Finite Group Databasebeta

← #37565420

Subgroups — #37565420

Order 24256.

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#27131377C211yesmaximal, characteristic
2#27131362C211yesmaximal, characteristic, complemented
3#27131320C211yesmaximal, characteristic, complemented
4#27131258C211yesmaximal, characteristic, complemented
5#27131257C211yesmaximal, characteristic, complemented
6#27131247C211yesmaximal, characteristic, complemented
7#137×C379C211yesmaximal, characteristic, complemented
8#20258607—21no—
9#20258606—21no—
10#20258606C2211yescharacteristic
11C2×D1516—21no—
12C2×D1516C2211yescharacteristic
13D4×D379—22no—
14D4×D379—21no—
15#20258596—21no—
16#20258593—21no—
17#20258590C2211yescharacteristic, complemented
18(C2×D379)⋊C4—22no—
19C4×Dic379C2211yescharacteristic, complemented
20#41×C379C2211yescharacteristic
21C2×Q8×C379C2211yescharacteristic, complemented
22Q16×C379—21no—
23SD16×C379—21no—
24M16×C379C2211yescharacteristic, complemented
25C22×D379—22no—
26C22×D379D412yes—
27Dic758D411yescharacteristic
28D1516—22no—
29D1516—21no—
30D1516—21no—
31D1516D411yescharacteristic
32C22⋊D379—22no—
33C22⋊D379—21no—
34C2×Dic379—21no—
35C4×D379—22no—
36C4×D379—21no—
37C379⋊C8—21no—
38Q8×C379—21no—
39Q8×C379—21no—
40Q8×C379D411yescharacteristic
41D4×C379—21no—
42D4×C379D411yescharacteristic
43C2×C4×C379—21no—
44C2×C4×C379—21no—
45C2×C4×C379C2311yescharacteristic
46C8×C379—21no—
47C22×C379—21no—
48C22×C379C2×D411yescharacteristic
49C2×D379—22no—
50C2×D379—21no—
51C2×D379—42no—
52C2×D379—82no—
53C4×C379—21no—
54C4×C379—41no—
55C4×C379C2×D411yescharacteristic
56C4×C379C2×D411yescharacteristic
57Dic379—21no—
58Dic379—41no—
59C2×C379—21no—
60C2×C379—41no—
61C2×C379#12011yescharacteristic
62D379—42no—
63D379—41no—
64C379#45511yescharacteristic, complemented, Sylow
65#455—3791nomaximal, Sylow
66H(5,𝔽2)—3791no—
67#137—3791no—
68#137D37911yescharacteristic, complemented
69#124—3791no—
70#104—3791no—
71#104—3791no—
72#100—3791no—
73#41—3791no—
74#41—7581no—
75#41—7581no—
76#41C2×D37911yescharacteristic
77C2×Q8C2×D37911yescharacteristic, complemented
78C2×D4—3791no—
79C2×D4—7581no—
80C2×D4—7582no—
81C2×D4—7581no—
82Q16—21no—
83Q16—7581no—
84SD16—21no—
85SD16—7581no—
86M16—3791no—
87M16C2×D37911yescharacteristic, complemented
88C22⋊C4—7582no—
89C42—3791no—
90C23—3792no—
91C23—7582no—
92Q8—21no—
93Q8—21no—
94Q8—3791no—
95Q8C22⋊D37911yescharacteristic
96D4—21no—
97D4—3791no—
98D4—7582no—
99D4—7582no—
100D4—7581no—
101D4—7581no—
102D4—7581no—
103D4C22⋊D37911yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—7582no—
107C2×C4—7581no—
108C2×C4—7581no—
109C2×C4C22×D37911yescharacteristic
110C8—21no—
111C8—7581no—
112C22—21no—
113C22—30322no—
114C22—15162no—
115C22—7581no—
116C22—7582no—
117C22D4×D37911yescharacteristic
118C4—21no—
119C4—41no—
120C4—15161no—
121C4—7581no—
122C4D4×D37911yescharacteristic
123C4C2×C22⋊D37911yescharacteristic
124C2—21no—
125C2—41no—
126C2—15161no—
127C2—15162no—
128C2#2713129911yescharacteristic
subgroupG/H
centerC2#27131299
commutator, cosocleC2×C4×C379C23
FrattiniC2×C4C22×D379
Fitting#137×C379C2
socleC2×C379#120
prime radical#27131377C2
hypercenter#137D379
hypocenterC379#455
LCSG ▷ C2×C4×C379 ▷ C2×C379 ▷ C379
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #27131362C2semidirect
3. #27131320C2semidirect
4. #27131258C2semidirect
5. #27131257C2semidirect
6. #27131247C2semidirect
7. #137×C379C2semidirect
17. #20258590C22semidirect
19. C4×Dic379C22semidirect
21. C2×Q8×C379C22semidirect
24. M16×C379C22semidirect
64. C379#455semidirect
68. #137D379semidirect
77. C2×Q8C2×D379semidirect
87. M16C2×D379semidirect

Maximal subgroups

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

subgroupindex
#271313772
#271313622
#271313202
#271312582
#271312572
#271312472
#137×C3792
#455379