Finite Group Databasebeta

← D109⋊#327 #28378398

Subgroups — D109⋊#327 #28378398

Order 13952.

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

normal subgroups only

This group has 388 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, complemented
2C2×(C2×#89137)⋊C4C211yesmaximal, characteristic, complemented
3#20295878C212yesmaximal, complemented
4#20295877C212yesmaximal, complemented
5#121×D109C211yesmaximal, characteristic
6C23×#89137—21no—
7C2×D109⋊C8—21no—
8#16812756C2212yescomplemented
9C22⋊#11724786—21no—
10(C2×#89137)⋊C4—21no—
11(C2×#89137)⋊C4C2212yescomplemented
12C2×D4×D109—21no—
13C2×D4×D109C411yescharacteristic, complemented
14C22×C4×D109C2211yescharacteristic
15#16812674C2211yescharacteristic
16#16812673C411yescharacteristic, complemented
17#16812672—21no—
18#16812671—21no—
19#16812670C2211yescharacteristic
20C4⋊C4×D109C411yescharacteristic, complemented
21C22⋊C4×D109—21no—
22#121×C109C411yescharacteristic, complemented
23C22×#89137—24no—
24C22×#89137—22no—
25C22×#89137—42no—
26C22×#89137—42no—
27D109⋊C8—42no—
28C2×#11724786—22no—
29C23×D109—21no—
30C23×D109D411yescharacteristic
31C2×D436—21no—
32C2×D436C2×C411yescharacteristic, complemented
33C2×C22⋊D109—21no—
34C2×C22⋊D109—21no—
35C2×C22⋊D109—21no—
36D4×D109—42no—
37D4×D109—42no—
38D4×D109—42no—
39C22×Dic109D411yescharacteristic
40C2×C4×D109—21no—
41C2×C4×D109—42no—
42C2×C4×D109C2×C411yescharacteristic
43C2×C4×D109D411yescharacteristic
44C2×C4×D109D411yescharacteristic
45C2×C4×D109C2311yescharacteristic
46C4⋊Dic109C2×C411yescharacteristic, complemented
47#13182008—21no—
48C22⋊Dic109—21no—
49#13182006—21no—
50#13182006—21no—
51C2×D4×C109—21no—
52C2×D4×C109C2×C411yescharacteristic, complemented
53C22×C4×C109C2×C411yescharacteristic
54C4⋊C4×C109C2×C411yescharacteristic, complemented
55C22⋊C4×C109—21no—
56C23×C109—21no—
57C23×C109C22⋊C411yescharacteristic
58C22×D109—22no—
59C22×D109—22no—
60C22×D109—21no—
61C22×D109—42no—
62C22×D109—42no—
63C22×D109—42no—
64C22×D109C22⋊C411yescharacteristic
65C22×D109D812yes—
66C22×D109SD1612yes—
67C22×D109C2×D411yescharacteristic
68C2×#89137—24no—
69C2×#89137—42no—
70C2×#89137—48no—
71C2×#89137—42no—
72D4×C109—41no—
73D4×C109—42no—
74C2×C4×C109—21no—
75C2×C4×C109—41no—
76C2×C4×C109C22⋊C411yescharacteristic
77C2×C4×C109C22×C411yescharacteristic
78C22⋊D109—42no—
79C22⋊D109—42no—
80C22⋊D109—41no—
81C22⋊D109—41no—
82C2×Dic109—21no—
83C2×Dic109—41no—
84C2×Dic109C22⋊C411yescharacteristic
85C2×Dic109C2×D411yescharacteristic
86D436—41no—
87D436—41no—
88D436—41no—
89C4×D109—22no—
90C4×D109—22no—
91C4×D109—41no—
92C4×D109—41no—
93C4×D109—42no—
94#11724786—42no—
95C22×C109—21no—
96C22×C109—21no—
97C22×C109—41no—
98C22×C109—41no—
99C22×C109—41no—
100C22×C109#10212yes—
101C22×C109C2×C22⋊C411yescharacteristic
102C2×D109—21no—
103C2×D109—21no—
104C2×D109—24no—
105C2×D109—24no—
106C2×D109—41no—
107C2×D109—41no—
108C2×D109—41no—
109C2×D109—84no—
110C2×D109#9912yes—
111C2×D109#10212yes—
112C2×D109#10412yes—
113C2×D109C2×D811yescharacteristic
114C2×D109C2×SD1611yescharacteristic
115#89137—44no—
116C4×C109—21no—
117C4×C109—21no—
118C4×C109—41no—
119Dic109—21no—
120Dic109—21no—
121Dic109—41no—
122C2×C109—22no—
123C2×C109—81no—
124C2×C109C2×#9911yescharacteristic
125C2×C109C2×#10211yescharacteristic
126C2×C109C2×#10411yescharacteristic
