Finite Group Databasebeta

← #13061584

Subgroups — #13061584

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
1C2×C3×C7×C22⋊C4C211yesmaximal, characteristic, complemented
2#10621436C212yesmaximal, complemented
3#10621413C211yesmaximal, characteristic, complemented
4C4×#4268C212yesmaximal, complemented
5#10621407C211yesmaximal, characteristic, complemented
6#10621368C212yesmaximal, complemented
7#10621367C212yesmaximal, complemented
8#10621366C212yesmaximal, complemented
9#10621362C212yesmaximal, complemented
10#90185—31nomaximal, Hall
11C24×C3×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×C3×C4×C7C2211yescharacteristic
14C2×#4268C2212yes—
15C2×#4268C2212yes—
16C2×#4268C2212yes—
17C2×D84C2211yescharacteristic, complemented
18C2×C4×D21C2212yescomplemented
19C2×Dic42C2211yescharacteristic, complemented
20C3×C7×C22⋊C4C2212yes—
21C3×C7×C22⋊C4C2212yes—
22#66636C2212yes—
23#66636C2212yes—
24#66636C2212yes—
25#66631C2212yescomplemented
26#66631C2212yescomplemented
27C4⋊Dic21C2211yescharacteristic, complemented
28C4⋊Dic21C2211yescharacteristic, complemented
29#66629C2212yescomplemented
30#66629C2212yescomplemented
31C4×Dic21C2212yescomplemented
32C2×C7×C22⋊C4D311yescharacteristic, complemented
33#6516—32no—
34#6493—31no—
35C4×C22⋊D7—32no—
36#6487—31no—
37#6448—32no—
38#6447—32no—
39#6446—32no—
40#6442—32no—
41#5494—71nomaximal, Hall
42C23×C3×C7—22no—
43C23×C3×C7—41no—
44C23×C3×C7—41no—
45C23×C3×C7C2312yes—
46C23×C3×C7C2311yescharacteristic
47C22×D21C2312yes—
48C2×C3×C4×C7—41no—
49C2×C3×C4×C7—41no—
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×C3×C4×C7C2311yescharacteristic
53#4268—24no—
54#4268—42no—
55#4268—42no—
56C2×Dic21C2312yes—
57C2×Dic21C2312yes—
58C2×Dic21C2312yes—
59D84—41no—
60C4×D21—42no—
61Dic42—41no—
62C24×C7C2×D311yescharacteristic
63C22×C4×C7C2×D311yescharacteristic
64C22×C4×C7C2×D311yescharacteristic
65C2×C22⋊D7—32no—
66C2×C22⋊D7—32no—
67C2×C22⋊D7—32no—
68C2×D28—31no—
69C2×C4×D7—32no—
70C2×Dic14—31no—
71C7×C22⋊C4C2×D312yes—
72C7×C22⋊C4C2×D312yes—
73C22⋊Dic7—32no—
74C22⋊Dic7—32no—
75C22⋊Dic7—32no—
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—32no—
78C4⋊Dic7—31no—
79C4⋊Dic7—31no—
80Dic7⋊C4—32no—
81Dic7⋊C4—32no—
82C4×Dic7—32no—
83C2×C3×C22⋊C4D711yescharacteristic, complemented
84#953—72no—
85#930—71no—
86C4×#67—72no—
87#924—71no—
88#885—72no—
89#884—72no—
90#883—72no—
91#879—72no—
92C22×C3×C7—21no—
93C22×C3×C7—21no—
94C22×C3×C7—22no—
95C22×C3×C7—22no—
96C22×C3×C7—22no—
97C22×C3×C7—44no—
98C22×C3×C7#4112yes—
99C22×C3×C7C2411yescharacteristic
100C2×D21—22no—
101C2×D21—22no—
102C2×D21—22no—
103C3×C4×C7—21no—
104C3×C4×C7—22no—
105C3×C4×C7—21no—
106Dic21—22no—
107Dic21—22no—
108Dic21—22no—
109#538—211noSylow
110C23×C7—22no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7C22×D312yes—
114C23×C7C22×D311yescharacteristic
115C22×D7—32no—
116C2×C4×C7—41no—
117C2×C4×C7—41no—
118C2×C4×C7C22×D312yes—
119C2×C4×C7C22×D311yescharacteristic
120C2×C4×C7C22×D311yescharacteristic
121C22⋊D7—64no—
122C22⋊D7—122no—
123C22⋊D7—122no—
124C2×Dic7—32no—
125C2×Dic7—32no—
126C2×Dic7—32no—
127D28—121no—
128C4×D7—122no—
129Dic14—121no—
130C24×C3C2×D711yescharacteristic
131C22×C3×C4C2×D711yescharacteristic
132C22×C3×C4C2×D711yescharacteristic
133C2×#67—72no—
134C2×#67—72no—
135C2×#67—72no—
136C2×D12—71no—
137C2×C4×D3—72no—
138C2×Dic6—71no—
139C3×C22⋊C4C2×D712yes—
140C3×C22⋊C4C2×D712yes—
141#217—72no—
142#217—72no—
143#217—72no—
144(C2×D3)⋊C4—72no—
145(C2×D3)⋊C4—72no—
146C4⋊Dic3—71no—
147C4⋊Dic3—71no—
148Dic3⋊C4—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—211no—
162#121—212no—
