Finite Group Databasebeta

← #28364702

Subgroups — #28364702

Order 13952.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#20295589C211yesmaximal, characteristic
2#20294998C212yesmaximal, complemented
3#20294819C211yesmaximal, characteristic, complemented
4#20294818C212yesmaximal, complemented
5#353×C109C211yesmaximal, characteristic, complemented
6C2×#13182028C2211yescharacteristic
7C2×D4×D109C411yescharacteristic, complemented
8#16812695C2211yescharacteristic
9#16812687C411yescharacteristic, complemented
10#16812686C411yescharacteristic, complemented
11#16812671—22no—
12#16812667—22no—
13#16812656C411yescharacteristic, complemented
14#16812655—22no—
15#16812614C2212yescomplemented
16#16812613—22no—
17#127×C109C2211yescharacteristic
18#99×C109C2212yescomplemented
19C23×D109—21no—
20#13182028—42no—
21C2×Dic218C2×C411yescharacteristic
22C2×D436C2×C411yescharacteristic
23C2×C22⋊D109—22no—
24C2×C22⋊D109—21no—
25C2×C22⋊D109D412yes—
26D4×D109—42no—
27C22×Dic109D412yes—
28C2×C4×D109C2×C411yescharacteristic
29#13182008—21no—
30#13182008—42no—
31C22⋊Dic109—22no—
32C22⋊Dic109—22no—
33#13182006—21no—
34#13182006—22no—
35C4×Dic109—42no—
36C4×Dic109C2×C411yescharacteristic
37C2×D4×C109—22no—
38C2×D4×C109C2×C411yescharacteristic
39C2×D4×C109C2311yescharacteristic
40C22⋊C4×C109—22no—
41C42×C109C2×C411yescharacteristic
42C23×C109—21no—
43C23×C109C2×D412yes—
44C22×D109—21no—
45C22×D109—42no—
46C22×D109—82no—
47C22×D109C22⋊C411yescharacteristic
48D4×C109—41no—
49D4×C109—41no—
50D4×C109—82no—
51C2×C4×C109—21no—
52C2×C4×C109—42no—
53C2×C4×C109C22×C411yescharacteristic
54C22⋊D109—42no—
55C22⋊D109—42no—
56C22⋊D109—82no—
57C2×Dic109—22no—
58C2×Dic109—22no—
59C2×Dic109—21no—
60C2×Dic109—42no—
61C2×Dic109C22⋊C412yes—
62C2×Dic109C22⋊C411yescharacteristic
63D436—41no—
64C4×D109—22no—
65Dic218—41no—
66C22×C109—22no—
67C22×C109—42no—
68C22×C109—42no—
69C22×C109—41no—
70C22×C109C2×C22⋊C411yescharacteristic
71C2×D109—21no—
72C2×D109—41no—
73C2×D109—42no—
74C2×D109—41no—
75C2×D109—84no—
76C4×C109—21no—
77C4×C109—41no—
78C4×C109—82no—
79Dic109—22no—
80Dic109—41no—
81Dic109—82no—
82Dic109#9912yes—
83C2×C109—21no—
84C2×C109—42no—
85C2×C109—81no—
86C2×C109C2×#9911yescharacteristic
87D109—41no—
88D109—81no—
89#2128—1091nomaximal, Sylow
90C109#212811yescharacteristic, complemented, Sylow
91#535—1091no—
92#410—1092no—
93#353—1091no—
94#353—1092no—
95#353D10911yescharacteristic, complemented
96C2×#41—1091no—
97C22×D4—1091no—
98#127—1091no—
99#127C2×D10911yescharacteristic
100#124—1091no—
101#124—1091no—
102#121—2182no—
103#120—2182no—
104#117—1091no—
105#117—2182no—
106#99—1092no—
107#99—2182no—
108#99C2×D10912yescomplemented
109C24—2181no—
110#41—4362no—
111C2×Q8—1091no—
112C2×D4—22no—
113C2×D4—1092no—
114C2×D4—1091no—
115C2×D4—4362no—
116C2×D4—2181no—
117C2×D4—2182no—
118C2×D4C4×D10911yescharacteristic
119C2×D4C22×D10911yescharacteristic
120C22×C4—1092no—
121C22×C4—1091no—
122C4⋊C4—4362no—
123C4⋊C4—2181no—
124C22⋊C4—22no—
125C22⋊C4—2182no—
126C22⋊C4—2181no—
127C22⋊C4—2182no—
128C22⋊C4—2182no—
129C42—1091no—
130C42—4362no—
131C42C4×D10911yescharacteristic
132C23—21no—
133C23—8722no—
134C23—1091no—
135C23—4362no—
136C23—2181no—
137C23D4×D10912yes—
138Q8—4361no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—8722no—
143D4—4361no—
144D4—4362no—
145D4—4362no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—1091no—
149C2×C4—1092no—
150C2×C4—4362no—
151C2×C4—2182no—
152C2×C4—2182no—
153C2×C4—2182no—
154C2×C4—2181no—
155C2×C4C2×C4×D10911yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—8724no—
161C22—4362no—
162C22—4361no—
163C22—4361no—
164C22—2181no—
165C22C22⋊C4×D10911yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—8722no—
170C4—1092no—
171C4—4361no—
172C4—2182no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—8721no—
177C2—4361no—
178C2#99×D10911yescharacteristic
subgroupG/H
centerC2#99×D109
commutatorC2×C4×C109C22×C4
FrattiniC2×D4C22×D109
Fitting#353×C109C2
socleC2×C109C2×#99
prime radical#20295589C2
cosocleC2×D4×C109C23
hypercenter#353D109
hypocenterC109#2128
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #20294998C2semidirect
3. #20294819C2semidirect
4. #20294818C2semidirect
5. #353×C109C2semidirect
7. C2×D4×D109C4semidirect
9. #16812687C4semidirect
10. #16812686C4semidirect
13. #16812656C4semidirect
15. #16812614C22semidirect
18. #99×C109C22semidirect
90. C109#2128semidirect
95. #353D109semidirect
108. #99C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955892
#202949982
#202948192
#202948182
#353×C1092
#2128109