Finite Group Databasebeta

← #13059588

Subgroups — #13059588

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621407C211yesmaximal, characteristic, complemented
2C7×#930C211yesmaximal, characteristic
3C2×C3×C4×Dic7C211yesmaximal, characteristic, complemented
4#67×Dic7C212yesmaximal, complemented
5#10621101C212yesmaximal, complemented
6D12×Dic7C211yesmaximal, characteristic, complemented
7#10620960C212yesmaximal, complemented
8#10620950C211yesmaximal, characteristic, complemented
9#10620935C211yesmaximal, characteristic, complemented
10#10620934C212yesmaximal, complemented
11#10620895C211yesmaximal, characteristic, complemented
12#90291—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C2×Dic42C2211yescharacteristic
15C2×C7×#67C2212yes—
16C2×C7×D12C2211yescharacteristic
17C22×C3×Dic7C2211yescharacteristic
18C22×C3×Dic7C2211yescharacteristic
19C2×D3×Dic7C2212yes—
20#66636C2212yes—
21C4⋊Dic21C2211yescharacteristic
22#66629C2212yes—
23C7×(C2×D3)⋊C4C2212yes—
24C7×C4⋊Dic3C2211yescharacteristic
25C3×C4×Dic7C2211yescharacteristic
26C3×C4×Dic7C2211yescharacteristic
27C3×C4×Dic7C2211yescharacteristic
28C3×C4×Dic7C2211yescharacteristic
29Dic3⋊Dic7C2212yes—
30#66577C2212yes—
31#66577C2212yes—
32(C2×D3)⋊Dic7C2212yes—
33(C2×D3)⋊Dic7C2212yes—
34(C2×D3)⋊Dic7C2212yes—
35Dic3×Dic7C2212yes—
36C7×#121—31no—
37#6499—31no—
38#6498—32no—
39D4×Dic7—32no—
40D4×Dic7—31no—
41#6487—31no—
42C2×C4×Dic7D311yescharacteristic, complemented
43#6453—31no—
44#6450—31no—
45#6442—32no—
46#6440—32no—
47#5479—71nomaximal, Hall
48C23×C3×C7C2311yescharacteristic
49C22×D3×C7C2312yes—
50C2×C3×C4×C7—22no—
51C2×C3×C4×C7C2311yescharacteristic
52C2×C3×C4×C7C2311yescharacteristic
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55Dic42—22no—
56C7×#67—24no—
57C2×C7×Dic3C2312yes—
58C7×D12—22no—
59C2×C3×Dic7—22no—
60C2×C3×Dic7—22no—
61C2×C3×Dic7C2311yescharacteristic
62C2×C3×Dic7C2311yescharacteristic
63C2×C3×Dic7C2311yescharacteristic
64C2×C3×Dic7C2311yescharacteristic
65D3×Dic7—24no—
66C2×C7×D4—32no—
67C2×C7×D4—31no—
68C22×C4×C7C2×D311yescharacteristic
69C22×Dic7—32no—
70C22×Dic7C2×D311yescharacteristic
71C22×Dic7C2×D311yescharacteristic
72C2×Dic14—31no—
73C7×C4⋊C4—31no—
74C7×C22⋊C4—32no—
75C22⋊Dic7—32no—
76C22⋊Dic7—32no—
77C22⋊Dic7—32no—
78C22⋊Dic7—32no—
79C4⋊Dic7—31no—
80C4⋊Dic7—32no—
81Dic7⋊C4—32no—
82Dic7⋊C4—32no—
83Dic7⋊C4—32no—
84C4×Dic7—32no—
85C4×Dic7C2×D311yescharacteristic
86C4×Dic7C2×D311yescharacteristic
87C4×Dic7C2×D311yescharacteristic
88C4×Dic7C2×D311yescharacteristic
89C2×C3×C42—71no—
90#930D711yescharacteristic
91#929—72no—
92C4×#67—72no—
93#924—71no—
94#876—72no—
95#875—71no—
96C4×D12—71no—
97#872—72no—
98#870—71no—
99C4×Dic6—71no—
100C22×C3×C7—21no—
101C22×C3×C7—21no—
102C22×C3×C7#4112yes—
103C22×C3×C7C2411yescharacteristic
104C2×D3×C7—22no—
105C2×D3×C7—22no—
106C2×D3×C7—22no—
107C3×C4×C7—21no—
108C3×C4×C7#4112yes—
109Dic21—22no—
110Dic21—22no—
111C3×Dic7—21no—
112C3×Dic7—21no—
113C3×Dic7—21no—
114C3×Dic7#4112yes—
115C7×Dic3—22no—
116#529—211noSylow
117C23×C7—32no—
118C23×C7C22×D311yescharacteristic
119C7×D4—64no—
120C7×D4—62no—
121C2×C4×C7—22no—
122C2×C4×C7—32no—
123C2×C4×C7C22×D311yescharacteristic
124C2×C4×C7C22×D311yescharacteristic
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—32no—
128C2×Dic7—32no—
129C2×Dic7—64no—
130C2×Dic7C22×D311yescharacteristic
131C2×Dic7C22×D311yescharacteristic
132C2×Dic7C22×D311yescharacteristic
133C2×Dic7C22×D311yescharacteristic
134Dic14—62no—
135C22×C3×C4—71no—
136C22×C3×C4—71no—
137C22×C3×C4C2×D711yescharacteristic
138C2×#67C2×D712yes—
139C2×D12C2×D711yescharacteristic
140C2×C4×D3—72no—
141C2×Dic6—71no—
142C3×C42—71no—
143C3×C42—71no—
144C3×C42—71no—
145C3×C42—71no—
146#217—72no—