163#120—212no—
164C4×D4—212no—
165C2×C22⋊C4D2111yescharacteristic, complemented
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7—22no—
171C22×C7—44no—
172C22×C7#23512yes—
173C22×C7C23×D311yescharacteristic
174C2×D7—62no—
175C2×D7—62no—
176C2×D7—62no—
177C4×C7—21no—
178C4×C7—22no—
179C4×C7—21no—
180Dic7—62no—
181Dic7—62no—
182Dic7—62no—
183C23×C3—22no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3C22×D712yes—
187C23×C3C22×D711yescharacteristic
188C22×D3—72no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4C22×D712yes—
192C2×C3×C4C22×D711yescharacteristic
193C2×C3×C4C22×D711yescharacteristic
194#67—144no—
195#67—282no—
196#67—282no—
197C2×Dic3—72no—
198C2×Dic3—72no—
199C2×Dic3—72no—
200D12—281no—
201C4×D3—282no—
202Dic6—281no—
203C3×C7#53811yescharacteristic, complemented, Hall
204C24C2×D2111yescharacteristic
205C2×Q8—211no—
206C2×D4—212no—
207C2×D4—212no—
208C2×D4—212no—
209C2×D4—211no—
210C22×C4—212no—
211C22×C4C2×D2111yescharacteristic
212C22×C4C2×D2111yescharacteristic
213C4⋊C4—212no—
214C4⋊C4—212no—
215C4⋊C4—211no—
216C4⋊C4—211no—
217C22⋊C4—212no—
218C22⋊C4—212no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—212no—
222C22⋊C4C2×D2112yes—
223C22⋊C4C2×D2112yes—
224C42—212no—
225C2×C7—22no—
226C2×C7—42no—
227C2×C7C2×#23511yescharacteristic
228C2×C7#100411yescharacteristic
229C2×C7#100411yescharacteristic
230D7—122no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—44no—
237C22×C3#115712yes—
238C22×C3C23×D711yescharacteristic
239C2×D3—142no—
240C2×D3—142no—
241C2×D3—142no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—21no—
245Dic3—142no—
246Dic3—142no—
247Dic3—142no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—212no—
252C23C22×D2112yes—
253C23C22×D2111yescharacteristic
254Q8—841no—
255D4—424no—
256D4—841no—
257D4—842no—
258D4—842no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—212no—
262C2×C4—212no—
263C2×C4—212no—
264C2×C4—842no—
265C2×C4C22×D2112yes—
266C2×C4C22×D2111yescharacteristic
267C2×C4C22×D2111yescharacteristic
268C7#549411yescharacteristic, complemented, Sylow
269C2×C3—22no—
270C2×C3—42no—
271C2×C3C2×#115711yescharacteristic
272C2×C3#654811yescharacteristic
273C2×C3#654811yescharacteristic
274D3—282no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—44no—
281C22—422no—
282C22—422no—
283C22—422no—
284C22#6672812yes—
285C22C23×D2111yescharacteristic
286C4—22no—
287C4—21no—
288C4—21no—
289C4—422no—
290C4—422no—
291C4—422no—
292C3#9018511yescharacteristic, complemented, Sylow
293C2—22no—
294C2—42no—
295C2—842no—
296C2C2×#6672811yescharacteristic
297C2#1062165711yescharacteristic
298C2#1062165711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×C22⋊C4C2
hypercenterC2×C22⋊C4D21
hypocenterC3×C7#538
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C7×C22⋊C4C2semidirect
2. #10621436C2semidirect
3. #10621413C2semidirect
4. C4×#4268C2semidirect
5. #10621407C2semidirect
6. #10621368C2semidirect
7. #10621367C2semidirect
8. #10621366C2semidirect
9. #10621362C2semidirect
17. C2×D84C22semidirect
18. C2×C4×D21C22semidirect
19. C2×Dic42C22semidirect
25. #66631C22semidirect
26. #66631C22semidirect
27. C4⋊Dic21C22semidirect
28. C4⋊Dic21C22semidirect
29. #66629C22semidirect
30. #66629C22semidirect
31. C4×Dic21C22semidirect
32. C2×C7×C22⋊C4D3semidirect
83. C2×C3×C22⋊C4D7semidirect
165. C2×C22⋊C4D21semidirect
203. C3×C7#538semidirect
268. C7#5494semidirect
292. C3#90185semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×C22⋊C42
#106214362
#106214132
C4×#42682
#106214072
#106213682
#106213672
#106213662
#106213622
#901853
#54947