Finite Group Databasebeta

← C3×#10621053 #72245841

Subgroups — C3×#10621053 #72245841

Order 2016.

326 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C32×D4×D7C211yesmaximal, characteristic, complemented
2C3×#66677C211yesmaximal, characteristic, complemented
3C3×#66632C211yesmaximal, characteristic, complemented
4C3×C7×#214C211yesmaximal, characteristic, complemented
5C3×#66563C211yesmaximal, characteristic, complemented
6C3×#66561C211yesmaximal, characteristic, complemented
7C3×#66557C211yesmaximal, characteristic, complemented
8C3×#6477—31nomaximal
9#10621053C311yesmaximal, characteristic, direct product factor
10C22×C32×D7—21no—
11C32×C7×D4C2211yescharacteristic
12C3×D84C2211yescharacteristic, complemented
13C3×C7×Dic6C2211yescharacteristic, complemented
14C32×C22⋊D7—21no—
15C32×D28C2211yescharacteristic
16C32×C4×D7C2211yescharacteristic
17C3×#4246—21no—
18C3×D21⋊C4—21no—
19C3×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
20C3×C7×C3⋊C8C2211yescharacteristic, complemented
21C3×#1161—31no—
22C3×D4×D7—21no—
23C3×D4×D7D311yescharacteristic, complemented
24C3×D4×D7C2×C311yescharacteristic, complemented
25#66677C2×C311yescharacteristic, complemented
26C3×C7×SD16—31no—
27#66632C2×C311yescharacteristic, complemented
28C7×#214C2×C311yescharacteristic, complemented
29C3×#1143—31no—
30C3×#1141—31no—
31C3×D56—31no—
32C3×#1131—31no—
33#66563C2×C311yescharacteristic, complemented
34#66561C2×C311yescharacteristic, complemented
35#66557C2×C311yescharacteristic, complemented
36C3×#932—71nomaximal, Hall
37C22×C32×C7—21no—
38C2×C3×D21—21no—
39C2×C32×D7—21no—
40C2×C32×D7—42no—
41C2×C32×D7D411yescharacteristic
42C32×C4×C7C2311yescharacteristic
43C3×C7×Dic3—21no—
44C32×Dic7D411yescharacteristic
45#6477—31noHall
46C22×C3×D7—21no—
47C22×C3×D7—21no—
48C22×C3×D7—21no—
49C22×C3×D7—21no—
50C3×C7×Q8—31no—
51C3×C7×D4—21no—
52C3×C7×D4C2×D311yescharacteristic
53C3×C7×D4C22×C311yescharacteristic
54D84C22×C311yescharacteristic, complemented
55C7×Dic6C22×C311yescharacteristic, complemented
56C3×C22⋊D7—21no—
57C3×C22⋊D7—21no—
58C3×C22⋊D7—21no—
59C3×C22⋊D7—21no—
60C3×D28—21no—
61C3×D28—31no—
62C3×D28—61no—
63C3×D28C2×D311yescharacteristic
64C3×D28C22×C311yescharacteristic
65C3×C4×D7—21no—
66C3×C4×D7—61no—
67C3×C4×D7C2×D311yescharacteristic
68C3×C4×D7C22×C311yescharacteristic
69#4246—21no—
70D21⋊C4—21no—
71C3×C7×C8—31no—
72C3⋊(C7⋊C8)C22×C311yescharacteristic, complemented
73C3×C7⋊C8—31no—
74C7×C3⋊C8C22×C311yescharacteristic, complemented
75C2×C32×D4—71no—
76C3×#235—71no—
77C3×#214—71no—
78C3×#214D711yescharacteristic, complemented
79C3×#213—71no—
80C3×#213—71no—
81C3×#208—71no—
82C2×C32×C7—41no—
83C2×C32×C7C2×D411yescharacteristic
84C3×D21—41no—
85C32×D7—21no—
86C32×D7—41no—
87#1161—31no—
88D4×D7C3×D311yescharacteristic, complemented
89C7×SD16—31no—
90#1143—31no—
91#1141—31no—
92D56—31no—
93#1131—31no—
94C3×AGL(1,ℤ/8)—211no—
95#932—71no—
96C22×C3×C7—21no—
