Finite Group Databasebeta

← #17993894

Subgroups — #17993894

Order 4736.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#14557923C211yesmaximal, characteristic
2#14557899C212yesmaximal, complemented
3#14557275C212yesmaximal, complemented
4#14557273C211yesmaximal, characteristic, complemented
5C37×#563C211yesmaximal, characteristic, complemented
6C2×#10596941—21no—
7C2×#10596939—21no—
8#11924533C2211yescharacteristic
9#11924456—22no—
10#11924456—22no—
11#11924447C2212yes—
12#11924447C2211yescharacteristic
13C4⋊Q8×C37—21no—
14#126×C37—22no—
15#125×C37C2212yes—
16#124×C37C2211yescharacteristic
17#123×C37—21no—
18C22⋊Q8×C37—22no—
19C2×Q8×C37—21no—
20C2×Q8×C37D411yescharacteristic
21C2×D4×C37D411yescharacteristic
22C22×C4×C37—21no—
23C2×D148—21no—
24C2×D148—22no—
25C4⋊C4×C37—22no—
26C4⋊C4×C37—22no—
27C4⋊C4×C37—22no—
28C4⋊C4×C37D412yes—
29C4⋊C4×C37D412yes—
30C22⋊C4×C37—22no—
31C22⋊C4×C37—21no—
32C22⋊C4×C37—22no—
33C42×C37—21no—
34C42×C37C2311yescharacteristic
35#10596941—42no—
36#10596939—42no—
37C2×#64436—21no—
38C2×#64436—22no—
39C23×C37—21no—
40C22×D37—41no—
41Q8×C37—41no—
42Q8×C37—81no—
43D4×C37—41no—
44C2×C4×C37—22no—
45C2×C4×C37—21no—
46C2×C4×C37—22no—
47C2×C4×C37—81no—
48C2×C4×C37C2×D412yes—
49C2×C4×C37C2×D411yescharacteristic
50D148—41no—
51D148—41no—
52D148—44no—
53#64436—22no—
54#64436—42no—
55C22×C37—41no—
56C22×C37—42no—
57C22×C37#12011yescharacteristic
58C2×D37—81no—
59C2×D37—82no—
60C4×C37—21no—
61C4×C37—22no—
62C4×C37—42no—
63C4×C37—41no—
64C4×C37—42no—
65#1657—371nomaximal, Sylow
66C2×C37—81no—
67C2×C37#45311yescharacteristic
68C2×C37#45412yes—
69D37—161no—
70#563D3711yescharacteristic, complemented
71#496—371no—
72#490—372no—
73#330—372no—
74#329—371no—
75C37#165711yescharacteristic, complemented, Sylow
76C2×SD16—741no—
77C2×D8—741no—
78C4⋊Q8—21no—
79#127—371no—
80#126—22no—
81#125C2×D3712yes—
82#124C2×D3711yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—742no—
86#102—742no—
87#97—372no—
88#97—371no—
89C2×Q8—21no—
90C2×Q8C22⋊D3711yescharacteristic
91C2×D4—742no—
92C2×D4—741no—
93C2×D4C22⋊D3711yescharacteristic
94C22×C4—21no—
95SD16—1482no—
96D8—1482no—
97C2×C8—741no—
98C2×C8—742no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D3712yes—
103C4⋊C4C22⋊D3712yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D3711yescharacteristic
109C23—21no—
110C23—1481no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—1481no—
115D4—1481no—
116D4—1484no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D3712yes—
122C2×C4C2×C22⋊D3711yescharacteristic
123C8—742no—
124C8—1482no—
125C22—41no—
126C22—42no—
127C22—2961no—
128C22—2962no—
129C22#1192450511yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—5921no—
137C2#1455770111yescharacteristic
138C2#1455770612yes—
subgroupG/H
centerC22#11924505
commutator, cosocleC42×C37C23
FrattiniC42C22×D37
FittingC37×#563C2
socleC22×C37#120
prime radical#14557923C2
hypercenter#563D37
hypocenterC37#1657
LCSG ▷ C42×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #14557899C2semidirect
3. #14557275C2semidirect
4. #14557273C2semidirect
5. C37×#563C2semidirect
70. #563D37semidirect
75. C37#1657semidirect

Maximal subgroups

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

subgroupindex
#145579232
#145578992
#145572752
#145572732
C37×#5632
#165737