Finite Group Databasebeta

← #28362001

Subgroups — #28362001

Order 13952.

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

normal subgroups only

This group has 282 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#20295164C212yesmaximal, complemented
2#20294946C211yesmaximal, characteristic, complemented
3#20294760C212yesmaximal, complemented
4#20294752C211yesmaximal, characteristic, complemented
5#537×C109C211yesmaximal, characteristic
6C22×Dic218C2211yescharacteristic
7C2×#13182017—22no—
8C2×#13182008—21no—
9#16812622—22no—
10C22⋊#11724784C2212yes—
11#16812606—21no—
12#16812606C2211yescharacteristic
13C2×#41×C109—21no—
14C22⋊Q8×C109—22no—
15#121×C109—22no—
16#121×C109—21no—
17#121×C109C2212yes—
18C2×C4⋊C4×C109C2211yescharacteristic
19C2×Dic218—22no—
20C2×Dic218—42no—
21C22×Dic109—21no—
22C22×Dic109—21no—
23#13182017—44no—
24C2×#11724784—22no—
25#13182008—42no—
26#41×C109—42no—
27#41×C109—42no—
28C2×Q8×C109—21no—
29C2×D4×C109—21no—
30C2×D4×C109—22no—
31C2×D4×C109—21no—
32C2×D4×C109D412yes—
33C22×C4×C109—22no—
34C22×C4×C109D411yescharacteristic
35C22×C4×C109D411yescharacteristic
36C22×C4×C109C2311yescharacteristic
37C4⋊C4×C109—21no—
38C4⋊C4×C109—22no—
39C4⋊C4×C109D412yes—
40C22⋊C4×C109—22no—
41C22⋊C4×C109—22no—
42C23×C109—22no—
43C23×C109C2×D411yescharacteristic
44Q8×C109—42no—
45D4×C109—42no—
46D4×C109—42no—
47D4×C109—42no—
48D4×C109—42no—
49D4×C109—81no—
50C2×C4×C109—21no—
51C2×C4×C109—21no—
52C2×C4×C109—21no—
53C2×C4×C109—22no—
54C2×C4×C109—41no—
55C2×C4×C109—44no—
56C2×C4×C109—41no—
57C2×C4×C109SD1612yescomplemented
58C2×C4×C109C2×D412yes—
59C2×Dic109—22no—
60C2×Dic109—22no—
61C2×Dic109—41no—
62C2×Dic109—41no—
63C2×Dic109—41no—
64C2×Dic109—41no—
65Dic218—24no—
66Dic218—42no—
67#11724784—42no—
68C22×C109—21no—
69C22×C109—21no—
70C22×C109—42no—
71C22×C109—42no—
72C22×C109—42no—
73C22×C109#12011yescharacteristic
74C22×C109C2×SD1611yescharacteristic
75C22×C109AGL(1,ℤ/8)11yescharacteristic
76C4×C109—22no—
77C4×C109—22no—
78C4×C109—42no—
79C4×C109—41no—
80Dic109—42no—
81Dic109—42no—
82C2×C109—21no—
83C2×C109—21no—
84C2×C109—82no—
85C2×C109#45011yescharacteristic
86C2×C109#45611yescharacteristic
87C2×C109B(4,𝔽2)11yescharacteristic
88#1604—1091nomaximal, Sylow
89C109#160411yescharacteristic, complemented, Sylow
90#537D10911yescharacteristic
91#448—1092no—
92#395—1091no—
93#327—1092no—
94#324—1091no—
95C2×#41—21no—
96C22×Q8—1091no—
97C2×SD16—2182no—
98C22⋊Q8—22no—
99#121—22no—
100#121—21no—
101#121C2×D10912yes—
102C2×C4⋊C4—2181no—
103C2×C4⋊C4C2×D10911yescharacteristic
104#103—2182no—
105C22⋊C8—1092no—
106#95—1091no—
107#95—2181no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—4362no—
112C2×Q8—2182no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4C22⋊D10912yes—
117C22×C4—22no—
118C22×C4—2181no—
119C22×C4—2181no—
120C22×C4C22⋊D10911yescharacteristic
121C22×C4C22⋊D10911yescharacteristic
122C22×C4C22×D10911yescharacteristic
123SD16—4364no—
124C2×C8—2182no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—4362no—
128C4⋊C4C22⋊D10912yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C23—22no—
132C23C2×C22⋊D10911yescharacteristic
133Q8—42no—
134Q8—4362no—
135Q8—2184no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—4361no—
149C2×C4—4361no—
150C2×C4—4361no—
151C2×C4—4361no—
152C2×C4—2182no—
153C2×C4—2182no—
154C2×C4#1318201712yescomplemented
155C2×C4C2×C22⋊D10912yes—
156C8—4362no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22#1681266811yescharacteristic
163C22C2×#1318201711yescharacteristic
164C22#1681272511yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—41no—
169C4—4362no—
170C4—4362no—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2029517711yescharacteristic
175C2#2029520911yescharacteristic
176C2#2029521311yescharacteristic
subgroupG/H
centerC22#16812668
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting, prime radical#537×C109C2
socleC22×C109#120
hypercenter#537D109
hypocenterC109#1604
LCSG ▷ C22×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20295164C2semidirect
2. #20294946C2semidirect
3. #20294760C2semidirect
4. #20294752C2semidirect
57. C2×C4×C109SD16semidirect
89. C109#1604semidirect
154. C2×C4#13182017semidirect

Maximal subgroups

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

subgroupindex
#202951642
#202949462
#202947602
#202947522
#537×C1092
#1604109