Finite Group Databasebeta

← #72246768

Subgroups — #72246768

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×C4⋊(C3⋊D21)C211yesmaximal, characteristic, complemented
3C2×#92678C211yesmaximal, characteristic, complemented
4C2×#92677C211yesmaximal, characteristic, complemented
5C7×#3752C211yesmaximal, characteristic, complemented
6C32×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
7#11759168C211yesmaximal, characteristic, complemented
8#10621038—34nomaximal
9C22×C3⋊D21C2211yescharacteristic
10C22×C3⋊D21C2211yescharacteristic, complemented
11C22×C7×C3⋊D3C2211yescharacteristic
12C22×C32×D7C2211yescharacteristic
13C2×D7×C3⋊D3—22no—
14C2×D7×C3⋊D3—22no—
15C2×D7×C3⋊D3—22no—
16C2×C32×C4×C7C2211yescharacteristic, complemented
17C4⋊(C3⋊D21)—22no—
18C2×C7×C3⋊Dic3C2211yescharacteristic, complemented
19C2×C32×Dic7C2211yescharacteristic, complemented
20#92678—22no—
21#92677—22no—
22C22×D3×D7—34no—
23C2×D84—34no—
24C2×#4247—34no—
25C2×#4246—34no—
26C7×(C2×D3)⋊C4—34no—
27C3×(C2×D7)⋊C4D314yescomplemented
28#66575—34no—
29#64091—71nomaximal, Hall
30C22×C32×C7C2311yescharacteristic
31C2×C3⋊D21—21no—
32C2×C3⋊D21—21no—
33C2×C3⋊D21—21no—
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—
44C32×C4×C7—21no—
45C7×C3⋊Dic3—21no—
46C32×Dic7—21no—
47#6445—91noHall
48C22×D21—34no—
49C22×D21—34no—
50C22×D3×C7—34no—
51C22×C3×D7C2×D314yes—
52C2×D3×D7—68no—
53C2×D3×D7—68no—
54C2×D3×D7—68no—
55C2×C3×C4×C7C2×D314yescomplemented
56D84—68no—
57C2×C7×Dic3—34no—
58C2×C3×Dic7C2×D314yescomplemented
59#4247—68no—
60#4246—68no—
61C23×C3⋊D3—71no—
62C2×#638—71no—
63C2×C4⋊(C3⋊D3)—71no—
64C2×C4⋊(C3⋊D3)—71no—
65C32×C22⋊C4—71no—
66#3752—71no—
67#3752D711yescharacteristic, complemented
68C2×C32×C7C2×D411yescharacteristic
69C2×C32×C7C2×D411yescharacteristic
70C2×C32×C7C2×D411yescharacteristic
71C3⋊D21—22no—
72C3⋊D21—41no—
73C7×C3⋊D3—22no—
74C32×D7—22no—
75C23×D7—91no—
76C2×C22⋊D7—91no—
77C2×D28—91no—
78C2×D28—91no—
79C7×C22⋊C4—91no—
80(C2×D7)⋊C4—91no—
81(C2×D7)⋊C4C3⋊D311yescharacteristic, complemented
82#882—214no—
83C22×C3×C7C22×D314yes—
84C2×D21—38no—
85C2×D21—64no—
86C2×D21—64no—
87C2×D21—64no—
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×D7D1218yescomplemented
96D3×D7—632no—
97C3×C4×C7—24no—
98C3×Dic7—24no—
99C7×Dic3—64no—
100C23×C32—71no—
101C22×C3⋊D3—71no—
102C22×C3⋊D3—71no—
103C22×C3⋊D3—142no—
104C22×C3⋊D3—142no—
105C22×C3⋊D3—142no—
106C22×C3⋊D3C2×D711yescharacteristic
107C2×C32×C4—71no—
108C2×C32×C4C2×D711yescharacteristic, complemented
109#638—142no—
110C2×C3⋊Dic3C2×D711yescharacteristic, complemented
111C4⋊(C3⋊D3)—142no—
112C4⋊(C3⋊D3)—142no—
113C32×C7#12011yescharacteristic, complemented, Hall
114C23×C7—91no—
115C22×D7—91no—
116C22×D7—91no—
117C22×D7—182no—
118C22×D7—182no—
119C22×D7—182no—
120C22×D7C2×C3⋊D311yescharacteristic
121C2×C4×C7—91no—
122C2×C4×C7C2×C3⋊D311yescharacteristic, complemented
123C22⋊D7—182no—
124C2×Dic7C2×C3⋊D311yescharacteristic, complemented
125D28—182no—
126D28—182no—
127C23×D3—214no—
128C2×#67—214no—
129C2×D12—214no—
130C2×D12—214no—
131C3×C22⋊C4—74no—
132(C2×D3)⋊C4—34no—
133(C2×D3)⋊C4—214no—
134C2×C3×C7C2×D1214yes—
135C2×C3×C7D3×D414yes—
136C2×C3×C7D3×D414yes—
137D21—68no—
138D21—124no—
139C3×D7—28no—
140D3×C7—68no—
141C22×C32—72no—
142C22×C32—141no—
143C22×C32—141no—
144C22×C32C22×D711yescharacteristic
145C2×C3⋊D3—21no—
146C2×C3⋊D3—21no—
147C2×C3⋊D3—72no—
148C2×C3⋊D3—148no—
149C2×C3⋊D3—141no—
150C2×C3⋊D3—141no—
151C2×C3⋊D3—141no—
152C2×C3⋊D3—141no—
153C2×C3⋊D3—141no—
154C2×C3⋊D3D2812yescomplemented
155C32×C4—21no—
156C32×C4—141no—
