Finite Group Databasebeta

← #13058001

Subgroups — #13058001

Order 1344.

504 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 120 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
1#10621359C211yesmaximal, characteristic, complemented
2C7×#876C211yesmaximal, characteristic, complemented
3C3×#6437C211yesmaximal, characteristic, complemented
4#10621036C213yesmaximal, complemented
5#10620971C213yesmaximal, complemented
6#10620970C213yesmaximal, complemented
7#10620969C213yesmaximal, complemented
8#90402—31nomaximal, Hall
9C22×D3×D7C2211yescharacteristic
10C2×C3×D28C2213yes—
11C2×C3×D28C2213yescomplemented
12C2×#4246C2213yes—
13C2×#4245C2213yes—
14C2×Dic3×D7C2213yescomplemented
15C3×C42×C7C2211yescharacteristic, complemented
16#66631C2213yes—
17#66629C2213yescomplemented
18C7×(C2×D3)⋊C4C2213yes—
19C7×Dic3⋊C4C2213yescomplemented
20(C2×D7)⋊Dic3C2213yes—
21(C2×D7)⋊Dic3C2213yescomplemented
22C7×#126—31no—
23#6458—33no—
24#6457—33no—
25#6447—33no—
26#6445—33no—
27#6439—31no—
28#6437D311yescharacteristic, complemented
29#5720—71nomaximal, Hall
30C22×D21C2311yescharacteristic
31C22×D3×C7C2311yescharacteristic
32C22×C3×D7C2311yescharacteristic
33C22×C3×D7C2313yes—
34C2×D3×D7—43no—
35C2×C3×C4×C7C2313yes—
36C2×Dic21C2313yes—
37C2×C7×Dic3C2313yes—
38C3×D28—43no—
39C3×D28—43no—
40#4246—43no—
41#4245—43no—
42Dic3×D7—43no—
43C23×D7—31no—
44C2×C22⋊D7—33no—
45C2×D28—33no—
46C2×D28C2×D313yes—
47C2×D28C2×D313yescomplemented
48C2×C4×D7—33no—
49C7×C4⋊C4—33no—
50C7×C22⋊C4—33no—
51C42×C7C2×D311yescharacteristic, complemented
52(C2×D7)⋊C4—33no—
53(C2×D7)⋊C4—33no—
54(C2×D7)⋊C4—33no—
55Dic7⋊C4—33no—
56C3×#127—71no—
57#939—73no—
58#939—73no—
59#937—73no—
60#935—73no—
61#876—71no—
62#876D711yescharacteristic, complemented
63C22×C3×C7C2411yescharacteristic
64C2×D21—23no—
65C2×D3×C7—23no—
66C2×C3×D7—23no—
67C2×C3×D7—23no—
68C2×C3×D7—23no—
69C2×C3×D7—23no—
70D3×D7—44no—
71C3×C4×C7—23no—
72Dic21—23no—
73C7×Dic3—23no—
74#560—211noSylow
75C23×C7—31no—
76C22×D7—31no—
77C22×D7—123no—
78C22×D7C22×D311yescharacteristic
79C22×D7C22×D313yes—
80C2×C4×C7—33no—
81C2×C4×C7C22×D313yes—
82C22⋊D7—123no—
83C2×Dic7—33no—
84D28—43no—
85D28—43no—
86D28—123no—
87C4×D7—123no—
88C23×D3—71no—
89C2×C3×D4—73no—
90C2×C3×D4—73no—
91C2×#67—73no—
92C2×#67—73no—
93C22×Dic3—73no—
94C3×C42C2×D711yescharacteristic, complemented
95#217—73no—
96#217—73no—
97(C2×D3)⋊C4—73no—
98(C2×D3)⋊C4C2×D713yes—
99Dic3⋊C4—73no—
100Dic3⋊C4C2×D713yescomplemented
101C2×C3×C7H(5,𝔽2)13yes—
102D21—41no—
103C3×D7—41no—
104C3×D7—43no—
105D3×C7—41no—
106#127—71no—
107#126—31no—
108#126—211no—
