Finite Group Databasebeta

← #72245865

Subgroups — #72245865

Order 2016.

326 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 62 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
1D7×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#66676C211yesmaximal, characteristic, complemented
3#11759220C211yesmaximal, characteristic, complemented
4#11759200C211yesmaximal, characteristic, complemented
5#11759196C211yesmaximal, characteristic, complemented
6C7×#3714C211yesmaximal, characteristic, complemented
7C3×#66557C211yesmaximal, characteristic, complemented
8#10621040—31nomaximal
9#10620812—31nomaximal
10C2×D7×C3⋊D3—21no—
11C4⋊(C3⋊D21)C2211yescharacteristic
12C7×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×Dic42C2211yescharacteristic, complemented
14C3×C7×D12C2211yescharacteristic, complemented
15C32×C4×D7C2211yescharacteristic
16#92678—21no—
17C3×#4245—21no—
18C3×D3×Dic7—21no—
19C3×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
20C3×C7×C3⋊C8C2211yescharacteristic, complemented
21C3×#1159—31no—
22D7×D12—31no—
23D7×D12—31no—
24D7×D12—61no—
25#66676D311yescharacteristic, complemented
26#66632—31no—
27#66623—31no—
28C7×#213—31no—
29C7×D24—31no—
30C3×#1131—31no—
31#66567—31no—
32#66566—31no—
33#66564—31no—
34#66562—31no—
35#66557D311yescharacteristic, complemented
36#63825—71nomaximal, Hall
37C2×C3⋊D21—21no—
38C2×C7×C3⋊D3—21no—
39C2×C3×D3×C7—21no—
40C2×C32×D7D411yescharacteristic
41D7×C3⋊D3—42no—
42C32×C4×C7C2311yescharacteristic
43C3×Dic21—21no—
44C32×Dic7D411yescharacteristic
45#6474—91noHall
46C2×D3×D7—61no—
47C2×D3×D7—61no—
48C2×D3×D7—61no—
49C2×D3×D7—61no—
50C3×C7×D4—31no—
51D84—31no—
52D84—31no—
53D84—61no—
54Dic42C2×D311yescharacteristic, complemented
55C7×D12—31no—
56C7×D12—31no—
57C7×D12—61no—
58C7×D12C2×D311yescharacteristic, complemented
59C3×C22⋊D7—61no—
60C2×C3×Dic7—61no—
61C3×C4×D7—21no—
62C3×C4×D7C2×D311yescharacteristic
63C3×C4×D7C2×D311yescharacteristic
64C3×Dic14—31no—
65#4247—61no—
66#4247—61no—
67#4247—61no—
68#4247—61no—
69#4245—21no—
70D3×Dic7—21no—
71C3×C7×C8—31no—
72C3⋊(C7⋊C8)C2×D311yescharacteristic, complemented
73C3×C7⋊C8—31no—
74C7×C3⋊C8C2×D311yescharacteristic, complemented
75C2×C4⋊(C3⋊D3)—71no—
76C3×#235—71no—
77C3×#208—71no—
78#3717—71no—
79#3717—71no—
80#3714—71no—
81#3714D711yescharacteristic, complemented
82C2×C32×C7C2×D411yescharacteristic
83C3⋊D21—41no—
84C7×C3⋊D3—41no—
85C3×D3×C7—41no—
86C32×D7—21no—
87#1159—31no—
88D4×D7—91no—
89C7×D8—91no—
90#1142—91no—
91#1141—91no—
92#1132—91no—
93#1131—31no—
94#949—211no—
95#908—211no—
96C22×C3×C7—61no—
97C2×D21—61no—
98C2×D21—61no—
99C2×D21—61no—
100C2×D21—61no—
