Finite Group Databasebeta

← #72246749

Subgroups — #72246749

Order 2016.

780 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C22×D7×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#92726C211yesmaximal, characteristic, complemented
3C2×C7×#638C211yesmaximal, characteristic, complemented
4C2×C32×C22⋊D7C211yesmaximal, characteristic, complemented
5#11759168C211yesmaximal, characteristic, complemented
6#11759167C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8#10621139—34nomaximal
9C23×C32×C7C2211yescharacteristic, complemented
10C22×C3⋊D21C2211yescharacteristic
11C22×C7×C3⋊D3C2211yescharacteristic
12C22×C32×D7C2211yescharacteristic
13C2×D7×C3⋊D3—22no—
14C2×D7×C3⋊D3—22no—
15C2×D7×C3⋊D3—22no—
16#92726—22no—
17C2×C3⋊Dic21C2211yescharacteristic, complemented
18C7×#638—22no—
19C2×C7×C3⋊Dic3C2211yescharacteristic, complemented
20C32×C22⋊D7—22no—
21C2×C32×Dic7C2211yescharacteristic, complemented
22C22×D3×D7—34no—
23C2×#4268—34no—
24C2×C7×#67—34no—
25C2×C3×C22⋊D7D314yescomplemented
26#66575—34no—
27(C2×D3)⋊Dic7—34no—
28(C2×D7)⋊Dic3—34no—
29#64146—71nomaximal, Hall
30C22×C32×C7—21no—
31C22×C32×C7—21no—
32C22×C32×C7—21no—
33C22×C32×C7C2311yescharacteristic
34C2×C3⋊D21—21no—
35C2×C3⋊D21—21no—
36C2×C3⋊D21D412yescomplemented
37C2×C7×C3⋊D3—21no—
38C2×C7×C3⋊D3—21no—
39C2×C7×C3⋊D3D412yescomplemented
40C2×C32×D7—21no—
41C2×C32×D7—21no—
42C2×C32×D7D412yescomplemented
43D7×C3⋊D3—28no—
44C3⋊Dic21—21no—
45C7×C3⋊Dic3—21no—
46C32×Dic7—21no—
47#6500—91noHall
48C23×C3×C7C2×D314yescomplemented
49C22×D21—34no—
50C22×D3×C7—34no—
51C22×C3×D7C2×D314yes—
52C2×D3×D7—68no—
53C2×D3×D7—68no—
54C2×D3×D7—68no—
55#4268—68no—
56C2×Dic21—34no—
57C7×#67—68no—
58C2×C7×Dic3—34no—
59C3×C22⋊D7—28no—
60C2×C3×Dic7C2×D314yescomplemented
61C23×C3⋊D3—71no—
62C2×C32×D4—71no—
63C2×#638—71no—
64C2×#638D711yescharacteristic, complemented
65#3757—71no—
66#3752—71no—
67#3752—71no—
68C2×C32×C7—41no—
69C2×C32×C7C2×D411yescharacteristic
70C2×C32×C7C2×D411yescharacteristic
71C2×C32×C7C2×D411yescharacteristic
72C3⋊D21—22no—
73C7×C3⋊D3—22no—
74C32×D7—22no—
75C23×D7—91no—
76C2×C7×D4—91no—
77C2×C22⋊D7—91no—
78C2×C22⋊D7C3⋊D311yescharacteristic, complemented
79C22⋊Dic7—91no—
80(C2×D7)⋊C4—91no—
81(C2×D7)⋊C4—91no—
82#937—214no—
83C22×C3×C7—24no—
84C22×C3×C7—24no—
85C22×C3×C7—24no—
86C22×C3×C7C22×D314yes—
87C2×D21—38no—
88C2×D21—64no—
89C2×D21—64no—
90C2×D3×C7—38no—
91C2×D3×C7—64no—
92C2×D3×C7—64no—
93C2×C3×D7—24no—
94C2×C3×D7—24no—
95C2×C3×D7#6718yescomplemented
96D3×D7—632no—
97Dic21—64no—
98C3×Dic7—24no—
99C7×Dic3—64no—
100C23×C32—71no—
101C23×C32C2×D711yescharacteristic, complemented
102C22×C3⋊D3—71no—
103C22×C3⋊D3—142no—
104C22×C3⋊D3—142no—
105C22×C3⋊D3—142no—
106C22×C3⋊D3C2×D711yescharacteristic
107C32×D4—142no—
108C2×C32×C4—71no—
109#638—22no—
110#638—142no—
111C2×C3⋊Dic3—71no—
112C2×C3⋊Dic3C2×D711yescharacteristic, complemented
113C32×C7#12011yescharacteristic, complemented, Hall
114C23×C7—91no—
115C23×C7C2×C3⋊D311yescharacteristic, complemented
116C22×D7—91no—
117C22×D7—182no—
118C22×D7—182no—
119C22×D7—182no—
120C22×D7C2×C3⋊D311yescharacteristic
121C7×D4—182no—
122C2×C4×C7—91no—
123C22⋊D7—22no—
124C22⋊D7—182no—
125C2×Dic7—91no—
126C2×Dic7C2×C3⋊D311yescharacteristic, complemented
127C23×D3—214no—
128C2×C3×D4—74no—
129C2×#67—34no—
130C2×#67—214no—
131#217—214no—
132(C2×D3)⋊C4—214no—
133(C2×D3)⋊C4—214no—
134C2×C3×C7—44no—
135C2×C3×C7D3×D414yes—
136C2×C3×C7D3×D414yes—
137C2×C3×C7C2×#6714yes—
138D21—68no—
139C3×D7—28no—
140D3×C7—68no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—72no—
145C22×C32—141no—
146C22×C32—141no—
147C22×C32C22×D711yescharacteristic
148C2×C3⋊D3—21no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3—72no—