97C22×C3×C7—21no—
98C22×C3×C7—21no—
99C22×C3×C7—21no—
100C2×D21—21no—
101C2×C3×D7—21no—
102C2×C3×D7—21no—
103C2×C3×D7—21no—
104C2×C3×D7—21no—
105C2×C3×D7—21no—
106C2×C3×D7—42no—
107C2×C3×D7—42no—
108C2×C3×D7—42no—
109C2×C3×D7—42no—
110C2×C3×D7—61no—
111C2×C3×D7#6711yescharacteristic
112C2×C3×D7C3×D411yescharacteristic
113C3×C4×C7—21no—
114C3×C4×C7—61no—
115C3×C4×C7C22×D311yescharacteristic
116C3×C4×C7C23×C311yescharacteristic
117C3×Dic7—21no—
118C3×Dic7#6711yescharacteristic
119C3×Dic7C3×D411yescharacteristic
120C7×Dic3—21no—
121C23×C32—141no—
122C32×D4—71no—
123C32×D4—141no—
124C32×D4C2×D711yescharacteristic
125C2×C32×C4—71no—
126C3×#67—141no—
127C3×D12—71no—
128C3×C4×D3—141no—
129C3×Dic6C2×D711yescharacteristic, complemented
130C3×C3⋊C8—71no—
131C3×C3⋊C8C2×D711yescharacteristic, complemented
132C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
133C22×D7—21no—
134C7×Q8—31no—
135C7×D4C2×C3×D311yescharacteristic
136C22⋊D7—21no—
137D28—31no—
138D28—61no—
139D28C2×C3×D311yescharacteristic
140C4×D7—61no—
141C4×D7C2×C3×D311yescharacteristic
142C7×C8—31no—
143C7⋊C8—31no—
144C3×#41—211no—
145C2×C3×D4—71no—
146C2×C3×D4—71no—
147C2×C3×D4—141no—
148#235—71no—
149C3×SD16—31no—
150C3×SD16—211no—
151C3×D8—211no—
152C3×D8—211no—
153C3×M16—211no—
154#214—71no—
155#214C3×D711yescharacteristic, complemented
156#213—71no—
157#213—71no—
158#208—71no—
159C2×C3×C7—21no—
160C2×C3×C7—41no—
161C2×C3×C7—41no—
162C2×C3×C7—41no—
163C2×C3×C7—41no—
164C2×C3×C7C2×#6711yescharacteristic
165C2×C3×C7C2×C3×D411yescharacteristic
166D21—41no—
167C3×D7—21no—
168C3×D7—21no—
169C3×D7—21no—
170C3×D7—21no—
171C3×D7—41no—
172C3×D7—41no—
173C3×D7—41no—
174C3×D7—41no—
175C3×D7—121no—
176C22×C32—21no—
177C22×C32—71no—
178C22×C32—141no—
179C22×C32—282no—
180C2×C3×D3—141no—
181C32×C4—71no—
182C32×C4C22×D711yescharacteristic
183C3×Dic3—21no—
184AGL(1,ℤ/8)—211noSylow
185C22×C7—21no—
186C2×D7—21no—
187C2×D7—42no—
188C2×D7—61no—
189C2×D7C3×#6711yescharacteristic
190C4×C7—61no—
191C4×C7C22×C3×D311yescharacteristic
192Dic7C3×#6711yescharacteristic
193C23×C3—141no—
194C23×C3—141no—
195C23×C3—141no—
196C23×C3—141no—
197C3×Q8—31no—
198C3×D4—21no—
199C3×D4—71no—
200C3×D4—71no—
201C3×D4—141no—
202C3×D4—141no—
203C3×D4—141no—
204C3×D4—141no—
205C3×D4—141no—
206C3×D4—211no—
207C3×D4—421no—
208C3×D4D3×D711yescharacteristic
209C3×D4C2×C3×D711yescharacteristic
210C2×C3×C4—71no—
211C2×C3×C4—71no—
212C2×C3×C4—141no—
213C2×C3×C4—421no—
214#67—141no—
215D12—71no—
216C4×D3—141no—
217Dic6C2×C3×D711yescharacteristic, complemented
218C3×C8—31no—
219C3×C8—211no—
220C3⋊C8—71no—
221C3⋊C8C2×C3×D711yescharacteristic, complemented
222C3×C7—21no—
223C3×C7#93211yescharacteristic, complemented
224C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
225C2×C32—41no—
226C2×C32—141no—
227C2×C32—281no—
228C2×C32D4×D711yescharacteristic
229C3×D3—281no—
230#41—211no—
231C2×D4—71no—
232SD16—31no—
233SD16—211no—
