Finite Group Databasebeta

← C3×D7×C22⋊D7 #72246497

Subgroups — C3×D7×C22⋊D7 #72246497

Order 2352.

264 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 72 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×C7×D7C211yesmaximal, characteristic, complemented
2C2×C3×D72C211yesmaximal, characteristic, complemented
3C3×C22⋊(C7⋊D7)C211yesmaximal, characteristic, complemented
4C3×C7×C22⋊D7C211yesmaximal, characteristic, complemented
5C3×#87641C211yesmaximal, characteristic, complemented
6C3×#87640C211yesmaximal, characteristic, complemented
7C3×D7×Dic7C211yesmaximal, characteristic, complemented
8D7×C22⋊D7C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C72C2211yescharacteristic, complemented
10C2×C3×C7⋊D7C2211yescharacteristic, complemented
11C2×C3×C7×D7—22no—
12C2×C3×C7×D7C2211yescharacteristic
13C2×C3×C7×D7C2211yescharacteristic, complemented
14C2×C3×C7×D7C2211yescharacteristic, complemented
15C3×D72—22no—
16C3×C7⋊Dic7C2211yescharacteristic, complemented
17C3×C7×Dic7C2211yescharacteristic, complemented
18C22×C7×D7C2×C311yescharacteristic, complemented
19C2×D72C2×C311yescharacteristic, complemented
20C22⋊(C7⋊D7)C2×C311yescharacteristic, complemented
21C7×C22⋊D7C2×C311yescharacteristic, complemented
22#87641C2×C311yescharacteristic, complemented
23#87640C2×C311yescharacteristic, complemented
24D7×Dic7C2×C311yescharacteristic, complemented
25C2×C3×C22⋊D7—71nomaximal
26C3×D4×D7—71nomaximal
27C2×C3×C72—21no—
28C2×C3×C72C2311yescharacteristic
29C3×C7⋊D7—21no—
30C3×C7×D7—21no—
31C3×C7×D7—21no—
32C3×C7×D7D412yescomplemented
33C22×C72C22×C311yescharacteristic, complemented
34C2×C7⋊D7C22×C311yescharacteristic, complemented
35C2×C7×D7—22no—
36C2×C7×D7C22×C311yescharacteristic
37C2×C7×D7C22×C311yescharacteristic, complemented
38C2×C7×D7C22×C311yescharacteristic, complemented
39D72—22no—
40C7⋊Dic7C22×C311yescharacteristic, complemented
41C7×Dic7C22×C311yescharacteristic, complemented
42C23×C3×C7—71no—
43C22×C3×D7—71no—
44C22×C3×D7—71no—
45C22×C3×D7D711yescharacteristic, complemented
46C3×C7×D4—71no—
47C3×C22⋊D7—71no—
48C3×C22⋊D7—71no—
49C3×C22⋊D7—71no—
50C3×C22⋊D7—71no—
51C3×C22⋊D7—71no—
52C3×C22⋊D7—143no—
53C3×C22⋊D7D711yescharacteristic, direct product factor
54C2×C3×Dic7—71no—
55C3×D28—71no—
56C3×C4×D7—71no—
57C3×C72C2×D411yescharacteristic, complemented, Hall
58C2×C22⋊D7—71no—
59D4×D7—71no—
60C2×C72—21no—
61C2×C72C23×C311yescharacteristic
62C7⋊D7—21no—
63C7×D7—21no—
64C7×D7—21no—
65C7×D7C3×D412yescomplemented
66C22×C3×C7—23no—
67C22×C3×C7—71no—
68C22×C3×C7—71no—
69C22×C3×C7—71no—
70C22×C3×C7—142no—
71C22×C3×C7C2×D711yescharacteristic, complemented
72C22×C3×C7C2×D711yescharacteristic, complemented
73C2×C3×D7—22no—
74C2×C3×D7—71no—
75C2×C3×D7—71no—
76C2×C3×D7—143no—
77C2×C3×D7—142no—
78C2×C3×D7—142no—
79C2×C3×D7C2×D711yescharacteristic
80C2×C3×D7C2×D711yescharacteristic, complemented
81C2×C3×D7C2×D711yescharacteristic, complemented
82C3×C4×C7—71no—
83C3×Dic7—71no—
84C3×Dic7—71no—
85C3×Dic7—143no—
86C3×Dic7C2×D711yescharacteristic, complemented
87C23×C7—71no—
88C22×D7—71no—
89C22×D7—71no—
90C22×D7C3×D711yescharacteristic, complemented
91C7×D4—71no—
92C22⋊D7—71no—
93C22⋊D7—71no—
94C22⋊D7—71no—
95C22⋊D7—71no—
96C22⋊D7—71no—
97C22⋊D7—143no—
98C22⋊D7C3×D711yescharacteristic, direct product factor
99C2×Dic7—71no—
100D28—71no—
101C4×D7—71no—
102C72C2×C3×D411yescharacteristic, complemented, Sylow
103C2×C3×D4—491noHall
104C2×C3×C7—21no—
105C2×C3×C7—21no—
106C2×C3×C7—23no—
107C2×C3×C7—43no—
108C2×C3×C7—72no—
109C2×C3×C7—141no—
110C2×C3×C7—141no—
111C2×C3×C7C22×D711yescharacteristic
112C2×C3×C7C22×D711yescharacteristic
113C3×D7—21no—
114C3×D7—21no—
115C3×D7—141no—
116C3×D7—141no—
117C3×D7—283no—
118C3×D7C22⋊D712yesdirect product factor
119C22×C7—23no—
