Finite Group Databasebeta

← #13061707

Subgroups — #13061707

Order 1344.

500 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 139 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×#121C211yesmaximal, characteristic, complemented
2#10621422C212yesmaximal, complemented
3#10621420C212yesmaximal, complemented
4D4×Dic21C212yesmaximal, complemented
5C2×#66631C211yesmaximal, characteristic, complemented
6#10621407C211yesmaximal, characteristic, complemented
7#10621381C212yesmaximal, complemented
8#10621369C212yesmaximal, complemented
9#10621368C212yesmaximal, complemented
10#90308—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C2×C3×C7×D4C2212yes—
13C2×C3×C7×D4C2211yescharacteristic, complemented
14C22×C3×C4×C7C2211yescharacteristic
15C2×#4268C2212yes—
16C2×#4268C2212yes—
17C22×Dic21C2211yescharacteristic
18C22×Dic21C2212yescomplemented
19C2×Dic42C2211yescharacteristic, complemented
20C3×C7×C4⋊C4C2211yescharacteristic, complemented
21C3×C7×C22⋊C4C2212yescomplemented
22#66636C2212yes—
23#66636C2212yescomplemented
24#66631C2212yes—
25#66631C2212yes—
26#66631C2212yes—
27#66631C2212yes—
28C4⋊Dic21C2211yescharacteristic, complemented
29C4⋊Dic21C2212yescomplemented
30#66629C2212yescomplemented
31C4×Dic21C2212yescomplemented
32C7×#121D311yescharacteristic, complemented
33#6502—32no—
34#6500—32no—
35D4×Dic7—32no—
36C2×(C2×D7)⋊C4—31no—
37#6487—31no—
38#6461—32no—
39#6449—32no—
40#6448—32no—
41#5617—71nomaximal, Hall
42C23×C3×C7C2312yes—
43C23×C3×C7C2311yescharacteristic
44C22×D21—22no—
45C22×D21—41no—
46C22×D21—41no—
47C22×D21C2312yes—
48C3×C7×D4—24no—
49C3×C7×D4—41no—
50C2×C3×C4×C7—41no—
51C2×C3×C4×C7C2312yes—
52C2×C3×C4×C7C2311yescharacteristic
53C2×C3×C4×C7C2311yescharacteristic
54#4268—42no—
55#4268—42no—
56C2×Dic21—41no—
57C2×Dic21—42no—
58C2×Dic21C2312yes—
59C2×Dic21C2312yes—
60C2×Dic21C2312yes—
61Dic42—41no—
62C23×D7—31no—
63C2×C7×D4C2×D312yes—
64C2×C7×D4C2×D311yescharacteristic, complemented
65C22×C4×C7C2×D311yescharacteristic
66C2×C22⋊D7—32no—
67C2×C22⋊D7—32no—
68C22×Dic7—32no—
69C22×Dic7—31no—
70C2×Dic14—31no—
71C7×C4⋊C4C2×D311yescharacteristic, complemented
72C7×C22⋊C4C2×D312yescomplemented
73C22⋊Dic7—32no—
74C22⋊Dic7—32no—
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79C4⋊Dic7—31no—
80C4⋊Dic7—32no—
81Dic7⋊C4—32no—
82C4×Dic7—32no—
83C3×#121D711yescharacteristic, complemented
84#939—72no—
85#937—72no—
86D4×Dic3—72no—
87C2×(C2×D3)⋊C4—71no—
88#924—71no—
89#898—72no—
90#886—72no—
91#885—72no—
92C22×C3×C7—21no—
93C22×C3×C7—21no—
94C22×C3×C7—22no—
95C22×C3×C7—22no—
96C22×C3×C7—22no—
97C22×C3×C7#4112yes—
98C22×C3×C7C2411yescharacteristic
99C2×D21—22no—
100C2×D21—22no—
101C2×D21—22no—
102C2×D21—44no—
103C3×C4×C7—22no—
104C3×C4×C7—21no—
105C3×C4×C7—21no—
106Dic21—22no—
107Dic21—22no—
108Dic21—22no—
109#538—211noSylow
110C23×C7C22×D312yes—
111C23×C7C22×D311yescharacteristic
112C22×D7—32no—
113C22×D7—62no—
114C22×D7—121no—
115C22×D7—121no—
116C7×D4—24no—
117C7×D4—41no—
118C2×C4×C7—41no—
119C2×C4×C7C22×D312yes—
120C2×C4×C7C22×D311yescharacteristic
121C2×C4×C7C22×D311yescharacteristic
122C22⋊D7—122no—
123C22⋊D7—122no—
124C2×Dic7—32no—
125C2×Dic7—32no—
126C2×Dic7—32no—
127C2×Dic7—121no—
128C2×Dic7—122no—
129Dic14—121no—
130C23×D3—71no—
131C2×C3×D4C2×D712yes—
132C2×C3×D4C2×D711yescharacteristic, complemented
133C22×C3×C4C2×D711yescharacteristic
134C2×#67—72no—
135C2×#67—72no—
136C22×Dic3—72no—
137C22×Dic3—71no—
138C2×Dic6—71no—
139C3×C4⋊C4C2×D711yescharacteristic, complemented
140C3×C22⋊C4C2×D712yescomplemented
141#217—72no—
142#217—72no—
143(C2×D3)⋊C4—72no—
144(C2×D3)⋊C4—72no—
145(C2×D3)⋊C4—72no—
146(C2×D3)⋊C4—72no—
147C4⋊Dic3—71no—
148C4⋊Dic3—72no—
149Dic3⋊C4—72no—
150C4×Dic3—72no—
151C2×C3×C7—22no—
152C2×C3×C7—42no—
153C2×C3×C7C2×#4111yescharacteristic
154C2×C3×C7H(5,𝔽2)11yescharacteristic
155C2×C3×C7H(5,𝔽2)11yescharacteristic
156D21—42no—