157C3⋊Dic3—21no—
158#120—631noSylow
159C22×C7—92no—
160C22×C7—181no—
161C22×C7—181no—
162C22×C7C22×C3⋊D311yescharacteristic
163C2×D7—21no—
164C2×D7—21no—
165C2×D7—92no—
166C2×D7—188no—
167C2×D7—181no—
168C2×D7—181no—
169C2×D7—181no—
170C2×D7—181no—
171C2×D7—181no—
172C2×D7C4⋊(C3⋊D3)12yescomplemented
173C4×C7—21no—
174C4×C7—181no—
175Dic7—21no—
176C23×C3—74no—
177C22×D3—34no—
178C22×D3—214no—
179C22×D3—214no—
180C22×D3—428no—
181C22×D3—428no—
182C22×D3—428no—
183C2×C3×C4—74no—
184C2×C3×C4D3×D714yescomplemented
185#67—428no—
186C2×Dic3—34no—
187D12—428no—
188D12—428no—
189C3×C7#88214yescomplemented
190C2×C32—142no—
191C2×C32C2×D2811yescharacteristic
192C2×C32D4×D711yescharacteristic
193C2×C32D4×D711yescharacteristic
194C3⋊D3—22no—
195C3⋊D3—142no—
196C3⋊D3—281no—
197C24—631no—
198C2×D4—631no—
199C2×D4—631no—
200C2×D4—631no—
201C22⋊C4—71no—
202C22⋊C4—91no—
203C22⋊C4—631no—
204C2×C7—182no—
205C2×C7C2×C4⋊(C3⋊D3)11yescharacteristic
206C2×C7D4×C3⋊D311yescharacteristic
207C2×C7D4×C3⋊D311yescharacteristic
208D7—22no—
209D7—182no—
210D7—361no—
211C22×C3—78no—
212C22×C3—144no—
213C22×C3—144no—
214C22×C3C2×D3×D714yes—
215C2×D3—38no—
216C2×D3—64no—
217C2×D3—64no—
218C2×D3—218no—
219C2×D3—4232no—
220C2×D3—424no—
221C2×D3—424no—
222C2×D3—424no—
223C2×D3—424no—
224C2×D3—424no—
225C3×C4—24no—
226C3×C4—144no—
227Dic3—64no—
228C32#644511yescharacteristic, complemented, Sylow
229C23—71no—
230C23—91no—
231C23—631no—
232C23—631no—
233C23—1262no—
234C23—1262no—
235C23—1262no—
236D4—1262no—
237D4—1262no—
238D4—1262no—
239C2×C4—71no—
240C2×C4—91no—
241C2×C4D7×C3⋊D311yescharacteristic, complemented
242C7#6409111yescharacteristic, complemented, Sylow
243C2×C3—148no—
244C2×C3D7×D1214yes—
245C2×C3D3×D2814yes—
246C2×C3#6669414yes—
247D3—68no—
248D3—428no—
249D3—844no—
250C22—72no—
251C22—92no—
252C22—141no—
253C22—141no—
254C22—181no—
255C22—181no—
256C22—632no—
257C22—1268no—
258C22—1261no—
259C22—1261no—
260C22—1261no—
261C22—1261no—
262C22—1261no—
263C22C2×D7×C3⋊D311yescharacteristic
264C4—21no—
265C4—141no—
266C4—181no—
267C3#1062103814yescomplemented
268C2—142no—
269C2—182no—
270C2—1262no—
271C2—2521no—
272C2D7×C4⋊(C3⋊D3)11yescharacteristic
273C2C3⋊D3×D2811yescharacteristic
274C2#1175958111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D7×C3⋊D3
commutator, socle, cosocleC22×C32×C7C23
FittingC2×C32×C4×C7C22
hypercenterC2×C4D7×C3⋊D3
hypocenterC32×C7#120
LCSG ▷ C22×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D7×C3⋊D3C2semidirect
2. C2×C4⋊(C3⋊D21)C2semidirect
3. C2×#92678C2semidirect
4. C2×#92677C2semidirect
5. C7×#3752C2semidirect
6. C32×(C2×D7)⋊C4C2semidirect
7. #11759168C2semidirect
10. C22×C3⋊D21C22semidirect
16. C2×C32×C4×C7C22semidirect
18. C2×C7×C3⋊Dic3C22semidirect
19. C2×C32×Dic7C22semidirect
27. C3×(C2×D7)⋊C4D3semidirect
36. C2×C3⋊D21D4semidirect
39. C2×C7×C3⋊D3D4semidirect
42. C2×C32×D7D4semidirect
55. C2×C3×C4×C7C2×D3semidirect
58. C2×C3×Dic7C2×D3semidirect
67. #3752D7semidirect
81. (C2×D7)⋊C4C3⋊D3semidirect
95. C2×C3×D7D12semidirect
108. C2×C32×C4C2×D7semidirect
110. C2×C3⋊Dic3C2×D7semidirect
113. C32×C7#120semidirect
122. C2×C4×C7C2×C3⋊D3semidirect
124. C2×Dic7C2×C3⋊D3semidirect
154. C2×C3⋊D3D28semidirect
172. C2×D7C4⋊(C3⋊D3)semidirect
184. C2×C3×C4D3×D7semidirect
189. C3×C7#882semidirect
228. C32#6445semidirect
241. C2×C4D7×C3⋊D3semidirect
242. C7#64091semidirect
267. C3#10621038semidirect

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×C4⋊(C3⋊D21)2
C2×#926782
C2×#926772
C7×#37522
C32×(C2×D7)⋊C42
#117591682
#106210383
#640917