Finite Group Databasebeta

← #13058000

Subgroups — #13058000

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#10621358C211yesmaximal, characteristic, complemented
2C7×#875C211yesmaximal, characteristic, complemented
3C3×#6438C211yesmaximal, characteristic, complemented
4#10621038C212yesmaximal, complemented
5#10621037C212yesmaximal, complemented
6#10621036C212yesmaximal, complemented
7#10621024C212yesmaximal, complemented
8#10621017C212yesmaximal, complemented
9#10620888C212yesmaximal, complemented
10#90374—31nomaximal, Hall
11C22×D3×D7C2212yescomplemented
12C2×D84C2211yescharacteristic
13C2×Dic42C2211yescharacteristic, complemented
14C2×C7×D12C2211yescharacteristic
15C2×C7×Dic6C2211yescharacteristic, complemented
16C2×C3×D28C2211yescharacteristic
17C2×C3×Dic14C2211yescharacteristic, complemented
18C2×#4247C2212yes—
19C2×#4246C2212yes—
20C2×#4245C2212yes—
21C3×C42×C7C2211yescharacteristic, complemented
22#66631C2212yescomplemented
23#66631C2212yescomplemented
24C7×(C2×D3)⋊C4C2212yescomplemented
25C7×(C2×D3)⋊C4C2212yescomplemented
26C3×(C2×D7)⋊C4C2212yescomplemented
27C3×(C2×D7)⋊C4C2212yescomplemented
28#66575C2212yescomplemented
29(C2×D3)⋊Dic7C2212yescomplemented
30(C2×D7)⋊Dic3C2212yescomplemented
31Dic3×Dic7C2212yescomplemented
32C7×#124—31no—
33#6510—32no—
34#6500—32no—
35#6448—32no—
36#6448—32no—
37#6445—32no—
38#6445—32no—
39#6438—31no—
40#6438D311yescharacteristic, complemented
41#5683—71nomaximal, Hall
42C22×D21C2312yes—
43C22×D3×C7C2312yes—
44C22×C3×D7C2312yes—
45C2×D3×D7—42no—
46C2×D3×D7—42no—
47C2×D3×D7—42no—
48C2×C3×C4×C7C2311yescharacteristic
49C2×C3×C4×C7C2311yescharacteristic
50C2×C3×C4×C7C2311yescharacteristic
51C2×Dic21C2312yes—
52D84—41no—
53Dic42—41no—
54C2×C7×Dic3C2312yes—
55C7×D12—41no—
56C7×Dic6—41no—
57C2×C3×Dic7C2312yes—
58C3×D28—41no—
59C3×Dic14—41no—
60#4247—42no—
61#4246—42no—
62#4245—42no—
63C23×D7—32no—
64C2×C7×Q8—31no—
65C2×C7×D4—31no—
66C2×C22⋊D7—32no—
67C2×C22⋊D7—32no—
68C2×D28—32no—
69C2×D28—31no—
70C2×D28C2×D311yescharacteristic
71C2×Dic14—31no—
72C2×Dic14C2×D311yescharacteristic, complemented
73C7×C22⋊C4—32no—
74C7×C22⋊C4—32no—
75C42×C7C2×D311yescharacteristic, complemented
76C22⋊Dic7—32no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4C2×D312yescomplemented
82(C2×D7)⋊C4C2×D312yescomplemented
83C4×Dic7—32no—
84C3×#124—71no—
85#947—72no—
86#937—72no—
87#885—72no—
88#885—72no—
89#882—72no—
90#882—72no—
91#875—71no—
92#875D711yescharacteristic, complemented
93C22×C3×C7C2411yescharacteristic
94C2×D21—22no—
95C2×D21—22no—
96C2×D21—22no—
97C2×D3×C7—22no—
98C2×D3×C7—22no—
99C2×D3×C7—22no—
100C2×C3×D7—22no—
101C2×C3×D7—22no—
102C2×C3×D7—22no—
103D3×D7—48no—
104C3×C4×C7—21no—
105C3×C4×C7—21no—
106C3×C4×C7—21no—
107Dic21—22no—
108C3×Dic7—22no—
109C7×Dic3—22no—
110H(3,𝔽4)—211noSylow
111C23×C7—32no—
112C22×D7—32no—
113C22×D7—122no—
114C22×D7—122no—
115C22×D7—122no—
116C22×D7C22×D312yes—
117C7×Q8—121no—
118C7×D4—121no—
119C2×C4×C7—32no—
120C2×C4×C7C22×D311yescharacteristic
121C2×C4×C7C22×D311yescharacteristic
122C2×C4×C7C22×D311yescharacteristic
123C22⋊D7—122no—
124C22⋊D7—122no—
125C2×Dic7—32no—
126C2×Dic7C22×D312yes—
127D28—41no—
128D28—121no—
129D28—122no—
130Dic14—41no—
131Dic14—121no—
132C23×D3—72no—
133C2×C3×Q8—71no—
134C2×C3×D4—71no—
135C2×#67—72no—
136C2×#67—72no—
137C2×D12—72no—
138C2×D12—71no—
139C2×D12C2×D711yescharacteristic
140C2×Dic6—71no—
141C2×Dic6C2×D711yescharacteristic, complemented
142C3×C22⋊C4—72no—
143C3×C22⋊C4—72no—
144C3×C42C2×D711yescharacteristic, complemented
145#217—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
151(C2×D3)⋊C4C2×D712yescomplemented
152C4×Dic3—72no—
153C2×C3×C7H(5,𝔽2)11yescharacteristic
154C2×C3×C7H(5,𝔽2)11yescharacteristic
155C2×C3×C7H(5,𝔽2)11yescharacteristic
156D21—42no—
157C3×D7—42no—
158D3×C7—42no—
159#124—31no—
160#124—71no—
161#124—212no—
162#124—212no—
163#124—212no—
164#124—211no—
165#120—212no—
