Finite Group Databasebeta

← D109⋊#328 #28378401

Subgroups — D109⋊#328 #28378401

Order 13952.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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)C211yesmaximal, characteristic
2C2×(C2×#89137)⋊C4C211yesmaximal, characteristic
3#20295880C212yesmaximal, complemented
4#20295879C212yesmaximal, complemented
5C22⋊Q8×D109C211yesmaximal, characteristic
6C23×#89137—21no—
7C2×D109⋊C8—21no—
8#16812756C2212yes—
9C22⋊#11724786—21no—
10(C2×#89137)⋊C4—21no—
11(C2×#89137)⋊C4C2212yes—
12C2×Q8×D109C411yescharacteristic, complemented
13C22×C4×D109C2211yescharacteristic
14#16812679C2211yescharacteristic
15#16812678C411yescharacteristic, complemented
16#16812677—21no—
17#16812676—21no—
18#16812675C2211yescharacteristic
19C4⋊C4×D109—21no—
20C4⋊C4×D109C411yescharacteristic, complemented
21C22⋊C4×D109—21no—
22C22⋊Q8×C109C411yescharacteristic, complemented
23C22×#89137—24no—
24C22×#89137—22no—
25C22×#89137—42no—
26C22×#89137—42no—
27D109⋊C8—42no—
28C2×#11724786—22no—
29C23×D109D411yescharacteristic
30C2×Dic218—21no—
31C2×Dic218C2×C411yescharacteristic, complemented
32Q8×D109—42no—
33C22×Dic109D411yescharacteristic
34C2×C4×D109—21no—
35C2×C4×D109—21no—
36C2×C4×D109—42no—
37C2×C4×D109C2×C411yescharacteristic
38C2×C4×D109D411yescharacteristic
39C2×C4×D109D411yescharacteristic
40C2×C4×D109C2311yescharacteristic
41C4⋊Dic109—21no—
42C4⋊Dic109C2×C411yescharacteristic, complemented
43#13182008—21no—
44#13182008—21no—
45#13182008—21no—
46C22⋊Dic109—21no—
47#13182006—21no—
48#13182006—21no—
49C2×Q8×C109C2×C411yescharacteristic, complemented
50C22×C4×C109C2×C411yescharacteristic
51C4⋊C4×C109—21no—
52C4⋊C4×C109C2×C411yescharacteristic, complemented
53C22⋊C4×C109—21no—
54C23×C109C22⋊C411yescharacteristic
55C22×D109—22no—
56C22×D109—22no—
57C22×D109C22⋊C411yescharacteristic
58C22×D109SD1612yes—
59C22×D109Q1612yes—
60C22×D109C2×D411yescharacteristic
61C2×#89137—24no—
62C2×#89137—42no—
63C2×#89137—48no—
64C2×#89137—42no—
65Q8×C109—41no—
66C2×C4×C109—21no—
67C2×C4×C109—21no—
68C2×C4×C109—41no—
69C2×C4×C109C22⋊C411yescharacteristic
70C2×C4×C109C22×C411yescharacteristic
71C2×Dic109—21no—
72C2×Dic109—21no—
73C2×Dic109—41no—
74C2×Dic109C22⋊C411yescharacteristic
75C2×Dic109C2×D411yescharacteristic
76C4×D109—22no—
77C4×D109—22no—
78C4×D109—41no—
79C4×D109—41no—
80C4×D109—42no—
81C4×D109—42no—
82Dic218—42no—
83Dic218—41no—
84#11724786—42no—
85C22×C109—21no—
86C22×C109—21no—
87C22×C109#10312yes—
88C22×C109C2×C22⋊C411yescharacteristic
89C2×D109—21no—
90C2×D109—21no—
91C2×D109—24no—
92C2×D109—24no—
93C2×D109#9912yes—
94C2×D109#10312yes—
95C2×D109#10412yes—
96C2×D109C2×SD1611yescharacteristic
97C2×D109C2×Q1611yescharacteristic
98#89137—44no—
99C4×C109—21no—
100C4×C109—21no—
101C4×C109—41no—
102C4×C109—41no—
103Dic109—21no—
104Dic109—21no—
105Dic109—41no—
106Dic109—41no—
107C2×C109—22no—
108C2×C109C2×#9911yescharacteristic
109C2×C109C2×#10311yescharacteristic
110C2×C109C2×#10411yescharacteristic
111D109—21no—
112D109—21no—
