Finite Group Databasebeta

← #67×D21 #11759624

Subgroups — #67×D21 #11759624

Order 1008.

232 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 54 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×C3×D21C211yesmaximal, characteristic, complemented
2C2×D3×D21C211yesmaximal, characteristic, complemented
3#92726C211yesmaximal, characteristic, complemented
4C3×C7×#67C211yesmaximal, characteristic, complemented
5#92689C211yesmaximal, characteristic, complemented
6#92688C211yesmaximal, characteristic, complemented
7Dic3×D21C211yesmaximal, characteristic, complemented
8D4×D21—31nomaximal
9D7×#67—31nomaximal
10C22×C32×C7C2211yescharacteristic, complemented
11C2×C3⋊D21C2211yescharacteristic, complemented
12C2×C3×D21—22no—
13C2×C3×D21C2211yescharacteristic
14C2×C3×D21C2211yescharacteristic, complemented
15C2×C3×D3×C7C2211yescharacteristic, complemented
16D3×D21—22no—
17C3⋊Dic21C2211yescharacteristic, complemented
18C3×C7×Dic3C2211yescharacteristic, complemented
19C22×D21—31no—
20C22×D21D311yescharacteristic, complemented
21C22×C3×D7—31no—
22C2×D3×D7—31no—
23C3×C7×D4—31no—
24#4268—31no—
25#4268—31no—
26#4268—31no—
27#4268—61no—
28D84—31no—
29C4×D21—31no—
30C7×#67D311yescharacteristic, complemented
31#4246—31no—
32#4245—31no—
33Dic3×D7—31no—
34D3×#67—71nomaximal, Hall
35C2×C32×C7—21no—
36C2×C32×C7C2311yescharacteristic
37C3⋊D21—21no—
38C3×D21—21no—
39C3×D21D412yescomplemented
40C3×D3×C7—21no—
41D4×D7—91noHall
42C22×C3×C7—21no—
43C22×C3×C7—31no—
44C22×C3×C7C2×D311yescharacteristic, complemented
45C22×C3×C7C2×D311yescharacteristic, complemented
46C2×D21—22no—
47C2×D21—31no—
48C2×D21—31no—
49C2×D21—61no—
50C2×D21—62no—
51C2×D21C2×D311yescharacteristic
52C2×D21C2×D311yescharacteristic, complemented
53C2×D3×C7C2×D311yescharacteristic, complemented
54C2×C3×D7—31no—
55C2×C3×D7—31no—
56C2×C3×D7—62no—
57D3×D7—62no—
58C3×C4×C7—31no—
59Dic21—31no—
60Dic21—31no—
61Dic21—61no—
62C7×Dic3C2×D311yescharacteristic, complemented
63C22×C3×D3—71no—
64C2×D32—71no—
65#638—71no—
66C3×#67D711yescharacteristic, complemented
67#626—71no—
68#625—71no—
69D3×Dic3—71no—
70C32×C7C2×D411yescharacteristic, complemented, Hall
71C22×D7—31no—
72C22×D7—91no—
73C7×D4—31no—
74C22⋊D7—91no—
75C22⋊D7—91no—
76D28—91no—
77C4×D7—91no—
78C2×#67—211no—
79D3×D4—211no—
80C2×C3×C7—21no—
81C2×C3×C7—21no—
82C2×C3×C7—21no—
83C2×C3×C7—41no—
84C2×C3×C7—61no—
85C2×C3×C7C22×D311yescharacteristic
86C2×C3×C7C22×D311yescharacteristic
87D21—21no—
88D21—61no—
89D21—61no—
90D21—121no—
91D21#6712yesdirect product factor
92C3×D7—32no—
93C3×D7—61no—
94D3×C7—21no—
95C22×C32C2×D711yescharacteristic, complemented
96C2×C3⋊D3—71no—
97C2×C3×D3—71no—
98C2×C3×D3—71no—
99C2×C3×D3—142no—
100C2×C3×D3C2×D711yescharacteristic, complemented
101D32—142no—
102C3⋊Dic3—71no—
103C3×Dic3C2×D711yescharacteristic, complemented
104C22×C7—31no—
105C22×C7D3211yescharacteristic, complemented
106C2×D7—31no—
107C2×D7—31no—
108C2×D7—62no—
109C2×D7—91no—
110C2×D7—182no—
111C4×C7—31no—
112Dic7—91no—
113C23×C3—211no—
114C22×D3—71no—
115C22×D3—211no—
116C22×D3—211no—
117C3×D4—31no—
118#67—211no—
119#67—211no—
120#67—211no—
121#67—211no—
