Finite Group Databasebeta

← #17996608

Subgroups — #17996608

Order 4736.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#14558283C211yesmaximal, characteristic, complemented
2#14557723C212yesmaximal
3#14557328C211yesmaximal, characteristic, complemented
4#14557324C212yesmaximal, complemented
5C37×#455C211yesmaximal, characteristic
6#11924534—21no—
7#11924529—22no—
8#11924528—21no—
9#11924517—22no—
10#11924517C2212yes—
11#11924513—22no—
12#11924451—22no—
13#11924451C2212yes—
14H(5,𝔽2)×C37C2211yescharacteristic
15#137×C37—21no—
16#124×C37C2211yescharacteristic
17#104×C37—21no—
18#100×C37C2211yescharacteristic
19#41×C37—21no—
20#41×C37—41no—
21C2×Q8×C37D411yescharacteristic
22C2×D4×C37—22no—
23C2×D4×C37—41no—
24C2×D4×C37C2311yescharacteristic
25C22×Dic37D412yes—
26C2×Dic74—21no—
27Q16×C37—41no—
28SD16×C37—41no—
29M16×C37—21no—
30C22⋊C4×C37—22no—
31C42×C37D411yescharacteristic
32C22⋊Dic37—22no—
33C22⋊Dic37—22no—
34C22⋊Dic37—42no—
35C22⋊Dic37D412yes—
36C4⋊Dic37—21no—
37C4⋊Dic37—21no—
38#10596936—22no—
39#10596936—22no—
40#10596936—22no—
41#10596936—42no—
42C4×Dic37—21no—
43C23×C37—22no—
44C23×C37C2×D412yes—
45Q8×C37—21no—
46Q8×C37—41no—
47D4×C37—21no—
48D4×C37—41no—
49D4×C37—42no—
50D4×C37—41no—
51C2×C4×C37—22no—
52C2×C4×C37—21no—
53C2×C4×C37—21no—
54C2×C4×C37—41no—
55C2×C4×C37C2×D411yescharacteristic
56C2×Dic37—22no—
57C2×Dic37—22no—
58C2×Dic37—42no—
59C2×Dic37—42no—
60Dic74—81no—
61C8×C37—41no—
62C22×C37—22no—
63C22×C37—42no—
64C22×C37—41no—
65C22×C37—82no—
66C22×C37#12011yescharacteristic
67C4×C37—21no—
68C4×C37—41no—
69C4×C37—41no—
70C4×C37—41no—
71Dic37—42no—
72Dic37—42no—
73Dic37—82no—
74#2201—371nomaximal, Sylow
75C2×C37—21no—
76C2×C37—42no—
77C2×C37—81no—
78C2×C37B(4,𝔽2)11yescharacteristic
79#563—371no—
80#458—372no—
81#455D3711yescharacteristic
82#354—371no—
83#352—372no—
84C37#220111yescharacteristic, complemented, Sylow
85H(5,𝔽2)C2×D3711yescharacteristic
86#137—21no—
87C4⋊Q8—741no—
88#126—742no—
89#125—741no—
90#124C2×D3711yescharacteristic
91#123—372no—
92#123—742no—
93C22⋊Q8—742no—
94#104—21no—
95#100C2×D3711yescharacteristic
96#99—372no—
97#99—742no—
98#41—21no—
99#41—41no—
100C2×Q8—741no—
101C2×Q8C22⋊D3711yescharacteristic
102C2×D4—22no—
103C2×D4—41no—
104C2×D4C22×D3711yescharacteristic
105C22×C4—372no—
106Q16—41no—
107SD16—41no—
108M16—21no—
109C4⋊C4—742no—
110C4⋊C4—742no—
111C4⋊C4—741no—
112C4⋊C4—741no—
113C4⋊C4—742no—
114C4⋊C4—1482no—
115C22⋊C4—22no—
116C22⋊C4—372no—
117C22⋊C4—742no—
118C22⋊C4—742no—
119C22⋊C4—1482no—
120C42—741no—
121C42C22⋊D3711yescharacteristic
122C23—22no—
123C23D4×D3712yes—
124Q8—21no—
125Q8—41no—
126Q8—2961no—
127D4—21no—
128D4—41no—
129D4—42no—
130D4—41no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—742no—
136C2×C4—742no—
137C2×C4—1482no—
138C2×C4—1482no—
139C2×C4C2×C22⋊D3711yescharacteristic
140C8—41no—
141C22—22no—
142C22—42no—
143C22—41no—
144C22—82no—
145C22#1192450411yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—41no—
150C4—2962no—
151C4—1482no—
152C4—1482no—
153C2—21no—
154C2—42no—
155C2—81no—
156C2#1455771911yescharacteristic
subgroupG/H
centerC2#14557719
commutator, cosocleC2×D4×C37C23
FrattiniC2×D4C22×D37
FittingC37×#455C2
socleC2×C37B(4,𝔽2)
prime radicalH(5,𝔽2)×C37C22
hypercenter#455D37
hypocenterC37#2201
LCSG ▷ C2×D4×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #14558283C2semidirect
3. #14557328C2semidirect
4. #14557324C2semidirect
84. C37#2201semidirect

Maximal subgroups

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

subgroupindex
#145582832
#145577232
#145573282
#145573242
C37×#4552
#220137