Finite Group Databasebeta

← C7×#3416 #11742725

Subgroups — C7×#3416 #11742725

Order 896.

360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C7×#578C211yesmaximal, characteristic, complemented
2C7×#576C211yesmaximal, characteristic, complemented
3C7×#510C213yesmaximal, complemented
4C7×#509C213yesmaximal, complemented
5C7×#508C213yesmaximal, complemented
6C2×C7×SD32C213yesmaximal, complemented
7C7×#504C211yesmaximal, characteristic, complemented
8C7×2−1+4—21no—
9C7×H(5,𝔽2)—21no—
10C7×#137—23no—
11C7×AGL(1,ℤ/8)—23no—
12C7×#135C2213yes—
13C7×#135C2213yescomplemented
14C2×C7×Q16C2213yescomplemented
15C2×C7×D8C2213yes—
16C7×#131C2211yescharacteristic
17C7×Q32C2213yescomplemented
18C7×SD32C2213yescomplemented
19C7×SD32C2216yescomplemented
20C7×SD32C2211yescharacteristic, complemented
21C7×D16C2213yescomplemented
22C7×M32C2213yescomplemented
23C2×C7×C16C2213yescomplemented
24#3416C711yesmaximal, characteristic, direct product factor, Sylow
25C7×#41—23no—
26C7×#41—23no—
27C7×#41—41no—
28C7×#41—43no—
29C7×#41D411yescharacteristic
30C2×C7×Q8—23no—
31C2×C7×Q8—41no—
32C2×C7×D4—23no—
33C2×C7×D4—43no—
34C7×Q16—23no—
35C7×Q16C2313yes—
36C7×Q16C2311yescharacteristic
37C7×SD16—23no—
38C7×SD16—23no—
39C7×D8—23no—
40C7×D8C2311yescharacteristic
41C7×D8C2313yes—
42C7×M16D413yes—
43C2×C7×C8C2313yes—
44C7×C16C2311yescharacteristic
45C7×C16C2313yes—
46#578C2×C711yescharacteristic, complemented
47#576C2×C711yescharacteristic, complemented
48#510C2×C713yescomplemented
49#509C2×C713yescomplemented
50#508C2×C713yescomplemented
51C2×SD32C2×C713yescomplemented
52#504C2×C711yescharacteristic, complemented
53C23×C7—43no—
54C7×Q8—21no—
55C7×Q8—23no—
56C7×Q8—43no—
57C7×Q8D811yescharacteristic
58C7×D4—23no—
59C7×D4—21no—
60C7×D4—43no—
61C7×D4—43no—
62C7×D4—43no—
63C7×D4D813yes—
64C2×C4×C7—43no—
65C2×C4×C7—43no—
66C2×C4×C7—43no—
67C2×C4×C7C2×D413yes—
68C7×C8C2×D413yes—
69C7×C8C2411yescharacteristic
702−1+4—21no—
71H(5,𝔽2)—21no—
72#137—23no—
73AGL(1,ℤ/8)—23no—
74#135C22×C713yes—
75#135C22×C713yescomplemented
76C2×Q16C22×C713yescomplemented
77C2×D8C22×C713yes—
78#131C22×C711yescharacteristic
79Q32C22×C713yescomplemented
80SD32C22×C713yescomplemented
81SD32C22×C716yescomplemented
82SD32C22×C711yescharacteristic, complemented
83D16C22×C713yescomplemented
84M32C22×C713yescomplemented
85C2×C16C22×C713yescomplemented
86C22×C7—41no—
87C22×C7—43no—
88C22×C7—163no—
89C22×C7C2×D813yes—
90C4×C7—41no—
91C4×C7—43no—
92C4×C7C2×D813yes—
93C4×C7C22×D411yescharacteristic
94#41—23no—
95#41—23no—
96#41—41no—
97#41—43no—
98#41C7×D411yescharacteristic
99C2×Q8—23no—
100C2×Q8—41no—
101C2×D4—23no—
102C2×D4—43no—
103Q16—23no—
104Q16C23×C713yes—
105Q16C23×C711yescharacteristic
106SD16—23no—
107SD16—23no—
108D8—23no—
109D8C23×C711yescharacteristic
110D8C23×C713yes—
111M16C7×D413yes—
112C2×C8C23×C713yes—
113C16C23×C711yescharacteristic
114C16C23×C713yes—
115C2×C7—23no—
116C2×C7—81no—
117C2×C7—83no—
118C2×C7C22×D811yescharacteristic
119C23—43no—
120Q8—21no—
121Q8—23no—
122Q8—43no—
123Q8C7×D811yescharacteristic
124D4—23no—
125D4—21no—
126D4—43no—
127D4—43no—
128D4—43no—
129D4C7×D813yes—
130C2×C4—43no—
131C2×C4—43no—
132C2×C4—43no—
133C2×C4C2×C7×D413yes—
134C8C2×C7×D413yes—
135C8C24×C711yescharacteristic
136C7#341611yescharacteristic, direct product factor, Sylow
137C22—41no—
138C22—43no—
139C22—163no—
140C22C2×C7×D813yes—
141C4—41no—
142C4—43no—
143C4C2×C7×D813yes—
144C4C22×C7×D411yescharacteristic
145C2—23no—
146C2—83no—
147C2—81no—
148C2C22×C7×D811yescharacteristic
subgroupG/H
center, socleC2×C7C22×D8
commutator, Frattini, cosocleC8C24×C7
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C7×#41 ◁ C7×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#578C2semidirect
2. C7×#576C2semidirect
3. C7×#510C2semidirect
4. C7×#509C2semidirect
5. C7×#508C2semidirect
6. C2×C7×SD32C2semidirect
7. C7×#504C2semidirect
13. C7×#135C22semidirect
14. C2×C7×Q16C22semidirect
17. C7×Q32C22semidirect
18. C7×SD32C22semidirect
19. C7×SD32C22semidirect
20. C7×SD32C22semidirect
21. C7×D16C22semidirect
22. C7×M32C22semidirect
23. C2×C7×C16C22semidirect
24. #3416C7direct (N × H)
46. #578C2×C7semidirect
47. #576C2×C7semidirect
48. #510C2×C7semidirect
49. #509C2×C7semidirect
50. #508C2×C7semidirect
51. C2×SD32C2×C7semidirect
52. #504C2×C7semidirect
75. #135C22×C7semidirect
76. C2×Q16C22×C7semidirect
79. Q32C22×C7semidirect
80. SD32C22×C7semidirect
81. SD32C22×C7semidirect
82. SD32C22×C7semidirect
83. D16C22×C7semidirect
84. M32C22×C7semidirect
85. C2×C16C22×C7semidirect
136. C7#3416direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5782
C7×#5762
C7×#5102
C7×#5092
C7×#5082
C2×C7×SD322
C7×#5042
#34167