Finite Group Databasebeta

← #13058655

Subgroups — #13058655

Order 1344.

468 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 136 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#10621373C211yesmaximal, characteristic, complemented
2C7×#893C211yesmaximal, characteristic, complemented
3C3×#6460C211yesmaximal, characteristic, complemented
4#10621011C211yesmaximal, characteristic
5#10621003C212yesmaximal, complemented
6#10620976C212yesmaximal, complemented
7C4×Dic3×D7C211yesmaximal, characteristic, complemented
8#10620932C211yesmaximal, characteristic, complemented
9#10620903C212yesmaximal, complemented
10#10620900C212yesmaximal, complemented
11Dic3×Dic14C211yesmaximal, characteristic, complemented
12#90248—31nomaximal, Hall
13C2×C7×D12C2211yescharacteristic
14C2×C4×D3×C7C2212yes—
15C2×C3×C4×D7C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C2×#4245C2212yes—
18C2×Dic3×D7C2211yescharacteristic
19C2×Dic3×D7C2211yescharacteristic
20C3×C7×C4⋊C4C2211yescharacteristic
21C4⋊Dic21C2212yes—
22C4⋊Dic21C2211yescharacteristic
23#66629C2212yes—
24C4×Dic21C2211yescharacteristic
25C7×(C2×D3)⋊C4C2212yes—
26C4×C7×Dic3C2211yescharacteristic
27C3×(C2×D7)⋊C4C2212yes—
28C3×C4⋊Dic7C2212yes—
29Dic7⋊Dic3C2212yes—
30(C2×D3)⋊Dic7C2212yes—
31(C2×D3)⋊Dic7C2212yes—
32(C2×D3)⋊Dic7C2212yes—
33Dic3×Dic7C2211yescharacteristic
34Dic3×Dic7C2211yescharacteristic
35Dic3×Dic7C2212yes—
36C4×C7×D4—31no—
37#6501—31no—
38#6499—31no—
39C4×C22⋊D7—32no—
40#6489—32no—
41#6460D311yescharacteristic, complemented
42#6453—31no—
43#6446—32no—
44#6442—32no—
45C42×D7—31no—
46C4×Dic14—31no—
47#5624—71nomaximal, Hall
48C22×D3×C7C2312yes—
49C22×C3×D7C2311yescharacteristic
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2311yescharacteristic
53C2×Dic21C2311yescharacteristic
54C2×Dic21C2312yes—
55C2×C7×Dic3C2311yescharacteristic
56C2×C7×Dic3C2311yescharacteristic
57C7×D12—22no—
58C4×D3×C7—24no—
59C2×C3×Dic7C2311yescharacteristic
60C2×C3×Dic7C2312yes—
61C3×C4×D7—22no—
62C3×Dic14—22no—
63#4245—24no—
64Dic3×D7—22no—
65Dic3×D7—22no—
66C2×C7×D4—31no—
67C22×C4×C7—32no—
68C2×C22⋊D7—32no—
69C2×C4×D7—31no—
70C2×C4×D7—31no—
71C2×C4×D7C2×D311yescharacteristic
72C2×Dic14C2×D311yescharacteristic
73C7×C4⋊C4C2×D311yescharacteristic
74C7×C22⋊C4—32no—
75C42×C7—31no—
76C22⋊Dic7—32no—
77C22⋊Dic7—32no—
78C22⋊Dic7—32no—
79(C2×D7)⋊C4C2×D312yes—
80C4⋊Dic7—31no—
81C4⋊Dic7—32no—
82C4⋊Dic7C2×D312yes—
83Dic7⋊C4—32no—
84Dic7⋊C4—32no—
85C4×Dic7—31no—
86C4×Dic7—32no—
87C4×Dic7—31no—
88C4×Dic7—31no—
89C3×C22⋊Q8—71no—
90#947—71no—
91Q8×Dic3—71no—
92#930—71no—
93C2×C4×Dic3—71no—
94#898—72no—
95#893D711yescharacteristic, complemented
96#892—72no—
97#890—71no—
98#886—72no—
99#67⋊C4—72no—
100C22×C3×C7C2411yescharacteristic
101C2×D3×C7—22no—
102C2×D3×C7—22no—
103C2×D3×C7—22no—
104C2×C3×D7—21no—
105C2×C3×D7—21no—
106C2×C3×D7#4112yes—
107C3×C4×C7—22no—
108C3×C4×C7#4112yes—
109Dic21—21no—
110Dic21—21no—
111Dic21—22no—
112C3×Dic7—22no—
113C3×Dic7—21no—
114C7×Dic3—21no—
115C7×Dic3#4112yes—
116#529—211noSylow
117C23×C7—32no—
118C22×D7C22×D311yescharacteristic
119C7×D4—62no—
120C2×C4×C7—31no—
121C2×C4×C7—31no—
122C2×C4×C7—64no—
123C2×C4×C7C22×D312yes—
124C2×C4×C7C22×D311yescharacteristic
125C22⋊D7—64no—
126C2×Dic7—31no—
127C2×Dic7—31no—
128C2×Dic7—32no—
129C2×Dic7C22×D311yescharacteristic
130C2×Dic7C22×D312yes—
131C4×D7—22no—
132C4×D7—62no—
133C4×D7—62no—
134Dic14—22no—
135C2×C3×Q8—71no—
136C22×C3×C4—71no—
137C2×#67—72no—
138C22×Dic3—71no—
139C22×Dic3—71no—
140C2×D12C2×D711yescharacteristic
141C2×C4×D3C2×D712yes—
142C3×C4⋊C4—72no—
143C3×C4⋊C4C2×D711yescharacteristic
144C3×C22⋊C4—72no—
145(C2×D3)⋊C4—72no—
146(C2×D3)⋊C4—72no—
147(C2×D3)⋊C4—72no—
