Finite Group Databasebeta

← #17996611

Subgroups — #17996611

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#14557724C212yesmaximal
3#14557710C211yesmaximal, characteristic
4#14557322C212yesmaximal, complemented
5C37×#354C211yesmaximal, characteristic, complemented
6#11924581C2211yescharacteristic
7#11924567—21no—
8#11924534—21no—
9#11924529—22no—
10#11924528—21no—
11#11924520—22no—
12#11924517C2212yes—
13#11924513—22no—
14#11924464—21no—
15#11924453C2211yescharacteristic
16#11924450—22no—
17#124×C37C2211yescharacteristic
18#99×C37C2212yes—
19C2×Q8×C37D411yescharacteristic
20C2×D4×C37C2311yescharacteristic
21C2×C22⋊D37—22no—
22C22×Dic37D412yes—
23#10596964—41no—
24D4×D37—41no—
25#10596962—21no—
26C2×Dic74—21no—
27C22⋊C4×C37—22no—
28C22⋊C4×C37—22no—
29C42×C37D411yescharacteristic
30C22⋊Dic37—22no—
31C22⋊Dic37D412yes—
32#10596942—41no—
33#10596941—41no—
34#10596938—42no—
35C4⋊Dic37—21no—
36C4⋊Dic37—21no—
37C4⋊Dic37—42no—
38#10596936—22no—
39#10596936—22no—
40#10596936—22no—
41C4×Dic37—21no—
42#10596934—21no—
43C23×C37C2×D412yes—
44C22×D37—22no—
45Q8×C37—41no—
46D4×C37—41no—
47C2×C4×C37—21no—
48C2×C4×C37—21no—
49C2×C4×C37—42no—
50C2×C4×C37C2×D411yescharacteristic
51C22⋊D37—42no—
52C22⋊D37—41no—
53C2×Dic37—22no—
54C2×Dic37—22no—
55C2×Dic37—22no—
56C2×Dic37—42no—
57D148—21no—
58C4×D37—41no—
59Dic74—21no—
60Dic74—81no—
61#64436—41no—
62C22×C37—22no—
63C22×C37—42no—
64C22×C37#12011yescharacteristic
65C2×D37—41no—
66C2×D37—82no—
67C4×C37—21no—
68C4×C37—41no—
69C4×C37—41no—
70C4×C37—82no—
71Dic37—42no—
72Dic37—41no—
73Dic37—42no—
74#2201—371nomaximal, Sylow
75C2×C37—21no—
76C2×C37—42no—
77C2×C37B(4,𝔽2)11yescharacteristic
78D37—81no—
79#563—371no—
80#458—372no—
81#455—371no—
82#354D3711yescharacteristic, complemented
83#352—372no—
84C37#220111yescharacteristic, complemented, Sylow
85H(5,𝔽2)—371no—
86#137—741no—
87C4⋊Q8—741no—
88#126—742no—
89#125—741no—
90#124C2×D3711yescharacteristic
91#123—372no—
92#123—742no—
93C22⋊Q8—742no—
94#104—741no—
95#100—371no—
96#99—742no—
97#99C2×D3712yes—
98#41—741no—
99#41—1481no—
100C2×Q8—741no—
101C2×Q8D14811yescharacteristic
102C2×D4—742no—
103C2×D4—1481no—
104C2×D4C22×D3711yescharacteristic
105C22×C4—372no—
106Q16—1481no—
107SD16—1481no—
108M16—741no—
109C4⋊C4—742no—
110C4⋊C4—742no—
111C4⋊C4—741no—
112C4⋊C4—741no—
113C4⋊C4—742no—
114C4⋊C4—1482no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—372no—
118C22⋊C4—742no—
119C22⋊C4—1482no—
120C42—741no—
121C42D14811yescharacteristic
122C23—742no—
123C23D4×D3712yes—
124Q8—41no—
125Q8—2961no—
126Q8—741no—
127D4—41no—
128D4—741no—
129D4—1482no—
130D4—1481no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—742no—
135C2×C4—742no—
136C2×C4—742no—
137C2×C4—1481no—
138C2×C4—1482no—
139C2×C4C2×D14811yescharacteristic
140C8—1481no—
141C22—22no—
142C22—42no—
143C22—2962no—
144C22—1481no—
145C22#1192450611yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—1482no—
151C4—1481no—
152C4—1482no—
153C2—21no—
154C2—42no—
155C2—2961no—
156C2#1455772111yescharacteristic
subgroupG/H
centerC2#14557721
commutator, cosocleC2×D4×C37C23
FrattiniC2×D4C22×D37
FittingC37×#354C2
socleC2×C37B(4,𝔽2)
prime radical#11924581C22
hypercenter#354D37
hypocenterC37#2201
LCSG ▷ C2×D4×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #14558283C2semidirect
4. #14557322C2semidirect
5. C37×#354C2semidirect
82. #354D37semidirect
84. C37#2201semidirect

Maximal subgroups

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

subgroupindex
#145582832
#145577242
#145577102
#145573222
C37×#3542
#220137