127D109—21no—
128D109—21no—
129D109—81no—
130D109#32714yescomplemented
131C2×#327—1091nomaximal, Sylow
132C109C2×#32711yescharacteristic, complemented, Sylow
133C2×#121—1091no—
134C2×C22⋊C8—1091no—
135C2×#95—1091no—
136#327—1092no—
137#327—1092no—
138C22×D4—1091no—
139C22×D4—2181no—
140C23×C4—1091no—
141C23×C4—2181no—
142C22×C8—2181no—
143#121—1091no—
144#121—1091no—
145#121—1091no—
146#121—2181no—
147#121—2181no—
148#121#8913711yescharacteristic, complemented
149C2×C4⋊C4—1091no—
150C2×C22⋊C4—2181no—
151C22⋊C8—1092no—
152C22⋊C8—2181no—
153#95—1092no—
154#95—2181no—
155C24—1091no—
156C24—2181no—
157C2×D4—21no—
158C2×D4—1091no—
159C2×D4—4362no—
160C2×D4—4362no—
161C2×D4—4362no—
162C2×D4—2181no—
163C2×D4—2181no—
164C2×D4—2181no—
165C2×D4—2181no—
166C2×D4C2×#8913711yescharacteristic, complemented
167C22×C4—1091no—
168C22×C4—1091no—
169C22×C4—1091no—
170C22×C4—1091no—
171C22×C4—1091no—
172C22×C4—4362no—
173C22×C4—4362no—
174C22×C4—4362no—
175C22×C4—2181no—
176C22×C4—2182no—
177C22×C4—2184no—
178C22×C4C2×#8913711yescharacteristic
179C2×C8—4362no—
180C2×C8—2182no—
181C4⋊C4—1091no—
182C4⋊C4—2181no—
183C4⋊C4C2×#8913711yescharacteristic, complemented
184C22⋊C4—21no—
185C22⋊C4—2181no—
186C22⋊C4—2181no—
187C22⋊C4—2181no—
188C23—21no—
189C23—1091no—
190C23—1092no—
191C23—1092no—
192C23—1091no—
193C23—4362no—
194C23—4362no—
195C23—4362no—
196C23—2182no—
197C23—2182no—
198C23—2181no—
199C23C22⋊#8913711yescharacteristic
200D4—41no—
201D4—42no—
202D4—4361no—
203D4—4361no—
204D4—4361no—
205D4—4361no—
206D4—4361no—
207D4—4362no—
208D4—4362no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—1091no—
212C2×C4—1091no—
213C2×C4—4361no—
214C2×C4—4361no—
215C2×C4—4362no—
216C2×C4—4362no—
217C2×C4—4368no—
218C2×C4—4362no—
219C2×C4—4361no—
220C2×C4—2182no—
221C2×C4—2182no—
222C2×C4—2184no—
223C2×C4—2181no—
224C2×C4C22⋊#8913711yescharacteristic
225C2×C4C22×#8913711yescharacteristic
226C8—4362no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—8724no—
233C22—1092no—
234C22—1092no—
235C22—1092no—
236C22—1091no—
237C22—1091no—
238C22—4361no—
239C22—4361no—
240C22—4361no—
241C22—2184no—
242C22—2184no—
243C22—2181no—
244C22—2181no—
245C22#1681276112yes—
246C22C2×C22⋊#8913711yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—4364no—
251C4—4361no—
252C4—2181no—
253C4—2181no—
254C2—22no—
255C2—81no—
256C2—8721no—
257C2—1094no—
258C2—2181no—
259C2—2181no—
260C2D109⋊#9911yescharacteristic
261C2C2×#1681276111yescharacteristic
262C2D109⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#121×C109C4
socleC22×C109C2×C22⋊C4
prime radical#121×D109C2
cosocleC2×C4×D109C23
hypercenter#121#89137
hypocenterC109C2×#327
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. D109⋊(C22⋊C8)C2semidirect
2. C2×(C2×#89137)⋊C4C2semidirect
3. #20295878C2semidirect
4. #20295877C2semidirect
8. #16812756C22semidirect
11. (C2×#89137)⋊C4C22semidirect
13. C2×D4×D109C4semidirect
16. #16812673C4semidirect
20. C4⋊C4×D109C4semidirect
22. #121×C109C4semidirect
32. C2×D436C2×C4semidirect
46. C4⋊Dic109C2×C4semidirect
52. C2×D4×C109C2×C4semidirect
54. C4⋊C4×C109C2×C4semidirect
130. D109#327semidirect
132. C109C2×#327semidirect
148. #121#89137semidirect
166. C2×D4C2×#89137semidirect
183. 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
#202958782
#202958772
#121×D1092
C2×#327109