Finite Group Databasebeta

← C97×#1404 #27137362

Subgroups — C97×#1404 #27137362

Order 12416.

112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 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#454×C97C211yesmaximal, characteristic
2#354×C97C211yesmaximal, characteristic, complemented
3#329×C97C211yesmaximal, characteristic, complemented
42−1+4×C97C411yescharacteristic, complemented
5AGL(1,ℤ/8)×C97—21no—
6#124×C97C2211yescharacteristic
7#104×C97—21no—
8#101×C97C411yescharacteristic, complemented
9#99×C97—21no—
10#97×C97—21no—
11#41×C97—21no—
12#41×C97—41no—
13#41×C97—41no—
14C2×Q8×C97—41no—
15C2×Q8×C97C2×C411yescharacteristic
16C2×D4×C97D411yescharacteristic
17SD16×C97—41no—
18D8×C97—41no—
19M16×C97—21no—
20C2×C8×C97—41no—
21C22⋊C4×C97—41no—
22C22⋊C4×C97—41no—
23C42×C97D411yescharacteristic
24C23×C97—21no—
25Q8×C97—21no—
26Q8×C97—41no—
27Q8×C97—41no—
28D4×C97—21no—
29D4×C97—41no—
30D4×C97—41no—
31D4×C97—41no—
32C2×C4×C97—21no—
33C2×C4×C97—21no—
34C2×C4×C97—41no—
35C2×C4×C97—42no—
36C2×C4×C97—81no—
37C2×C4×C97C22⋊C411yescharacteristic
38C8×C97—41no—
39C8×C97—42no—
40C22×C97—41no—
41C22×C97—41no—
42C22×C97—81no—
43C22×C97#9911yescharacteristic
44C4×C97—21no—
45C4×C97—41no—
46C4×C97—41no—
47C4×C97—41no—
48C4×C97—161no—
49C2×C97—21no—
50C2×C97—81no—
51C2×C97—81no—
52C2×C97AΣL(1,𝔽16)11yescharacteristic
53#1404C9711yesmaximal, characteristic, direct product factor, Sylow
54C97#140411yescharacteristic, direct product factor, Sylow
55#454C2×C9711yescharacteristic
56#354C2×C9711yescharacteristic, complemented
57#329C2×C9711yescharacteristic, complemented
582−1+4C4×C9711yescharacteristic, complemented
59AGL(1,ℤ/8)—21no—
60#124C22×C9711yescharacteristic
61#104—21no—
62#101C4×C9711yescharacteristic, complemented
63#99—21no—
64#97—21no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×C4×C9711yescharacteristic
70C2×D4D4×C9711yescharacteristic
71SD16—41no—
72D8—41no—
73M16—21no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C42D4×C9711yescharacteristic
78C23—21no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—41no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—41no—
89C2×C4—42no—
90C2×C4—81no—
91C2×C4C22⋊C4×C9711yescharacteristic
92C8—41no—
93C8—42no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22#99×C9711yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—41no—
102C4—161no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2AΣL(1,𝔽16)×C9711yescharacteristic
subgroupG/H
center, socleC2×C97AΣL(1,𝔽16)
commutatorC2×Q8C2×C4×C97
Frattini, cosocle#124C22×C97
prime radical#454×C97C2
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C97 ◁ C22×C97 ◁ C2×C4×C97 ◁ #124×C97 ◁ G

Complements

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

NH ≅ G/Ntype
2. #354×C97C2semidirect
3. #329×C97C2semidirect
4. 2−1+4×C97C4semidirect
8. #101×C97C4semidirect
53. #1404C97direct (N × H)
54. C97#1404direct (N × H)
56. #354C2×C97semidirect
57. #329C2×C97semidirect
58. 2−1+4C4×C97semidirect
62. #101C4×C97semidirect

Maximal subgroups

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

subgroupindex
#454×C972
#354×C972
#329×C972
#140497