Finite Group Databasebeta

← #72245742

Subgroups — #72245742

Order 2016.

710 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 144 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#11759635C211yesmaximal, characteristic, complemented
2#11759602C212yesmaximal, complemented
3C7×#3797C211yesmaximal, characteristic, complemented
4C4×D7×C3⋊D3C211yesmaximal, characteristic, complemented
5C3×#66677C212yesmaximal, complemented
6#11759449C212yesmaximal, complemented
7#11759444C212yesmaximal, complemented
8#11759428C212yesmaximal, complemented
9#11759419C212yesmaximal, complemented
10D3×#1161—32nomaximal
11C2×(C3⋊D3)⋊D7C2212yes—
12C2×D7×C3⋊D3C2211yescharacteristic
13C4×C3⋊D21C2211yescharacteristic, complemented
14C4×C7×C3⋊D3C2211yescharacteristic
15C3×D84C2212yescomplemented
16C3×C7×Dic6C2212yescomplemented
17C32×C4×D7C2211yescharacteristic, complemented
18#92695C2212yescomplemented
19#92689C2214yescomplemented
20Dic3×D21C2214yescomplemented
21C7×#627C2212yescomplemented
22C7×#624C2212yes—
23#92675C2211yescharacteristic, complemented
24C3⋊D3×Dic7C2211yescharacteristic
25D7×C3⋊Dic3C2211yescharacteristic, complemented
26C3×#4246C2214yescomplemented
27C3×D21⋊C4C2214yescomplemented
28#66732—32no—
29D3×C7×Q8—32no—
30C3×#1161—32no—
31D3×D28—32no—
32D3×D28—34no—
33C4×D3×D7—32no—
34C4×D3×D7—34no—
35C4×D3×D7—61no—
36#66677—34no—
37#66677D312yescomplemented
38#66670—32no—
39#66670—34no—
40#64306—71nomaximal, Hall
41C2×C3⋊D21C2311yescharacteristic
42C2×C7×C3⋊D3C2311yescharacteristic
43C2×C3×D21C2314yes—
44C2×C32×D7C2311yescharacteristic
45(C3⋊D3)⋊D7—24no—
46D7×C3⋊D3—22no—
47C32×C4×C7C2311yescharacteristic
48C3⋊Dic21C2311yescharacteristic
49C7×C3⋊Dic3C2311yescharacteristic
50C3×C7×Dic3C2314yes—
51C32×Dic7C2311yescharacteristic
52C2×#1161—91noHall
53C2×D3×D7—34no—
54C2×D3×D7—32no—
55C2×D3×D7—61no—
56C3×C7×Q8—32no—
57D84—34no—
58D84C2×D312yescomplemented
59C4×D21—34no—
60C4×D21—32no—
61C4×D21—61no—
62C4×D3×C7—32no—
63C4×D3×C7—34no—
64C4×D3×C7—61no—
65C7×Dic6—34no—
66C7×Dic6C2×D312yescomplemented
67C3×D28—34no—
68C3×D28—32no—
69C3×C4×D7—21no—
70C3×C4×D7—34no—
71C3×C4×D7C2×D312yescomplemented
72#4246—34no—
73#4246—34no—
74#4246C2×D314yescomplemented
75D21⋊C4—32no—
76D21⋊C4—61no—
77D21⋊C4C2×D314yescomplemented
78D3×Dic7—32no—
79D3×Dic7—61no—
80Dic3×D7—32no—
81Dic3×D7—34no—
82Dic3×D7—61no—
83C2×C4×C3⋊D3—71no—
84C3×#235—72no—
85#3811—72no—
86#3805—72no—
87#3802—71no—
88C4×D32—72no—
89#3798—72no—
90#3797D711yescharacteristic, complemented
91#3796—72no—
92C2×C32×C7C2411yescharacteristic
93C3⋊D21—21no—
94C7×C3⋊D3#4112yescomplemented
95C3×D21—24no—
96C32×D7—21no—
97C2×C7×Q8—91no—
98#1161—32no—
99#1161—92no—
100#1161—92no—
101#1161—92no—
102C2×D28—92no—
103C2×D28—91no—
104C2×C4×D7—92no—
