Finite Group Databasebeta

← D109⋊#323 #28378258

Subgroups — D109⋊#323 #28378258

Order 13952.

456 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D109⋊(C22⋊C8)C212yesmaximal, complemented
2#20295870C212yesmaximal, complemented
3#20295869C212yesmaximal, complemented
4C2×C22⋊C4×D109C211yesmaximal, characteristic
5C2×D109⋊C8—22no—
6#16812756—22no—
7C22⋊#11724786C2214yescomplemented
8C24×D109C411yescharacteristic
9C22×C4×D109C411yescharacteristic
10C22×C4×D109C2211yescharacteristic
11C2×C22⋊Dic109C2211yescharacteristic
12C2×#13182006C411yescharacteristic
13C2×#13182006C2211yescharacteristic
14C22⋊C4×D109—22no—
15C22⋊C4×D109—22no—
16C2×C22⋊C4×C109C411yescharacteristic
17D109⋊C8—44no—
18C2×#11724786—24no—
19C23×D109—21no—
20C23×D109—21no—
21C23×D109—22no—
22C23×D109—42no—
23C23×D109—42no—
24C23×D109C814yescomplemented
25C23×D109C2×C411yescharacteristic
26C23×D109C2×C411yescharacteristic
27C22×Dic109C2×C411yescharacteristic
28C22×Dic109C2311yescharacteristic
29C2×C4×D109—21no—
30C2×C4×D109—21no—
31C2×C4×D109—42no—
32C2×C4×D109—42no—
33C2×C4×D109D412yes—
34C2×C4×D109D412yes—
35C22⋊Dic109—22no—
36#13182006—22no—
37#13182006—22no—
38C24×C109C2×C411yescharacteristic
39C22×C4×C109C2×C411yescharacteristic
40C22×C4×C109C2×C411yescharacteristic
41C22⋊C4×C109—22no—
42C23×C109—21no—
43C23×C109—21no—
44C23×C109—41no—
45C23×C109—41no—
46C23×C109C2×C812yescomplemented
47C23×C109C22×C411yescharacteristic
48C22×D109—22no—
49C22×D109—22no—
50C22×D109—21no—
51C22×D109—21no—
52C22×D109—24no—
53C22×D109—24no—
54C22×D109—42no—
55C22×D109—42no—
56C22×D109—48no—
57C22×D109—48no—
58C22×D109—42no—
59C22×D109—42no—
60C22×D109—41no—
61C22×D109—41no—
62C22×D109C22⋊C411yescharacteristic
63C22×D109C22⋊C411yescharacteristic
64C22×D109C2×C812yes—
65C22×D109C2×C812yescomplemented
66C22×D109M1612yes—
67C2×C4×C109—21no—
68C2×C4×C109—41no—
69C2×C4×C109—41no—
70C2×C4×C109C22⋊C412yes—
71C2×Dic109—21no—
72C2×Dic109—41no—
73C2×Dic109—41no—
74C2×Dic109C2×D412yes—
75C4×D109—24no—
76C4×D109—42no—
77C4×D109—42no—
78C4×D109—42no—
79#11724786—44no—
80C22×C109—21no—
81C22×C109—21no—
82C22×C109—22no—
83C22×C109—42no—
84C22×C109—42no—
85C22×C109—41no—
86C22×C109—41no—
87C22×C109C2×C22⋊C411yescharacteristic
88C22×C109C22×C811yescharacteristic
89C22×C109C2×M1611yescharacteristic
90C2×D109—21no—
91C2×D109—21no—
92C2×D109—24no—
93C2×D109—24no—
94C2×D109—22no—
95C2×D109—48no—
96C2×D109—42no—
97C2×D109—41no—
98C2×D109—41no—
99C2×D109—48no—
100C2×D109—42no—
101C2×D109C22⋊C812yes—
102C2×D109C22⋊C812yes—
103C2×D109#9912yes—
104C2×D109#10012yes—
105C4×C109—22no—
106C4×C109—41no—
107Dic109—22no—
108Dic109—41no—
109C2×C109—21no—
110C2×C109—21no—
111C2×C109—42no—
112C2×C109C2×C22⋊C811yescharacteristic
113C2×C109C2×#9911yescharacteristic
114C2×C109C2×#10011yescharacteristic
115D109—22no—
116D109—42no—
117D109#32314yescomplemented
