Finite Group Databasebeta

← #37570670

Subgroups — #37570670

Order 24448.

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#27132835C211yesmaximal, characteristic
2#27132403C211yesmaximal, characteristic, complemented
3#354×C191C211yesmaximal, characteristic, complemented
4#20259370C411yescharacteristic, complemented
5#20259348—21no—
6#20259250—21no—
7#20259241C411yescharacteristic, complemented
8#20259233—21no—
9#124×C191C2211yescharacteristic
10#99×C191—21no—
11#15690288—41no—
12#15690287—21no—
13#15690287—41no—
14Q8×D191—41no—
15#15690275—41no—
16#15690272—41no—
17#15690271—21no—
18C2×C191⋊C8—41no—
19C2×Q8×C191C2×C411yescharacteristic
20C2×D4×C191D411yescharacteristic
21C22⋊C4×C191—41no—
22C22⋊C4×C191—41no—
23C42×C191D411yescharacteristic
24Dic382—21no—
25Dic382—41no—
26D764—21no—
27D764—41no—
28C22⋊D191—41no—
29C4×D191—41no—
30C4×D191—42no—
31C191⋊C8—41no—
32C191⋊C8—42no—
33C23×C191—21no—
34Q8×C191—41no—
35D4×C191—41no—
36C2×C4×C191—21no—
37C2×C4×C191—21no—
38C2×C4×C191—81no—
39C2×C4×C191C22⋊C411yescharacteristic
40C22×C191—41no—
41C22×C191—81no—
42C22×C191#9911yescharacteristic
43C2×D191—41no—
44C4×C191—21no—
45C4×C191—41no—
46C4×C191—41no—
47C4×C191—161no—
48Dic191—41no—
49C2×C191—21no—
50C2×C191—81no—
51C2×C191AΣL(1,𝔽16)11yescharacteristic
52D191—81no—
53C191#140411yescharacteristic, complemented, Sylow
54#1404—1911nomaximal, Sylow
55#454—1911no—
56#354D19111yescharacteristic, complemented
57#329—1911no—
582−1+4—1911no—
59AGL(1,ℤ/8)—3821no—
60#124C2×D19111yescharacteristic
61#104—3821no—
62#101—1911no—
63#99—21no—
64#97—3821no—
65#41—3821no—
66#41—7641no—
67#41—7641no—
68C2×Q8—7641no—
69C2×Q8C4×D19111yescharacteristic
70C2×D4C22⋊D19111yescharacteristic
71SD16—7641no—
72D8—7641no—
73M16—3821no—
74C2×C8—7641no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C42D76411yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—3821no—
81Q8—7641no—
82D4—41no—
83D4—3821no—
84D4—7641no—
85D4—7641no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—81no—
89C2×C4—7641no—
90C2×C4—7642no—
91C2×C4(C2×D191)⋊C411yescharacteristic
92C8—7641no—
93C8—7642no—
94C22—41no—
95C22—81no—
96C22—7641no—
97C22#2025923611yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—161no—
102C4—7641no—
103C2—21no—
104C2—81no—
105C2—15281no—
106C2#2713244911yescharacteristic
subgroupG/H
centerC2#27132449
commutatorC2×Q8×C191C2×C4
Frattini#124C2×D191
Fitting#354×C191C2
socleC2×C191AΣL(1,𝔽16)
prime radical#27132835C2
cosocle#124×C191C22
hypercenter#354D191
hypocenterC191#1404
LCSG ▷ C2×Q8×C191 ▷ C2×C4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #27132403C2semidirect
3. #354×C191C2semidirect
4. #20259370C4semidirect
7. #20259241C4semidirect
53. C191#1404semidirect
56. #354D191semidirect

Maximal subgroups

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

subgroupindex
#271328352
#271324032
#354×C1912
#1404191