234D8—211no—
235D8—211no—
236M16—211no—
237C2×C7—41no—
238C2×C7C2×C3×#6711yescharacteristic
239D7—21no—
240D7—41no—
241D7—121no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—71no—
247C22×C3—71no—
248C22×C3—141no—
249C22×C3—141no—
250C22×C3—141no—
251C22×C3—141no—
252C22×C3—141no—
253C22×C3—282no—
254C22×C3—282no—
255C22×C3—282no—
256C22×C3—282no—
257C22×C3—421no—
258C2×D3—141no—
259C3×C4—21no—
260C3×C4—61no—
261C3×C4—71no—
262C3×C4—71no—
263C3×C4—141no—
264C3×C4C2×D3×D711yescharacteristic
265C3×C4C22×C3×D711yescharacteristic
266Dic3—21no—
267C32#647711yescharacteristic, complemented, Sylow
268C23—141no—
269Q8—31no—
270D4—71no—
271D4—141no—
272D4—211no—
273D4—421no—
274D4C3×D3×D711yescharacteristic
275C2×C4—71no—
276C2×C4—421no—
277C8—31no—
278C8—211no—
279C7C3×#93211yescharacteristic, complemented, Sylow
280C2×C3—21no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—141no—
286C2×C3—141no—
287C2×C3—141no—
288C2×C3—141no—
289C2×C3—281no—
290C2×C3—281no—
291C2×C3—281no—
292C2×C3—281no—
293C2×C3—841no—
294C2×C3D7×#6711yescharacteristic
295C2×C3C3×D4×D711yescharacteristic
296D3—281no—
297C22—21no—
298C22—71no—
299C22—141no—
300C22—282no—
301C22—421no—
302C4—61no—
303C4—71no—
304C4C2×C3×D3×D711yescharacteristic
305C3—21no—
306C3#1062105311yescharacteristic, direct product factor
307C3C3×#647711yescharacteristic, complemented
308C2—41no—
309C2—141no—
310C2—281no—
311C2—841no—
312C2C3×D7×#6711yescharacteristic
subgroupG/H
centerC2×C3D7×#67
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C2×C3×D3×D7
FittingC32×C7×D4C22
socleC2×C32×C7C2×D4
hypercenterC3×D4D3×D7
hypocenterC3×C7C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×D4×D7C2semidirect
2. C3×#66677C2semidirect
3. C3×#66632C2semidirect
4. C3×C7×#214C2semidirect
5. C3×#66563C2semidirect
6. C3×#66561C2semidirect
7. C3×#66557C2semidirect
9. #10621053C3direct (N × H)
12. C3×D84C22semidirect
13. C3×C7×Dic6C22semidirect
19. C3×C3⋊(C7⋊C8)C22semidirect
20. C3×C7×C3⋊C8C22semidirect
23. C3×D4×D7D3semidirect
24. C3×D4×D7C2×C3semidirect
25. #66677C2×C3semidirect
27. #66632C2×C3semidirect
28. C7×#214C2×C3semidirect
33. #66563C2×C3semidirect
34. #66561C2×C3semidirect
35. #66557C2×C3semidirect
54. D84C22×C3semidirect
55. C7×Dic6C22×C3semidirect
72. C3⋊(C7⋊C8)C22×C3semidirect
74. C7×C3⋊C8C22×C3semidirect
78. C3×#214D7semidirect
88. D4×D7C3×D3semidirect
129. C3×Dic6C2×D7semidirect
131. C3×C3⋊C8C2×D7semidirect
132. C32×C7AGL(1,ℤ/8)semidirect
155. #214C3×D7semidirect
217. Dic6C2×C3×D7semidirect
221. C3⋊C8C2×C3×D7semidirect
223. C3×C7#932semidirect
224. C3×C7C3×AGL(1,ℤ/8)semidirect
267. C32#6477semidirect
279. C7C3×#932semidirect
306. C3#10621053direct (N × H)
307. C3C3×#6477semidirect

Maximal subgroups

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

subgroupindex
C32×D4×D72
C3×#666772
C3×#666322
C3×C7×#2142
C3×#665632
C3×#665612
C3×#665572
C3×#64773
#106210533
C3×#9327