101C2×D3×C7—21no—
102C2×D3×C7—61no—
103C2×D3×C7—61no—
104C2×D3×C7—61no—
105C2×D3×C7—61no—
106C2×C3×D7—21no—
107C2×C3×D7D1211yescharacteristic
108C2×C3×D7#6711yescharacteristic
109D3×D7—122no—
110D3×D7—122no—
111D3×D7—122no—
112D3×D7—122no—
113C3×C4×C7—21no—
114C3×C4×C7C22×D311yescharacteristic
115C3×C4×C7C22×D311yescharacteristic
116Dic21—21no—
117C3×Dic7—21no—
118C3×Dic7—61no—
119C3×Dic7D1211yescharacteristic
120C3×Dic7#6711yescharacteristic
121C22×C3⋊D3—141no—
122C2×C32×C4—71no—
123C4⋊(C3⋊D3)—71no—
124C4⋊(C3⋊D3)—141no—
125C4⋊(C3⋊D3)C2×D711yescharacteristic
126C3×#67—141no—
127C3×D12C2×D711yescharacteristic, complemented
128C3×C4×D3—141no—
129C3×Dic6—71no—
130C3×C3⋊C8—71no—
131C3×C3⋊C8C2×D711yescharacteristic, complemented
132C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
133C22×D7—181no—
134C7×D4—31no—
135C7×D4—91no—
136C22⋊D7—61no—
137C22⋊D7—181no—
138C2×Dic7—61no—
139D28—91no—
140C4×D7D3211yescharacteristic
141Dic14—31no—
142C7×C8—31no—
143C7⋊C8—31no—
144C3×#41—211no—
145#235—71no—
146C2×D12—211no—
147C2×D12—211no—
148C2×D12—421no—
149C3×M16—211no—
150#215—211no—
151#215—211no—
152#213—31no—
153#213—211no—
154#208—71no—
155D24—31no—
156D24—211no—
157#204—211no—
158#204—211no—
159C2×C3×C7—21no—
160C2×C3×C7—121no—
161C2×C3×C7C2×D1211yescharacteristic
162C2×C3×C7C2×#6711yescharacteristic
163D21—121no—
164D21—121no—
165D21—121no—
166D21—121no—
167C3×D7—21no—
168C3×D7—21no—
169C3×D7—21no—
170C3×D7—21no—
171D3×C7—41no—
172D3×C7—121no—
173D3×C7—121no—
174D3×C7—121no—
175D3×C7—121no—
176C22×C32—71no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3—141no—
179C2×C3⋊D3—282no—
180C2×C3×D3—21no—
181C32×C4—71no—
182C32×C4C22×D711yescharacteristic
183C3×Dic3—141no—
184AGL(1,ℤ/8)—631noSylow
185C22×C7—61no—
186C22×C7—181no—
187C2×D7—181no—
188C2×D7—362no—
189C2×D7#62611yescharacteristic
190C4×C7C2×D3211yescharacteristic
191Dic7—61no—
192Dic7#62611yescharacteristic
193C22×D3—421no—
194C22×D3—421no—
195C22×D3—421no—
196C22×D3—421no—
197C3×Q8—211no—
198C3×D4—31no—
199C3×D4—421no—
200C2×C3×C4—71no—
201C2×C3×C4—71no—
202C2×C3×C4—141no—
203C2×C3×C4—421no—
204#67—141no—
205D12—31no—
206D12—31no—
207D12—61no—
208D12—211no—
209D12—211no—
210D12—421no—
211D12—421no—
212D12—421no—
213D12—421no—
214D12—421no—
215D12D3×D711yescharacteristic, complemented
216C4×D3—141no—
217Dic6—71no—
218C3×C8—31no—
219C3×C8—211no—
220C3⋊C8—71no—
221C3⋊C8D3×D711yescharacteristic, complemented
222C3×C7—21no—
223C3×C7#90811yescharacteristic, complemented
224C3×C7#94911yescharacteristic, complemented
225C2×C32—141no—
226C2×C32D4×D711yescharacteristic
227C3⋊D3—41no—
228C3⋊D3—281no—
229C3×D3—41no—