151C2×C3⋊D3—148no—
152C2×C3⋊D3—141no—
153C2×C3⋊D3—141no—
154C2×C3⋊D3C22⋊D712yescomplemented
155C32×C4—141no—
156C3⋊Dic3—21no—
157C3⋊Dic3—141no—
158#120—631noSylow
159C22×C7—21no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7—92no—
163C22×C7—181no—
164C22×C7—181no—
165C22×C7C22×C3⋊D311yescharacteristic
166C2×D7—21no—
167C2×D7—21no—
168C2×D7—92no—
169C2×D7—188no—
170C2×D7—181no—
171C2×D7—181no—
172C2×D7#63812yescomplemented
173C4×C7—181no—
174Dic7—21no—
175Dic7—181no—
176C23×C3—74no—
177C23×C3D3×D714yescomplemented
178C22×D3—34no—
179C22×D3—214no—
180C22×D3—428no—
181C22×D3—428no—
182C22×D3—428no—
183C3×D4—148no—
184C2×C3×C4—74no—
185#67—68no—
186#67—428no—
187C2×Dic3—34no—
188C2×Dic3—214no—
189C3×C7#93714yescomplemented
190C2×C32—41no—
191C2×C32—142no—
192C2×C32D4×D711yescharacteristic
193C2×C32D4×D711yescharacteristic
194C2×C32C2×C22⋊D711yescharacteristic
195C3⋊D3—22no—
196C3⋊D3—142no—
197C24—631no—
198C2×D4—71no—
199C2×D4—91no—
200C2×D4—631no—
201C22⋊C4—631no—
202C22⋊C4—631no—
203C22⋊C4—631no—
204C2×C7—41no—
205C2×C7—182no—
206C2×C7D4×C3⋊D311yescharacteristic
207C2×C7D4×C3⋊D311yescharacteristic
208C2×C7C2×#63811yescharacteristic
209D7—22no—
210D7—182no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—78no—
215C22×C3—144no—
216C22×C3—144no—
217C22×C3C2×D3×D714yes—
218C2×D3—38no—
219C2×D3—64no—
220C2×D3—64no—
221C2×D3—218no—
222C2×D3—4232no—
223C2×D3—424no—
224C2×D3—424no—
225C3×C4—144no—
226Dic3—64no—
227Dic3—424no—
228C32#650011yescharacteristic, complemented, Sylow
229C23—71no—
230C23—91no—
231C23—631no—
232C23—1262no—
233C23—1262no—
234C23—1262no—
235C23D7×C3⋊D311yescharacteristic, complemented
236D4—142no—
237D4—182no—
238D4—1262no—
239C2×C4—71no—
240C2×C4—91no—
241C2×C4—631no—
242C7#6414611yescharacteristic, complemented, Sylow
243C2×C3—44no—
244C2×C3—148no—
245C2×C3D7×#6714yes—
246C2×C3D3×C22⋊D714yes—
247C2×C3#6669414yes—
248D3—68no—
249D3—428no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—72no—
254C22—92no—
255C22—141no—
256C22—141no—
257C22—181no—
258C22—181no—
259C22—632no—
260C22—1268no—
261C22—1261no—
262C22—1261no—
263C22C2×D7×C3⋊D311yescharacteristic
264C4—141no—
265C4—181no—
266C4—1261no—
267C3#1062113914yescomplemented
268C2—41no—
269C2—142no—
270C2—182no—
271C2—1262no—
272C2D7×#63811yescharacteristic
273C2C3⋊D3×C22⋊D711yescharacteristic
274C2#1175958111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D7×C3⋊D3
commutator, socle, cosocleC22×C32×C7C23
FittingC23×C32×C7C22
hypercenterC23D7×C3⋊D3
hypocenterC32×C7#120
LCSG ▷ C22×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×D7×C3⋊D3C2semidirect
2. C2×#92726C2semidirect
3. C2×C7×#638C2semidirect
4. C2×C32×C22⋊D7C2semidirect
5. #11759168C2semidirect
6. #11759167C2semidirect
7. (C2×D7)⋊(C3⋊Dic3)C2semidirect
9. C23×C32×C7C22semidirect
17. C2×C3⋊Dic21C22semidirect
19. C2×C7×C3⋊Dic3C22semidirect
21. C2×C32×Dic7C22semidirect
25. C2×C3×C22⋊D7D3semidirect
36. C2×C3⋊D21D4semidirect
39. C2×C7×C3⋊D3D4semidirect
42. C2×C32×D7D4semidirect
48. C23×C3×C7C2×D3semidirect
60. C2×C3×Dic7C2×D3semidirect
64. C2×#638D7semidirect
78. C2×C22⋊D7C3⋊D3semidirect
95. C2×C3×D7#67semidirect
101. C23×C32C2×D7semidirect
112. C2×C3⋊Dic3C2×D7semidirect
113. C32×C7#120semidirect
115. C23×C7C2×C3⋊D3semidirect
126. C2×Dic7C2×C3⋊D3semidirect
154. C2×C3⋊D3C22⋊D7semidirect
172. C2×D7#638semidirect
177. C23×C3D3×D7semidirect
189. C3×C7#937semidirect
228. C32#6500semidirect
235. C23D7×C3⋊D3semidirect
242. C7#64146semidirect
267. C3#10621139semidirect

Maximal subgroups

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

subgroupindex
C22×D7×C3⋊D32
C2×#927262
C2×C7×#6382
C2×C32×C22⋊D72
#117591682
#117591672
(C2×D7)⋊(C3⋊Dic3)2
#106211393
#641467