Finite Group Databasebeta

← #72246705

Subgroups — #72246705

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×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#66728C212yesmaximal, complemented
3#11759644C214yesmaximal, complemented
4#11759634C214yesmaximal, complemented
5#11759632C214yesmaximal, complemented
6#10621565—32nomaximal
7C22×C7×C3⋊D3C2212yescomplemented
8C2×(C3⋊D3)⋊D7C2214yescomplemented
9C2×C32×C4×C7C2211yescharacteristic, complemented
10C7×C4⋊(C3⋊D3)C2214yescomplemented
11C3×#4268C2214yescomplemented
12C3×D84C2212yescomplemented
13C3×C4×D21C2214yescomplemented
14C3×Dic42C2212yescomplemented
15#92694C2218yescomplemented
16#92693C2214yescomplemented
17#66728D312yescomplemented
18C2×C7×D12—32no—
19C2×C7×D12—61no—
20C3×#1157—32no—
21D3×C22⋊D7—38no—
22#66681—34no—
23D7×D12—34no—
24#66676—34no—
25#66672—34no—
26#64308—71nomaximal, Hall
27C22×C32×C7C2311yescharacteristic
28C2×C7×C3⋊D3—42no—
29C2×C7×C3⋊D3C2314yes—
30C2×C3×D21C2314yes—
31(C3⋊D3)⋊D7—44no—
32C32×C4×C7C2312yes—
33C3×Dic21C2314yes—
34#6548—91noHall
35C22×D3×C7—34no—
36C22×D3×C7—62no—
37C2×D3×D7—38no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7C2×D312yescomplemented
40#4268C2×D314yescomplemented
41D84C2×D312yescomplemented
42C4×D21C2×D314yescomplemented
43Dic42C2×D312yescomplemented
44C7×D12—34no—
45C7×D12—34no—
46C7×D12—64no—
47C3×C22⋊D7—34no—
48C3×D28—32no—
49C3×C4×D7—34no—
50C3×Dic14—32no—
51#4247—38no—
52#4245—38no—
53D3×Dic7—38no—
54C2×C4⋊(C3⋊D3)D711yescharacteristic, complemented
55C3×#235—72no—
56#3811—74no—
57D3×D12—74no—
58#3799—74no—
59C2×C32×C7—21no—
60C2×C32×C7C2411yescharacteristic
61C7×C3⋊D3—24no—
62C3×D21—24no—
63C2×C7×D4—91no—
64C2×C22⋊D7—94no—
65#1159—94no—
66D4×D7—94no—
67#1157—32no—
68#1009—212no—
69C22×C3×C7—21no—
70C22×C3×C7C22×D312yes—
71C2×D21C22×D314yes—
72C2×D3×C7—38no—
73C2×D3×C7—64no—
74C2×D3×C7—124no—
75C2×D3×C7—122no—
76C2×D3×C7—122no—
77C2×C3×D7—34no—
78D3×D7—128no—
79C3×C4×C7—22no—
80C3×C4×C7C22×D312yes—
81C3×C4×C7C22×D312yes—
82Dic21C22×D314yes—
83C3×Dic7—34no—
84C22×C3⋊D3C2×D712yescomplemented
85C2×D32—74no—
86C2×C32×C4C2×D711yescharacteristic, complemented
87C4⋊(C3⋊D3)C2×D714yescomplemented
88C3×#67—74no—
89C3×D12—72no—
90C3×C4×D3—74no—
91C3×Dic6—72no—
92#626—78no—
93#624—74no—
94C32×C7H(5,𝔽2)11yescharacteristic, complemented, Hall
95C23×C7—92no—
96C22×D7—94no—
97C7×D4—94no—
98C2×C4×C7D3211yescharacteristic, complemented
99C22⋊D7—34no—
100C22⋊D7—98no—
101C2×Dic7—94no—
102D28—32no—
103C4×D7—34no—
104Dic14—32no—
105C3×#41—212no—
106#239—214no—
107D3×D4—214no—
108D3×D4—218no—
109#235—72no—
110#235—214no—
111C2×D12—32no—
112C2×D12—61no—
113C2×D12—214no—
114C2×C3×C7—21no—
115C2×C3×C7—21no—
116C2×C3×C7—22no—
117C2×C3×C7—21no—
118C2×C3×C7C23×D312yes—
119D21—24no—
120C3×D7—64no—
121D3×C7—68no—
122D3×C7—124no—
123C22×C32C22×D711yescharacteristic
124C2×C3⋊D3—42no—
125C2×C3⋊D3C22×D714yes—
126C2×C3×D3—74no—
127D32—284no—
128C32×C4C22×D712yes—