148(C2×D3)⋊C4C2×D712yes—
149C4⋊Dic3—72no—
150C4⋊Dic3—71no—
151C4⋊Dic3—72no—
152Dic3⋊C4—72no—
153C4×Dic3—71no—
154C4×Dic3—72no—
155C4×Dic3—71no—
156C4×Dic3—71no—
157C4×Dic3C2×D711yescharacteristic
158C2×C3×C7C2×#4111yescharacteristic
159C2×C3×C7C2×#4111yescharacteristic
160C2×C3×C7C2×#4111yescharacteristic
161C3×D7—22no—
162D3×C7—42no—
163#126—212no—
164#125—211no—
165#124—211no—
166#123—212no—
167C22⋊Q8—71no—
168#121—211no—
169C4×Q8—211no—
170C4×D4—31no—
171C4×D4—212no—
172#117—212no—
173C2×C42—211no—
174C22×C7—62no—
175C22×C7—62no—
176C22×C7—62no—
177C22×C7C23×D311yescharacteristic
178C2×D7—21no—
179C2×D7—21no—
180C2×D7#23912yes—
181C4×C7—22no—
182C4×C7—32no—
183C4×C7—61no—
184C4×C7#23712yes—
185Dic7—22no—
186Dic7—21no—
187Dic7—61no—
188Dic7—62no—
189Dic7—61no—
190C23×C3—71no—
191C22×D3C22×D712yes—
192C3×Q8—142no—
193C2×C3×C4—71no—
194C2×C3×C4—72no—
195C2×C3×C4—142no—
196C2×C3×C4C22×D712yes—
197C2×C3×C4C22×D711yescharacteristic
198#67—144no—
199C2×Dic3—71no—
200C2×Dic3—71no—
201C2×Dic3—72no—
202C2×Dic3—142no—
203C2×Dic3—142no—
204C2×Dic3C22×D711yescharacteristic
205C2×Dic3C22×D711yescharacteristic
206D12—22no—
207C4×D3—24no—
208C3×C7#52911yescharacteristic, complemented, Hall
209C2×Q8—71no—
210C2×D4—31no—
211C2×D4—212no—
212C22×C4—32no—
213C22×C4—71no—
214C22×C4—211no—
215C22×C4—211no—
216C4⋊C4—72no—
217C4⋊C4—212no—
218C4⋊C4—212no—
219C4⋊C4—212no—
220C4⋊C4—211no—
221C4⋊C4D3×D711yescharacteristic
222C22⋊C4—32no—
223C22⋊C4—72no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C22⋊C4—212no—
227C42—31no—
228C42—211no—
229C42—211no—
230C42—212no—
231C42—211no—
232C2×C7—122no—
233C2×C7C2×#23711yescharacteristic
234C2×C7C2×#23911yescharacteristic
235C2×C7D3×#4111yescharacteristic
236D7—22no—
237C22×C3—72no—
238C22×C3—141no—
239C22×C3—141no—
240C22×C3C23×D711yescharacteristic
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C3×C4—22no—
245C3×C4—141no—
246C3×C4—142no—
247C3×C4#115712yes—
248Dic3—21no—
249Dic3—141no—
250Dic3—141no—
251Dic3—142no—
252Dic3#115912yes—
253C23—32no—
254C23—71no—
255Q8—142no—
256D4—62no—
257D4—424no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—64no—
261C2×C4—71no—
262C2×C4—72no—
263C2×C4—142no—
264C2×C4—211no—
265C2×C4—211no—
266C2×C4—212no—
267C2×C4—422no—
268C2×C4—422no—
269C2×C4C2×D3×D712yes—
270C2×C4C2×D3×D711yescharacteristic
271C7#562411yescharacteristic, complemented, Sylow
272C2×C3—142no—
273C2×C3C2×#115711yescharacteristic
274C2×C3C2×#115911yescharacteristic
275C2×C3D7×#4111yescharacteristic
276D3—42no—
277C22—62no—
278C22—62no—
279C22—62no—
280C22—72no—
281C22—141no—
282C22—141no—
283C22C22×D3×D711yescharacteristic
284C4—22no—
285C4—32no—
286C4—61no—
287C4—141no—
288C4—142no—
289C4—421no—
290C4—421no—
291C4—422no—
292C4#6666912yes—
293C3#9024811yescharacteristic, complemented, Sylow
294C2—122no—
295C2—142no—
296C2C2×#6666911yescharacteristic
297C2D3×#115911yescharacteristic
298C2D7×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radical#10621011C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#529
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621373C2semidirect
2. C7×#893C2semidirect
3. C3×#6460C2semidirect
5. #10621003C2semidirect
6. #10620976C2semidirect
7. C4×Dic3×D7C2semidirect
8. #10620932C2semidirect
9. #10620903C2semidirect
10. #10620900C2semidirect
11. Dic3×Dic14C2semidirect
41. #6460D3semidirect
95. #893D7semidirect
208. C3×C7#529semidirect
271. C7#5624semidirect
293. C3#90248semidirect

Maximal subgroups

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

subgroupindex
#106213732
C7×#8932
C3×#64602
#106210112
#106210032
#106209762
C4×Dic3×D72
#106209322
#106209032
#106209002
Dic3×Dic142
#902483
#56247