113D109#32814yescomplemented
114C2×#328—1091nomaximal, Sylow
115C109C2×#32811yescharacteristic, complemented, Sylow
116C2×C22⋊Q8—1091no—
117C2×C22⋊C8—1091no—
118C2×#95—1091no—
119#328—1092no—
120#328—1092no—
121C22×Q8—1091no—
122C23×C4—1091no—
123C23×C4—2181no—
124C22×C8—2181no—
125C22⋊Q8—1091no—
126C22⋊Q8—1091no—
127C22⋊Q8—1091no—
128C22⋊Q8—2181no—
129C22⋊Q8—2181no—
130C22⋊Q8#8913711yescharacteristic, complemented
131C2×C4⋊C4—1091no—
132C2×C4⋊C4—2181no—
133C2×C22⋊C4—2181no—
134C22⋊C8—1092no—
135C22⋊C8—2181no—
136#95—1092no—
137#95—2181no—
138C24—1091no—
139C2×Q8—1091no—
140C2×Q8—4362no—
141C2×Q8—2181no—
142C2×Q8C2×#8913711yescharacteristic, complemented
143C22×C4—1091no—
144C22×C4—1091no—
145C22×C4—1091no—
146C22×C4—1091no—
147C22×C4—1091no—
148C22×C4—4362no—
149C22×C4—4362no—
150C22×C4—4362no—
151C22×C4—2181no—
152C22×C4—2181no—
153C22×C4—2182no—
154C22×C4—2184no—
155C22×C4C2×#8913711yescharacteristic
156C2×C8—4362no—
157C2×C8—2182no—
158C4⋊C4—21no—
159C4⋊C4—1091no—
160C4⋊C4—2181no—
161C4⋊C4—2181no—
162C4⋊C4—2181no—
163C4⋊C4—2181no—
164C4⋊C4C2×#8913711yescharacteristic, complemented
165C22⋊C4—21no—
166C22⋊C4—2181no—
167C22⋊C4—2181no—
168C22⋊C4—2181no—
169C23—1091no—
170C23—1092no—
171C23—1092no—
172C23—1091no—
173C23—2182no—
174C23—2182no—
175C23C22⋊#8913711yescharacteristic
176Q8—41no—
177Q8—4361no—
178Q8—4362no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—1091no—
183C2×C4—1091no—
184C2×C4—4361no—
185C2×C4—4361no—
186C2×C4—4362no—
187C2×C4—4362no—
188C2×C4—4362no—
189C2×C4—4368no—
190C2×C4—4362no—
191C2×C4—4361no—
192C2×C4—2182no—
193C2×C4—2182no—
194C2×C4—2184no—
195C2×C4—2181no—
196C2×C4—2181no—
197C2×C4C22⋊#8913711yescharacteristic
198C2×C4C22×#8913711yescharacteristic
199C8—4362no—
200C22—21no—
201C22—21no—
202C22—1092no—
203C22—1092no—
204C22—1092no—
205C22—1091no—
206C22—1091no—
207C22—2184no—
208C22—2184no—
209C22—2181no—
210C22—2181no—
211C22#1681276212yes—
212C22C2×C22⋊#8913711yescharacteristic
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—4364no—
218C4—4361no—
219C4—4361no—
220C4—2181no—
221C4—2181no—
222C2—22no—
223C2—1094no—
224C2—2181no—
225C2—2181no—
226C2D109⋊#9911yescharacteristic
227C2C2×#1681276211yescharacteristic
228C2D109⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC22⋊Q8×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC23×D109D4
cosocleC2×C4×D109C23
hypercenterC22⋊Q8#89137
hypocenterC109C2×#328
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20295880C2semidirect
4. #20295879C2semidirect
12. C2×Q8×D109C4semidirect
15. #16812678C4semidirect
20. C4⋊C4×D109C4semidirect
22. C22⋊Q8×C109C4semidirect
31. C2×Dic218C2×C4semidirect
42. C4⋊Dic109C2×C4semidirect
49. C2×Q8×C109C2×C4semidirect
52. C4⋊C4×C109C2×C4semidirect
113. D109#328semidirect
115. C109C2×#328semidirect
130. C22⋊Q8#89137semidirect
142. C2×Q8C2×#89137semidirect
164. C4⋊C4C2×#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊(C22⋊C8)2
C2×(C2×#89137)⋊C42
#202958802
#202958792
C22⋊Q8×D1092
C2×#328109