Finite Group Databasebeta

← #17996619

Subgroups — #17996619

Order 4736.

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

normal subgroups only

This group has 216 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#14557718C212yesmaximal, complemented
3#14557705C211yesmaximal, characteristic
4#14557331C212yesmaximal, complemented
5C37×#355C211yesmaximal, characteristic, complemented
6#11924584C2211yescharacteristic
7#11924569—22no—
8#11924562—21no—
9#11924534C2212yes—
10#11924529—22no—
11#11924528—21no—
12#11924517—21no—
13#11924513—22no—
14#11924464—21no—
15#11924462—22no—
16#11924455C2211yescharacteristic
17#124×C37C2211yescharacteristic
18#101×C37C2212yes—
19C2×Q8×C37C2311yescharacteristic
20C2×D4×C37D411yescharacteristic
21C22×Dic37—21no—
22#10596966—41no—
23Q8×D37—41no—
24#10596962—21no—
25#10596962—22no—
26C2×Dic74D412yes—
27M16×C37—22no—
28C22⋊C4×C37—22no—
29C42×C37D411yescharacteristic
30C22⋊Dic37—22no—
31C22⋊Dic37—21no—
32#10596940—41no—
33#10596939—41no—
34C4⋊Dic37—21no—
35C4⋊Dic37—21no—
36#10596936—22no—
37#10596936—22no—
38#10596936—22no—
39C4×Dic37D412yes—
40#10596934—21no—
41Dic148—42no—
42#10596930—42no—
43C23×C37—21no—
44Q8×C37—41no—
45D4×C37—41no—
46C2×C4×C37—21no—
47C2×C4×C37C2×D411yescharacteristic
48C2×C4×C37C2×D412yes—
49C22⋊D37—41no—
50C2×Dic37—22no—
51C2×Dic37—22no—
52C2×Dic37—81no—
53D148—22no—
54D148—21no—
55C4×D37—41no—
56C4×D37—42no—
57Dic74—22no—
58Dic74—21no—
59Dic74—42no—
60C8×C37—42no—
61#64436—41no—
62C22×C37—42no—
63C22×C37—41no—
64C22×C37#12011yescharacteristic
65C2×D37—41no—
66C4×C37—21no—
67C4×C37—22no—
68C4×C37—41no—
69Dic37—42no—
70Dic37—41no—
71Dic37—42no—
72#2203—371nomaximal, Sylow
73C2×C37—21no—
74C2×C37—81no—
75C2×C37B(4,𝔽2)11yescharacteristic
76D37—81no—
77#563—371no—
78#456—372no—
79#454—371no—
80#356—372no—
81#355D3711yescharacteristic, complemented
82C37#220311yescharacteristic, complemented, Sylow
832−1+4—371no—
84#137—742no—
85AGL(1,ℤ/8)—741no—
86C4⋊Q8—372no—
87#126—742no—
88#125—741no—
89#124C2×D3711yescharacteristic
90#123—741no—
91C22⋊Q8—742no—
92#104—742no—
93#104—741no—
94#101—371no—
95#101C2×D3712yes—
96#41—741no—
97#41—742no—
98#41—1481no—
99C2×Q8—372no—
100C2×Q8—1481no—
101C2×Q8C22×D3711yescharacteristic
102C2×D4D14811yescharacteristic
103C22×C4—741no—
104Q16—1482no—
105SD16—1482no—
106SD16—1481no—
107D8—1481no—
108M16—22no—
109M16—741no—
110C4⋊C4—742no—
111C4⋊C4—741no—
112C4⋊C4—741no—
113C4⋊C4—742no—
114C4⋊C4—742no—
115C22⋊C4—22no—
116C22⋊C4—742no—
117C22⋊C4—741no—
118C42—372no—
119C42D14811yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—741no—
123Q8—742no—
124Q8—1482no—
125D4—41no—
126D4—741no—
127D4—742no—
128D4—1481no—
129C2×C4—21no—
130C2×C4—2961no—
131C2×C4—742no—
132C2×C4—742no—
133C2×C4—1482no—
134C2×C4—1481no—
135C2×C4C2×D14811yescharacteristic
136C2×C4D4×D3712yes—
137C8—42no—
138C8—1481no—
139C22—42no—
140C22—41no—
141C22—1481no—
142C22#1192450611yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—1482no—
147C4—1482no—
148C4—1481no—
149C2—21no—
150C2—81no—
151C2—2961no—
152C2#1455772111yescharacteristic
subgroupG/H
centerC2#14557721
commutator, cosocleC2×Q8×C37C23
FrattiniC2×Q8C22×D37
FittingC37×#355C2
socleC2×C37B(4,𝔽2)
prime radical#14557705C2
hypercenter#355D37
hypocenterC37#2203
LCSG ▷ C2×Q8×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #14558283C2semidirect
2. #14557718C2semidirect
4. #14557331C2semidirect
5. C37×#355C2semidirect
81. #355D37semidirect
82. C37#2203semidirect

Maximal subgroups

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

subgroupindex
#145582832
#145577182
#145577052
#145573312
C37×#3552
#220337