118C2×#323—1091nomaximal, Sylow
119C109C2×#32311yescharacteristic, complemented, Sylow
120C22×C22⋊C4—1091no—
121C2×C22⋊C8—1092no—
122#323—1092no—
123#323—1092no—
124C25—1091no—
125C23×C4—1091no—
126C23×C4—1091no—
127C22×C8—2182no—
128C2×C22⋊C4—1091no—
129C2×C22⋊C4—1091no—
130C2×C22⋊C4—1091no—
131C2×C22⋊C4—2182no—
132C2×C22⋊C4—2182no—
133C2×C22⋊C4#8913711yescharacteristic
134C22⋊C8—1094no—
135C22⋊C8—2182no—
136C24—1091no—
137C24—1094no—
138C24—1091no—
139C24—4362no—
140C24—4362no—
141C24—2181no—
142C24—2181no—
143C24—2182no—
144C24C2×#8913711yescharacteristic
145C22×C4—1091no—
146C22×C4—1092no—
147C22×C4—1092no—
148C22×C4—1091no—
149C22×C4—4362no—
150C22×C4—4362no—
151C22×C4—2181no—
152C22×C4—2181no—
153C22×C4C2×#8913711yescharacteristic
154C22×C4C2×#8913711yescharacteristic
155C2×C8—4364no—
156C2×C8—2184no—
157C22⋊C4—22no—
158C22⋊C4—2182no—
159C22⋊C4—2182no—
160C22⋊C4—2182no—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23—1091no—
166C23—1091no—
167C23—1092no—
168C23—1092no—
169C23—1092no—
170C23—4361no—
171C23—4361no—
172C23—4362no—
173C23—4362no—
174C23—4368no—
175C23—4362no—
176C23—4362no—
177C23—4368no—
178C23—2184no—
179C23—2184no—
180C23—2182no—
181C23—2182no—
182C23—2181no—
183C23—2181no—
184C23D109⋊C812yescomplemented
185C23C22×#8913711yescharacteristic
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1092no—
190C2×C4—4362no—
191C2×C4—4362no—
192C2×C4—4362no—
193C2×C4—4361no—
194C2×C4—4361no—
195C2×C4—2184no—
196C2×C4—2181no—
197C2×C4C22⋊#8913712yes—
198C8—4364no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—1092no—
207C22—1092no—
208C22—1092no—
209C22—1092no—
210C22—4362no—
211C22—4362no—
212C22—4368no—
213C22—4368no—
214C22—4361no—
215C22—4361no—
216C22—2184no—
217C22—2184no—
218C22—2182no—
219C22—2181no—
220C22—2181no—
221C22C2×C22⋊#8913711yescharacteristic
222C22C2×D109⋊C811yescharacteristic
223C22D109⋊M1611yescharacteristic
224C4—22no—
225C4—41no—
226C4—4361no—
227C4—2182no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2—1094no—
232C2—4362no—
233C2—2182no—
234C2D109⋊(C22⋊C8)11yescharacteristic
235C2D109⋊#9911yescharacteristic
236C2D109⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC22×C109C22×C8
FrattiniC23C22×#89137
FittingC2×C22⋊C4×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC24×D109C4
cosocleC22×Dic109C23
hypercenterC2×C22⋊C4#89137
hypocenterC109C2×#323
LCSG ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. D109⋊(C22⋊C8)C2semidirect
2. #20295870C2semidirect
3. #20295869C2semidirect
7. C22⋊#11724786C22semidirect
24. C23×D109C8semidirect
46. C23×C109C2×C8semidirect
65. C22×D109C2×C8semidirect
117. D109#323semidirect
119. C109C2×#323semidirect
184. C23D109⋊C8semidirect

Maximal subgroups

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

subgroupindex
D109⋊(C22⋊C8)2
#202958702
#202958692
C2×C22⋊C4×D1092
C2×#323109