Finite Group Databasebeta

← #72246551

Subgroups — #72246551

Order 2352.

264 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 51 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×C7×D21C211yesmaximal, characteristic, complemented
2C2×D21⋊D7C211yesmaximal, characteristic, complemented
3C7×#4268C211yesmaximal, characteristic, complemented
4C3×C22⋊(C7⋊D7)C211yesmaximal, characteristic, complemented
5#11924254C211yesmaximal, characteristic, complemented
6#11924253C211yesmaximal, characteristic, complemented
7D21⋊Dic7C211yesmaximal, characteristic, complemented
8D7×C22⋊D7—31nomaximal, Hall
9C22×C3×C72C2211yescharacteristic, complemented
10C2×C7×D21—22no—
11C2×C7×D21C2211yescharacteristic
12C2×C7×D21C2211yescharacteristic, complemented
13C2×C7×D21C2211yescharacteristic, complemented
14C2×C3×C7⋊D7C2211yescharacteristic, complemented
15D21⋊D7—22no—
16C7×Dic21C2211yescharacteristic, complemented
17C3×C7⋊Dic7C2211yescharacteristic, complemented
18C22×C7×D7—31no—
19C2×D72—31no—
20C22⋊(C7⋊D7)D311yescharacteristic, complemented
21C7×C22⋊D7—31no—
22#87641—31no—
23#87640—31no—
24D7×Dic7—31no—
25#66694—71nomaximal
26D3×C22⋊D7—71nomaximal
27C2×C3×C72—21no—
28C2×C3×C72C2311yescharacteristic
29C7×D21—21no—
30C7×D21—21no—
31C7×D21D412yescomplemented
32C3×C7⋊D7—21no—
33C22×C72C2×D311yescharacteristic, complemented
34C2×C7⋊D7C2×D311yescharacteristic, complemented
35C2×C7×D7—31no—
36C2×C7×D7—31no—
37C2×C7×D7—31no—
38C2×C7×D7—62no—
39D72—62no—
40C7⋊Dic7C2×D311yescharacteristic, complemented
41C7×Dic7—31no—
42C22×D21D711yescharacteristic, complemented
43C22×D3×C7—71no—
44C2×D3×D7—71no—
45C2×D3×D7—71no—
46#4268D711yescharacteristic, complemented
47C7×#67—71no—
48C3×C22⋊D7—71no—
49C3×C22⋊D7—71no—
50C3×C22⋊D7—143no—
51#4247—71no—
52#4247—71no—
53#4246—71no—
54#4245—71no—
55D21⋊C4—71no—
56D3×Dic7—71no—
57C3×C72C2×D411yescharacteristic, complemented, Hall
58C2×C22⋊D7—211no—
59D4×D7—211no—
60C2×C72—21no—
61C2×C72C22×D311yescharacteristic
62C7⋊D7—21no—
63C7×D7—32no—
64C7×D7—61no—
65C7×D7—61no—
66C22×C3×C7—23no—
67C22×C3×C7C2×D711yescharacteristic, complemented
68C22×C3×C7C2×D711yescharacteristic, complemented
69C2×D21—22no—
70C2×D21C2×D711yescharacteristic
71C2×D21C2×D711yescharacteristic, complemented
72C2×D21C2×D711yescharacteristic, complemented
73C2×D3×C7—71no—
74C2×D3×C7—71no—
75C2×D3×C7—71no—
76C2×D3×C7—142no—
77C2×C3×D7—71no—
78C2×C3×D7—71no—
79C2×C3×D7—143no—
80D3×D7—142no—
81D3×D7—142no—
82Dic21C2×D711yescharacteristic, complemented
83C3×Dic7—71no—
84C3×Dic7—71no—
85C3×Dic7—143no—
86C7×Dic3—71no—
87C23×C7—211no—
88C22×D7—31no—
89C22×D7—211no—
90C22×D7—211no—
91C7×D4—211no—
92C22⋊D7—31no—
93C22⋊D7—71no—
94C22⋊D7—71no—
95C22⋊D7—143no—
96C22⋊D7—211no—
97C22⋊D7—211no—
98C22⋊D7—211no—
99C2×Dic7—211no—
100D28—211no—
101C4×D7—211no—
102C72D3×D411yescharacteristic, complemented, Sylow
103D3×D4—491noHall
104C2×C3×C7—21no—
105C2×C3×C7—21no—
106C2×C3×C7—23no—
107C2×C3×C7—43no—
108C2×C3×C7C22×D711yescharacteristic
109C2×C3×C7C22×D711yescharacteristic
110D21—21no—
111D21—21no—
112D21C22⋊D712yescomplemented
113C3×D7—141no—
114C3×D7—141no—
115C3×D7—283no—
