Finite Group Databasebeta

← #13060198

Subgroups — #13060198

Order 1344.

520 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 120 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#10621420C211yesmaximal, characteristic, complemented
2C7×#937C211yesmaximal, characteristic, complemented
3C3×#6499C211yesmaximal, characteristic, complemented
4#10621135C212yesmaximal, complemented
5#10621132C212yesmaximal, complemented
6#10621088C212yesmaximal, complemented
7#10621087C212yesmaximal, complemented
8#10620912C212yesmaximal, complemented
9#10620888C212yesmaximal, complemented
10#90285—31nomaximal, Hall
11C23×D21C2211yescharacteristic, complemented
12C23×D3×C7C2211yescharacteristic, complemented
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C2×C7×#67C2212yes—
16C2×C3×Dic14C2211yescharacteristic, complemented
17C2×#4248C2212yescomplemented
18C2×#4247C2212yes—
19C2×#4247C2212yes—
20#66636C2211yescharacteristic, complemented
21#66631C2212yescomplemented
22C7×#217C2211yescharacteristic, complemented
23C7×(C2×D3)⋊C4C2212yescomplemented
24C3×C22⋊Dic7C2212yescomplemented
25C3×C22⋊Dic7C2212yescomplemented
26C3×C4×Dic7C2211yescharacteristic, complemented
27#66575C2212yescomplemented
28#66575C2212yescomplemented
29(C2×D3)⋊Dic7C2212yescomplemented
30(C2×D3)⋊Dic7C2212yescomplemented
31Dic3×Dic7C2212yescomplemented
32C7×#120—31no—
33#6516—32no—
34#6500—31no—
35#6500—32no—
36#6499—32no—
37#6499D311yescharacteristic, complemented
38#6448—32no—
39#6448—32no—
40#6448—32no—
41#5683—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C23×C3×C7C2311yescharacteristic
44C22×D21—41no—
45C22×D21—42no—
46C22×D21C2312yes—
47C22×D3×C7—41no—
48C22×D3×C7—42no—
49C22×D3×C7C2312yes—
50C3×C7×D4—41no—
51C2×C3×C4×C7C2311yescharacteristic
52#4268—42no—
53C2×Dic21C2312yes—
54C7×#67—42no—
55C2×C7×Dic3C2312yes—
56C2×C3×Dic7C2312yes—
57C2×C3×Dic7C2312yes—
58C3×Dic14—41no—
59#4248—42no—
60#4247—42no—
61#4247—42no—
62C24×C7—31no—
63C23×D7—31no—
64C2×C7×D4—32no—
65C2×C7×D4C2×D311yescharacteristic
66C2×C22⋊D7—32no—
67C2×C22⋊D7—32no—
68C2×C22⋊D7—32no—
69C2×Dic14—32no—
70C2×Dic14C2×D311yescharacteristic, complemented
71C7×C22⋊C4—31no—
72C7×C22⋊C4—32no—
73C22⋊Dic7—31no—
74C22⋊Dic7—32no—
75C22⋊Dic7—32no—
76C22⋊Dic7C2×D312yescomplemented
77C22⋊Dic7C2×D312yescomplemented
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80(C2×D7)⋊C4—32no—
81C4×Dic7—32no—
82C4×Dic7C2×D311yescharacteristic, complemented
83C3×#124—71no—
84#947—72no—
85#937—71no—
86#937D711yescharacteristic, complemented
87#885—72no—
88#885—72no—
89#882—72no—
90#882—72no—
91#875—72no—
92C22×C3×C7—22no—
93C22×C3×C7—22no—
94C22×C3×C7—21no—
95C22×C3×C7—21no—
96C22×C3×C7C2411yescharacteristic
97C2×D21—22no—
98C2×D21—22no—
99C2×D21—22no—
100C2×D21—44no—
101C2×D3×C7—22no—
102C2×D3×C7—22no—
103C2×D3×C7—22no—
104C2×D3×C7—44no—
105C3×C4×C7—21no—
106Dic21—22no—
107C3×Dic7—22no—
108C3×Dic7—22no—
109C7×Dic3—22no—
110H(3,𝔽4)—211noSylow
111C23×C7—32no—
112C23×C7—121no—
113C23×C7—122no—
114C23×C7C22×D311yescharacteristic
115C23×C7C22×D311yescharacteristic
116C22×D7—32no—
117C22×D7—121no—
118C22×D7—122no—
119C7×D4—41no—
120C7×D4—122no—
121C2×C4×C7—32no—
122C2×C4×C7C22×D311yescharacteristic
123C22⋊D7—122no—
124C22⋊D7—122no—
125C22⋊D7—122no—
126C2×Dic7—32no—
127C2×Dic7C22×D312yes—
128C2×Dic7C22×D312yes—
129Dic14—41no—
130Dic14—122no—
131C23×D3—71no—
132C23×D3C2×D711yescharacteristic, complemented
133C2×C3×Q8—71no—
134C2×C3×D4C2×D711yescharacteristic
135C2×#67—72no—
136C2×#67C2×D712yes—
137C2×D12—72no—
138C2×D12—72no—
139C2×Dic6—72no—
140C3×C22⋊C4—72no—
141C3×C22⋊C4—72no—
142C3×C42—71no—
143#217—71no—
144#217C2×D711yescharacteristic, complemented
145(C2×D3)⋊C4—72no—
146(C2×D3)⋊C4—72no—
147(C2×D3)⋊C4—72no—
148(C2×D3)⋊C4—72no—
149(C2×D3)⋊C4—72no—
150(C2×D3)⋊C4C2×D712yescomplemented
151C4×Dic3—72no—
152C2×C3×C7—41no—
153C2×C3×C7—41no—
