Finite Group Databasebeta

← #13061734

Subgroups — #13061734

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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#10621430C212yesmaximal, complemented
3#10621413C211yesmaximal, characteristic, complemented
4C2×#66631C211yesmaximal, characteristic, complemented
5#10621381C212yesmaximal, complemented
6#10621378C212yesmaximal, complemented
7#10621377C212yesmaximal, complemented
8#10621365C212yesmaximal, complemented
9#10621364C212yesmaximal, complemented
10#90335—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C2×C3×C7×Q8C2211yescharacteristic, complemented
13C22×C3×C4×C7C2211yescharacteristic
14C2×#4268C2212yes—
15C22×Dic21C2211yescharacteristic
16C2×D84C2212yes—
17C2×D84C2212yes—
18C2×D84C2211yescharacteristic, complemented
19C2×C4×D21C2212yescomplemented
20C3×C7×C4⋊C4C2211yescharacteristic, complemented
21C3×C7×C4⋊C4C2212yescomplemented
22C3×C7×C22⋊C4C2212yes—
23#66631C2212yes—
24#66631C2212yes—
25#66631C2212yes—
26#66631C2212yes—
27#66631C2212yescomplemented
28#66631C2212yescomplemented
29C4⋊Dic21C2211yescharacteristic, complemented
30#66629C2212yescomplemented
31C4×Dic21C2212yescomplemented
32C7×C22⋊Q8D311yescharacteristic, complemented
33#6510—32no—
34#6493—31no—
35C2×(C2×D7)⋊C4—31no—
36#6461—32no—
37#6458—32no—
38#6457—32no—
39#6445—32no—
40(C22⋊D7)⋊C4—32no—
41#5644—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C22×D21—22no—
44C22×D21—41no—
45C22×D21—41no—
46C22×D21C2312yes—
47C22×D21C2312yes—
48C3×C7×Q8—41no—
49C2×C3×C4×C7—41no—
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2312yes—
52C2×C3×C4×C7C2311yescharacteristic
53C2×C3×C4×C7C2311yescharacteristic
54#4268—24no—
55C2×Dic21—41no—
56C2×Dic21C2312yes—
57C2×Dic21C2312yes—
58D84—42no—
59D84—42no—
60D84—41no—
61C4×D21—42no—
62C23×D7—31no—
63C2×C7×Q8C2×D311yescharacteristic, complemented
64C22×C4×C7C2×D311yescharacteristic
65C2×C22⋊D7—32no—
66C22×Dic7—31no—
67C2×D28—32no—
68C2×D28—32no—
69C2×D28—31no—
70C2×C4×D7—32no—
71C7×C4⋊C4C2×D311yescharacteristic, complemented
72C7×C4⋊C4C2×D312yescomplemented
73C7×C22⋊C4C2×D312yes—
74(C2×D7)⋊C4—32no—
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80C4⋊Dic7—31no—
81Dic7⋊C4—32no—
82C4×Dic7—32no—
83C3×C22⋊Q8D711yescharacteristic, complemented
84#947—72no—
85#930—71no—
86C2×(C2×D3)⋊C4—71no—
87#898—72no—
88#895—72no—
89#894—72no—
90#882—72no—
91#67⋊C4—72no—
92C22×C3×C7—21no—
93C22×C3×C7—21no—
94C22×C3×C7#4112yes—
95C22×C3×C7C2411yescharacteristic
96C2×D21—22no—
97C2×D21—22no—
98C2×D21—22no—
99C2×D21—22no—
100C2×D21—22no—
101C2×D21—22no—
102C2×D21—44no—
103C3×C4×C7—22no—
104C3×C4×C7—21no—
105C3×C4×C7—22no—
106C3×C4×C7—21no—
107Dic21—22no—
108Dic21—22no—
109#538—211noSylow
110C23×C7C22×D311yescharacteristic
111C22×D7—32no—
112C22×D7—32no—
113C22×D7—62no—
114C22×D7—121no—
115C22×D7—121no—
116C7×Q8—41no—
117C2×C4×C7—41no—
118C2×C4×C7C22×D312yes—
119C2×C4×C7C22×D312yes—
120C2×C4×C7C22×D311yescharacteristic
121C2×C4×C7C22×D311yescharacteristic
122C22⋊D7—64no—
123C2×Dic7—32no—
124C2×Dic7—32no—
125C2×Dic7—121no—
126D28—121no—
127D28—122no—
128D28—122no—
129C4×D7—122no—
130C23×D3—71no—
131C2×C3×Q8C2×D711yescharacteristic, complemented
132C22×C3×C4C2×D711yescharacteristic
133C2×#67—72no—
134C22×Dic3—71no—
135C2×D12—72no—
136C2×D12—72no—
137C2×D12—71no—
138C2×C4×D3—72no—
139C3×C4⋊C4C2×D711yescharacteristic, complemented
140C3×C4⋊C4C2×D712yescomplemented
141C3×C22⋊C4C2×D712yes—
142(C2×D3)⋊C4—72no—
143(C2×D3)⋊C4—72no—
144(C2×D3)⋊C4—72no—
145(C2×D3)⋊C4—72no—
146(C2×D3)⋊C4—72no—
147(C2×D3)⋊C4—72no—
148C4⋊Dic3—71no—
149Dic3⋊C4—72no—
150C4×Dic3—72no—
151C2×C3×C7—22no—
152C2×C3×C7C2×#4111yescharacteristic
153C2×C3×C7H(5,𝔽2)11yescharacteristic
154C2×C3×C7H(5,𝔽2)11yescharacteristic
155D21—42no—
156D21—42no—
157#126—212no—
158#124—212no—
159#123—212no—
160C22⋊Q8D2111yescharacteristic, complemented
161#121—211no—
162#121—212no—
