Finite Group Databasebeta

← D109⋊#352 #28379281

Subgroups — D109⋊#352 #28379281

Order 13952.

278 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 54 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⋊#100C211yesmaximal, characteristic
2D109⋊#99C211yesmaximal, characteristic
3#20295929C212yesmaximal, complemented
4#20295928C212yesmaximal, complemented
5#123×D109C211yesmaximal, characteristic
6D109⋊M16—21no—
7C2×C22⋊#89137—21no—
8#16812759—21no—
9#16812759C2212yes—
10#16812758—21no—
11#16812757C2212yes—
12C2×D4×D109C2211yescharacteristic
13C22×C4×D109C411yescharacteristic, complemented
14#16812684—21no—
15#16812683C2211yescharacteristic
16#16812682C2211yescharacteristic
17#16812681—21no—
18#16812680C411yescharacteristic, complemented
19C4⋊C4×D109—21no—
20C22⋊C4×D109—21no—
21C22⋊C4×D109C411yescharacteristic, complemented
22#123×C109C411yescharacteristic, complemented
23C22×#89137—41no—
24#13182039—41no—
25#13182038—22no—
26D109⋊C8—41no—
27C22⋊#89137—22no—
28C22⋊#89137—41no—
29C23×D109C2×C411yescharacteristic
30C23×D109D411yescharacteristic
31C2×D436D411yescharacteristic
32C2×C22⋊D109D411yescharacteristic
33C2×C22⋊D109C2311yescharacteristic
34D4×D109—41no—
35D4×D109—41no—
36C22×Dic109C2×C411yescharacteristic, complemented
37C2×C4×D109—21no—
38C2×C4×D109—21no—
39C2×C4×D109—21no—
40C2×C4×D109—41no—
41C2×C4×D109—41no—
42C2×C4×D109D411yescharacteristic
43C4⋊Dic109—21no—
44#13182008—21no—
45#13182008—21no—
46C22⋊Dic109—21no—
47C22⋊Dic109C2×C411yescharacteristic, complemented
48#13182006—21no—
49#13182006—21no—
50#13182006—21no—
51C2×D4×C109C2×C411yescharacteristic
52C22×C4×C109C2×C411yescharacteristic, complemented
53C4⋊C4×C109—21no—
54C22⋊C4×C109—21no—
55C22⋊C4×C109C2×C411yescharacteristic, complemented
56C23×C109C22⋊C411yescharacteristic
57C23×C109C22×C411yescharacteristic
58C22×D109—21no—
59C22×D109—21no—
60C22×D109—21no—
61C22×D109—21no—
62C22×D109—42no—
63C22×D109—42no—
64C22×D109C22⋊C411yescharacteristic
65C22×D109C22⋊C411yescharacteristic
66C22×D109C2×D411yescharacteristic
67C2×#89137—42no—
68C2×#89137—82no—
69D4×C109—41no—
70C2×C4×C109—21no—
71C2×C4×C109—21no—
72C2×C4×C109—41no—
73C2×C4×C109C22⋊C411yescharacteristic
74C22⋊D109—41no—
75C22⋊D109—41no—
76C2×Dic109—21no—
77C2×Dic109—21no—
78C2×Dic109—41no—
79C2×Dic109C2×D411yescharacteristic
80D436—41no—
81C4×D109—21no—
82C4×D109—21no—
83C4×D109—42no—
84C4×D109—42no—
85C4×D109—41no—
86C4×D109—41no—
87#11724786—42no—
88C22×C109—21no—
89C22×C109—21no—
90C22×C109—41no—
91C22×C109—41no—
92C22×C109C2×C22⋊C411yescharacteristic
93C2×D109—22no—
94C2×D109—21no—
95C2×D109—21no—
96C2×D109—42no—
97C2×D109—42no—
98C2×D109—42no—
99C2×D109—41no—
100C2×D109—41no—
101C2×D109—81no—
102C2×D109#9911yescharacteristic
103C2×D109#9911yescharacteristic
104#89137—82no—
105C4×C109—21no—
106C4×C109—41no—
107C4×C109—41no—
108Dic109—21no—
109Dic109—41no—
110Dic109—41no—
111C2×C109—21no—
112C2×C109—41no—
113C2×C109—41no—
114C2×C109C2×#9911yescharacteristic
115D109—21no—
116D109—41no—
