Finite Group Databasebeta

← #28378117

Subgroups — #28378117

Order 13952.

234 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C2×#16812759C211yesmaximal, characteristic
2C2×#16812757C211yesmaximal, characteristic
3#20294903C211yesmaximal, characteristic
4C2×#13182038—21no—
5C2×C22⋊#89137—21no—
6#16812759—21no—
7#16812759C412yescomplemented
8#16812757—21no—
9#16812757C412yescomplemented
10C22×C22⋊D109C2211yescharacteristic
11C23×Dic109C411yescharacteristic, complemented
12C2×#13182006—21no—
13#16812606—21no—
14C2×C22⋊C4×C109C411yescharacteristic, complemented
15C22×#89137—41no—
16#13182038—22no—
17#13182038—41no—
18C2×#11724786—41no—
19C22⋊#89137—22no—
20C22⋊#89137—41no—
21C23×D109D411yescharacteristic
22C2×C22⋊D109—41no—
23C2×C22⋊D109—41no—
24C2×C22⋊D109C2×C411yescharacteristic
25C2×C22⋊D109C2×C411yescharacteristic
26C2×C22⋊D109D411yescharacteristic
27C2×C22⋊D109Q811yescharacteristic
28C22×Dic109—21no—
29C22×Dic109—42no—
30C22×Dic109—41no—
31C22×Dic109D411yescharacteristic
32#13182006—42no—
33C24×C109C2×C411yescharacteristic
34C22×C4×C109—21no—
35C22⋊C4×C109—41no—
36C23×C109—21no—
37C23×C109—21no—
38C23×C109—41no—
39C23×C109—41no—
40C23×C109C4211yescharacteristic
41C23×C109C22⋊C411yescharacteristic
42C23×C109C4⋊C411yescharacteristic
43C22×D109—21no—
44C22×D109—21no—
45C22×D109—41no—
46C22×D109—41no—
47C22×D109C22⋊C411yescharacteristic
48C22×D109C4⋊C411yescharacteristic
49C2×#89137—42no—
50C2×#89137—81no—
51C2×#89137—81no—
52C2×C4×C109—41no—
53C2×C4×C109—41no—
54C2×C4×C109—41no—
55C22⋊D109—41no—
56C22⋊D109—41no—
57C22⋊D109—41no—
58C22⋊D109—41no—
59C2×Dic109—21no—
60C2×Dic109—21no—
61C2×Dic109—22no—
62C2×Dic109—41no—
63C2×Dic109—42no—
64C2×Dic109—41no—
65C2×Dic109—42no—
66C2×Dic109C22⋊C411yescharacteristic
67C2×Dic109C4⋊C411yescharacteristic
68#11724786—42no—
69C22×C109—21no—
70C22×C109—21no—
71C22×C109—21no—
72C22×C109—21no—
73C22×C109—41no—
74C22×C109—41no—
75C22×C109—42no—
76C22×C109—41no—
77C22×C109—41no—
78C22×C109#9511yescharacteristic
79C22×C109#9911yescharacteristic
80C22×C109#9911yescharacteristic
81C2×D109—21no—
82C2×D109—21no—
83C2×D109—41no—
84C2×D109—41no—
85C2×D109—41no—
86C2×D109—41no—
87C2×D109—81no—
88#89137—82no—
89C4×C109—81no—
90Dic109—21no—
91Dic109—21no—
92Dic109—42no—
93C2×C109—21no—
94C2×C109—21no—
95C2×C109—41no—
96C2×C109—41no—
97C2×C109#34211yescharacteristic
98C2×C109AΣL(1,𝔽16)11yescharacteristic
99C2×C109#35211yescharacteristic
100D109—41no—
101D109—41no—
102#1390—1091nomaximal, Sylow
103C109#139011yescharacteristic, complemented, Sylow
104C2×#100—1091no—
105C2×#99—1091no—
106#386—1091no—
107C22×D4—1091no—
108C23×C4—1091no—
109C2×M16—2181no—
110C2×C22⋊C4—2181no—
111C2×C22⋊C4—2181no—
112C2×C22⋊C4#8913711yescharacteristic, complemented
113#100—1092no—
114#100—2181no—
115#99—1092no—
116#99—2181no—
117#95—2181no—
118C24—1091no—
119C24C2×#8913711yescharacteristic
120C2×D4—1091no—
121C2×D4—1091no—
122C2×D4—1091no—
123C2×D4—1091no—
124C2×D4—4361no—
125C2×D4—4361no—
126C22×C4—21no—
127C22×C4—1091no—
128C22×C4—4361no—
129C22×C4—4361no—
130C22×C4—4362no—
131C22×C4—2181no—
132M16—4361no—
133M16—2182no—
134C2×C8—4361no—
135C22⋊C4—41no—
136C22⋊C4—4362no—
137C22⋊C4—4361no—
138C22⋊C4—2182no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—1091no—
144C23—1091no—
145C23—4361no—
146C23—4361no—
147C23—2181no—
148C23—2181no—
149C23C4×#8913711yescharacteristic
150C23C22⋊#8913711yescharacteristic
151C23C4⋊#8913711yescharacteristic
152D4—4361no—
153D4—4361no—
154D4—4361no—
155D4—4361no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—8721no—
160C2×C4—8721no—
161C2×C4—1091no—
162C2×C4—1091no—
163C2×C4—4362no—
164C2×C4—4361no—
165C2×C4—4362no—
166C2×C4—4361no—
167C2×C4—4362no—
168C2×C4—2182no—
169C2×C4—2181no—
170C2×C4—2181no—
171C8—4362no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—8721no—
182C22—4361no—
183C22—4361no—
184C22—4361no—
185C22—4361no—
186C22—2181no—
187C22—2181no—
188C22(C2×#89137)⋊C411yescharacteristic
189C22#1681275711yescharacteristic
190C22#1681275811yescharacteristic
191C4—81no—
192C4—8722no—
193C4—4362no—
194C4—2181no—
195C4—2181no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—4361no—
201C2—4361no—
202C2#2029590411yescharacteristic
203C2#2029592711yescharacteristic
204C2#2029592911yescharacteristic
subgroupG/H
centerC22#16812758
commutatorC23×C109C42
FrattiniC24C2×#89137
FittingC2×C22⋊C4×C109C4
socleC22×C109#99
prime radical, cosocleC22×C22⋊D109C22
hypercenterC2×C22⋊C4#89137
hypocenterC109#1390
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. #16812759C4semidirect
9. #16812757C4semidirect
11. C23×Dic109C4semidirect
14. C2×C22⋊C4×C109C4semidirect
103. C109#1390semidirect
112. C2×C22⋊C4#89137semidirect

Maximal subgroups

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

subgroupindex
C2×#168127592
C2×#168127572
#202949032
#1390109