166#120—212no—
167#120—212no—
168C22×C7—62no—
169C22×C7—62no—
170C22×C7—62no—
171C22×C7C23×D311yescharacteristic
172C2×D7—22no—
173C2×D7—22no—
174C2×D7—22no—
175C2×D7—62no—
176C2×D7—62no—
177C2×D7—62no—
178C2×D7—128no—
179C4×C7—21no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7—62no—
183Dic7—22no—
184Dic7—62no—
185C23×C3—72no—
186C22×D3—72no—
187C22×D3—282no—
188C22×D3—282no—
189C22×D3—282no—
190C22×D3C22×D712yes—
191C3×Q8—281no—
192C3×D4—281no—
193C2×C3×C4—72no—
194C2×C3×C4C22×D711yescharacteristic
195C2×C3×C4C22×D711yescharacteristic
196C2×C3×C4C22×D711yescharacteristic
197#67—282no—
198#67—282no—
199C2×Dic3—72no—
200C2×Dic3C22×D712yes—
201D12—41no—
202D12—281no—
203D12—282no—
204Dic6—41no—
205Dic6—281no—
206C3×C7H(3,𝔽4)11yescharacteristic, complemented, Hall
207C24—212no—
208C2×Q8—31no—
209C2×Q8—71no—
210C2×Q8—211no—
211C2×D4—31no—
212C2×D4—71no—
213C2×D4—212no—
214C2×D4—212no—
215C2×D4—212no—
216C2×D4—211no—
217C22⋊C4—32no—
218C22⋊C4—32no—
219C22⋊C4—72no—
220C22⋊C4—72no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C42—212no—
227C42D3×D711yescharacteristic, complemented
228C2×C7—122no—
229C2×C7#100411yescharacteristic
230C2×C7#100911yescharacteristic
231C2×C7#100911yescharacteristic
232D7—42no—
233D7—122no—
234C22×C3—142no—
235C22×C3—142no—
236C22×C3—142no—
237C22×C3C23×D711yescharacteristic
238C2×D3—22no—
239C2×D3—22no—
240C2×D3—22no—
241C2×D3—142no—
242C2×D3—142no—
243C2×D3—142no—
244C2×D3—288no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—142no—
249Dic3—22no—
250Dic3—142no—
251C23—32no—
252C23—72no—
253C23—212no—
254C23—842no—
255C23—842no—
256C23—842no—
257Q8—121no—
258Q8—281no—
259Q8—841no—
260D4—121no—
261D4—281no—
262D4—841no—
263D4—842no—
264D4—842no—
265D4—842no—
266C2×C4—32no—
267C2×C4—72no—
268C2×C4—212no—
269C2×C4C2×D3×D711yescharacteristic
270C2×C4C2×D3×D711yescharacteristic
271C2×C4C2×D3×D711yescharacteristic
272C7#568311yescharacteristic, complemented, Sylow
273C2×C3—282no—
274C2×C3#654811yescharacteristic
275C2×C3#655311yescharacteristic
276C2×C3#655311yescharacteristic
277D3—42no—
278D3—282no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22—142no—
283C22—142no—
284C22—142no—
285C22—422no—
286C22—422no—
287C22—422no—
288C22—848no—
289C22C22×D3×D711yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—62no—
294C4—142no—
295C4—422no—
296C3#9037411yescharacteristic, complemented, Sylow
297C2—122no—
298C2—282no—
299C2—842no—
300C2#1062156411yescharacteristic
301C2#1062156511yescharacteristic
302C2#1062156611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C42×C7C22
hypercenterC42D3×D7
hypocenterC3×C7H(3,𝔽4)
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #10621358C2semidirect
2. C7×#875C2semidirect
3. C3×#6438C2semidirect
4. #10621038C2semidirect
5. #10621037C2semidirect
6. #10621036C2semidirect
7. #10621024C2semidirect
8. #10621017C2semidirect
9. #10620888C2semidirect
11. C22×D3×D7C22semidirect
13. C2×Dic42C22semidirect
15. C2×C7×Dic6C22semidirect
17. C2×C3×Dic14C22semidirect
21. C3×C42×C7C22semidirect
22. #66631C22semidirect
23. #66631C22semidirect
24. C7×(C2×D3)⋊C4C22semidirect
25. C7×(C2×D3)⋊C4C22semidirect
26. C3×(C2×D7)⋊C4C22semidirect
27. C3×(C2×D7)⋊C4C22semidirect
28. #66575C22semidirect
29. (C2×D3)⋊Dic7C22semidirect
30. (C2×D7)⋊Dic3C22semidirect
31. Dic3×Dic7C22semidirect
40. #6438D3semidirect
72. C2×Dic14C2×D3semidirect
75. C42×C7C2×D3semidirect
81. (C2×D7)⋊C4C2×D3semidirect
82. (C2×D7)⋊C4C2×D3semidirect
92. #875D7semidirect
141. C2×Dic6C2×D7semidirect
144. C3×C42C2×D7semidirect
150. (C2×D3)⋊C4C2×D7semidirect
151. (C2×D3)⋊C4C2×D7semidirect
206. C3×C7H(3,𝔽4)semidirect
227. C42D3×D7semidirect
272. C7#5683semidirect
296. C3#90374semidirect

Maximal subgroups

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

subgroupindex
#106213582
C7×#8752
C3×#64382
#106210382
#106210372
#106210362
#106210242
#106210172
#106208882
#903743
#56837