Finite Group Databasebeta

← #72246706

Subgroups — #72246706

Order 2016.

718 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×C7×#638C211yesmaximal, characteristic, complemented
2C2×C3×#4268C212yesmaximal, complemented
3#11759644C212yesmaximal, complemented
4#11759643C214yesmaximal, complemented
5#11759638C214yesmaximal, complemented
6#11759637C212yesmaximal, complemented
7#10621596—32nomaximal
8C23×C32×C7C2211yescharacteristic, complemented
9C22×C7×C3⋊D3C2211yescharacteristic, complemented
10C22×C3×D21C2212yescomplemented
11C2×(C3⋊D3)⋊D7C2212yescomplemented
12C7×#638C2214yescomplemented
13C2×C7×C3⋊Dic3C2211yescharacteristic, complemented
14C3×#4268C2214yescomplemented
15C3×#4268C2214yescomplemented
16C2×C3×Dic21C2212yescomplemented
17#92696C2212yescomplemented
18#92695C2212yescomplemented
19#92694C2214yescomplemented
20#92693C2212yescomplemented
21D21⋊Dic3C2214yescomplemented
22C2×#4268D312yescomplemented
23C2×C7×#67—32no—
24C2×C7×#67—61no—
25C2×C3×C22⋊D7—32no—
26#66694—34no—
27D3×C22⋊D7—34no—
28D7×#67—34no—
29#66686—34no—
30#66684—34no—
31#66683—34no—
32#64330—71nomaximal, Hall
33C22×C32×C7—41no—
34C22×C32×C7C2312yes—
35C22×C32×C7C2311yescharacteristic
36C2×C7×C3⋊D3—41no—
37C2×C7×C3⋊D3C2312yes—
38C2×C3×D21—42no—
39C2×C3×D21C2314yes—
40(C3⋊D3)⋊D7—42no—
41C7×C3⋊Dic3C2312yes—
42C3×Dic21C2314yes—
43#6548—91noHall
44C23×C3×C7—21no—
45C23×C3×C7C2×D312yescomplemented
46C22×D21C2×D312yescomplemented
47C22×D3×C7—32no—
48C22×D3×C7—61no—
49C22×C3×D7—32no—
50C2×D3×D7—34no—
51#4268C2×D314yescomplemented
52#4268C2×D314yescomplemented
53C2×Dic21C2×D312yescomplemented
54C7×#67—34no—
55C7×#67—34no—
56C7×#67—64no—
57C2×C7×Dic3—32no—
58C2×C7×Dic3—61no—
59C3×C22⋊D7—34no—
60C3×C22⋊D7—34no—
61C2×C3×Dic7—32no—
62#4248—34no—
63#4247—34no—
64#4246—34no—
65#4245—34no—
66D21⋊C4—34no—
67D3×Dic7—34no—
68Dic3×D7—34no—
69C2×#638D711yescharacteristic, complemented
70C2×C3×#67—72no—
71#3811—72no—
72D3×#67—74no—
73#3805—72no—
74#3804—74no—
75C2×C32×C7—21no—
76C2×C32×C7—22no—
77C2×C32×C7C2411yescharacteristic
78C7×C3⋊D3—22no—
79C3×D21—24no—
80C2×C7×D4—91no—
81C2×C22⋊D7—32no—
82C2×C22⋊D7—92no—
83#1159—94no—
84D4×D7—94no—
85#1157—92no—
86#1004—212no—
87C22×C3×C7—21no—
88C22×C3×C7—22no—
89C22×C3×C7—42no—
90C22×C3×C7—41no—
91C22×C3×C7—41no—
92C22×C3×C7C22×D312yes—
93C22×C3×C7C22×D312yes—
94C22×C3×C7C22×D312yes—
95C2×D21—42no—
96C2×D21C22×D314yes—
97C2×D3×C7—34no—
98C2×D3×C7—62no—
99C2×D3×C7—122no—
100C2×D3×C7—121no—
101C2×D3×C7—121no—
102C2×C3×D7—34no—
103C2×C3×D7—122no—
104D3×D7—124no—
105Dic21C22×D314yes—
106C3×Dic7—34no—
107C7×Dic3—34no—
108C7×Dic3—62no—
109C23×C32C2×D711yescharacteristic, complemented
110C22×C3⋊D3C2×D711yescharacteristic, complemented
111C22×C3×D3—72no—
112C2×D32—72no—
113#638C2×D714yescomplemented
114C2×C3⋊Dic3C2×D711yescharacteristic, complemented
115C3×#67—74no—
116C3×#67—74no—
117C2×C3×Dic3—72no—
118#627—72no—
119#626—74no—
120#625—72no—
121#624—72no—
122D3×Dic3—74no—
123C32×C7H(5,𝔽2)11yescharacteristic, complemented, Hall
124C23×C7—91no—
125C23×C7D3211yescharacteristic, complemented
126C22×D7—32no—
127C22×D7—92no—
128C7×D4—94no—
129C2×C4×C7—91no—
130C22⋊D7—34no—
131C22⋊D7—34no—
132C22⋊D7—94no—
133C2×Dic7—32no—
134C2×Dic7—92no—
135D28—92no—
136C4×D7—94no—
137Dic14—92no—
138C2×C3×D4—212no—
139C2×#67—32no—
140C2×#67—61no—
141C2×#67—72no—
142C2×#67—214no—
143#237—214no—
144#237—214no—
145D3×D4—214no—
146D3×D4—214no—
147#235—214no—
148C2×C3×C7—21no—
149C2×C3×C7—21no—
150C2×C3×C7—22no—
151C2×C3×C7—22no—
152C2×C3×C7—22no—
153C2×C3×C7—21no—
154C2×C3×C7—42no—
155C2×C3×C7C23×D312yes—
156D21—24no—
157C3×D7—64no—
158D3×C7—64no—
159D3×C7—122no—
160C22×C32—41no—
161C22×C32C22×D712yes—