230#41—211no—
231C2×D4—631no—
232SD16—631no—
233SD16—631no—
234D8—91no—
235D8—631no—
236M16—211no—
237C2×C7—121no—
238C2×C7—361no—
239C2×C7C2×#62611yescharacteristic
240D7—21no—
241D7—361no—
242C22×C3—61no—
243C22×C3—71no—
244C22×C3—71no—
245C22×C3—141no—
246C2×D3—21no—
247C2×D3—61no—
248C2×D3—61no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—421no—
252C2×D3—421no—
253C2×D3—421no—
254C2×D3—421no—
255C2×D3—842no—
256C2×D3—842no—
257C2×D3—842no—
258C2×D3—842no—
259C3×C4—21no—
260C3×C4—71no—
261C3×C4—71no—
262C3×C4—141no—
263C3×C4—421no—
264C3×C4C2×D3×D711yescharacteristic
265C3×C4C2×D3×D711yescharacteristic
266Dic3—141no—
267C32#647411yescharacteristic, complemented, Sylow
268C23—1261no—
269Q8—211no—
270D4—31no—
271D4—91no—
272D4—421no—
273D4—631no—
274D4—1261no—
275C2×C4—71no—
276C2×C4—421no—
277C8—31no—
278C8—211no—
279C7#6382511yescharacteristic, complemented, Sylow
280C2×C3—21no—
281C2×C3—121no—
282C2×C3—141no—
283C2×C3—141no—
284C2×C3—141no—
285C2×C3—141no—
286C2×C3D7×D1211yescharacteristic
287C2×C3D7×#6711yescharacteristic
288D3—41no—
289D3—121no—
290D3—121no—
291D3—121no—
292D3—121no—
293D3—841no—
294D3—841no—
295D3—841no—
296D3—841no—
297C22—61no—
298C22—71no—
299C22—181no—
300C22—1261no—
301C22—2522no—
302C4—71no—
303C4—421no—
304C4D32×D711yescharacteristic
305C3—21no—
306C3#1062081211yescharacteristic, complemented
307C3#1062104011yescharacteristic, complemented
308C2—121no—
309C2—141no—
310C2—361no—
311C2—2521no—
312C2D7×#62611yescharacteristic
subgroupG/H
centerC2D7×#626
commutator, Fitting, cosocleC32×C4×C7C23
Frattini, hypercenterC4D32×D7
socleC2×C32×C7C2×D4
hypocenterC32×C7AGL(1,ℤ/8)
LCSG ▷ C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. D7×C4⋊(C3⋊D3)C2semidirect
2. C3×#66676C2semidirect
3. #11759220C2semidirect
4. #11759200C2semidirect
5. #11759196C2semidirect
6. C7×#3714C2semidirect
7. C3×#66557C2semidirect
13. C3×Dic42C22semidirect
14. C3×C7×D12C22semidirect
19. C3×C3⋊(C7⋊C8)C22semidirect
20. C3×C7×C3⋊C8C22semidirect
25. #66676D3semidirect
35. #66557D3semidirect
54. Dic42C2×D3semidirect
58. C7×D12C2×D3semidirect
72. C3⋊(C7⋊C8)C2×D3semidirect
74. C7×C3⋊C8C2×D3semidirect
81. #3714D7semidirect
127. C3×D12C2×D7semidirect
131. C3×C3⋊C8C2×D7semidirect
132. C32×C7AGL(1,ℤ/8)semidirect
215. D12D3×D7semidirect
221. C3⋊C8D3×D7semidirect
223. C3×C7#908semidirect
224. C3×C7#949semidirect
267. C32#6474semidirect
279. C7#63825semidirect
306. C3#10620812semidirect
307. C3#10621040semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊(C3⋊D3)2
C3×#666762
#117592202
#117592002
#117591962
C7×#37142
C3×#665572
#106210403
#106208123
#638257