122#67—211no—
123#67—421no—
124#67D2111yescharacteristic, direct product factor
125C2×Dic3—211no—
126D12—211no—
127C4×D3—211no—
128C3×C7—21no—
129C3×C7D3×D411yescharacteristic, complemented
130C3×C7C2×#6711yescharacteristic, complemented
131C2×C32—21no—
132C2×C32C22×D711yescharacteristic
133C3⋊D3—141no—
134C3×D3—21no—
135C3×D3—72no—
136C3×D3—141no—
137C2×D4—631noSylow
138C2×C7—21no—
139C2×C7—61no—
140C2×C7C2×D3211yescharacteristic
141D7—32no—
142D7—61no—
143D7—181no—
144C22×C3—21no—
145C22×C3—31no—
146C22×C3—211no—
147C22×C3—211no—
148C22×C3—422no—
149C22×C3D3×D711yescharacteristic, complemented
150C22×C3C2×D2111yescharacteristic, complemented
151C2×D3—71no—
152C2×D3—71no—
153C2×D3—142no—
154C2×D3—211no—
155C2×D3—211no—
156C2×D3—421no—
157C2×D3—422no—
158C2×D3—422no—
159C2×D3C2×D2111yescharacteristic, complemented
160C3×C4—31no—
161Dic3—211no—
162Dic3—211no—
163Dic3—421no—
164Dic3C2×D2111yescharacteristic, complemented
165C32D4×D711yescharacteristic, complemented, Sylow
166C23—211no—
167C23—631no—
168D4—31no—
169D4—631no—
170D4—631no—
171D4—631no—
172C2×C4—631no—
173C7D3×#6711yescharacteristic, complemented, Sylow
174C2×C3—21no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3—41no—
178C2×C3—61no—
179C2×C3—212no—
180C2×C3—421no—
181C2×C3C2×D3×D711yescharacteristic
182C2×C3C22×D2111yescharacteristic
183D3—21no—
184D3—72no—
185D3—141no—
186D3—421no—
187D3—421no—
188D3—841no—
189C22—31no—
190C22—211no—
191C22—211no—
192C22—422no—
193C22—631no—
194C22—1262no—
195C22D3×D2111yescharacteristic, complemented
196C4—31no—
197C4—631no—
198C3—21no—
199C3D7×#6711yescharacteristic, complemented
200C3D4×D2111yescharacteristic, complemented
201C2—21no—
202C2—61no—
203C2—212no—
204C2—421no—
205C2—1261no—
206C2C2×D3×D2111yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×D21
commutator, socle, cosocleC2×C32×C7C23
FittingC22×C32×C7C22
hypercenterC22D3×D21
hypocenterC32×C7C2×D4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×D21C2semidirect
2. C2×D3×D21C2semidirect
3. #92726C2semidirect
4. C3×C7×#67C2semidirect
5. #92689C2semidirect
6. #92688C2semidirect
7. Dic3×D21C2semidirect
10. C22×C32×C7C22semidirect
11. C2×C3⋊D21C22semidirect
14. C2×C3×D21C22semidirect
15. C2×C3×D3×C7C22semidirect
17. C3⋊Dic21C22semidirect
18. C3×C7×Dic3C22semidirect
20. C22×D21D3semidirect
30. C7×#67D3semidirect
39. C3×D21D4semidirect
44. C22×C3×C7C2×D3semidirect
45. C22×C3×C7C2×D3semidirect
52. C2×D21C2×D3semidirect
53. C2×D3×C7C2×D3semidirect
62. C7×Dic3C2×D3semidirect
66. C3×#67D7semidirect
70. C32×C7C2×D4semidirect
91. D21#67direct (N × H)
95. C22×C32C2×D7semidirect
100. C2×C3×D3C2×D7semidirect
103. C3×Dic3C2×D7semidirect
105. C22×C7D32semidirect
124. #67D21direct (N × H)
129. C3×C7D3×D4semidirect
130. C3×C7C2×#67semidirect
149. C22×C3D3×D7semidirect
150. C22×C3C2×D21semidirect
159. C2×D3C2×D21semidirect
164. Dic3C2×D21semidirect
165. C32D4×D7semidirect
173. C7D3×#67semidirect
195. C22D3×D21semidirect
199. C3D7×#67semidirect
200. C3D4×D21semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D212
C2×D3×D212
#927262
C3×C7×#672
#926892
#926882
Dic3×D212
D4×D213
D7×#673
D3×#677