Finite Group Databasebeta

← C7×D7×#67 #72246523

Subgroups — C7×D7×#67 #72246523

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×D3×C7×D7C211yesmaximal, characteristic, complemented
3C7×#4268C211yesmaximal, characteristic, complemented
4C72×#67C211yesmaximal, characteristic, complemented
5C7×#4246C211yesmaximal, characteristic, complemented
6C7×#4245C211yesmaximal, characteristic, complemented
7C7×Dic3×D7C211yesmaximal, characteristic, complemented
8C7×D4×D7—31nomaximal, Hall
9C22×C3×C72C2211yescharacteristic, complemented
10C2×C7×D21C2211yescharacteristic, complemented
11C2×D3×C72C2211yescharacteristic, complemented
12C2×C3×C7×D7—22no—
13C2×C3×C7×D7C2211yescharacteristic
14C2×C3×C7×D7C2211yescharacteristic, complemented
15D3×C7×D7—22no—
16C7×Dic21C2211yescharacteristic, complemented
17C72×Dic3C2211yescharacteristic, complemented
18C22×C7×D7—31no—
19C22×C7×D7D311yescharacteristic, complemented
20C72×D4—31no—
21C7×C22⋊D7—31no—
22C7×C22⋊D7—31no—
23C7×D28—31no—
24C4×C7×D7—31no—
25C2×C7×#67—71nomaximal
26D7×#67C711yesmaximal, characteristic, direct product factor
27C2×C3×C72—21no—
28C2×C3×C72C2311yescharacteristic
29C7×D21—21no—
30D3×C72—21no—
31C3×C7×D7—21no—
32C3×C7×D7D412yescomplemented
33C22×C72—31no—
34C22×C72C2×D311yescharacteristic, complemented
35C2×C7×D7—22no—
36C2×C7×D7—31no—
37C2×C7×D7—62no—
38C2×C7×D7C2×D311yescharacteristic
39C2×C7×D7C2×D311yescharacteristic, complemented
40C4×C72—31no—
41C7×Dic7—31no—
42C23×C3×C7—71no—
43C22×D3×C7—71no—
44C22×C3×D7C2×C711yescharacteristic, complemented
45C2×D3×D7C2×C711yescharacteristic, complemented
46#4268C2×C711yescharacteristic, complemented
47C7×#67—23no—
48C7×#67—71no—
49C7×#67—71no—
50C7×#67—71no—
51C7×#67D711yescharacteristic, direct product factor
52C7×#67C2×C711yescharacteristic, complemented
53C2×C7×Dic3—71no—
54#4246C2×C711yescharacteristic, complemented
55#4245C2×C711yescharacteristic, complemented
56Dic3×D7C2×C711yescharacteristic, complemented
57C3×C72C2×D411yescharacteristic, complemented, Hall
58C2×C7×D4—211no—
59D4×D7—31no—
60C2×C72—21no—
61C2×C72—61no—
62C2×C72C22×D311yescharacteristic
63C7×D7—21no—
64C7×D7—61no—
65C7×D7#6712yesdirect product factor
66C22×C3×C7—23no—
67C22×C3×C7—71no—
68C22×C3×C7—71no—
69C22×C3×C7—142no—
70C22×C3×C7C2×D711yescharacteristic, complemented
71C22×C3×C7C22×C711yescharacteristic, complemented
72C2×D21C22×C711yescharacteristic, complemented
73C2×D3×C7—23no—
74C2×D3×C7—71no—
75C2×D3×C7—142no—
76C2×D3×C7C2×D711yescharacteristic, complemented
77C2×D3×C7C22×C711yescharacteristic, complemented
78C2×C3×D7—22no—
79C2×C3×D7C22×C711yescharacteristic
80C2×C3×D7C22×C711yescharacteristic, complemented
81D3×D7—22no—
82Dic21C22×C711yescharacteristic, complemented
83C7×Dic3—23no—
84C7×Dic3—71no—
85C7×Dic3C2×D711yescharacteristic, complemented
86C7×Dic3C22×C711yescharacteristic, complemented
87C23×C7—71no—
88C23×C7—211no—
89C22×D7—31no—
90C22×D7D3×C711yescharacteristic, complemented
91C7×D4—31no—
92C7×D4—31no—
93C7×D4—63no—
94C7×D4—211no—
95C7×D4—211no—
96C7×D4—211no—
97C2×C4×C7—211no—
98C22⋊D7—31no—
99C22⋊D7—31no—
100D28—31no—
101C4×D7—31no—
102C72C2×#6711yescharacteristic, complemented, Sylow
103C2×#67—71noHall
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×C7C22×D711yescharacteristic
111C2×C3×C7C23×C711yescharacteristic
112D21—21no—
113C3×D7—21no—
114C3×D7C7×D412yescomplemented
115D3×C7—21no—
116D3×C7—21no—
117D3×C7—43no—
118D3×C7—141no—
119C22×C7—23no—
120C22×C7—31no—
121C22×C7—31no—
122C22×C7—63no—
123C22×C7—71no—
124C22×C7—71no—
125C22×C7—142no—
126C22×C7—211no—
127C22×C7—422no—
128C22×C7D3×D711yescharacteristic, complemented
129C22×C7C2×D3×C711yescharacteristic, complemented
130C2×D7—22no—
131C2×D7—31no—
132C2×D7—62no—
133C2×D7C2×D3×C711yescharacteristic
134C2×D7C2×D3×C711yescharacteristic, complemented
135C4×C7—31no—
136C4×C7—31no—
137C4×C7—63no—
138C4×C7—211no—
139Dic7—31no—
140C23×C3—71no—
141C22×D3—71no—
142#67—71no—
143#67—71no—
144#67—71no—
145#67C7×D711yescharacteristic, direct product factor
146C2×Dic3—71no—
147C3×C7—23no—
148C3×C7D4×D711yescharacteristic, complemented
149C3×C7C2×C7×D411yescharacteristic, complemented
150C2×D4—211noSylow
151C2×C7—21no—
152C2×C7—21no—
153C2×C7—23no—
154C2×C7—43no—
155C2×C7—61no—
156C2×C7—61no—
157C2×C7—72no—
158C2×C7—123no—
159C2×C7—141no—
160C2×C7—421no—
161C2×C7C2×D3×D711yescharacteristic
162C2×C7C22×D3×C711yescharacteristic
163D7—21no—
164D7—61no—
165D7C7×#6712yesdirect product factor
166C22×C3—71no—
167C22×C3—71no—
168C22×C3—142no—
169C22×C3C2×C7×D711yescharacteristic, complemented
170C2×D3—71no—
171C2×D3—142no—
172C2×D3C2×C7×D711yescharacteristic, complemented
173Dic3—71no—
174Dic3C2×C7×D711yescharacteristic, complemented
175C23—71no—
176C23—211no—
177D4—31no—
178D4—211no—
179D4—211no—
180D4—211no—
181C2×C4—211no—
182C7—23no—
183C7D7×#6711yescharacteristic, direct product factor
184C7C2×C7×#6711yescharacteristic, complemented
185C2×C3—21no—
186C2×C3—72no—
187C2×C3—141no—
188C2×C3C22×C7×D711yescharacteristic
189D3—21no—
190D3—141no—
191C22—31no—
192C22—71no—
193C22—71no—
194C22—142no—
195C22—211no—
196C22—422no—
197C22D3×C7×D711yescharacteristic, complemented
198C4—31no—
199C4—211no—
200C3C7×D4×D711yescharacteristic, complemented, Sylow
201C2—21no—
202C2—61no—
203C2—72no—
204C2—141no—
205C2—421no—
206C2C2×D3×C7×D711yescharacteristic
subgroupG/H
centerC2×C7C2×D3×D7
commutator, cosocleC2×C3×C7C23×C7
FrattiniC2C2×D3×C7×D7
FittingC22×C3×C72C22
socleC2×C3×C72C23
hypercenterC22×C7D3×D7
hypocenterC3×C7C2×C7×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C7 ◁ C22×C7

