Finite Group Databasebeta

← D21×C22⋊D7 #72246512

Subgroups — D21×C22⋊D7 #72246512

Order 2352.

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

normal subgroups only

This group has 264 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×C7×D21C211yesmaximal, characteristic, complemented
2C2×D7×D21C211yesmaximal, characteristic, complemented
3#11924293C211yesmaximal, characteristic, complemented
4C3×C7×C22⋊D7C211yesmaximal, characteristic, complemented
5#11924248C211yesmaximal, characteristic, complemented
6#11924247C211yesmaximal, characteristic, complemented
7Dic7×D21C211yesmaximal, characteristic, complemented
8D7×C22⋊D7—31nomaximal, Hall
9C22×C3×C72C2211yescharacteristic, complemented
10C2×C3⋊(C7⋊D7)C2211yescharacteristic, complemented
11C2×C7×D21—22no—
12C2×C7×D21C2211yescharacteristic
13C2×C7×D21C2211yescharacteristic, complemented
14C2×C3×C7×D7C2211yescharacteristic, complemented
15D7×D21—22no—
16C3⋊(C7⋊Dic7)C2211yescharacteristic, complemented
17C3×C7×Dic7C2211yescharacteristic, complemented
18C22×C7×D7—31no—
19C2×D72—31no—
20C22⋊(C7⋊D7)—31no—
21C7×C22⋊D7D311yescharacteristic, complemented
22#87641—31no—
23#87640—31no—
24D7×Dic7—31no—
25D4×D21—71nomaximal
26D3×C22⋊D7—71nomaximal
27C2×C3×C72—21no—
28C2×C3×C72C2311yescharacteristic
29C3⋊(C7⋊D7)—21no—
30C7×D21—21no—
31C7×D21D412yescomplemented
32C3×C7×D7—21no—
33C22×C72C2×D311yescharacteristic, complemented
34C2×C7⋊D7—31no—
35C2×C7×D7—31no—
36C2×C7×D7—31no—
37C2×C7×D7—62no—
38C2×C7×D7C2×D311yescharacteristic, complemented
39D72—62no—
40C7⋊Dic7—31no—
41C7×Dic7C2×D311yescharacteristic, complemented
42C22×D21—71no—
43C22×D21D711yescharacteristic, complemented
44C22×D3×C7—71no—
45C2×D3×D7—71no—
46C3×C7×D4—71no—
47#4268—71no—
48#4268—71no—
49#4268—71no—
50#4268—143no—
51D84—71no—
52C4×D21—71no—
53C3×C22⋊D7D711yescharacteristic, complemented
54#4247—71no—
55#4245—71no—
56D3×Dic7—71no—
57C3×C72C2×D411yescharacteristic, complemented, Hall
58C2×C22⋊D7—211no—
59D4×D7—211no—
60C2×C72—21no—
61C2×C72C22×D311yescharacteristic
62C7⋊D7—61no—
63C7×D7—21no—
64C7×D7—32no—
65C7×D7—61no—
66C22×C3×C7—23no—
67C22×C3×C7—71no—
68C22×C3×C7C2×D711yescharacteristic, complemented
69C22×C3×C7C2×D711yescharacteristic, complemented
70C2×D21—22no—
71C2×D21—71no—
72C2×D21—71no—
73C2×D21—143no—
74C2×D21—142no—
75C2×D21C2×D711yescharacteristic
76C2×D21C2×D711yescharacteristic, complemented
77C2×D3×C7—71no—
78C2×D3×C7—71no—
79C2×D3×C7—142no—
80C2×C3×D7C2×D711yescharacteristic, complemented
81D3×D7—142no—
82C3×C4×C7—71no—
83Dic21—71no—
84Dic21—71no—
85Dic21—143no—
86C3×Dic7C2×D711yescharacteristic, complemented
87C23×C7—211no—
88C22×D7—31no—
89C22×D7—211no—
90C22×D7—211no—
91C7×D4—71no—
92C22⋊D7—211no—
93C22⋊D7—211no—
94C22⋊D7—211no—
95C22⋊D7—211no—
96C22⋊D7—211no—
97C22⋊D7—423no—
98C22⋊D7D2111yescharacteristic, direct product factor
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×C7—141no—
109C2×C3×C7C22×D711yescharacteristic
110C2×C3×C7C22×D711yescharacteristic
111D21—21no—
112D21—141no—
113D21—141no—
