Finite Group Databasebeta

← #72246518

Subgroups — #72246518

Order 2352.

264 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 50 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×D21⋊D7C211yesmaximal, characteristic, complemented
2C2×D7×D21C211yesmaximal, characteristic, complemented
3#11924248C211yesmaximal, characteristic, complemented
4#11924241C211yesmaximal, characteristic, complemented
5C7×#4246C211yesmaximal, characteristic, complemented
6C7×D21⋊C4C211yesmaximal, characteristic, complemented
7C3×#87641C211yesmaximal, characteristic, complemented
8D7×D28—31nomaximal, Hall
9C2×C3⋊(C7⋊D7)C2211yescharacteristic, complemented
10C2×C7×D21C2211yescharacteristic
11C2×C7×D21C2211yescharacteristic, complemented
12C2×C3×C7⋊D7C2211yescharacteristic, complemented
13C2×C3×C7×D7C2211yescharacteristic, complemented
14D21⋊D7—22no—
15D7×D21—22no—
16C72×Dic3C2211yescharacteristic, complemented
17C3×C7×Dic7C2211yescharacteristic, complemented
18C2×D72—31no—
19C2×D72—31no—
20C4⋊(C7⋊D7)—31no—
21C7×D28—31no—
22C4×C7×D7—31no—
23#87641—31no—
24#87641D311yescharacteristic, complemented
25#66694—71nomaximal
26D3×D28—71nomaximal
27C2×C3×C72C2311yescharacteristic
28C3⋊(C7⋊D7)—21no—
29C7×D21—21no—
30C7×D21D412yescomplemented
31C3×C7⋊D7—21no—
32C3×C7×D7—21no—
33C2×C7⋊D7—31no—
34C2×C7⋊D7C2×D311yescharacteristic, complemented
35C2×C7×D7—31no—
36C2×C7×D7—31no—
37C2×C7×D7C2×D311yescharacteristic, complemented
38D72—62no—
39D72—62no—
40C4×C72—31no—
41C7×Dic7C2×D311yescharacteristic, complemented
42C22×D21—71no—
43C2×D3×D7—71no—
44C2×D3×D7—71no—
45C2×D3×D7—71no—
46D84—71no—
47C7×#67—71no—
48C4×D3×C7—71no—
49C3×C22⋊D7—71no—
50C3×D28—71no—
51#4247—71no—
52#4246—71no—
53#4246—71no—
54#4246—143no—
55#4246D711yescharacteristic, complemented
56D21⋊C4D711yescharacteristic, complemented
57C3×C72C2×D411yescharacteristic, complemented, Hall
58D4×D7—211no—
59C2×D28—211no—
60C2×C72C22×D311yescharacteristic
61C7⋊D7—21no—
62C7⋊D7—61no—
63C7×D7—21no—
64C7×D7—32no—
65C7×D7—61no—
66C22×C3×C7—71no—
67C2×D21—71no—
68C2×D21—71no—
69C2×D21—143no—
70C2×D21—142no—
71C2×D21C2×D711yescharacteristic
72C2×D21C2×D711yescharacteristic, complemented
73C2×D3×C7—71no—
74C2×D3×C7—71no—
75C2×C3×D7—71no—
76C2×C3×D7—71no—
77C2×C3×D7—143no—
78C2×C3×D7C2×D711yescharacteristic, complemented
79D3×D7—142no—
80D3×D7—142no—
81D3×D7—142no—
82C3×C4×C7—71no—
83C3×Dic7C2×D711yescharacteristic, complemented
84C7×Dic3—23no—
85C7×Dic3C2×D711yescharacteristic, complemented
86C7×Dic3C2×D711yescharacteristic, complemented
87C22×D7—211no—
88C22×D7—211no—
89C22×D7—211no—
90C22×D7—211no—
91C7×D4—211no—
92C2×C4×C7—211no—
93C22⋊D7—71no—
94C22⋊D7—211no—
95D28—31no—
96D28—71no—
97D28—211no—
98D28—211no—
99D28—211no—
100D28—423no—
101C4×D7—31no—
102C72D3×D411yescharacteristic, complemented, Sylow
103D3×D4—491noHall
104C2×C3×C7—23no—
105C2×C3×C7—141no—
106C2×C3×C7C22×D711yescharacteristic
107C2×C3×C7C22×D711yescharacteristic
108D21—21no—
109D21—141no—
110D21—141no—
111D21—283no—
112D21D2812yescomplemented
