Finite Group Databasebeta

← #21422246

Subgroups — #21422246

Order 7552.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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#16837574C211yesmaximal, characteristic
2#16837142C211yesmaximal, characteristic, complemented
3C59×#354C211yesmaximal, characteristic, complemented
4#14273775C411yescharacteristic, complemented
5#14273753—21no—
6#14273655—21no—
7#14273646C411yescharacteristic, complemented
8#14273638—21no—
9#124×C59C2211yescharacteristic
10#99×C59—21no—
11C2×Q8×C59C2×C411yescharacteristic
12C2×D4×C59D411yescharacteristic
13#11745689—41no—
14Q8×D59—41no—
15#11745685—21no—
16#11745685—41no—
17C22⋊C4×C59—41no—
18C22⋊C4×C59—41no—
19C42×C59D411yescharacteristic
20#11745670—41no—
21#11745669—41no—
22#11745664—21no—
23C2×C59⋊C8—41no—
24C23×C59—21no—
25Q8×C59—41no—
26D4×C59—41no—
27C2×C4×C59—21no—
28C2×C4×C59—21no—
29C2×C4×C59—81no—
30C2×C4×C59C22⋊C411yescharacteristic
31C22⋊D59—41no—
32D236—21no—
33D236—41no—
34C4×D59—41no—
35C4×D59—42no—
36Dic118—21no—
37Dic118—41no—
38C59⋊C8—41no—
39C59⋊C8—42no—
40C22×C59—41no—
41C22×C59—81no—
42C22×C59#9911yescharacteristic
43C2×D59—41no—
44C4×C59—21no—
45C4×C59—41no—
46C4×C59—41no—
47C4×C59—161no—
48Dic59—41no—
49#1404—591nomaximal, Sylow
50C2×C59—21no—
51C2×C59—81no—
52C2×C59AΣL(1,𝔽16)11yescharacteristic
53D59—81no—
54#454—591no—
55#354D5911yescharacteristic, complemented
56#329—591no—
57C59#140411yescharacteristic, complemented, Sylow
582−1+4—591no—
59AGL(1,ℤ/8)—1181no—
60#124C2×D5911yescharacteristic
61#104—1181no—
62#101—591no—
63#99—21no—
64#97—1181no—
65#41—1181no—
66#41—2361no—
67#41—2361no—
68C2×Q8—2361no—
69C2×Q8C4×D5911yescharacteristic
70C2×D4C22⋊D5911yescharacteristic
71SD16—2361no—
72D8—2361no—
73M16—1181no—
74C2×C8—2361no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C42D23611yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—1181no—
81Q8—2361no—
82D4—41no—
83D4—1181no—
84D4—2361no—
85D4—2361no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—81no—
89C2×C4—2361no—
90C2×C4—2362no—
91C2×C4(C2×D59)⋊C411yescharacteristic
92C8—2361no—
93C8—2362no—
94C22—41no—
95C22—81no—
96C22—2361no—
97C22#1427364111yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—161no—
102C4—2361no—
103C2—21no—
104C2—81no—
105C2—4721no—
106C2#1683718811yescharacteristic
subgroupG/H
centerC2#16837188
commutatorC2×Q8×C59C2×C4
Frattini#124C2×D59
FittingC59×#354C2
socleC2×C59AΣL(1,𝔽16)
hypercenter#354D59
hypocenterC59#1404
LCSG ▷ C2×Q8×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #16837142C2semidirect
3. C59×#354C2semidirect
4. #14273775C4semidirect
7. #14273646C4semidirect
55. #354D59semidirect
57. C59#1404semidirect