105C2×C4×D7—91no—
106D3×#41—212no—
107C2×D21—32no—
108C2×D21—61no—
109C2×D21C22×D314yes—
110C2×D3×C7—32no—
111C2×D3×C7—61no—
112C2×C3×D7—21no—
113C2×C3×D7—34no—
114C2×C3×D7C22×D312yes—
115D3×D7—64no—
116D3×D7—68no—
117D3×D7—62no—
118D3×D7—62no—
119C3×C4×C7—21no—
120C3×C4×C7—34no—
121C3×C4×C7C22×D312yes—
122Dic21—32no—
123Dic21—61no—
124C3×Dic7—21no—
125C3×Dic7C22×D312yes—
126C7×Dic3—32no—
127C7×Dic3—61no—
128C7×Dic3C22×D314yes—
129C22×C3⋊D3—71no—
130C2×D32—72no—
131C2×C32×C4—71no—
132C2×C3⋊Dic3—71no—
133C4×C3⋊D3—71no—
134C4×C3⋊D3—71no—
135C4×C3⋊D3—71no—
136C4×C3⋊D3C2×D711yescharacteristic
137C3×#67—74no—
138C3×D12—72no—
139C3×C4×D3—74no—
140C3×Dic6C2×D712yescomplemented
141#627C2×D712yescomplemented
142#626—74no—
143#625—72no—
144#624C2×D712yes—
145D3×Dic3—74no—
146C32×C7C2×#4111yescharacteristic, complemented, Hall
147C22×D7—92no—
148C22×D7—91no—
149C7×Q8—32no—
150C7×Q8—92no—
151C2×C4×C7—92no—
152C2×C4×C7—91no—
153C2×Dic7—91no—
154D28—34no—
155D28—32no—
156D28—94no—
157D28—92no—
158C4×D7—34no—
159C4×D7—94no—
160C4×D7—91no—
161C4×D7—91no—
162C4×D7—91no—
163C4×D7D3211yescharacteristic, complemented
164C3×#41—212no—
165#239—212no—
166D3×Q8—32no—
167#237—214no—
168#237—212no—
169D3×D4—212no—
170D3×D4—214no—
171#235—72no—
172#235—214no—
173C2×C4×D3—214no—
174C2×C4×D3—212no—
175C2×C4×D3—421no—
176C2×C3×C7—21no—
177C2×C3×C7C23×D312yes—
178D21—24no—
179D21—61no—
180D21—61no—
181D21—62no—
182C3×D7—21no—
183C3×D7—21no—
184C3×D7—22no—
185C3×D7—64no—
186D3×C7—34no—
187D3×C7—62no—
188C22×C32—71no—
189C2×C3⋊D3—71no—
190C2×C3⋊D3—142no—
191C2×C3⋊D3C22×D711yescharacteristic
192C2×C3×D3—74no—
193D32—144no—
194C32×C4—71no—
195C32×C4C22×D711yescharacteristic
196C3⋊Dic3—71no—
197C3⋊Dic3C22×D711yescharacteristic
198C3×Dic3C22×D714yes—
199C2×#41—631noSylow
200C22×C7—91no—
201C2×D7—34no—
202C2×D7—91no—
203C2×D7—182no—
204C2×D7—184no—
205C2×D7C2×D3211yescharacteristic
206C4×C7—34no—
207C4×C7—91no—
208C4×C7C2×D3211yescharacteristic
209Dic7—91no—
210Dic7C2×D3211yescharacteristic
211C22×D3—214no—
212C22×D3—212no—
213C22×D3—421no—
214C3×Q8—32no—
215C3×D4—214no—
216C3×D4—212no—
217C2×C3×C4—72no—
218C2×C3×C4—141no—
219C2×C3×C4—214no—
220#67—74no—
221#67—214no—
222#67—214no—
223C2×Dic3—212no—
224C2×Dic3—214no—
225C2×Dic3—421no—
226D12—72no—
227D12—214no—
228C4×D3—32no—
229C4×D3—34no—
230C4×D3—61no—
231C4×D3—74no—
232C4×D3—212no—
233C4×D3—214no—
234C4×D3—212no—
235C4×D3—212no—
236C4×D3—421no—
237C4×D3—421no—
238C4×D3—421no—
239Dic6—34no—
240Dic6D3×D712yescomplemented
241C3×C7—21no—
242C3×C7D3×#4112yescomplemented
243C2×C32—141no—
244C2×C32C23×D711yescharacteristic
245C3⋊D3—141no—
246C3⋊D3#116112yescomplemented
247C3×D3—144no—
248#41—212no—
249#41—632no—
250#41—632no—
251#41—632no—
252C2×Q8—91no—