129C3×Dic3—74no—
130H(5,𝔽2)—631noSylow
131C22×C7—94no—
132C22×C7—362no—
133C22×C7C2×D3211yescharacteristic
134C2×D7—34no—
135C2×D7—364no—
136C4×C7C2×D3212yes—
137Dic7—34no—
138C22×D3—34no—
139C22×D3—62no—
140C22×D3—218no—
141C3×Q8—212no—
142C3×D4—214no—
143C3×D4—212no—
144C2×C3×C4—21no—
145C2×C3×C4—214no—
146C2×C3×C4D3×D712yescomplemented
147#67—74no—
148#67—218no—
149D12—34no—
150D12—34no—
151D12—64no—
152D12—72no—
153D12—218no—
154C4×D3—74no—
155C4×D3—218no—
156Dic6—72no—
157C3×C7—21no—
158C3×C7#100912yescomplemented
159C2×C32—21no—
160C2×C32C23×D711yescharacteristic
161C3⋊D3—24no—
162C3×D3—144no—
163#41—212no—
164#41—634no—
165C2×D4—91no—
166C2×D4—634no—
167C2×D4—634no—
168C2×C7—21no—
169C2×C7—184no—
170C2×C7C22×D3211yescharacteristic
171D7—64no—
172C22×C3—21no—
173C22×C3—214no—
174C22×C3C2×D3×D712yes—
175C2×D3—38no—
176C2×D3—64no—
177C2×D3—74no—
178C2×D3—124no—
179C2×D3—122no—
180C2×D3—122no—
181C2×D3—848no—
182C3×C4—22no—
183C3×C4—214no—
184C3×C4C2×D3×D712yes—
185C3×C4C2×D3×D712yes—
186Dic3—74no—
187C32#654811yescharacteristic, complemented, Sylow
188C23—92no—
189C23—634no—
190Q8—212no—
191D4—94no—
192D4—214no—
193D4—212no—
194D4—638no—
195C2×C4—214no—
196C2×C4—634no—
197C2×C4(C3⋊D3)⋊D711yescharacteristic, complemented
198C7#6430811yescharacteristic, complemented, Sylow
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—22no—
202C2×C3—21no—
203C2×C3—424no—
204C2×C3C22×D3×D712yes—
205D3—68no—
206D3—124no—
207D3—144no—
208C22—94no—
209C22—214no—
210C22—362no—
211C22—2524no—
212C22C2×(C3⋊D3)⋊D711yescharacteristic
213C4—214no—
214C4C2×(C3⋊D3)⋊D712yes—
215C3—21no—
216C3#1062156512yescomplemented
217C2—21no—
218C2—184no—
219C2—424no—
220C2C22×(C3⋊D3)⋊D711yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D7
commutator, socle, cosocleC2×C32×C7C24
FittingC2×C32×C4×C7C22
hypercenterC2×C4(C3⋊D3)⋊D7
hypocenterC32×C7H(5,𝔽2)
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C7×C4⋊(C3⋊D3)C2semidirect
2. C3×#66728C2semidirect
3. #11759644C2semidirect
4. #11759634C2semidirect
5. #11759632C2semidirect
7. C22×C7×C3⋊D3C22semidirect
8. C2×(C3⋊D3)⋊D7C22semidirect
9. C2×C32×C4×C7C22semidirect
10. C7×C4⋊(C3⋊D3)C22semidirect
11. C3×#4268C22semidirect
12. C3×D84C22semidirect
13. C3×C4×D21C22semidirect
14. C3×Dic42C22semidirect
15. #92694C22semidirect
16. #92693C22semidirect
17. #66728D3semidirect
39. C2×C3×C4×C7C2×D3semidirect
40. #4268C2×D3semidirect
41. D84C2×D3semidirect
42. C4×D21C2×D3semidirect
43. Dic42C2×D3semidirect
54. C2×C4⋊(C3⋊D3)D7semidirect
84. C22×C3⋊D3C2×D7semidirect
86. C2×C32×C4C2×D7semidirect
87. C4⋊(C3⋊D3)C2×D7semidirect
94. C32×C7H(5,𝔽2)semidirect
98. C2×C4×C7D32semidirect
146. C2×C3×C4D3×D7semidirect
158. C3×C7#1009semidirect
187. C32#6548semidirect
197. C2×C4(C3⋊D3)⋊D7semidirect
198. C7#64308semidirect
216. C3#10621565semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C4⋊(C3⋊D3)2
C3×#667282
#117596442
#117596342
#117596322
#106215653
#643087