116D3×C7—72no—
117D3×C7—141no—
118D3×C7—141no—
119C22×C7—23no—
120C22×C7—211no—
121C22×C7—211no—
122C22×C7—211no—
123C22×C7—422no—
124C22×C7D3×D711yescharacteristic, complemented
125C22×C7D3×D711yescharacteristic, complemented
126C2×D7—31no—
127C2×D7—31no—
128C2×D7—31no—
129C2×D7—62no—
130C2×D7—71no—
131C2×D7—71no—
132C2×D7—143no—
133C2×D7—422no—
134C2×D7—422no—
135C4×C7—211no—
136Dic7—31no—
137Dic7—71no—
138Dic7—71no—
139Dic7—143no—
140C22×D3—71no—
141C22×D3—491no—
142C3×D4—491no—
143#67—71no—
144#67—491no—
145D12—491no—
146C4×D3—491no—
147C3×C7—23no—
148C3×C7D4×D711yescharacteristic, complemented
149C3×C7C2×C22⋊D711yescharacteristic, complemented
150C2×D4—1471noSylow
151C2×C7—21no—
152C2×C7—21no—
153C2×C7—23no—
154C2×C7—43no—
155C2×C7—212no—
156C2×C7—421no—
157C2×C7—421no—
158C2×C7C2×D3×D711yescharacteristic
159C2×C7C2×D3×D711yescharacteristic
160D7—32no—
161D7—61no—
162D7—61no—
163D7—141no—
164D7—141no—
165D7—283no—
166C22×C3—491no—
167C22×C3D7211yescharacteristic, complemented
168C2×D3—71no—
169C2×D3—71no—
170C2×D3—71no—
171C2×D3—142no—
172C2×D3—982no—
173C3×C4—491no—
174Dic3—71no—
175C23—211no—
176C23—1471no—
177D4—211no—
178D4—491no—
179D4—1471no—
180D4—1471no—
181C2×C4—1471no—
182C7—23no—
183C7D3×C22⋊D711yescharacteristic, complemented
184C7#6669411yescharacteristic, complemented
185C2×C3—21no—
186C2×C3—981no—
187C2×C3C2×D7211yescharacteristic
188D3—72no—
189D3—141no—
190D3—141no—
191C22—211no—
192C22—211no—
193C22—211no—
194C22—2942no—
195C22—422no—
196C22—491no—
197C22D21⋊D711yescharacteristic, complemented
198C4—211no—
199C4—491no—
200C3D7×C22⋊D711yescharacteristic, complemented, Sylow
201C2—21no—
202C2—212no—
203C2—421no—
204C2—421no—
205C2—981no—
206C2C2×D21⋊D711yescharacteristic
subgroupG/H
center, FrattiniC2C2×D21⋊D7
commutator, socle, cosocleC2×C3×C72C23
FittingC22×C3×C72C22
hypercenterC22D21⋊D7
hypocenterC3×C72C2×D4
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C7×D21C2semidirect
2. C2×D21⋊D7C2semidirect
3. C7×#4268C2semidirect
4. C3×C22⋊(C7⋊D7)C2semidirect
5. #11924254C2semidirect
6. #11924253C2semidirect
7. D21⋊Dic7C2semidirect
9. C22×C3×C72C22semidirect
12. C2×C7×D21C22semidirect
13. C2×C7×D21C22semidirect
14. C2×C3×C7⋊D7C22semidirect
16. C7×Dic21C22semidirect
17. C3×C7⋊Dic7C22semidirect
20. C22⋊(C7⋊D7)D3semidirect
31. C7×D21D4semidirect
33. C22×C72C2×D3semidirect
34. C2×C7⋊D7C2×D3semidirect
40. C7⋊Dic7C2×D3semidirect
42. C22×D21D7semidirect
46. #4268D7semidirect
57. C3×C72C2×D4semidirect
67. C22×C3×C7C2×D7semidirect
68. C22×C3×C7C2×D7semidirect
71. C2×D21C2×D7semidirect
72. C2×D21C2×D7semidirect
82. Dic21C2×D7semidirect
102. C72D3×D4semidirect
112. D21C22⋊D7semidirect
124. C22×C7D3×D7semidirect
125. C22×C7D3×D7semidirect
148. C3×C7D4×D7semidirect
149. C3×C7C2×C22⋊D7semidirect
167. C22×C3D72semidirect
183. C7D3×C22⋊D7semidirect
184. C7#66694semidirect
197. C22D21⋊D7semidirect
200. C3D7×C22⋊D7semidirect

Maximal subgroups

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

subgroupindex
C22×C7×D212
C2×D21⋊D72
C7×#42682
C3×C22⋊(C7⋊D7)2
#119242542
#119242532
D21⋊Dic72
D7×C22⋊D73
#666947
D3×C22⋊D77