Finite Group Databasebeta

← #17996609

Subgroups — #17996609

Order 4736.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#14557722C212yesmaximal
2#14557710C211yesmaximal, characteristic
3#14557328C211yesmaximal, characteristic, complemented
4#14557323C212yesmaximal, complemented
5C37×#563C211yesmaximal, characteristic, complemented
6#11924581C2211yescharacteristic
7#11924567—21no—
8#11924519—22no—
9#11924464—21no—
10#11924453C2211yescharacteristic
11#11924451C2212yes—
12#11924450—22no—
13C4⋊Q8×C37—21no—
14#126×C37—22no—
15#125×C37—21no—
16#124×C37C2211yescharacteristic
17#123×C37C2212yes—
18C22⋊Q8×C37—22no—
19C2×Q8×C37—21no—
20C2×Q8×C37D411yescharacteristic
21C2×D4×C37C2311yescharacteristic
22C22×C4×C37D412yes—
23C2×C22⋊D37—22no—
24#10596964—41no—
25D4×D37—41no—
26#10596962—21no—
27C4⋊C4×C37—21no—
28C4⋊C4×C37—22no—
29C4⋊C4×C37—22no—
30C4⋊C4×C37—21no—
31C4⋊C4×C37—22no—
32C22⋊C4×C37—22no—
33C22⋊C4×C37—22no—
34C22⋊C4×C37D412yes—
35C42×C37—21no—
36C42×C37D411yescharacteristic
37C22⋊Dic37—22no—
38#10596942—41no—
39#10596941—41no—
40#10596938—42no—
41#10596936—42no—
42#10596934—21no—
43C23×C37C2×D412yes—
44C22×D37—22no—
45Q8×C37—41no—
46Q8×C37—81no—
47D4×C37—41no—
48C2×C4×C37—21no—
49C2×C4×C37—22no—
50C2×C4×C37—21no—
51C2×C4×C37—22no—
52C2×C4×C37—42no—
53C2×C4×C37C2×D411yescharacteristic
54C22⋊D37—42no—
55C22⋊D37—41no—
56C2×Dic37—22no—
57C2×Dic37—42no—
58D148—21no—
59C4×D37—41no—
60Dic74—21no—
61#64436—41no—
62C22×C37—22no—
63C22×C37—42no—
64C22×C37#12011yescharacteristic
65C2×D37—41no—
66C2×D37—82no—
67C4×C37—21no—
68C4×C37—42no—
69C4×C37—41no—
70C4×C37—42no—
71C4×C37—41no—
72Dic37—41no—
73Dic37—82no—
74#2201—371nomaximal, Sylow
75C2×C37—21no—
76C2×C37—42no—
77C2×C37B(4,𝔽2)11yescharacteristic
78D37—81no—
79#563D3711yescharacteristic, complemented
80#458—372no—
81#455—371no—
82#354—371no—
83#352—372no—
84C37#220111yescharacteristic, complemented, Sylow
85H(5,𝔽2)—371no—
86#137—741no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D3711yescharacteristic
91#123—742no—
92#123C2×D3712yes—
93C22⋊Q8—22no—
94#104—741no—
95#100—371no—
96#99—372no—
97#99—742no—
98#41—741no—
99#41—1481no—
100C2×Q8—21no—
101C2×Q8C22⋊D3711yescharacteristic
102C2×D4—742no—
103C2×D4—1481no—
104C2×D4C22×D3711yescharacteristic
105C22×C4C22⋊D3712yes—
106Q16—1481no—
107SD16—1481no—
108M16—741no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—1482no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—742no—
118C22⋊C4—1482no—
119C22⋊C4C22⋊D3712yes—
120C42—21no—
121C42C22⋊D3711yescharacteristic
122C23—742no—
123C23C2×C22⋊D3712yes—
124Q8—41no—
125Q8—81no—
126Q8—741no—
127D4—41no—
128D4—741no—
129D4—1482no—
130D4—1481no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—742no—
137C2×C4—1481no—
138C2×C4—1482no—
139C2×C4C2×C22⋊D3711yescharacteristic
140C8—1481no—
141C22—22no—
142C22—42no—
143C22—2962no—
144C22—1481no—
145C22#1192450511yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—2962no—
152C4—1481no—
153C2—21no—
154C2—42no—
155C2—2961no—
156C2#1455772011yescharacteristic
subgroupG/H
centerC2#14557720
commutator, cosocleC2×D4×C37C23
FrattiniC2×D4C22×D37
FittingC37×#563C2
socleC2×C37B(4,𝔽2)
prime radical#11924581C22
hypercenter#563D37
hypocenterC37#2201
LCSG ▷ C2×D4×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #14557328C2semidirect
4. #14557323C2semidirect
5. C37×#563C2semidirect
79. #563D37semidirect
84. C37#2201semidirect

Maximal subgroups

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

subgroupindex
#145577222
#145577102
#145573282
#145573232
C37×#5632
#220137