114D21—283no—
115D21C22⋊D712yesdirect product factor
116C3×D7—21no—
117D3×C7—72no—
118D3×C7—141no—
119C22×C7—23no—
120C22×C7—71no—
121C22×C7—211no—
122C22×C7—211no—
123C22×C7—422no—
124C22×C7D3×D711yescharacteristic, complemented
125C22×C7C2×D2111yescharacteristic, complemented
126C2×D7—31no—
127C2×D7—31no—
128C2×D7—62no—
129C2×D7—211no—
130C2×D7—211no—
131C2×D7—422no—
132C2×D7—422no—
133C2×D7—423no—
134C2×D7C2×D2111yescharacteristic, complemented
135C4×C7—71no—
136Dic7—211no—
137Dic7—211no—
138Dic7—423no—
139Dic7C2×D2111yescharacteristic, complemented
140C22×D3—71no—
141C22×D3—491no—
142C3×D4—71no—
143#67—491no—
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—141no—
156C2×C7—212no—
157C2×C7—421no—
158C2×C7C2×D3×D711yescharacteristic
159C2×C7C22×D2111yescharacteristic
160D7—21no—
161D7—32no—
162D7—61no—
163D7—421no—
164D7—421no—
165D7—843no—
166C22×C3—71no—
167C22×C3D7211yescharacteristic, complemented
168C2×D3—71no—
169C2×D3—71no—
170C2×D3—142no—
171C2×D3—491no—
172C2×D3—982no—
173C3×C4—71no—
174Dic3—491no—
175C23—211no—
176C23—1471no—
177D4—71no—
178D4—1471no—
179D4—1471no—
180D4—1471no—
181C2×C4—1471no—
182C7—23no—
183C7D3×C22⋊D711yescharacteristic, complemented
184C7D4×D2111yescharacteristic, complemented
185C2×C3—21no—
186C2×C3—141no—
187C2×C3C2×D7211yescharacteristic
188D3—72no—
189D3—141no—
190D3—981no—
191C22—71no—
192C22—211no—
193C22—211no—
194C22—2942no—
195C22—422no—
196C22—1471no—
197C22D7×D2111yescharacteristic, complemented
198C4—71no—
199C4—1471no—
200C3D7×C22⋊D711yescharacteristic, complemented, Sylow
201C2—21no—
202C2—141no—
203C2—212no—
204C2—2941no—
205C2—421no—
206C2C2×D7×D2111yescharacteristic
subgroupG/H
center, FrattiniC2C2×D7×D21
commutator, socle, cosocleC2×C3×C72C23
FittingC22×C3×C72C22
hypercenterC22D7×D21
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×D7×D21C2semidirect
3. #11924293C2semidirect
4. C3×C7×C22⋊D7C2semidirect
5. #11924248C2semidirect
6. #11924247C2semidirect
7. Dic7×D21C2semidirect
9. C22×C3×C72C22semidirect
10. C2×C3⋊(C7⋊D7)C22semidirect
13. C2×C7×D21C22semidirect
14. C2×C3×C7×D7C22semidirect
16. C3⋊(C7⋊Dic7)C22semidirect
17. C3×C7×Dic7C22semidirect
21. C7×C22⋊D7D3semidirect
31. C7×D21D4semidirect
33. C22×C72C2×D3semidirect
38. C2×C7×D7C2×D3semidirect
41. C7×Dic7C2×D3semidirect
43. C22×D21D7semidirect
53. C3×C22⋊D7D7semidirect
57. C3×C72C2×D4semidirect
68. C22×C3×C7C2×D7semidirect
69. C22×C3×C7C2×D7semidirect
76. C2×D21C2×D7semidirect
80. C2×C3×D7C2×D7semidirect
86. C3×Dic7C2×D7semidirect
98. C22⋊D7D21direct (N × H)
102. C72D3×D4semidirect
115. D21C22⋊D7direct (N × H)
124. C22×C7D3×D7semidirect
125. C22×C7C2×D21semidirect
134. C2×D7C2×D21semidirect
139. Dic7C2×D21semidirect
148. C3×C7D4×D7semidirect
149. C3×C7C2×C22⋊D7semidirect
167. C22×C3D72semidirect
183. C7D3×C22⋊D7semidirect
184. C7D4×D21semidirect
197. C22D7×D21semidirect
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×D7×D212
#119242932
C3×C7×C22⋊D72
#119242482
#119242472
Dic7×D212
D7×C22⋊D73
D4×D217
D3×C22⋊D77