Finite Group Databasebeta

← #17996597

Subgroups — #17996597

Order 4736.

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

normal subgroups only

This group has 270 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#14557719C212yesmaximal, complemented
2#14557710C211yesmaximal, characteristic, complemented
3#14557328C211yesmaximal, characteristic, complemented
4#14557320C212yesmaximal, complemented
5C37×H(3,𝔽4)C211yesmaximal, characteristic, complemented
6#11924581C2211yescharacteristic
7#11924567—21no—
8#11924505—22no—
9#11924464—21no—
10#11924453C2211yescharacteristic
11#11924451C2212yes—
12#11924450—22no—
13#124×C37—21no—
14#124×C37—22no—
15#124×C37—21no—
16#124×C37C2211yescharacteristic
17#120×C37—22no—
18#120×C37C2212yes—
19C24×C37D412yescomplemented
20C2×Q8×C37—21no—
21C2×Q8×C37D411yescharacteristic, complemented
22C2×D4×C37—21no—
23C2×D4×C37—21no—
24C2×D4×C37—22no—
25C2×D4×C37C2311yescharacteristic
26C2×C22⋊D37—22no—
27C2×C22⋊D37—42no—
28#10596964—41no—
29D4×D37—41no—
30#10596962—21no—
31C22⋊C4×C37—22no—
32C22⋊C4×C37—22no—
33C22⋊C4×C37—22no—
34C22⋊C4×C37—22no—
35C22⋊C4×C37D412yescomplemented
36C42×C37—21no—
37C42×C37D411yescharacteristic, complemented
38C22⋊Dic37—22no—
39C22⋊Dic37—42no—
40#10596942—41no—
41#10596941—41no—
42#10596934—21no—
43C23×C37—22no—
44C23×C37—42no—
45C23×C37—82no—
46C23×C37C2×D412yes—
47C22×D37—22no—
48Q8×C37—41no—
49Q8×C37—81no—
50D4×C37—41no—
51D4×C37—82no—
52D4×C37—81no—
53D4×C37—81no—
54C2×C4×C37—21no—
55C2×C4×C37—22no—
56C2×C4×C37—21no—
57C2×C4×C37C2×D411yescharacteristic
58C22⋊D37—42no—
59C22⋊D37—41no—
60C22⋊D37—84no—
61C2×Dic37—22no—
62C2×Dic37—42no—
63D148—21no—
64C4×D37—41no—
65Dic74—21no—
66#64436—41no—
67C22×C37—22no—
68C22×C37—42no—
69C22×C37—42no—
70C22×C37—42no—
71C22×C37—44no—
72C22×C37—42no—
73C22×C37—82no—
74C22×C37#12011yescharacteristic
75C2×D37—41no—
76C2×D37—82no—
77C4×C37—21no—
78C4×C37—42no—
79C4×C37—41no—
80C4×C37—41no—
81Dic37—41no—
82Dic37—82no—
83#2199—371nomaximal, Sylow
84C2×C37—21no—
85C2×C37—42no—
86C2×C37—82no—
87C2×C37B(4,𝔽2)11yescharacteristic
88D37—81no—
89H(3,𝔽4)D3711yescharacteristic, complemented
90B(4,𝔽2)—372no—
91#455—371no—
92#354—371no—
93AΣL(1,𝔽16)—372no—
94C37#219911yescharacteristic, complemented, Sylow
95H(5,𝔽2)—371no—
96#137—741no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D3711yescharacteristic
101#120—22no—
102#120—742no—
103#120C2×D3712yes—
104#104—741no—
105#100—371no—
106#99—372no—
107#99—742no—
108C24C22⋊D3712yescomplemented
109#41—741no—
110#41—1481no—
111C2×Q8—21no—
112C2×Q8C22⋊D3711yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—742no—
117C2×D4—1481no—
118C2×D4—1482no—
119C2×D4C22×D3711yescharacteristic
120Q16—1481no—
121SD16—1481no—
122M16—741no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—742no—
128C22⋊C4—1482no—
129C22⋊C4C22⋊D3712yescomplemented
130C42—21no—
131C42C22⋊D3711yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—742no—
136C23C2×C22⋊D3712yes—
137Q8—41no—
138Q8—81no—
139Q8—741no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—2964no—
145D4—741no—
146D4—1482no—
147D4—1481no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—742no—
152C2×C4—1481no—
153C2×C4—1482no—
154C2×C4C2×C22⋊D3711yescharacteristic
155C8—1481no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—2962no—
164C22—1481no—
165C22#1192450511yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—2962no—
171C4—1481no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—2961no—
176C2#1455772011yescharacteristic
subgroupG/H
centerC2#14557720
commutator, cosocleC2×D4×C37C23
FrattiniC2×D4C22×D37
FittingC37×H(3,𝔽4)C2
socleC2×C37B(4,𝔽2)
hypercenterH(3,𝔽4)D37
hypocenterC37#2199
LCSG ▷ C2×D4×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #14557719C2semidirect
2. #14557710C2semidirect
3. #14557328C2semidirect
4. #14557320C2semidirect
5. C37×H(3,𝔽4)C2semidirect
19. C24×C37D4semidirect
21. C2×Q8×C37D4semidirect
35. C22⋊C4×C37D4semidirect
37. C42×C37D4semidirect
89. H(3,𝔽4)D37semidirect
94. C37#2199semidirect
108. C24C22⋊D37semidirect
112. C2×Q8C22⋊D37semidirect
129. C22⋊C4C22⋊D37semidirect
131. C42C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
#145577192
#145577102
#145573282
#145573202
C37×H(3,𝔽4)2
#219937