253C2×D4—632no—
254C2×D4—631no—
255C22×C4—631no—
256C22×C4—632no—
257C2×C7—92no—
258C2×C7C22×D3211yescharacteristic
259D7—21no—
260D7—64no—
261D7—181no—
262C22×C3—72no—
263C22×C3—141no—
264C22×C3—214no—
265C2×D3—32no—
266C2×D3—61no—
267C2×D3—74no—
268C2×D3—212no—
269C2×D3—421no—
270C2×D3—428no—
271C2×D3—424no—
272C2×D3—422no—
273C2×D3—422no—
274C3×C4—21no—
275C3×C4—34no—
276C3×C4—72no—
277C3×C4—141no—
278C3×C4C2×D3×D712yes—
279Dic3—32no—
280Dic3—61no—
281Dic3—212no—
282Dic3—421no—
283Dic3C2×D3×D714yes—
284C32C2×#116111yescharacteristic, complemented, Sylow
285C23—631no—
286C23—632no—
287Q8—32no—
288Q8—92no—
289D4—214no—
290D4—212no—
291D4—634no—
292D4—632no—
293C2×C4—71no—
294C2×C4—92no—
295C2×C4—91no—
296C2×C4—214no—
297C2×C4—634no—
298C2×C4—631no—
299C2×C4—631no—
300C2×C4—631no—
301C2×C4—631no—
302C7#6430611yescharacteristic, complemented, Sylow
303C2×C3—21no—
304C2×C3—141no—
305C2×C3—141no—
306C2×C3—142no—
307C2×C3—424no—
308C2×C3C22×D3×D712yes—
309D3—34no—
310D3—62no—
311D3—144no—
312D3—421no—
313D3—421no—
314D3—422no—
315C22—71no—
316C22—91no—
317C22—214no—
318C22—631no—
319C22—1262no—
320C22—1264no—
321C4—34no—
322C4—71no—
323C4—91no—
324C4—631no—
325C4D32×D711yescharacteristic
326C3—21no—
327C3D3×#116112yescomplemented
328C2—92no—
329C2—141no—
330C2—424no—
331C2—1261no—
332C2C2×D32×D711yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D7
commutator, socle, cosocleC2×C32×C7C24
FittingC32×C4×C7C23
hypercenterC4D32×D7
hypocenterC32×C7C2×#41
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11759635C2semidirect
2. #11759602C2semidirect
3. C7×#3797C2semidirect
4. C4×D7×C3⋊D3C2semidirect
5. C3×#66677C2semidirect
6. #11759449C2semidirect
7. #11759444C2semidirect
8. #11759428C2semidirect
9. #11759419C2semidirect
13. C4×C3⋊D21C22semidirect
15. C3×D84C22semidirect
16. C3×C7×Dic6C22semidirect
17. C32×C4×D7C22semidirect
18. #92695C22semidirect
19. #92689C22semidirect
20. Dic3×D21C22semidirect
21. C7×#627C22semidirect
23. #92675C22semidirect
25. D7×C3⋊Dic3C22semidirect
26. C3×#4246C22semidirect
27. C3×D21⋊C4C22semidirect
37. #66677D3semidirect
58. D84C2×D3semidirect
66. C7×Dic6C2×D3semidirect
71. C3×C4×D7C2×D3semidirect
74. #4246C2×D3semidirect
77. D21⋊C4C2×D3semidirect
90. #3797D7semidirect
94. C7×C3⋊D3#41semidirect
140. C3×Dic6C2×D7semidirect
141. #627C2×D7semidirect
146. C32×C7C2×#41semidirect
163. C4×D7D32semidirect
240. Dic6D3×D7semidirect
242. C3×C7D3×#41semidirect
246. C3⋊D3#1161semidirect
284. C32C2×#1161semidirect
302. C7#64306semidirect
327. C3D3×#1161semidirect

Maximal subgroups

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

subgroupindex
#117596352
#117596022
C7×#37972
C4×D7×C3⋊D32
C3×#666772
#117594492
#117594442
#117594282
#117594192
D3×#11613
#643067