154C2×C3×C7H(5,𝔽2)11yescharacteristic
155C2×C3×C7H(5,𝔽2)12yes—
156D21—42no—
157D3×C7—42no—
158#124—71no—
159#124—212no—
160#124—212no—
161#124—212no—
162#124—212no—
163#120—31no—
164#120—212no—
165#120—212no—
166#120—211no—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7—22no—
170C22×C7—22no—
171C22×C7—62no—
172C22×C7—62no—
173C22×C7—62no—
174C22×C7—124no—
175C22×C7C23×D311yescharacteristic
176C2×D7—62no—
177C2×D7—62no—
178C2×D7—62no—
179C2×D7—124no—
180C4×C7—21no—
181C4×C7—62no—
182Dic7—22no—
183Dic7—22no—
184Dic7—62no—
185C23×C3C22×D711yescharacteristic
186C23×C3C22×D711yescharacteristic
187C22×D3—41no—
188C22×D3—42no—
189C22×D3—72no—
190C22×D3—281no—
191C22×D3—282no—
192C22×D3C22×D712yes—
193C3×Q8—281no—
194C3×D4—41no—
195C2×C3×C4—72no—
196C2×C3×C4—72no—
197C2×C3×C4C22×D711yescharacteristic
198#67—42no—
199#67—282no—
200C2×Dic3—72no—
201C2×Dic3C22×D712yes—
202D12—282no—
203D12—282no—
204Dic6—282no—
205C3×C7H(3,𝔽4)11yescharacteristic, complemented, Hall
206C24—31no—
207C24—211no—
208C2×Q8—71no—
209C2×Q8—212no—
210C2×D4—32no—
211C2×D4—212no—
212C2×D4—212no—
213C2×D4—212no—
214C2×D4D3×D711yescharacteristic
215C22⋊C4—31no—
216C22⋊C4—32no—
217C22⋊C4—72no—
218C22⋊C4—72no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4—211no—
224C22⋊C4—212no—
225C42—71no—
226C42—212no—
227C2×C7—41no—
228C2×C7—41no—
229C2×C7—122no—
230C2×C7#100411yescharacteristic
231C2×C7#100912yes—
232D7—122no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3C23×D711yescharacteristic
238C2×D3—22no—
239C2×D3—22no—
240C2×D3—22no—
241C2×D3—44no—
242C2×D3—142no—
243C2×D3—142no—
244C2×D3—142no—
245C2×D3—284no—
246C3×C4—21no—
247C3×C4—142no—
248C3×C4—142no—
249Dic3—22no—
250Dic3—142no—
251C23—32no—
252C23—122no—
253C23—121no—
254C23—212no—
255C23—842no—
256C23—841no—
257C23C2×D3×D711yescharacteristic
258C23C2×D3×D711yescharacteristic
259Q8—281no—
260Q8—842no—
261D4—41no—
262D4—122no—
263D4—842no—
264D4—842no—
265D4—842no—
266C2×C4—32no—
267C2×C4—72no—
268C2×C4—72no—
269C2×C4—212no—
270C2×C4C2×D3×D711yescharacteristic
271C7#568311yescharacteristic, complemented, Sylow
272C2×C3—41no—
273C2×C3—41no—
274C2×C3#654812yes—
275C2×C3#654811yescharacteristic
276D3—42no—
277D3—282no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—124no—
286C22—422no—
287C22—422no—
288C22—422no—
289C22—844no—
290C22C22×D3×D711yescharacteristic
291C4—21no—
292C4—62no—
293C4—142no—
294C4—142no—
295C4—422no—
296C3#9028511yescharacteristic, complemented, Sylow
297C2—41no—
298C2—41no—
299C2—122no—
300C2—842no—
301C2#1062157412yes—
302C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7H(3,𝔽4)
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621420C2semidirect
2. C7×#937C2semidirect
3. C3×#6499C2semidirect
4. #10621135C2semidirect
5. #10621132C2semidirect
6. #10621088C2semidirect
7. #10621087C2semidirect
8. #10620912C2semidirect
9. #10620888C2semidirect
11. C23×D21C22semidirect
12. C23×D3×C7C22semidirect
16. C2×C3×Dic14C22semidirect
17. C2×#4248C22semidirect
20. #66636C22semidirect
21. #66631C22semidirect
22. C7×#217C22semidirect
23. C7×(C2×D3)⋊C4C22semidirect
24. C3×C22⋊Dic7C22semidirect
25. C3×C22⋊Dic7C22semidirect
26. C3×C4×Dic7C22semidirect
27. #66575C22semidirect
28. #66575C22semidirect
29. (C2×D3)⋊Dic7C22semidirect
30. (C2×D3)⋊Dic7C22semidirect
31. Dic3×Dic7C22semidirect
37. #6499D3semidirect
70. C2×Dic14C2×D3semidirect
76. C22⋊Dic7C2×D3semidirect
77. C22⋊Dic7C2×D3semidirect
82. C4×Dic7C2×D3semidirect
86. #937D7semidirect
132. C23×D3C2×D7semidirect
144. #217C2×D7semidirect
150. (C2×D3)⋊C4C2×D7semidirect
205. C3×C7H(3,𝔽4)semidirect
271. C7#5683semidirect
296. C3#90285semidirect

Maximal subgroups

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

subgroupindex
#106214202
C7×#9372
C3×#64992
#106211352
#106211322
#106210882
#106210872
#106209122
#106208882
#902853
#56837