117D109—41no—
118D109#35212yescomplemented
119C2×#352—1091nomaximal, Sylow
120C109C2×#35211yescharacteristic, complemented, Sylow
121C2×#123—1091no—
122C2×#100—1091no—
123C2×#99—1091no—
124#352—1092no—
125#352—1092no—
126C22×D4—1091no—
127C23×C4—1091no—
128C2×M16—2181no—
129#123—1091no—
130#123—1091no—
131#123—1091no—
132#123—2181no—
133#123—2181no—
134#123#8913711yescharacteristic, complemented
135C2×C4⋊C4—2181no—
136C2×C22⋊C4—1091no—
137C2×C22⋊C4—2181no—
138C2×C22⋊C4—2181no—
139#100—1092no—
140#100—2181no—
141#99—1092no—
142#99—2181no—
143C24—1091no—
144C24—1091no—
145C2×D4—1091no—
146C2×D4—1091no—
147C2×D4—1091no—
148C2×D4—4361no—
149C2×D4—4361no—
150C2×D4C2×#8913711yescharacteristic
151C22×C4—1091no—
152C22×C4—1091no—
153C22×C4—4361no—
154C22×C4—4361no—
155C22×C4—4361no—
156C22×C4—2181no—
157C22×C4—2181no—
158C22×C4—2181no—
159C22×C4C2×#8913711yescharacteristic, complemented
160M16—4361no—
161M16—2182no—
162C2×C8—4361no—
163C4⋊C4—21no—
164C4⋊C4—2181no—
165C4⋊C4—2181no—
166C4⋊C4—2181no—
167C22⋊C4—21no—
168C22⋊C4—1091no—
169C22⋊C4—4361no—
170C22⋊C4—2181no—
171C22⋊C4—2181no—
172C22⋊C4—2181no—
173C22⋊C4—2182no—
174C22⋊C4—2181no—
175C22⋊C4C2×#8913711yescharacteristic, complemented
176C23—1091no—
177C23—1091no—
178C23—1091no—
179C23—4362no—
180C23—4362no—
181C23—2181no—
182C23—2181no—
183C23—2181no—
184C23—2181no—
185C23C22⋊#8913711yescharacteristic
186C23C22×#8913711yescharacteristic
187D4—41no—
188D4—4361no—
189D4—4361no—
190D4—4361no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—8722no—
195C2×C4—1091no—
196C2×C4—4361no—
197C2×C4—4361no—
198C2×C4—4362no—
199C2×C4—4362no—
200C2×C4—4362no—
201C2×C4—4361no—
202C2×C4—2181no—
203C2×C4—2181no—
204C2×C4—2181no—
205C2×C4—2181no—
206C2×C4C22⋊#8913711yescharacteristic
207C8—4362no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—8721no—
213C22—1091no—
214C22—1091no—
215C22—4361no—
216C22—4361no—
217C22—4362no—
218C22—4362no—
219C22—4362no—
220C22—2182no—
221C22—2181no—
222C22—2181no—
223C22C2×C22⋊#8913711yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—8722no—
228C4—4361no—
229C4—4361no—
230C4—2181no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—1092no—
235C2—4361no—
236C2—4361no—
237C2—2181no—
238C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
Fitting#123×C109C4
socleC2×C109C2×#99
prime radicalC2×D4×D109C22
cosocleC2×C22⋊D109C23
hypercenter#123#89137
hypocenterC109C2×#352
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #20295929C2semidirect
4. #20295928C2semidirect
13. C22×C4×D109C4semidirect
18. #16812680C4semidirect
21. C22⋊C4×D109C4semidirect
22. #123×C109C4semidirect
36. C22×Dic109C2×C4semidirect
47. C22⋊Dic109C2×C4semidirect
52. C22×C4×C109C2×C4semidirect
55. C22⋊C4×C109C2×C4semidirect
118. D109#352semidirect
120. C109C2×#352semidirect
134. #123#89137semidirect
159. C22×C4C2×#89137semidirect
175. C22⋊C4C2×#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊#1002
D109⋊#992
#202959292
#202959282
#123×D1092
C2×#352109