7×Q8—121no—
111C7×D4—41no—
112C2×C4×C7—32no—
113C2×C4×C7—32no—
114C2×C4×C7C22×D311yescharacteristic
115C22⋊D7—64no—
116C2×Dic7—32no—
117C2×Dic7—41no—
118C2×Dic7C22×D312yes—
119C2×Dic7C22×D312yes—
120C4×D7—64no—
121C4×D7—122no—
122C23×D3—71no—
123C2×C3×D4C2×D711yescharacteristic
124C22×C3×C4—71no—
125C2×#67—72no—
126C2×C4×D3—72no—
127C2×C4×D3—72no—
128C2×Dic6C2×D711yescharacteristic, complemented
129C3×C4⋊C4—72no—
130C3×C22⋊C4—72no—
131C3×C22⋊C4—71no—
132#217—71no—
133#217C2×D712yes—
134#217C2×D712yescomplemented
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×Dic3C2×D711yescharacteristic, complemented
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—31no—
150#123—71no—
151#123—212no—
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—62no—
168C4×C7—62no—
169Dic7—22no—
170Dic7—22no—
171Dic7—62no—
172C23×C3C22×D711yescharacteristic
173C23×C3C22×D711yescharacteristic
174C22×D3—72no—
175C22×D3—144no—
176C22×D3—281no—
177C3×D4—41no—
178C2×C3×C4—72no—
179C2×C3×C4—72no—
180C2×C3×C4—281no—
181C2×C3×C4C22×D711yescharacteristic
182#67—144no—
183C2×Dic3—72no—
184C2×Dic3C22×D712yes—
185C2×Dic3C22×D712yes—
186C4×D3—144no—
187C4×D3—282no—
188Dic6—41no—
189C3×C7#55111yescharacteristic, complemented, Hall
190C24—211no—
191C2×Q8—31no—
192C2×D4—212no—
193C2×D4D3×D711yescharacteristic
194C22×C4—71no—
195C22×C4—212no—
196C22×C4—212no—
197C4⋊C4—72no—
198C4⋊C4—212no—
199C4⋊C4—212no—
200C4⋊C4—212no—
201C22⋊C4—32no—
202C22⋊C4—32no—
203C22⋊C4—72no—
204C22⋊C4—71no—
205C22⋊C4—212no—
206C22⋊C4—212no—
207C22⋊C4—211no—
208C22⋊C4—212no—
209C42—31no—
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—142no—
227C3×C4—142no—
228Dic3—22no—
229Dic3—22no—
230Dic3—142no—
231C23—212no—
232C23—424no—
233C23—841no—
234C23C2×D3×D711yescharacteristic
235C23C2×D3×D711yescharacteristic
236Q8—121no—
237D4—41no—
238D4—424no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—72no—
242C2×C4—72no—
243C2×C4—212no—
244C2×C4—281no—
245C2×C4—424no—
246C2×C4—842no—
247C2×C4C2×D3×D711yescharacteristic
248C7#565911yescharacteristic, complemented, Sylow
249C2×C3—41no—
250C2×C3—41no—
251C2×C3#654811yescharacteristic
252C2×C3D7×#4112yes—
253D3—144no—
254C22—22no—
255C22—22no—
256C22—21no—
257C22—21no—
258C22—214no—
259C22—422no—
260C22—422no—
261C22—844no—
262C22C22×D3×D711yescharacteristic
263C4—21no—
264C4—62no—
265C4—62no—
266C4—142no—
267C4—142no—
268C4—422no—
269C3#9037011yescharacteristic, complemented, Sylow
270C2—41no—
271C2—41no—
272C2—424no—
273C2#1062157112yes—
274C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
prime radical#10621420C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#551
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C7×#936C2semidirect
3. C3×#6498C2semidirect
4. #10621124C2semidirect
5. #10621123C2semidirect
6. #10621114C2semidirect
7. #10621093C2semidirect
8. #10620986C2semidirect
9. #10620965C2semidirect
14. C2×C7×Dic6C22semidirect
15. C22×C3×Dic7C22semidirect
21. C7×#217C22semidirect
22. C4×C7×Dic3C22semidirect
24. C3×C22⋊Dic7C22semidirect
25. C3×Dic7⋊C4C22semidirect
26. Dic3⋊Dic7C22semidirect
27. #66577C22semidirect
28. Dic7⋊Dic3C22semidirect
31. Dic3×Dic7C22semidirect
35. #6498D3semidirect
63. C22×Dic7C2×D3semidirect
71. C22⋊Dic7C2×D3semidirect
78. Dic7⋊C4C2×D3semidirect
82. #936D7semidirect
128. C2×Dic6C2×D7semidirect
134. #217C2×D7semidirect
142. C4×Dic3C2×D7semidirect
189. C3×C7#551semidirect
248. C7#5659semidirect
269. C3#90370semidirect

Maximal subgroups

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

subgroupindex
#106214202
C7×#9362
C3×#64982
#106211242
#106211232
#106211142
#106210932
#106209862
#106209652
#903703
#56597