113C3×D7—21no—
114C3×D7—141no—
115C3×D7—141no—
116C3×D7—283no—
117D3×C7—72no—
118D3×C7—141no—
119C22×C7—71no—
120C22×C7—211no—
121C22×C7—211no—
122C2×D7—31no—
123C2×D7—31no—
124C2×D7—71no—
125C2×D7—71no—
126C2×D7—143no—
127C2×D7—211no—
128C2×D7—211no—
129C2×D7—422no—
130C2×D7—422no—
131C2×D7—422no—
132C2×D7—422no—
133C2×D7—423no—
134C2×D7D3×D711yescharacteristic, complemented
135C4×C7—31no—
136C4×C7—31no—
137C4×C7—63no—
138C4×C7—71no—
139Dic7D3×D711yescharacteristic, complemented
140C22×D3—491no—
141C22×D3—491no—
142C3×D4—491no—
143#67—71no—
144#67—491no—
145D12—491no—
146C4×D3—71no—
147C3×C7—23no—
148C3×C7C2×D2811yescharacteristic, complemented
149C3×C7D4×D711yescharacteristic, complemented
150C2×D4—1471noSylow
151C2×C7—23no—
152C2×C7—141no—
153C2×C7—212no—
154C2×C7—421no—
155C2×C7C2×D3×D711yescharacteristic
156C2×C7C2×D3×D711yescharacteristic
157D7—21no—
158D7—32no—
159D7—61no—
160D7—141no—
161D7—141no—
162D7—283no—
163D7—421no—
164D7—421no—
165D7—843no—
166C22×C3—71no—
167C22×C3—491no—
168C2×D3—71no—
169C2×D3—71no—
170C2×D3—491no—
171C2×D3—982no—
172C2×D3—982no—
173C3×C4—71no—
174Dic3D7211yescharacteristic, complemented
175C23—1471no—
176C23—1471no—
177D4—211no—
178D4—491no—
179D4—1471no—
180D4—1471no—
181C2×C4—211no—
182C7—23no—
183C7D3×D2811yescharacteristic, complemented
184C7#6669411yescharacteristic, complemented
185C2×C3—141no—
186C2×C3—981no—
187C2×C3C2×D7211yescharacteristic
188D3—72no—
189D3—141no—
190D3—981no—
191C22—71no—
192C22—211no—
193C22—211no—
194C22—2942no—
195C22—2942no—
196C22—491no—
197C22—1471no—
198C4—31no—
199C4—71no—
200C3D7×D2811yescharacteristic, complemented, Sylow
201C2—141no—
202C2—212no—
203C2—2941no—
204C2—421no—
205C2—981no—
206C2D3×D7211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D72
commutator, Fitting, socle, cosocleC2×C3×C72C23
hypocenterC3×C72C2×D4
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×D21⋊D7C2semidirect
2. C2×D7×D21C2semidirect
3. #11924248C2semidirect
4. #11924241C2semidirect
5. C7×#4246C2semidirect
6. C7×D21⋊C4C2semidirect
7. C3×#87641C2semidirect
9. C2×C3⋊(C7⋊D7)C22semidirect
11. C2×C7×D21C22semidirect
12. C2×C3×C7⋊D7C22semidirect
13. C2×C3×C7×D7C22semidirect
16. C72×Dic3C22semidirect
17. C3×C7×Dic7C22semidirect
24. #87641D3semidirect
30. C7×D21D4semidirect
34. C2×C7⋊D7C2×D3semidirect
37. C2×C7×D7C2×D3semidirect
41. C7×Dic7C2×D3semidirect
55. #4246D7semidirect
56. D21⋊C4D7semidirect
57. C3×C72C2×D4semidirect
72. C2×D21C2×D7semidirect
78. C2×C3×D7C2×D7semidirect
83. C3×Dic7C2×D7semidirect
85. C7×Dic3C2×D7semidirect
86. C7×Dic3C2×D7semidirect
102. C72D3×D4semidirect
112. D21D28semidirect
134. C2×D7D3×D7semidirect
139. Dic7D3×D7semidirect
148. C3×C7C2×D28semidirect
149. C3×C7D4×D7semidirect
174. Dic3D72semidirect
183. C7D3×D28semidirect
184. C7#66694semidirect
200. C3D7×D28semidirect

Maximal subgroups

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

subgroupindex
C2×D21⋊D72
C2×D7×D212
#119242482
#119242412
C7×#42462
C7×D21⋊C42
C3×#876412
D7×D283
#666947
D3×D287