Finite Group Databasebeta

← #72245864

Subgroups — #72245864

Order 2016.

326 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 56 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#11759604C211yesmaximal, characteristic, complemented
2#11759568C211yesmaximal, characteristic, complemented
3#11759220C211yesmaximal, characteristic, complemented
4#11759193C211yesmaximal, characteristic, complemented
5C7×#3714C211yesmaximal, characteristic, complemented
6C3×#66565C211yesmaximal, characteristic, complemented
7C3×#66560C211yesmaximal, characteristic, complemented
8#10621044—31nomaximal
9#10620819—31nomaximal
10C2×D7×C3⋊D3—21no—
11C4×C3⋊D21C2211yescharacteristic
12C7×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×Dic42C2211yescharacteristic, complemented
14C3×C7×D12C2211yescharacteristic, complemented
15C32×D28C2211yescharacteristic
16#92690—21no—
17D3×Dic21—21no—
18#92676—21no—
19C3×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
20C3×C7×C3⋊C8C2211yescharacteristic, complemented
21#66730—31no—
22#66681—31no—
23#66681—31no—
24#66681—61no—
25#66672—31no—
26#66622—31no—
27C7×#213—31no—
28C7×D24—31no—
29C3×#1141—31no—
30C3×#1132—31no—
31#66567—31no—
32#66565D311yescharacteristic, complemented
33#66564—31no—
34#66560D311yescharacteristic, complemented
35#66559—31no—
36#63937—71nomaximal, Hall
37C2×C3⋊D21D411yescharacteristic
38C2×C7×C3⋊D3—21no—
39C2×C3×D3×C7—21no—
40C2×C32×D7—21no—
41D7×C3⋊D3—42no—
42C32×C4×C7C2311yescharacteristic
43C3⋊Dic21D411yescharacteristic
44C3×Dic21—21no—
45#6474—91noHall
46C2×D3×D7—61no—
47C2×D3×D7—61no—
48C2×D3×D7—61no—
49C2×D3×D7—61no—
50C3×C7×D4—31no—
51#4268—61no—
52C2×Dic21—61no—
53C4×D21—31no—
54C4×D21—31no—
55C4×D21—61no—
56Dic42C2×D311yescharacteristic, complemented
57C7×D12—31no—
58C7×D12—31no—
59C7×D12—61no—
60C7×D12C2×D311yescharacteristic, complemented
61C3×D28—21no—
62C3×D28C2×D311yescharacteristic
63C3×D28C2×D311yescharacteristic
64C3×Dic14—31no—
65#4247—61no—
66#4245—61no—
67#4245—61no—
68#4245—61no—
69#4245—61no—
70D3×Dic7—61no—
71C3×C7×C8—31no—
72C3⋊(C7⋊C8)C2×D311yescharacteristic, complemented
73C3×C7⋊C8—31no—
74C7×C3⋊C8C2×D311yescharacteristic, complemented
75D4×C3⋊D3—71no—
76#3796—71no—
77C3×#214—71no—
78C3×#213—71no—
79#3717—71no—
80#3714D711yescharacteristic, complemented
81#3712—71no—
82C2×C32×C7C2×D411yescharacteristic
83C3⋊D21—21no—
84C7×C3⋊D3—41no—
85C3×D3×C7—41no—
86C32×D7—41no—
87#1159—91no—
88D4×D7—91no—
89C7×D8—91no—
90#1142—91no—
91#1141—31no—
92#1132—31no—
93#1131—91no—
94#914—211no—
95#911—211no—
96C22×C3×C7—61no—
97C2×D21—31no—
98C2×D21—31no—
99C2×D21—61no—
100C2×D3×C7—21no—
101C2×D3×C7—61no—
102C2×D3×C7—61no—
103C2×D3×C7—61no—
104C2×D3×C7—61no—
105C2×C3×D7—21no—
106C2×C3×D7—21no—
107C2×C3×D7—21no—
108C2×C3×D7—21no—
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—
117Dic21—31no—
118Dic21—31no—
119Dic21—61no—
120C3×Dic7—61no—
121C22×C3⋊D3—141no—
122C32×D4—71no—
123#638—141no—
124C4⋊(C3⋊D3)C2×D711yescharacteristic
125C4×C3⋊D3—71no—
126C3×D12C2×D711yescharacteristic, complemented
127C3×Dic6—71no—
128#626—141no—
129D3×Dic3—141no—
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—181no—
137C22⋊D7—181no—
138C2×Dic7—181no—
139D28D3211yescharacteristic
140C4×D7—91no—
141Dic14—31no—
142C7×C8—31no—
143C7⋊C8—31no—
144#239—211no—
145#237—211no—
146D3×D4—211no—
147D3×D4—211no—
148D3×D4—421no—
149C3×SD16—211no—
150C3×D8—211no—
151#215—211no—
152#214—71no—
153#213—31no—
154#213—71no—
155D24—31no—
156#204—211no—
157ΓL(1,𝔽25)—211no—
158ΓL(1,𝔽25)—211no—
159C2×C3×C7—21no—
160C2×C3×C7—121no—
161C2×C3×C7D3×D411yescharacteristic
162C2×C3×C7D3×D411yescharacteristic
163D21—61no—
164D21—61no—
165D21—61no—
166D21—61no—
167C3×D7—41no—
168C3×D7—41no—
169C3×D7—41no—