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×D7C2semidirect
2. C2×D3×C7×D7C2semidirect
3. C7×#4268C2semidirect
4. C72×#67C2semidirect
5. C7×#4246C2semidirect
6. C7×#4245C2semidirect
7. C7×Dic3×D7C2semidirect
9. C22×C3×C72C22semidirect
10. C2×C7×D21C22semidirect
11. C2×D3×C72C22semidirect
14. C2×C3×C7×D7C22semidirect
16. C7×Dic21C22semidirect
17. C72×Dic3C22semidirect
19. C22×C7×D7D3semidirect
26. D7×#67C7direct (N × H)
32. C3×C7×D7D4semidirect
34. C22×C72C2×D3semidirect
39. C2×C7×D7C2×D3semidirect
44. C22×C3×D7C2×C7semidirect
45. C2×D3×D7C2×C7semidirect
46. #4268C2×C7semidirect
51. C7×#67D7direct (N × H)
52. C7×#67C2×C7semidirect
54. #4246C2×C7semidirect
55. #4245C2×C7semidirect
56. Dic3×D7C2×C7semidirect
57. C3×C72C2×D4semidirect
65. C7×D7#67direct (N × H)
70. C22×C3×C7C2×D7semidirect
71. C22×C3×C7C22×C7semidirect
72. C2×D21C22×C7semidirect
76. C2×D3×C7C2×D7semidirect
77. C2×D3×C7C22×C7semidirect
80. C2×C3×D7C22×C7semidirect
82. Dic21C22×C7semidirect
85. C7×Dic3C2×D7semidirect
86. C7×Dic3C22×C7semidirect
90. C22×D7D3×C7semidirect
102. C72C2×#67semidirect
114. C3×D7C7×D4semidirect
128. C22×C7D3×D7semidirect
129. C22×C7C2×D3×C7semidirect
134. C2×D7C2×D3×C7semidirect
145. #67C7×D7direct (N × H)
148. C3×C7D4×D7semidirect
149. C3×C7C2×C7×D4semidirect
165. D7C7×#67direct (N × H)
169. C22×C3C2×C7×D7semidirect
172. C2×D3C2×C7×D7semidirect
174. Dic3C2×C7×D7semidirect
183. C7D7×#67direct (N × H)
184. C7C2×C7×#67semidirect
197. C22D3×C7×D7semidirect
200. C3C7×D4×D7semidirect

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×D3×C7×D72
C7×#42682
C72×#672
C7×#42462
C7×#42452
C7×Dic3×D72
C7×D4×D73
C2×C7×#677
D7×#677