Finite Group Databasebeta

← #17993555

Subgroups — #17993555

Order 4736.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#14558159C211yesmaximal, characteristic
2#14557691C212yesmaximal, complemented
3#14557445C211yesmaximal, characteristic
4#14557271C212yesmaximal, complemented
5C37×#323C211yesmaximal, characteristic
6C22×Dic74C2211yescharacteristic
7C2×Dic148—22no—
8#11924515—21no—
9#11924515C2212yes—
10#11924513—22no—
11#11924513—22no—
12C2×C22⋊Dic37—21no—
13C2×C22⋊Dic37—21no—
14#11924461—22no—
15#11924443—21no—
16#11924443C2211yescharacteristic
17C2×C22⋊C4×C37C2211yescharacteristic
18C22⋊C8×C37C2212yes—
19C24×C37D411yescharacteristic
20C22×C4×C37D411yescharacteristic
21C22×C4×C37C2311yescharacteristic
22C22×Dic37—21no—
23C22×Dic37—21no—
24C22×Dic37—21no—
25C22×Dic37—21no—
26C2×Dic74—22no—
27C2×Dic74—21no—
28C2×Dic74—42no—
29C2×Dic74D412yes—
30C2×C8×C37—22no—
31C22⋊C4×C37—22no—
32C22⋊Dic37—22no—
33C22⋊Dic37—22no—
34C22⋊Dic37—42no—
35C4⋊Dic37—21no—
36C4⋊Dic37D412yes—
37#10596936—22no—
38#10596936—22no—
39Dic148—44no—
40C23×C37—21no—
41C23×C37—21no—
42C23×C37—41no—
43C23×C37—41no—
44C23×C37Q1612yescomplemented
45C23×C37C2×D411yescharacteristic
46C2×C4×C37—21no—
47C2×C4×C37—41no—
48C2×C4×C37—41no—
49C2×C4×C37C2×D412yes—
50C2×Dic37—22no—
51C2×Dic37—22no—
52C2×Dic37—22no—
53C2×Dic37—41no—
54C2×Dic37—41no—
55C2×Dic37—41no—
56C2×Dic37—41no—
57C2×Dic37—41no—
58C2×Dic37—81no—
59C2×Dic37—81no—
60Dic74—24no—
61Dic74—42no—
62Dic74—42no—
63Dic74—81no—
64C8×C37—42no—
65C22×C37—21no—
66C22×C37—21no—
67C22×C37—22no—
68C22×C37—42no—
69C22×C37—41no—
70C22×C37—41no—
71C22×C37—81no—
72C22×C37#12011yescharacteristic
73C22×C37C2×Q1611yescharacteristic
74C22×C37#13711yescharacteristic
75C4×C37—22no—
76C4×C37—41no—
77Dic37—42no—
78Dic37—42no—
79Dic37—42no—
80Dic37—81no—
81#1602—371nomaximal, Sylow
82C2×C37—21no—
83C2×C37—21no—
84C2×C37—42no—
85C2×C37#45111yescharacteristic
86C2×C37#45511yescharacteristic
87C2×C37B(4,𝔽2)11yescharacteristic
88#543—371no—
89#451—372no—
90#393—371no—
91#328—372no—
92#323D3711yescharacteristic
93C37#160211yescharacteristic, complemented, Sylow
94C22×Q8—371no—
95C2×Q16—742no—
96C22⋊Q8—372no—
97C22⋊Q8—742no—
98C22⋊Q8—742no—
99C22⋊Q8—741no—
100C2×C22⋊C4—741no—
101C2×C22⋊C4—741no—
102C2×C22⋊C4C2×D3711yescharacteristic
103#103—742no—
104C22⋊C8C2×D3712yes—
105#95—371no—
106#95—741no—
107C24D14811yescharacteristic
108C2×Q8—372no—
109C2×Q8—741no—
110C2×Q8—742no—
111C2×Q8—1482no—
112C22×C4—741no—
113C22×C4—741no—
114C22×C4—741no—
115C22×C4—741no—
116C22×C4D14811yescharacteristic
117C22×C4C22×D3711yescharacteristic
118Q16—1484no—
119C2×C8—22no—
120C4⋊C4—372no—
121C4⋊C4—742no—
122C4⋊C4—741no—
123C4⋊C4—742no—
124C22⋊C4—22no—
125C22⋊C4—742no—
126C22⋊C4—742no—
127C22⋊C4—1482no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic14812yescomplemented
133C23C2×D14811yescharacteristic
134Q8—2961no—
135Q8—744no—
136Q8—1482no—
137Q8—1482no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—2961no—
142C2×C4—2961no—
143C2×C4—742no—
144C2×C4—742no—
145C2×C4—742no—
146C2×C4—1481no—
147C2×C4—1481no—
148C2×C4—1481no—
149C2×C4—1481no—
150C2×C4—1481no—
151C2×C4D4×D3712yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1192450611yescharacteristic
161C22C2×Dic14811yescharacteristic
162C22#1192456911yescharacteristic
163C4—22no—
164C4—41no—
165C4—2961no—
166C4—1482no—
167C4—1482no—
168C4—1482no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#1455769111yescharacteristic
173C2#1455771311yescharacteristic
174C2#1455772111yescharacteristic
subgroupG/H
centerC22#11924506
commutator, cosocleC22×C4×C37C23
FrattiniC22×C4C22×D37
FittingC37×#323C2
socleC22×C37#120
prime radicalC24×C37D4
hypercenter#323D37
hypocenterC37#1602
LCSG ▷ C22×C4×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #14557691C2semidirect
4. #14557271C2semidirect
44. C23×C37Q16semidirect
93. C37#1602semidirect
132. C23Dic148semidirect

Maximal subgroups

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

subgroupindex
#145581592
#145576912
#145574452
#145572712
C37×#3232
#160237