Finite Group Databasebeta

← #17996565

Subgroups — #17996565

Order 4736.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#14558338C211yesmaximal, characteristic
2#14557974C211yesmaximal, characteristic, complemented
3C2×#11924479C211yesmaximal, characteristic, complemented
4#14557345C211yesmaximal, characteristic, complemented
5#14557339C211yesmaximal, characteristic, complemented
6#14557317C211yesmaximal, characteristic, complemented
7C37×#473C211yesmaximal, characteristic, complemented
8#11924582—21no—
9#11924564—21no—
10#11924564—22no—
11#11924558—21no—
12#11924558C2211yescharacteristic
13C2×D296—21no—
14C2×D296C2211yescharacteristic
15#11924482—21no—
16#11924479—22no—
17#11924479—21no—
18#11924476C2211yescharacteristic, complemented
19C2×#10596925C2211yescharacteristic, complemented
20#137×C37—21no—
21C2×Q16×C37C2211yescharacteristic, complemented
22#131×C37C2211yescharacteristic
23#108×C37C2211yescharacteristic, complemented
24#101×C37—21no—
25#41×C37D411yescharacteristic
26C2×Q8×C37—21no—
27#10596966—41no—
28D4×D37—42no—
29D4×D37—41no—
30#10596962—22no—
31#10596962—21no—
32C2×D148—22no—
33C2×D148—21no—
34Q16×C37—21no—
35Q16×C37—21no—
36Q16×C37—41no—
37SD16×C37—41no—
38M16×C37—21no—
39M16×C37—21no—
40M16×C37D411yescharacteristic
41C2×C8×C37—21no—
42C2×C8×C37C2311yescharacteristic
43Dic148D411yescharacteristic
44D296—22no—
45D296—21no—
46D296D411yescharacteristic
47D296D412yes—
48#10596930—22no—
49#10596930—21no—
50#10596925—21no—
51#10596925—21no—
52C22×D37—42no—
53C22×D37—41no—
54Q8×C37—41no—
55Q8×C37—41no—
56Q8×C37D811yescharacteristic
57D4×C37—21no—
58D4×C37D811yescharacteristic
59C2×C4×C37—21no—
60C2×C4×C37—41no—
61C2×C4×C37C2×D411yescharacteristic
62C22⋊D37—42no—
63C22⋊D37—41no—
64D148—22no—
65D148—21no—
66D148—42no—
67D148—41no—
68C4×D37—42no—
69C4×D37—41no—
70Dic74—21no—
71C8×C37—21no—
72C8×C37—41no—
73C8×C37C2×D411yescharacteristic
74C8×C37C2×D411yescharacteristic
75C22×C37—21no—
76C22×C37AGL(1,ℤ/8)11yescharacteristic
77C2×D37—42no—
78C2×D37—41no—
79C2×D37—161no—
80C2×D37—162no—
81C4×C37—21no—
82C4×C37—81no—
83C4×C37#12011yescharacteristic
84C4×C37C2×D811yescharacteristic
85Dic37—41no—
86#2194—371nomaximal, Sylow
87C2×C37—21no—
88C2×C37—41no—
89C2×C37#44711yescharacteristic
90D37—82no—
91D37—81no—
92#576—371no—
93#510—371no—
94C2×SD32—371no—
95#473D3711yescharacteristic, complemented
96#361—371no—
97#359—371no—
98#350—371no—
99C37#219411yescharacteristic, complemented, Sylow
100H(5,𝔽2)—741no—
101#137—21no—
102AGL(1,ℤ/8)—742no—
103AGL(1,ℤ/8)—741no—
104#135—371no—
105#135—741no—
106C2×Q16C2×D3711yescharacteristic, complemented
107C2×D8—371no—
108C2×D8—741no—
109#131C2×D3711yescharacteristic
110Q32—741no—
111SD32—742no—
112SD32—741no—
113M32—371no—
114C2×C16—371no—
115#108C2×D3711yescharacteristic, complemented
116#101—21no—
117#41—742no—
118#41—741no—
119#41—1481no—
120#41C22⋊D3711yescharacteristic
121C2×Q8—21no—
122C2×D4—742no—
123C2×D4—741no—
124C2×D4—1482no—
125C2×D4—1481no—
126Q16—21no—
127Q16—21no—
128Q16—41no—
129Q16—371no—
130SD16—41no—
131SD16—742no—
132SD16—741no—
133D8—372no—
134D8—371no—
135D8—742no—
136D8—741no—
137M16—21no—
138M16—21no—
139M16C22⋊D3711yescharacteristic
140C2×C8—21no—
141C2×C8C22×D3711yescharacteristic
142C16—741no—
143C16—741no—
144C23—1482no—
145C23—1481no—
146Q8—41no—
147Q8—41no—
148Q8—741no—
149Q8#1059693911yescharacteristic
150D4—21no—
151D4—742no—
152D4—741no—
153D4—1482no—
154D4—1482no—
155D4—1481no—
156D4—1481no—
157D4#1059693911yescharacteristic
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—1482no—
161C2×C4—1481no—
162C2×C4C2×C22⋊D3711yescharacteristic
163C8—21no—
164C8—41no—
165C8D4×D3711yescharacteristic
166C8D4×D3711yescharacteristic
167C22—21no—
168C22—5921no—
169C22—5922no—
170C22—1482no—
171C22—1481no—
172C22#1192456111yescharacteristic
173C4—21no—
174C4—81no—
175C4—1481no—
176C4#1192450411yescharacteristic
177C4C2×#1059693911yescharacteristic
178C2—21no—
179C2—41no—
180C2—2962no—
181C2—2961no—
182C2#1455766611yescharacteristic
subgroupG/H
centerC2#14557666
commutator, cosocleC2×C8×C37C23
FrattiniC2×C8C22×D37
FittingC37×#473C2
socleC2×C37#447
prime radical#14558338C2
hypercenter#473D37
hypocenterC37#2194
LCSG ▷ C2×C8×C37 ▷ C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
2. #14557974C2semidirect
3. C2×#11924479C2semidirect
4. #14557345C2semidirect
5. #14557339C2semidirect
6. #14557317C2semidirect
7. C37×#473C2semidirect
18. #11924476C22semidirect
19. C2×#10596925C22semidirect
21. C2×Q16×C37C22semidirect
23. #108×C37C22semidirect
95. #473D37semidirect
99. C37#2194semidirect
106. C2×Q16C2×D37semidirect
115. #108C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145583382
#145579742
C2×#119244792
#145573452
#145573392
#145573172
C37×#4732
#219437