Finite Group Databasebeta

← #46830351

Subgroups — #46830351

Order 34624.

152 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 39 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#31772674C211yesmaximal, characteristic
2#31772659C211yesmaximal, characteristic, complemented
3#31772617C211yesmaximal, characteristic, complemented
4#31772555C211yesmaximal, characteristic, complemented
5#31772554C211yesmaximal, characteristic, complemented
6#31772544C211yesmaximal, characteristic, complemented
7#137×C541C211yesmaximal, characteristic, complemented
8#22575837—21no—
9#22575836—21no—
10#22575836C2211yescharacteristic
11C2×D2164—21no—
12C2×D2164C2211yescharacteristic
13D4×D541—21no—
14D4×D541—22no—
15#22575826—21no—
16#22575823—21no—
17#22575820C2211yescharacteristic, complemented
18#22575813—22no—
19C4×Dic541C2211yescharacteristic, complemented
20#41×C541C2211yescharacteristic
21C2×Q8×C541C2211yescharacteristic, complemented
22Q16×C541—21no—
23SD16×C541—21no—
24M16×C541C2211yescharacteristic, complemented
25C22×D541—22no—
26C22×D541D412yes—
27Dic1082D411yescharacteristic
28D2164—22no—
29D2164—21no—
30D2164—21no—
31D2164D411yescharacteristic
32C22⋊D541—22no—
33C22⋊D541—21no—
34C2×Dic541—21no—
35C4×D541—22no—
36C4×D541—21no—
37#16864513—21no—
38Q8×C541—21no—
39Q8×C541—21no—
40Q8×C541D411yescharacteristic
41D4×C541—21no—
42D4×C541D411yescharacteristic
43C2×C4×C541—21no—
44C2×C4×C541—21no—
45C2×C4×C541C2311yescharacteristic
46C8×C541—21no—
47C22×C541—21no—
48C22×C541C2×D411yescharacteristic
49C2×D541—22no—
50C2×D541—21no—
51C2×D541—42no—
52C2×D541—82no—
53C4×C541—21no—
54C4×C541—41no—
55C4×C541C2×D411yescharacteristic
56C4×C541C2×D411yescharacteristic
57Dic541—21no—
58Dic541—41no—
59C2×C541—21no—
60C2×C541—41no—
61C2×C541#12011yescharacteristic
62D541—42no—
63D541—41no—
64C541#45511yescharacteristic, complemented, Sylow
65#455—5411nomaximal, Sylow
66H(5,𝔽2)—5411no—
67#137—5411no—
68#137D54111yescharacteristic, complemented
69#124—5411no—
70#104—5411no—
71#104—5411no—
72#100—5411no—
73#41—5411no—
74#41—10821no—
75#41—10821no—
76#41C2×D54111yescharacteristic
77C2×Q8C2×D54111yescharacteristic, complemented
78C2×D4—5411no—
79C2×D4—10821no—
80C2×D4—10822no—
81C2×D4—10821no—
82Q16—21no—
83Q16—10821no—
84SD16—21no—
85SD16—10821no—
86M16—5411no—
87M16C2×D54111yescharacteristic, complemented
88C22⋊C4—10822no—
89C42—5411no—
90C23—5412no—
91C23—10822no—
92Q8—21no—
93Q8—21no—
94Q8—5411no—
95Q8C22⋊D54111yescharacteristic
96D4—21no—
97D4—5411no—
98D4—10822no—
99D4—10822no—
100D4—10821no—
101D4—10821no—
102D4—10821no—
103D4C22⋊D54111yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—10822no—
107C2×C4—10821no—
108C2×C4—10821no—
109C2×C4C22×D54111yescharacteristic
110C8—21no—
111C8—10821no—
112C22—21no—
113C22—10821no—
114C22—10822no—
115C22—21642no—
116C22—43282no—
117C22D4×D54111yescharacteristic
118C4—21no—
119C4—41no—
120C4—10821no—
121C4—21641no—
122C4D4×D54111yescharacteristic
123C4C2×C22⋊D54111yescharacteristic
124C2—21no—
125C2—41no—
126C2—21642no—
127C2—21641no—
128C2#3177259611yescharacteristic
subgroupG/H
centerC2#31772596
commutator, cosocleC2×C4×C541C23
FrattiniC2×C4C22×D541
Fitting#137×C541C2
socleC2×C541#120
prime radical#31772674C2
hypercenter#137D541
hypocenterC541#455
LCSG ▷ C2×C4×C541 ▷ C2×C541 ▷ C541
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #31772659C2semidirect
3. #31772617C2semidirect
4. #31772555C2semidirect
5. #31772554C2semidirect
6. #31772544C2semidirect
7. #137×C541C2semidirect
17. #22575820C22semidirect
19. C4×Dic541C22semidirect
21. C2×Q8×C541C22semidirect
24. M16×C541C22semidirect
64. C541#455semidirect
68. #137D541semidirect
77. C2×Q8C2×D541semidirect
87. M16C2×D541semidirect

Maximal subgroups

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

subgroupindex
#317726742
#317726592
#317726172
#317725552
#317725542
#317725442
#137×C5412
#455541