170C3×D7—41no—
171D3×C7—41no—
172D3×C7—121no—
173D3×C7—121no—
174D3×C7—121no—
175D3×C7—121no—
176C22×C32—141no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3—71no—
179C2×C3⋊D3—282no—
180C2×C3×D3—21no—
181C32×C4C22×D711yescharacteristic
182C3⋊Dic3—71no—
183C3×Dic3—141no—
184AGL(1,ℤ/8)—631noSylow
185C22×C7—61no—
186C22×C7—181no—
187C2×D7—21no—
188C2×D7—91no—
189C2×D7—362no—
190C4×C7C2×D3211yescharacteristic
191Dic7—61no—
192Dic7—91no—
193C22×D3—421no—
194C22×D3—421no—
195C22×D3—421no—
196C22×D3—421no—
197C3×Q8—211no—
198C3×D4—31no—
199C3×D4—71no—
200C3×D4—71no—
201C3×D4—141no—
202#67—421no—
203#67—421no—
204#67—421no—
205#67—421no—
206#67—421no—
207C2×Dic3—421no—
208D12—31no—
209D12—31no—
210D12—61no—
211D12—421no—
212D12D3×D711yescharacteristic, complemented
213C4×D3—211no—
214C4×D3—211no—
215C4×D3—421no—
216C4×D3—421no—
217Dic6—71no—
218C3×C8—31no—
219C3×C8—211no—
220C3⋊C8—71no—
221C3⋊C8D3×D711yescharacteristic, complemented
222C3×C7—21no—
223C3×C7#91111yescharacteristic, complemented
224C3×C7#91411yescharacteristic, complemented
225C2×C32—281no—
226C2×C32D4×D711yescharacteristic
227C3⋊D3—41no—
228C3⋊D3—141no—
229C3×D3—41no—
230#41—631no—
231C2×D4—631no—
232SD16—211no—
233SD16—631no—
234D8—91no—
235D8—211no—
236M16—631no—
237C2×C7—121no—
238C2×C7—361no—
239C2×C7#381111yescharacteristic
240D7—41no—
241D7—181no—
242C22×C3—61no—
243C22×C3—141no—
244C22×C3—141no—
245C22×C3—141no—
246C22×C3—141no—
247C2×D3—21no—
248C2×D3—61no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—61no—
252C2×D3—211no—
253C2×D3—211no—
254C2×D3—421no—
255C2×D3—842no—
256C2×D3—842no—
257C2×D3—842no—
258C2×D3—842no—
259C3×C4—21no—
260C3×C4—421no—
261C3×C4C2×D3×D711yescharacteristic
262C3×C4C2×D3×D711yescharacteristic
263Dic3—141no—
264Dic3—211no—
265Dic3—211no—
266Dic3—421no—
267C32#647411yescharacteristic, complemented, Sylow
268C23—1261no—
269Q8—211no—
270D4—31no—
271D4—71no—
272D4—91no—
273D4—1261no—
274D4—1261no—
275C2×C4—631no—
276C2×C4—1261no—
277C8—31no—
278C8—211no—
279C7#6393711yescharacteristic, complemented, Sylow
280C2×C3—21no—
281C2×C3—121no—
282C2×C3—281no—
283C2×C3—281no—
284C2×C3—281no—
285C2×C3—281no—
286C2×C3#6668111yescharacteristic
287C2×C3#6669411yescharacteristic
288D3—41no—
289D3—121no—
290D3—121no—
291D3—121no—
292D3—121no—
293D3—421no—
294D3—421no—
295D3—421no—
296D3—421no—
297C22—61no—
298C22—141no—
299C22—181no—
300C22—631no—
301C22—2522no—
302C4—421no—
303C4—631no—
304C4D32×D711yescharacteristic
305C3—21no—
306C3#1062081911yescharacteristic, complemented
307C3#1062104411yescharacteristic, complemented
308C2—121no—
309C2—281no—
310C2—361no—
311C2—1261no—
312C2#1175942711yescharacteristic
subgroupG/H
centerC2#11759427
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. #11759604C2semidirect
2. #11759568C2semidirect
3. #11759220C2semidirect
4. #11759193C2semidirect
5. C7×#3714C2semidirect
6. C3×#66565C2semidirect
7. C3×#66560C2semidirect
13. C3×Dic42C22semidirect
14. C3×C7×D12C22semidirect
19. C3×C3⋊(C7⋊C8)C22semidirect
20. C3×C7×C3⋊C8C22semidirect
32. #66565D3semidirect
34. #66560D3semidirect
56. Dic42C2×D3semidirect
60. C7×D12C2×D3semidirect
72. C3⋊(C7⋊C8)C2×D3semidirect
74. C7×C3⋊C8C2×D3semidirect
80. #3714D7semidirect
126. C3×D12C2×D7semidirect
131. C3×C3⋊C8C2×D7semidirect
132. C32×C7AGL(1,ℤ/8)semidirect
212. D12D3×D7semidirect
221. C3⋊C8D3×D7semidirect
223. C3×C7#911semidirect
224. C3×C7#914semidirect
267. C32#6474semidirect
279. C7#63937semidirect
306. C3#10620819semidirect
307. C3#10621044semidirect

Maximal subgroups

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

subgroupindex
#117596042
#117595682
#117592202
#117591932
C7×#37142
C3×#665652
C3×#665602
#106210443
#106208193
#639377