Finite Group Databasebeta

← #17993763

Subgroups — #17993763

Order 4736.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D148C211yesmaximal, characteristic
2#14557884C211yesmaximal, characteristic, complemented
3#14557828C211yesmaximal, characteristic, complemented
4#14557824C211yesmaximal, characteristic
5#14557278C211yesmaximal, characteristic, complemented
6#14557262C211yesmaximal, characteristic
7C37×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D37—21no—
9#11924536—21no—
10#11924536C2211yescharacteristic
11#11924536C2211yescharacteristic
12#11924516—21no—
13#11924516—22no—
14C4×D148—21no—
15C4×D148C2211yescharacteristic
16#11924469—21no—
17#11924468—21no—
18C4⋊#64436C2211yescharacteristic
19#11924461—22no—
20#11924460—21no—
21#11924458—21no—
22C4×Q8×C37C2211yescharacteristic
23C4⋊C8×C37C2211yescharacteristic
24C4×C8×C37C2211yescharacteristic
25C2×Q8×C37D411yescharacteristic
26Q8×D37—42no—
27Q8×D37—42no—
28C2×D148D411yescharacteristic
29C2×C4×D37—22no—
30C2×C4×D37—21no—
31C2×Dic74—22no—
32C2×Dic74—21no—
33C2×Dic74—21no—
34C2×C8×C37—21no—
35C2×C8×C37—21no—
36C4⋊C4×C37—21no—
37C4⋊C4×C37D411yescharacteristic
38C42×C37—21no—
39C42×C37C2311yescharacteristic
40#10596938—22no—
41#10596938—21no—
42C4⋊Dic37—22no—
43C4⋊Dic37—21no—
44C4⋊Dic37—21no—
45C4⋊Dic37—21no—
46C4⋊Dic37D411yescharacteristic
47C4⋊Dic37D412yes—
48C2×#64436—21no—
49C22×D37—21no—
50Q8×C37—21no—
51Q8×C37SD1612yes—
52C2×C4×C37—21no—
53C2×C4×C37—21no—
54C2×C4×C37C2×D411yescharacteristic
55C2×C4×C37C2×D411yescharacteristic
56C2×C4×C37C2×D411yescharacteristic
57C2×Dic37—21no—
58C2×Dic37—22no—
59C2×Dic37—41no—
60D148—21no—
61D148Q1612yes—
62C4×D37—44no—
63C4×D37—41no—
64C4×D37—41no—
65Dic74—44no—
66Dic74—41no—
67Dic74—41no—
68Dic74—42no—
69C8×C37—41no—
70C8×C37—41no—
71#64436—41no—
72C22×C37#12011yescharacteristic
73C2×D37—22no—
74C2×D37—41no—
75C2×D37—41no—
76C4×C37—22no—
77C4×C37—21no—
78C4×C37—41no—
79C4×C37C2×SD1611yescharacteristic
80C4×C37C2×Q1611yescharacteristic
81C4×C37AGL(1,ℤ/8)11yescharacteristic
82C4×C37#13711yescharacteristic
83Dic37—41no—
84Dic37—42no—
85Dic37—81no—
86#1637—371nomaximal, Sylow
87C2×C37#45011yescharacteristic
88C2×C37#45111yescharacteristic
89C2×C37#45411yescharacteristic
90D37—42no—
91D4×Q8—371no—
92#487—371no—
93#477—371no—
94#476—371no—
95#332—371no—
96Q8⋊C8D3711yescharacteristic, complemented
97D4⋊C8—371no—
98C37#163711yescharacteristic, complemented, Sylow
99C22×Q8—741no—
100C4⋊Q8—371no—
101C4⋊Q8—371no—
102C4⋊Q8—741no—
103C22⋊Q8—742no—
104C22⋊Q8—741no—
105C4×Q8C2×D3711yescharacteristic
106C4×D4—371no—
107C4×D4—741no—
108#107—741no—
109#106—741no—
110C4⋊C8—371no—
111C4⋊C8C2×D3711yescharacteristic
112#103—742no—
113#103—741no—
114#102—741no—
115C4×C8C2×D3711yescharacteristic
116C2×Q8—742no—
117C2×Q8—741no—
118C2×Q8—741no—
119C2×Q8—1482no—
120C2×Q8—1482no—
121C2×Q8D14811yescharacteristic
122C2×D4—371no—
123C22×C4—742no—
124C22×C4—741no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—741no—
128C4⋊C4—21no—
129C4⋊C4—372no—
130C4⋊C4—371no—
131C4⋊C4—742no—
132C4⋊C4—741no—
133C4⋊C4—741no—
134C4⋊C4—741no—
135C4⋊C4D14811yescharacteristic
136C22⋊C4—742no—
137C22⋊C4—741no—
138C42—21no—
139C42C22×D3711yescharacteristic
140C23—741no—
141Q8—21no—
142Q8—1482no—
143Q8—1484no—
144Q8—1481no—
145Q8—1481no—
146Q8#1059693012yes—
147D4—372no—
148D4—741no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—742no—
152C2×C4—741no—
153C2×C4—1484no—
154C2×C4—1481no—
155C2×C4—1481no—
156C2×C4—1481no—
157C2×C4C2×D14811yescharacteristic
158C2×C4D4×D3711yescharacteristic
159C2×C4D4×D3711yescharacteristic
160C8—41no—
161C8—41no—
162C8—1481no—
163C22—742no—
164C22—1481no—
165C22—1481no—
166C22#1192450611yescharacteristic
167C4—21no—
168C4—22no—
169C4—41no—
170C4—2961no—
171C4—1482no—
172C4—1481no—
173C4C2×#1059693011yescharacteristic
174C4Q16×D3711yescharacteristic
175C4#1192456411yescharacteristic
176C4#1192456511yescharacteristic
177C2—1482no—
178C2#1455768611yescharacteristic
179C2#1455769011yescharacteristic
180C2#1455770711yescharacteristic
subgroupG/H
centerC22#11924506
commutator, cosocleC42×C37C23
FrattiniC42C22×D37
FittingC37×Q8⋊C8C2
socleC22×C37#120
prime radicalC2×D148D4
hypercenterQ8⋊C8D37
hypocenterC37#1637
LCSG ▷ C42×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #14557884C2semidirect
3. #14557828C2semidirect
5. #14557278C2semidirect
7. C37×Q8⋊C8C2semidirect
96. Q8⋊C8D37semidirect
98. C37#1637semidirect

Maximal subgroups

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

subgroupindex
Q8×D1482
#145578842
#145578282
#145578242
#145572782
#145572622
C37×Q8⋊C82
#163737