162C22×C32C22×D711yescharacteristic
163C2×C3⋊D3—41no—
164C2×C3⋊D3C22×D712yes—
165C2×C3×D3—74no—
166C2×C3×D3—282no—
167D32—282no—
168C3⋊Dic3C22×D712yes—
169C3×Dic3—74no—
170H(5,𝔽2)—631noSylow
171C22×C7—41no—
172C22×C7—92no—
173C22×C7—361no—
174C22×C7C2×D3212yes—
175C22×C7C2×D3211yescharacteristic
176C2×D7—34no—
177C2×D7—122no—
178C2×D7—362no—
179C4×C7—92no—
180Dic7—34no—
181C23×C3—21no—
182C23×C3—212no—
183C23×C3D3×D712yescomplemented
184C22×D3—32no—
185C22×D3—61no—
186C22×D3—72no—
187C22×D3—214no—
188C3×D4—214no—
189C3×D4—214no—
190C2×C3×C4—212no—
191#67—34no—
192#67—34no—
193#67—64no—
194#67—74no—
195#67—74no—
196#67—214no—
197#67—214no—
198C2×Dic3—32no—
199C2×Dic3—61no—
200C2×Dic3—72no—
201C2×Dic3—214no—
202D12—214no—
203C4×D3—214no—
204C4×D3—214no—
205Dic6—214no—
206C3×C7—21no—
207C3×C7#100412yescomplemented
208C2×C32—21no—
209C2×C32—22no—
210C2×C32C23×D711yescharacteristic
211C3⋊D3—22no—
212C3×D3—144no—
213#41—632no—
214#41—634no—
215C2×D4—91no—
216C2×D4—212no—
217C2×D4—634no—
218C2×D4—632no—
219C2×C7—22no—
220C2×C7—21no—
221C2×C7—182no—
222C2×C7C22×D3211yescharacteristic
223D7—64no—
224C22×C3—21no—
225C22×C3—22no—
226C22×C3—42no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—214no—
230C22×C3—842no—
231C22×C3C2×D3×D712yes—
232C22×C3C2×D3×D712yes—
233C22×C3C2×D3×D712yes—
234C2×D3—34no—
235C2×D3—62no—
236C2×D3—74no—
237C2×D3—122no—
238C2×D3—121no—
239C2×D3—121no—
240C2×D3—282no—
241C2×D3—844no—
242C3×C4—214no—
243Dic3—34no—
244Dic3—62no—
245Dic3—74no—
246C32#654811yescharacteristic, complemented, Sylow
247C23—91no—
248C23—212no—
249C23—632no—
250C23(C3⋊D3)⋊D711yescharacteristic, complemented
251Q8—632no—
252D4—94no—
253D4—214no—
254D4—214no—
255D4—632no—
256D4—634no—
257C2×C4—91no—
258C2×C4—212no—
259C2×C4—634no—
260C2×C4—632no—
261C7#6433011yescharacteristic, complemented, Sylow
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—21no—
268C2×C3—42no—
269C2×C3—424no—
270C2×C3C22×D3×D712yes—
271D3—64no—
272D3—122no—
273D3—144no—
274C22—41no—
275C22—92no—
276C22—214no—
277C22—361no—
278C22—842no—
279C22—2522no—
280C22C2×(C3⋊D3)⋊D711yescharacteristic
281C22C2×(C3⋊D3)⋊D712yes—
282C4—92no—
283C4—214no—
284C3—21no—
285C3#1062159612yescomplemented
286C2—21no—
287C2—22no—
288C2—182no—
289C2—424no—
290C2C22×(C3⋊D3)⋊D711yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D7
commutator, socle, cosocleC2×C32×C7C24
FittingC23×C32×C7C22
hypercenterC23(C3⋊D3)⋊D7
hypocenterC32×C7H(5,𝔽2)
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C7×#638C2semidirect
2. C2×C3×#4268C2semidirect
3. #11759644C2semidirect
4. #11759643C2semidirect
5. #11759638C2semidirect
6. #11759637C2semidirect
8. C23×C32×C7C22semidirect
9. C22×C7×C3⋊D3C22semidirect
10. C22×C3×D21C22semidirect
11. C2×(C3⋊D3)⋊D7C22semidirect
12. C7×#638C22semidirect
13. C2×C7×C3⋊Dic3C22semidirect
14. C3×#4268C22semidirect
15. C3×#4268C22semidirect
16. C2×C3×Dic21C22semidirect
17. #92696C22semidirect
18. #92695C22semidirect
19. #92694C22semidirect
20. #92693C22semidirect
21. D21⋊Dic3C22semidirect
22. C2×#4268D3semidirect
45. C23×C3×C7C2×D3semidirect
46. C22×D21C2×D3semidirect
51. #4268C2×D3semidirect
52. #4268C2×D3semidirect
53. C2×Dic21C2×D3semidirect
69. C2×#638D7semidirect
109. C23×C32C2×D7semidirect
110. C22×C3⋊D3C2×D7semidirect
113. #638C2×D7semidirect
114. C2×C3⋊Dic3C2×D7semidirect
123. C32×C7H(5,𝔽2)semidirect
125. C23×C7D32semidirect
183. C23×C3D3×D7semidirect
207. C3×C7#1004semidirect
246. C32#6548semidirect
250. C23(C3⋊D3)⋊D7semidirect
261. C7#64330semidirect
285. C3#10621596semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#6382
C2×C3×#42682
#117596442
#117596432
#117596382
#117596372
#106215963
#643307