Finite Group Databasebeta

← #17996627

Subgroups — #17996627

Order 4736.

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

normal subgroups only

This group has 224 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#14558263C211yesmaximal, characteristic, complemented
2#14557718C212yesmaximal
3#14557715C211yesmaximal, characteristic
4#14557331C212yesmaximal, complemented
5C37×#356C211yesmaximal, characteristic, complemented
6#11924584C2211yescharacteristic
7#11924569—22no—
8#11924567—21no—
9#11924535—22no—
10#11924534—21no—
11#11924534C2212yes—
12C4×Dic74—21no—
13#11924501—22no—
14#11924464—21no—
15#11924462—22no—
16#11924455C2211yescharacteristic
17C4⋊Q8×C37C2211yescharacteristic
18#101×C37C2212yes—
19C2×Q8×C37D411yescharacteristic
20C2×Q8×C37C2311yescharacteristic
21#10596966—41no—
22Q8×D37—41no—
23#10596962—21no—
24#10596962—22no—
25C2×Dic74—21no—
26C2×Dic74—21no—
27C2×Dic74D412yes—
28M16×C37—22no—
29C4⋊C4×C37—22no—
30C42×C37D411yescharacteristic
31#10596942—41no—
32#10596941—41no—
33C4⋊Dic37—21no—
34#10596936—22no—
35#10596936—22no—
36#10596936—22no—
37C4×Dic37—21no—
38C4×Dic37—22no—
39C4×Dic37D412yes—
40#10596934—21no—
41Dic148—42no—
42#10596930—42no—
43Q8×C37—41no—
44Q8×C37—41no—
45C2×C4×C37—21no—
46C2×C4×C37—21no—
47C2×C4×C37C2×D411yescharacteristic
48C2×C4×C37C2×D412yes—
49C22⋊D37—41no—
50C2×Dic37—22no—
51C2×Dic37—22no—
52D148—22no—
53D148—21no—
54C4×D37—41no—
55C4×D37—42no—
56Dic74—24no—
57Dic74—22no—
58Dic74—21no—
59Dic74—42no—
60Dic74—81no—
61C8×C37—42no—
62#64436—41no—
63C22×C37#12011yescharacteristic
64C2×D37—41no—
65C4×C37—22no—
66C4×C37—21no—
67C4×C37—22no—
68C4×C37—41no—
69Dic37—24no—
70Dic37—41no—
71Dic37—42no—
72#2205—371nomaximal, Sylow
73C2×C37—21no—
74C2×C37B(4,𝔽2)11yescharacteristic
75D37—81no—
76Q82—371no—
77#456—371no—
78#456—372no—
79#356—372no—
80#356D3711yescharacteristic, complemented
81C37#220511yescharacteristic, complemented, Sylow
822−1+4—371no—
83#137—742no—
84#137—741no—
85C4⋊Q8—372no—
86C4⋊Q8—742no—
87C4⋊Q8—741no—
88C4⋊Q8C2×D3711yescharacteristic
89C4×Q8—742no—
90C4×Q8—741no—
91#104—742no—
92#104—741no—
93#101—371no—
94#101C2×D3712yes—
95#41—741no—
96#41—742no—
97#41—1481no—
98C2×Q8—372no—
99C2×Q8—741no—
100C2×Q8—741no—
101C2×Q8—1481no—
102C2×Q8D14811yescharacteristic
103C2×Q8C22×D3711yescharacteristic
104Q16—1482no—
105Q16—1481no—
106SD16—1482no—
107SD16—1481no—
108M16—22no—
109M16—741no—
110C4⋊C4—22no—
111C4⋊C4—742no—
112C4⋊C4—741no—
113C4⋊C4—742no—
114C4⋊C4—742no—
115C42—372no—
116C42—742no—
117C42—741no—
118C42D14811yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—2961no—
122Q8—744no—
123Q8—742no—
124Q8—741no—
125Q8—1482no—
126D4—742no—
127D4—741no—
128D4—1481no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—742no—
132C2×C4—742no—
133C2×C4—1482no—
134C2×C4—1481no—
135C2×C4C2×D14811yescharacteristic
136C2×C4D4×D3712yes—
137C8—42no—
138C8—1481no—
139C22—1481no—
140C22#1192450611yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—744no—
146C4—1482no—
147C4—1481no—
148C2—21no—
149C2—2961no—
150C2#1455772111yescharacteristic
subgroupG/H
centerC2#14557721
commutator, cosocleC2×Q8×C37C23
FrattiniC2×Q8C22×D37
FittingC37×#356C2
socleC2×C37B(4,𝔽2)
prime radical#11924584C22
hypercenter#356D37
hypocenterC37#2205
LCSG ▷ C2×Q8×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #14558263C2semidirect
4. #14557331C2semidirect
5. C37×#356C2semidirect
80. #356D37semidirect
81. C37#2205semidirect

Maximal subgroups

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

subgroupindex
#145582632
#145577182
#145577152
#145573312
C37×#3562
#220537