Finite Group Databasebeta

← #28389531

Subgroups — #28389531

Order 14464.

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#20298753C212yesmaximal, complemented
2#20298510C211yesmaximal, characteristic
3#20298335C212yesmaximal, complemented
4#20298322C211yesmaximal, characteristic
5#543×C113C211yesmaximal, characteristic
6C22×Dic226C2211yescharacteristic
7C2×#14268637—22no—
8C2×C22⋊Dic113—21no—
9#16813545—22no—
10C22⋊#11744751C2212yes—
11#16813529—21no—
12#16813529C2211yescharacteristic
13C22⋊Q8×C113—22no—
14C22⋊Q8×C113—21no—
15C22⋊Q8×C113—22no—
16C22⋊Q8×C113C2212yes—
17C2×C22⋊C4×C113—21no—
18C2×C22⋊C4×C113C2211yescharacteristic
19C2×Dic226—22no—
20C2×Dic226—42no—
21C22×Dic113—21no—
22C22×Dic113—21no—
23#14268637—44no—
24C2×#11744751—22no—
25C22⋊Dic113—42no—
26C24×C113D411yescharacteristic
27C2×Q8×C113—21no—
28C2×Q8×C113D412yes—
29C22×C4×C113—21no—
30C22×C4×C113—21no—
31C22×C4×C113D411yescharacteristic
32C22×C4×C113C2311yescharacteristic
33C4⋊C4×C113—21no—
34C4⋊C4×C113—22no—
35C4⋊C4×C113—22no—
36C4⋊C4×C113D412yes—
37C22⋊C4×C113—22no—
38C22⋊C4×C113—22no—
39C22⋊C4×C113—22no—
40C23×C113—21no—
41C23×C113—21no—
42C23×C113—41no—
43C23×C113—41no—
44C23×C113Q1612yescomplemented
45C23×C113C2×D411yescharacteristic
46Q8×C113—42no—
47Q8×C113—81no—
48C2×C4×C113—22no—
49C2×C4×C113—21no—
50C2×C4×C113—22no—
51C2×C4×C113—41no—
52C2×C4×C113—41no—
53C2×C4×C113—81no—
54C2×C4×C113—81no—
55C2×C4×C113C2×D412yes—
56C2×Dic113—22no—
57C2×Dic113—41no—
58C2×Dic113—41no—
59C2×Dic113—41no—
60C2×Dic113—41no—
61C2×Dic113—41no—
62Dic226—24no—
63Dic226—42no—
64#11744751—42no—
65C22×C113—21no—
66C22×C113—21no—
67C22×C113—22no—
68C22×C113—42no—
69C22×C113—41no—
70C22×C113—41no—
71C22×C113—81no—
72C22×C113#12011yescharacteristic
73C22×C113C2×Q1611yescharacteristic
74C22×C113#13711yescharacteristic
75C4×C113—22no—
76C4×C113—41no—
77C4×C113—42no—
78C4×C113—42no—
79Dic113—42no—
80Dic113—81no—
81C2×C113—21no—
82C2×C113—21no—
83C2×C113—42no—
84C2×C113#45111yescharacteristic
85C2×C113#45511yescharacteristic
86C2×C113B(4,𝔽2)11yescharacteristic
87#1602—1131nomaximal, Sylow
88C113#160211yescharacteristic, complemented, Sylow
89#543D11311yescharacteristic
90#451—1132no—
91#393—1131no—
92#328—1132no—
93#323—1131no—
94C22×Q8—1131no—
95C2×Q16—2262no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D11312yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—2261no—
102C2×C22⋊C4C2×D11311yescharacteristic
103#103—2262no—
104C22⋊C8—1132no—
105#95—1131no—
106#95—2261no—
107C24C22⋊D11311yescharacteristic
108C2×Q8—21no—
109C2×Q8—4522no—
110C2×Q8—2262no—
111C2×Q8C22⋊D11312yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—2261no—
115C22×C4—2261no—
116C22×C4C22⋊D11311yescharacteristic
117C22×C4C22×D11311yescharacteristic
118Q16—4524no—
119C2×C8—2262no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D11312yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—4522no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1426863712yescomplemented
133C23C2×C22⋊D11311yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—4522no—
137Q8—2264no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—4521no—
146C2×C4—4521no—
147C2×C4—4521no—
148C2×C4—4521no—
149C2×C4—4521no—
150C2×C4—2262no—
151C2×C4C2×C22⋊D11312yes—
152C8—4522no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1681359111yescharacteristic
161C22C2×#1426863711yescharacteristic
162C22#1681365311yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—9041no—
168C4—4522no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2029875411yescharacteristic
173C2#2029877611yescharacteristic
174C2#2029878511yescharacteristic
subgroupG/H
centerC22#16813591
commutator, cosocleC22×C4×C113C23
FrattiniC22×C4C22×D113
Fitting#543×C113C2
socleC22×C113#120
prime radicalC24×C113D4
hypercenter#543D113
hypocenterC113#1602
LCSG ▷ C22×C4×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20298753C2semidirect
3. #20298335C2semidirect
44. C23×C113Q16semidirect
88. C113#1602semidirect
132. C23#14268637semidirect

Maximal subgroups

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

subgroupindex
#202987532
#202985102
#202983352
#202983222
#543×C1132
#1602113