109#123—213no—
110#121—213no—
111#121—213no—
112#120—213no—
113C22×C7—63no—
114C22×C7C23×D311yescharacteristic
115C2×D7—23no—
116C2×D7—23no—
117C2×D7—23no—
118C2×D7—23no—
119C2×D7—63no—
120C2×D7—124no—
121C4×C7—23no—
122C4×C7—63no—
123Dic7—63no—
124C23×C3—71no—
125C23×C3—73no—
126C22×D3—71no—
127C22×D3—283no—
128C22×D3C22×D711yescharacteristic
129C3×D4—283no—
130C3×D4—283no—
131C2×C3×C4C22×D713yes—
132#67—283no—
133#67—283no—
134C2×Dic3—73no—
135C2×Dic3—283no—
136C2×Dic3C22×D713yes—
137C3×C7#56011yescharacteristic, complemented, Hall
138C24—211no—
139C2×D4—73no—
140C2×D4—73no—
141C2×D4—213no—
142C2×D4—213no—
143C22×C4—213no—
144C4⋊C4—33no—
145C4⋊C4—213no—
146C22⋊C4—33no—
147C22⋊C4—213no—
148C22⋊C4—213no—
149C22⋊C4—213no—
150C42D3×D711yescharacteristic, complemented
151C2×C7—121no—
152C2×C7#100413yes—
153D7—41no—
154D7—43no—
155D7—121no—
156C22×C3—143no—
157C22×C3—143no—
158C22×C3—143no—
159C22×C3—143no—
160C22×C3C23×D711yescharacteristic
161C2×D3—23no—
162C2×D3—143no—
163C2×D3—284no—
164C3×C4—23no—
165Dic3—23no—
166Dic3—143no—
167C23—31no—
168C23—71no—
169C23—73no—
170C23—211no—
171C23—843no—
172D4—283no—
173D4—283no—
174D4—843no—
175D4—843no—
176C2×C4—33no—
177C2×C4—213no—
178C2×C4—843no—
179C2×C4C2×D3×D713yes—
180C7#572011yescharacteristic, complemented, Sylow
181C2×C3—281no—
182C2×C3—283no—
183C2×C3#655313yes—
184D3—41no—
185D3—281no—
186C22—63no—
187C22—143no—
188C22—143no—
189C22—143no—
190C22—143no—
191C22—423no—
192C22—844no—
193C22C22×D3×D711yescharacteristic
194C4—23no—
195C4—63no—
196C4—423no—
197C3#9040211yescharacteristic, complemented, Sylow
198C2—121no—
199C2—281no—
200C2—283no—
201C2—841no—
202C2#1062156413yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C42×C7C22
hypercenterC42D3×D7
hypocenterC3×C7#560
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #10621359C2semidirect
2. C7×#876C2semidirect
3. C3×#6437C2semidirect
4. #10621036C2semidirect
5. #10620971C2semidirect
6. #10620970C2semidirect
7. #10620969C2semidirect
11. C2×C3×D28C22semidirect
14. C2×Dic3×D7C22semidirect
15. C3×C42×C7C22semidirect
17. #66629C22semidirect
19. C7×Dic3⋊C4C22semidirect
21. (C2×D7)⋊Dic3C22semidirect
28. #6437D3semidirect
47. C2×D28C2×D3semidirect
51. C42×C7C2×D3semidirect
62. #876D7semidirect
94. C3×C42C2×D7semidirect
100. Dic3⋊C4C2×D7semidirect
137. C3×C7#560semidirect
150. C42D3×D7semidirect
180. C7#5720semidirect
197. C3#90402semidirect

Maximal subgroups

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

subgroupindex
#106213592
C7×#8762
C3×#64372
#106210362
#106209712
#106209702
#106209692
#904023
#57207