120C22×C7—71no—
121C22×C7—71no—
122C22×C7—71no—
123C22×C7—142no—
124C22×C7C2×C3×D711yescharacteristic, complemented
125C22×C7C2×C3×D711yescharacteristic, complemented
126C2×D7—22no—
127C2×D7—71no—
128C2×D7—71no—
129C2×D7—143no—
130C2×D7—142no—
131C2×D7—142no—
132C2×D7C2×C3×D711yescharacteristic
133C2×D7C2×C3×D711yescharacteristic, complemented
134C2×D7C2×C3×D711yescharacteristic, complemented
135C4×C7—71no—
136Dic7—71no—
137Dic7—71no—
138Dic7—143no—
139Dic7C2×C3×D711yescharacteristic, complemented
140C23×C3—71no—
141C23×C3—491no—
142C3×D4—71no—
143C3×D4—491no—
144C3×D4—491no—
145C3×D4—491no—
146C2×C3×C4—491no—
147C3×C7—23no—
148C3×C7D4×D711yescharacteristic, complemented
149C3×C7C2×C22⋊D711yescharacteristic, complemented
150C2×D4—491noSylow
151C2×C7—21no—
152C2×C7—21no—
153C2×C7—23no—
154C2×C7—43no—
155C2×C7—72no—
156C2×C7—141no—
157C2×C7—141no—
158C2×C7C22×C3×D711yescharacteristic
159C2×C7C22×C3×D711yescharacteristic
160D7—21no—
161D7—21no—
162D7—141no—
163D7—141no—
164D7—283no—
165D7C3×C22⋊D712yesdirect product factor
166C22×C3—71no—
167C22×C3—71no—
168C22×C3—71no—
169C22×C3—142no—
170C22×C3—491no—
171C22×C3—982no—
172C22×C3D7211yescharacteristic, complemented
173C3×C4—71no—
174C3×C4—491no—
175C23—71no—
176C23—491no—
177D4—71no—
178D4—491no—
179D4—491no—
180D4—491no—
181C2×C4—491no—
182C7—23no—
183C7C3×D4×D711yescharacteristic, complemented
184C7C2×C3×C22⋊D711yescharacteristic, complemented
185C2×C3—21no—
186C2×C3—72no—
187C2×C3—141no—
188C2×C3—141no—
189C2×C3—981no—
190C2×C3C2×D7211yescharacteristic
191C22—71no—
192C22—71no—
193C22—71no—
194C22—142no—
195C22—491no—
196C22—982no—
197C22C3×D7211yescharacteristic, complemented
198C4—71no—
199C4—491no—
200C3D7×C22⋊D711yescharacteristic, direct product factor, Sylow
201C2—21no—
202C2—72no—
203C2—141no—
204C2—141no—
205C2—981no—
206C2C2×C3×D7211yescharacteristic
subgroupG/H
centerC2×C3C2×D72
commutator, cosocleC2×C72C23×C3
FrattiniC2C2×C3×D72
FittingC22×C3×C72C22
socleC2×C3×C72C23
hypercenterC22×C3D72
hypocenterC72C2×C3×D4
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×D7C2semidirect
2. C2×C3×D72C2semidirect
3. C3×C22⋊(C7⋊D7)C2semidirect
4. C3×C7×C22⋊D7C2semidirect
5. C3×#87641C2semidirect
6. C3×#87640C2semidirect
7. C3×D7×Dic7C2semidirect
8. D7×C22⋊D7C3direct (N × H)
9. C22×C3×C72C22semidirect
10. C2×C3×C7⋊D7C22semidirect
13. C2×C3×C7×D7C22semidirect
14. C2×C3×C7×D7C22semidirect
16. C3×C7⋊Dic7C22semidirect
17. C3×C7×Dic7C22semidirect
18. C22×C7×D7C2×C3semidirect
19. C2×D72C2×C3semidirect
20. C22⋊(C7⋊D7)C2×C3semidirect
21. C7×C22⋊D7C2×C3semidirect
22. #87641C2×C3semidirect
23. #87640C2×C3semidirect
24. D7×Dic7C2×C3semidirect
32. C3×C7×D7D4semidirect
33. C22×C72C22×C3semidirect
34. C2×C7⋊D7C22×C3semidirect
37. C2×C7×D7C22×C3semidirect
38. C2×C7×D7C22×C3semidirect
40. C7⋊Dic7C22×C3semidirect
41. C7×Dic7C22×C3semidirect
45. C22×C3×D7D7semidirect
53. C3×C22⋊D7D7direct (N × H)
57. C3×C72C2×D4semidirect
65. C7×D7C3×D4semidirect
71. C22×C3×C7C2×D7semidirect
72. C22×C3×C7C2×D7semidirect
80. C2×C3×D7C2×D7semidirect
81. C2×C3×D7C2×D7semidirect
86. C3×Dic7C2×D7semidirect
90. C22×D7C3×D7semidirect
98. C22⋊D7C3×D7direct (N × H)
102. C72C2×C3×D4semidirect
118. C3×D7C22⋊D7direct (N × H)
124. C22×C7C2×C3×D7semidirect
125. C22×C7C2×C3×D7semidirect
133. C2×D7C2×C3×D7semidirect
134. C2×D7C2×C3×D7semidirect
139. Dic7C2×C3×D7semidirect
148. C3×C7D4×D7semidirect
149. C3×C7C2×C22⋊D7semidirect
165. D7C3×C22⋊D7direct (N × H)
172. C22×C3D72semidirect
183. C7C3×D4×D7semidirect
184. C7C2×C3×C22⋊D7semidirect
197. C22C3×D72semidirect
200. C3D7×C22⋊D7direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D72
C2×C3×D722
C3×C22⋊(C7⋊D7)2
C3×C7×C22⋊D72
C3×#876412
C3×#876402
C3×D7×Dic72
D7×C22⋊D73
C2×C3×C22⋊D77
C3×D4×D77