163#120—212no—
164C4×D4—212no—
165C2×C22⋊C4—211no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7#23912yes—
169C22×C7C23×D311yescharacteristic
170C2×D7—62no—
171C2×D7—62no—
172C2×D7—62no—
173C2×D7—62no—
174C2×D7—62no—
175C2×D7—62no—
176C2×D7—124no—
177C4×C7—21no—
178C4×C7—22no—
179C4×C7—21no—
180C4×C7—22no—
181Dic7—62no—
182Dic7—62no—
183C23×C3C22×D711yescharacteristic
184C22×D3—72no—
185C22×D3—72no—
186C22×D3—142no—
187C22×D3—281no—
188C22×D3—281no—
189C3×Q8—41no—
190C2×C3×C4—41no—
191C2×C3×C4C22×D712yes—
192C2×C3×C4C22×D712yes—
193C2×C3×C4C22×D711yescharacteristic
194C2×C3×C4C22×D711yescharacteristic
195#67—144no—
196C2×Dic3—72no—
197C2×Dic3—72no—
198C2×Dic3—281no—
199D12—281no—
200D12—282no—
201D12—282no—
202C4×D3—282no—
203C3×C7#53811yescharacteristic, complemented, Hall
204C24—211no—
205C2×Q8C2×D2111yescharacteristic, complemented
206C2×D4—212no—
207C2×D4—212no—
208C2×D4—212no—
209C2×D4—211no—
210C22×C4—211no—
211C22×C4—212no—
212C22×C4C2×D2111yescharacteristic
213C4⋊C4—212no—
214C4⋊C4—211no—
215C4⋊C4C2×D2112yescomplemented
216C4⋊C4C2×D2111yescharacteristic, complemented
217C22⋊C4—212no—
218C22⋊C4—212no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4C2×D2112yes—
224C42—212no—
225C2×C7—22no—
226C2×C7#100411yescharacteristic
227C2×C7C2×#23911yescharacteristic
228C2×C7#100911yescharacteristic
229D7—122no—
230D7—122no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3#116112yes—
234C22×C3C23×D711yescharacteristic
235C2×D3—142no—
236C2×D3—142no—
237C2×D3—142no—
238C2×D3—142no—
239C2×D3—142no—
240C2×D3—142no—
241C2×D3—284no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—21no—
245C3×C4—22no—
246Dic3—142no—
247Dic3—142no—
248C23—212no—
249C23—212no—
250C23—422no—
251C23—841no—
252C23—841no—
253C23C22×D2111yescharacteristic
254Q8—41no—
255D4—424no—
256D4—841no—
257D4—842no—
258D4—842no—
259C2×C4—41no—
260C2×C4—212no—
261C2×C4—212no—
262C2×C4—842no—
263C2×C4—841no—
264C2×C4C22×D2112yes—
265C2×C4C22×D2111yescharacteristic
266C2×C4C22×D2111yescharacteristic
267C2×C4C22×D2112yes—
268C7#564411yescharacteristic, complemented, Sylow
269C2×C3—22no—
270C2×C3#654811yescharacteristic
271C2×C3C2×#116111yescharacteristic
272C2×C3#655311yescharacteristic
273D3—282no—
274D3—282no—
275C22—21no—
276C22—21no—
277C22—422no—
278C22—422no—
279C22—422no—
280C22—422no—
281C22—422no—
282C22—422no—
283C22—844no—
284C22#6673212yes—
285C22C23×D2111yescharacteristic
286C4—21no—
287C4—21no—
288C4—22no—
289C4—22no—
290C4—422no—
291C4—422no—
292C3#9033511yescharacteristic, complemented, Sylow
293C2—22no—
294C2—842no—
295C2—842no—
296C2#1062165711yescharacteristic
297C2C2×#6673211yescharacteristic
298C2#1062166211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊Q8C2
hypercenterC22⋊Q8D21
hypocenterC3×C7#538
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C7×C22⋊Q8C2semidirect
2. #10621430C2semidirect
3. #10621413C2semidirect
4. C2×#66631C2semidirect
5. #10621381C2semidirect
6. #10621378C2semidirect
7. #10621377C2semidirect
8. #10621365C2semidirect
9. #10621364C2semidirect
12. C2×C3×C7×Q8C22semidirect
18. C2×D84C22semidirect
19. C2×C4×D21C22semidirect
20. C3×C7×C4⋊C4C22semidirect
21. C3×C7×C4⋊C4C22semidirect
27. #66631C22semidirect
28. #66631C22semidirect
29. C4⋊Dic21C22semidirect
30. #66629C22semidirect
31. C4×Dic21C22semidirect
32. C7×C22⋊Q8D3semidirect
63. C2×C7×Q8C2×D3semidirect
71. C7×C4⋊C4C2×D3semidirect
72. C7×C4⋊C4C2×D3semidirect
83. C3×C22⋊Q8D7semidirect
131. C2×C3×Q8C2×D7semidirect
139. C3×C4⋊C4C2×D7semidirect
140. C3×C4⋊C4C2×D7semidirect
160. C22⋊Q8D21semidirect
203. C3×C7#538semidirect
205. C2×Q8C2×D21semidirect
215. C4⋊C4C2×D21semidirect
216. C4⋊C4C2×D21semidirect
268. C7#5644semidirect
292. C3#90335semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
#106214302
#106214132
C2×#666312
#106213812
#106213782
#106213772
#106213652
#106213642
#903353
#56447