Finite Group Databasebeta

← #17993721

Subgroups — #17993721

Order 4736.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#14558205C211yesmaximal, characteristic
2#14557907C211yesmaximal, characteristic, complemented
3#14557820C211yesmaximal, characteristic, complemented
4#14557276C211yesmaximal, characteristic, complemented
5Q8⋊#64436C211yesmaximal, characteristic, complemented
6#14557260C211yesmaximal, characteristic, complemented
7C37×#477C211yesmaximal, characteristic, complemented
8C2×#10596966—21no—
9C2×#10596941—22no—
10#11924533—21no—
11#11924533C2211yescharacteristic
12#11924511—22no—
13#11924511—21no—
14C4×D148—21no—
15C4×D148C2211yescharacteristic
16#11924467—21no—
17C4⋊#64436C2211yescharacteristic
18#11924456—21no—
19C4×#64436C2211yescharacteristic
20C4⋊Q8×C37C2211yescharacteristic
21C4×Q8×C37C2211yescharacteristic
22#107×C37—21no—
23C4⋊C8×C37C2211yescharacteristic
24#103×C37—21no—
25C2×Q8×C37—21no—
26C2×Q8×C37D411yescharacteristic
27#10596966—42no—
28#10596966—42no—
29C2×D148—22no—
30C2×D148—21no—
31C2×D148—21no—
32C2×D148—22no—
33C2×D148D411yescharacteristic
34C2×D148D412yes—
35C2×C4×D37—22no—
36C2×C4×D37—21no—
37C2×C8×C37—21no—
38C4⋊C4×C37—21no—
39C4⋊C4×C37—21no—
40C4⋊C4×C37—21no—
41C4⋊C4×C37D411yescharacteristic
42C42×C37—21no—
43C42×C37C2311yescharacteristic
44#10596941—44no—
45#10596938—22no—
46#10596938—21no—
47C4⋊Dic37D411yescharacteristic
48C2×#64436—21no—
49C2×#64436—21no—
50C22×D37—21no—
51C22×D37—22no—
52Q8×C37—21no—
53Q8×C37—42no—
54Q8×C37SD1612yescomplemented
55C2×C4×C37—21no—
56C2×C4×C37—21no—
57C2×C4×C37—41no—
58C2×C4×C37C2×D411yescharacteristic
59C2×C4×C37C2×D411yescharacteristic
60C2×C4×C37C2×D411yescharacteristic
61C2×Dic37—21no—
62D148—21no—
63D148—21no—
64D148—42no—
65D148—44no—
66D148—41no—
67D148—41no—
68D148—41no—
69D148—41no—
70D148—82no—
71C4×D37—44no—
72C4×D37—41no—
73C4×D37—41no—
74C8×C37—41no—
75#64436—22no—
76#64436—41no—
77C22×C37#12011yescharacteristic
78C2×D37—42no—
79C2×D37—41no—
80C2×D37—41no—
81C2×D37—42no—
82C2×D37—42no—
83C2×D37—41no—
84C4×C37—22no—
85C4×C37—21no—
86C4×C37—41no—
87C4×C37—81no—
88C4×C37C2×SD1611yescharacteristic
89C4×C37#13511yescharacteristic
90C4×C37AGL(1,ℤ/8)11yescharacteristic
91C4×C37#13711yescharacteristic
92Dic37—41no—
93#1631—371nomaximal, Sylow
94C2×C37#45011yescharacteristic
95C2×C37#45211yescharacteristic
96C2×C37#45311yescharacteristic
97D37—81no—
98D37—82no—
99#550—371no—
100#492—371no—
101#477D3711yescharacteristic, complemented
102#476—371no—
103#331—371no—
104Q8⋊C8—371no—
105D4⋊C8—371no—
106C37#163111yescharacteristic, complemented, Sylow
107C2×#41—741no—
108C2×SD16—742no—
109C4⋊Q8C2×D3711yescharacteristic
110#127—371no—
111#127—741no—
112#121—742no—
113#121—741no—
114C4×Q8C2×D3711yescharacteristic
115C4×D4—371no—
116C4×D4—741no—
117#107—21no—
118#106—741no—
119C4⋊C8—371no—
120C4⋊C8C2×D3711yescharacteristic
121#103—21no—
122#102—741no—
123C4×C8—371no—
124#41—1482no—
125#41—1482no—
126C2×Q8—21no—
127C2×Q8C22⋊D3711yescharacteristic
128C2×D4—371no—
129C2×D4—372no—
130C2×D4—742no—
131C2×D4—742no—
132C2×D4—741no—
133C2×D4—741no—
134C22×C4—742no—
135C22×C4—741no—
136SD16—1484no—
137C2×C8—21no—
138C2×C8—741no—
139C2×C8—741no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—371no—
144C4⋊C4C22⋊D3711yescharacteristic
145C22⋊C4—742no—
146C22⋊C4—741no—
147C42—21no—
148C42C22×D3711yescharacteristic
149C23—741no—
150C23—742no—
151Q8—21no—
152Q8—42no—
153Q8#1059694112yescomplemented
154D4—2962no—
155D4—741no—
156D4—741no—
157D4—1482no—
158D4—1484no—
159D4—1481no—
160D4—1481no—
161D4—1481no—
162D4—1481no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—741no—
167C2×C4—1484no—
168C2×C4—1481no—
169C2×C4—1481no—
170C2×C4D4×D3711yescharacteristic
171C2×C4D4×D3711yescharacteristic
172C2×C4C2×C22⋊D3711yescharacteristic
173C8—41no—
174C8—742no—
175C8—1481no—
176C22—1481no—
177C22—1482no—
178C22—1482no—
179C22—1481no—
180C22—1481no—
181C22—1482no—
182C22#1192450411yescharacteristic
183C4—21no—
184C4—22no—
185C4—41no—
186C4—81no—
187C4—1481no—
188C4C2×#1059694111yescharacteristic
189C4#1192455911yescharacteristic
190C4#1192456111yescharacteristic
191C4#1192456811yescharacteristic
192C2—2961no—
193C2—2962no—
194C2#1455768311yescharacteristic
195C2#1455769411yescharacteristic
196C2#1455769911yescharacteristic
subgroupG/H
centerC22#11924504
commutator, cosocleC42×C37C23
FrattiniC42C22×D37
FittingC37×#477C2
socleC22×C37#120
prime radical#14558205C2
hypercenter#477D37
hypocenterC37#1631
LCSG ▷ C42×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #14557907C2semidirect
3. #14557820C2semidirect
4. #14557276C2semidirect
5. Q8⋊#64436C2semidirect
6. #14557260C2semidirect
7. C37×#477C2semidirect
54. Q8×C37SD16semidirect
101. #477D37semidirect
106. C37#1631semidirect
153. Q8#10596941semidirect

Maximal subgroups

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

subgroupindex
#145582052
#145579072
#145578202
#145572762
Q8⋊#644362
#145572602
C37×#4772
#163137