157#126—212no—
158#124—212no—
159#123—212no—
160C22⋊Q8—211no—
161#121—212no—
162#121D2111yescharacteristic, complemented
163#120—212no—
164C4×D4—212no—
165C2×C22⋊C4—211no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7—22no—
171C22×C7#23712yes—
172C22×C7C23×D311yescharacteristic
173C2×D7—62no—
174C2×D7—62no—
175C2×D7—62no—
176C2×D7—124no—
177C4×C7—22no—
178C4×C7—21no—
179C4×C7—21no—
180Dic7—62no—
181Dic7—62no—
182Dic7—62no—
183C23×C3C22×D712yes—
184C23×C3C22×D711yescharacteristic
185C22×D3—72no—
186C22×D3—142no—
187C22×D3—281no—
188C22×D3—281no—
189C3×D4—24no—
190C3×D4—41no—
191C2×C3×C4—41no—
192C2×C3×C4C22×D712yes—
193C2×C3×C4C22×D711yescharacteristic
194C2×C3×C4C22×D711yescharacteristic
195#67—282no—
196#67—282no—
197C2×Dic3—72no—
198C2×Dic3—72no—
199C2×Dic3—72no—
200C2×Dic3—281no—
201C2×Dic3—282no—
202Dic6—281no—
203C3×C7#53811yescharacteristic, complemented, Hall
204C24—211no—
205C2×Q8—211no—
206C2×D4—212no—
207C2×D4—212no—
208C2×D4C2×D2112yes—
209C2×D4C2×D2111yescharacteristic, complemented
210C22×C4—211no—
211C22×C4—212no—
212C22×C4C2×D2111yescharacteristic
213C4⋊C4—212no—
214C4⋊C4—212no—
215C4⋊C4—211no—
216C4⋊C4C2×D2111yescharacteristic, complemented
217C22⋊C4—212no—
218C22⋊C4—212no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4C2×D2112yescomplemented
224C42—212no—
225C2×C7—22no—
226C2×C7—42no—
227C2×C7C2×#23711yescharacteristic
228C2×C7#100411yescharacteristic
229C2×C7#100911yescharacteristic
230D7—122no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3#115912yes—
237C22×C3C23×D711yescharacteristic
238C2×D3—142no—
239C2×D3—142no—
240C2×D3—142no—
241C2×D3—284no—
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—21no—
245Dic3—142no—
246Dic3—142no—
247Dic3—142no—
248C23—212no—
249C23—422no—
250C23—841no—
251C23—841no—
252C23C22×D2112yes—
253C23C22×D2111yescharacteristic
254Q8—841no—
255D4—24no—
256D4—41no—
257D4—842no—
258D4—842no—
259C2×C4—41no—
260C2×C4—212no—
261C2×C4—212no—
262C2×C4—212no—
263C2×C4—841no—
264C2×C4—842no—
265C2×C4C22×D2112yes—
266C2×C4C22×D2111yescharacteristic
267C2×C4C22×D2111yescharacteristic
268C7#561711yescharacteristic, complemented, Sylow
269C2×C3—22no—
270C2×C3—42no—
271C2×C3C2×#115911yescharacteristic
272C2×C3#654811yescharacteristic
273C2×C3#655311yescharacteristic
274D3—282no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—422no—
281C22—422no—
282C22—422no—
283C22—844no—
284C22#6673012yes—
285C22C23×D2111yescharacteristic
286C4—22no—
287C4—21no—
288C4—21no—
289C4—422no—
290C4—422no—
291C4—422no—
292C3#9030811yescharacteristic, complemented, Sylow
293C2—22no—
294C2—42no—
295C2—842no—
296C2C2×#6673011yescharacteristic
297C2#1062165711yescharacteristic
298C2#1062166211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#538
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C3×C7×#121C2semidirect
2. #10621422C2semidirect
3. #10621420C2semidirect
4. D4×Dic21C2semidirect
5. C2×#66631C2semidirect
6. #10621407C2semidirect
7. #10621381C2semidirect
8. #10621369C2semidirect
9. #10621368C2semidirect
13. C2×C3×C7×D4C22semidirect
18. C22×Dic21C22semidirect
19. C2×Dic42C22semidirect
20. C3×C7×C4⋊C4C22semidirect
21. C3×C7×C22⋊C4C22semidirect
23. #66636C22semidirect
28. C4⋊Dic21C22semidirect
29. C4⋊Dic21C22semidirect
30. #66629C22semidirect
31. C4×Dic21C22semidirect
32. C7×#121D3semidirect
64. C2×C7×D4C2×D3semidirect
71. C7×C4⋊C4C2×D3semidirect
72. C7×C22⋊C4C2×D3semidirect
83. C3×#121D7semidirect
132. C2×C3×D4C2×D7semidirect
139. C3×C4⋊C4C2×D7semidirect
140. C3×C22⋊C4C2×D7semidirect
162. #121D21semidirect
203. C3×C7#538semidirect
209. C2×D4C2×D21semidirect
216. C4⋊C4C2×D21semidirect
223. C22⋊C4C2×D21semidirect
268. C7#5617semidirect
292. C3#90308semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
#106214222
#106214202
D4×Dic212
C2×#666312
#106214072
#106213812
#106213692
#106213682
#903083
#56177