147(C2×D3)⋊C4—72no—
148(C2×D3)⋊C4—72no—
149(C2×D3)⋊C4—72no—
150(C2×D3)⋊C4C2×D712yes—
151C4⋊Dic3—71no—
152C4⋊Dic3C2×D711yescharacteristic
153Dic3⋊C4—72no—
154Dic3⋊C4—72no—
155Dic3⋊C4—72no—
156Dic3⋊C4—72no—
157C4×Dic3—72no—
158C2×C3×C7—22no—
159C2×C3×C7C2×#4111yescharacteristic
160C2×C3×C7C2×#4111yescharacteristic
161C2×C3×C7C2×#4111yescharacteristic
162D3×C7—42no—
163#126—212no—
164#125—211no—
165#124—211no—
166#123—212no—
167C22⋊Q8—211no—
168#121—31no—
169C4×Q8—211no—
170C4×D4—211no—
171C4×D4—212no—
172#117—212no—
173C2×C42—71no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—62no—
177C22×C7—62no—
178C22×C7—62no—
179C22×C7#23512yes—
180C22×C7C23×D311yescharacteristic
181C4×C7—21no—
182C4×C7—62no—
183C4×C7#23512yes—
184Dic7—21no—
185Dic7—21no—
186Dic7—21no—
187Dic7—62no—
188Dic7—62no—
189Dic7#23512yes—
190C23×C3C22×D711yescharacteristic
191C22×D3C22×D712yes—
192C2×C3×C4—22no—
193C2×C3×C4—71no—
194C2×C3×C4—71no—
195C2×C3×C4—71no—
196C2×C3×C4—71no—
197C2×C3×C4—142no—
198C2×C3×C4—142no—
199C2×C3×C4C22×D711yescharacteristic
200C2×C3×C4C22×D711yescharacteristic
201#67—24no—
202C2×Dic3—72no—
203C2×Dic3—72no—
204C2×Dic3C22×D712yes—
205D12—22no—
206C4×D3—144no—
207Dic6—142no—
208C3×C7#52911yescharacteristic, complemented, Hall
209C2×Q8—211no—
210C2×D4—32no—
211C2×D4—31no—
212C22×C4—71no—
213C22×C4—71no—
214C22×C4—212no—
215C22×C4D3×D711yescharacteristic
216C4⋊C4—31no—
217C4⋊C4—212no—
218C4⋊C4—212no—
219C4⋊C4—212no—
220C4⋊C4—212no—
221C4⋊C4—211no—
222C22⋊C4—32no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C22⋊C4—212no—
227C42—71no—
228C42—71no—
229C42—71no—
230C42—71no—
231C42—212no—
232C2×C7—22no—
233C2×C7—122no—
234C2×C7C2×#23511yescharacteristic
235C2×C7C2×#23511yescharacteristic
236C2×C7C2×#23511yescharacteristic
237C22×C3—21no—
238C22×C3—21no—
239C22×C3#115912yes—
240C22×C3C23×D711yescharacteristic
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C3×C4—21no—
245C3×C4—72no—
246C3×C4—141no—
247C3×C4—141no—
248C3×C4—141no—
249C3×C4#115912yes—
250Dic3—22no—
251Dic3—142no—
252Dic3—142no—
253C23—32no—
254C23C2×D3×D711yescharacteristic
255Q8—422no—
256D4—64no—
257D4—62no—
258C2×C4—22no—
259C2×C4—32no—
260C2×C4—71no—
261C2×C4—71no—
262C2×C4—71no—
263C2×C4—71no—
264C2×C4—142no—
265C2×C4—142no—
266C2×C4—212no—
267C2×C4—212no—
268C2×C4—424no—
269C2×C4C2×D3×D711yescharacteristic
270C2×C4C2×D3×D711yescharacteristic
271C7#547911yescharacteristic, complemented, Sylow
272C2×C3—22no—
273C2×C3C2×#115911yescharacteristic
274C2×C3C2×#115911yescharacteristic
275C2×C3D7×#4111yescharacteristic
276D3—42no—
277C22—21no—
278C22—21no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22#6667612yes—
283C22C22×D3×D711yescharacteristic
284C4—21no—
285C4—62no—
286C4—72no—
287C4—141no—
288C4—141no—
289C4—141no—
290C4—422no—
291C4—422no—
292C4#6668612yes—
293C3#9029111yescharacteristic, complemented, Sylow
294C2—22no—
295C2—122no—
296C2C2×#6667611yescharacteristic
297C2D7×#23511yescharacteristic
298C2C2×#6668611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC7×#930C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#529
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #10621407C2semidirect
3. C2×C3×C4×Dic7C2semidirect
4. #67×Dic7C2semidirect
5. #10621101C2semidirect
6. D12×Dic7C2semidirect
7. #10620960C2semidirect
8. #10620950C2semidirect
9. #10620935C2semidirect
10. #10620934C2semidirect
11. #10620895C2semidirect
42. C2×C4×Dic7D3semidirect
208. C3×C7#529semidirect
271. C7#5479semidirect
293. C3#90291semidirect

Maximal subgroups

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

subgroupindex
#106214072
C7×#9302
C2×C3×C4×Dic72
#67×Dic72
#106211012
D12×Dic72
#106209602
#106209502
#106209352
#106209342
#106208952
#902913
#54797