Finite Group Databasebeta

← #28379290

Subgroups — #28379290

Order 13952.

280 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 52 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, complemented
2D109⋊#99C211yesmaximal, characteristic, complemented
3#20295928C212yesmaximal, complemented
4#20295927C212yesmaximal, complemented
5#20295587C211yesmaximal, characteristic
6D109⋊M16—21no—
7C2×C22⋊#89137—21no—
8#16812759—21no—
9#16812759C2212yes—
10#16812758—21no—
11#16812757C2212yes—
12C22×D436C411yescharacteristic, complemented
13C2×D4×D109C2211yescharacteristic
14#16812684—21no—
15#16812682C2211yescharacteristic
16#16812674—21no—
17#16812672—21no—
18#16812669—21no—
19#16812669C2211yescharacteristic
20#16812667C411yescharacteristic, complemented
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×D109C2×C411yescharacteristic, complemented
31C23×D109D411yescharacteristic
32C2×D436—21no—
33C2×D436—21no—
34C2×D436—21no—
35C2×D436—41no—
36C2×D436—41no—
37C2×D436D411yescharacteristic
38C2×C22⋊D109—21no—
39C2×C22⋊D109D411yescharacteristic
40C2×C22⋊D109C2311yescharacteristic
41D4×D109—41no—
42D4×D109—41no—
43C2×C4×D109—21no—
44C2×C4×D109D411yescharacteristic
45#13182008—21no—
46C22⋊Dic109C2×C411yescharacteristic, complemented
47#13182006—21no—
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—21no—
63C22×D109—41no—
64C22×D109—81no—
65C22×D109—81no—
66C22×D109—81no—
67C22×D109C22⋊C411yescharacteristic
68C22×D109C22⋊C411yescharacteristic
69C22×D109C2×D411yescharacteristic
70C2×#89137—42no—
71C2×#89137—82no—
72D4×C109—41no—
73C2×C4×C109—21no—
74C2×C4×C109—21no—
75C2×C4×C109—41no—
76C2×C4×C109C22⋊C411yescharacteristic
77C22⋊D109—41no—
78C22⋊D109—41no—
79C22⋊D109—81no—
80C2×Dic109—21no—
81C2×Dic109C2×D411yescharacteristic
82D436—41no—
83D436—42no—
84D436—41no—
85D436—41no—
86D436—81no—
87C4×D109—21no—
88C4×D109—21no—
89C4×D109—81no—
90#11724786—42no—
91C22×C109—21no—
92C22×C109—21no—
93C22×C109—41no—
94C22×C109—41no—
95C22×C109C2×C22⋊C411yescharacteristic
96C2×D109—21no—
97C2×D109—21no—
98C2×D109—41no—
99C2×D109—42no—
100C2×D109—41no—
101C2×D109—41no—
102C2×D109—41no—
103C2×D109—41no—
104C2×D109—41no—
105C2×D109—81no—
106C2×D109—81no—
107C2×D109—81no—
108C2×D109—82no—
109C2×D109#9911yescharacteristic
110C2×D109#9911yescharacteristic
111#89137—82no—
112C4×C109—21no—
113C4×C109—41no—
114C4×C109—41no—
115Dic109—21no—
116Dic109—41no—
117C2×C109—21no—
118C2×C109—41no—
119C2×C109—41no—
120C2×C109C2×#9911yescharacteristic
121D109—21no—
122D109—21no—
123D109—41no—
124D109—41no—
125D109—81no—
126#2121—1091nomaximal, Sylow
127C109#212111yescharacteristic, complemented, Sylow
128#534—1091no—
129C2×#100—1091no—
130C2×#99—1091no—
131#352—1092no—
132AΣL(1,𝔽16)—1092no—
133C22×D4—1091no—
134C22×D4—1091no—
135C2×M16—2181no—
136#123—1091no—
137#123—2181no—
138#123#8913711yescharacteristic, complemented
139#121—2181no—
140#121—2181no—
141#120—1091no—
142#120—1091no—
143#120—2181no—
144C2×C22⋊C4—1091no—
145C2×C22⋊C4—2181no—
146#100—1092no—
147#100—2181no—
148#99—1092no—
149#99—2181no—
150C24—1091no—
151C24—1091no—
152C24—1091no—
153C2×D4—1091no—
154C2×D4—1091no—
155C2×D4—1091no—
156C2×D4—4361no—
157C2×D4—4361no—
158C2×D4—4361no—
159C2×D4—4361no—
160C2×D4—2181no—
161C2×D4—2181no—
162C2×D4—2181no—
163C2×D4—2181no—
164C2×D4C2×#8913711yescharacteristic
165C22×C4—1091no—
166C22×C4—4361no—
167C22×C4—2181no—
168C22×C4C2×#8913711yescharacteristic, complemented
169M16—4361no—
170M16—2182no—
171C2×C8—4361no—
172C4⋊C4—21no—
173C4⋊C4—2181no—
174C22⋊C4—21no—
175C22⋊C4—1091no—
176C22⋊C4—4361no—
177C22⋊C4—2181no—
178C22⋊C4—2181no—
179C22⋊C4—2181no—
180C22⋊C4—2182no—
181C22⋊C4—2181no—
182C22⋊C4C2×#8913711yescharacteristic, complemented
183C23—8721no—
184C23—8721no—
185C23—8721no—
186C23—1091no—
187C23—1091no—
188C23—1091no—
189C23—4361no—
190C23—2181no—
191C23—2181no—
192C23—2181no—
193C23—2181no—
194C23—2181no—
195C23C22⋊#8913711yescharacteristic
196C23C22×#8913711yescharacteristic
197D4—41no—
198D4—8721no—
199D4—8721no—
200D4—4361no—
201D4—4361no—
202D4—4361no—
203D4—4362no—
204D4—4361no—
205D4—4361no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—8721no—
210C2×C4—8722no—
211C2×C4—1091no—
212C2×C4—4362no—
213C2×C4—2181no—
214C2×C4—2181no—
215C2×C4—2181no—
216C2×C4C22⋊#8913711yescharacteristic
217C8—4362no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—8722no—
223C22—8721no—
224C22—8721no—
225C22—8721no—
226C22—1091no—
227C22—1091no—
228C22—4361no—
229C22—4361no—
230C22—4361no—
231C22—4361no—
232C22—4361no—
233C22—4362no—
234C22—4361no—
235C22—2181no—
236C22—2181no—
237C22C2×C22⋊#8913711yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—8722no—
242C4—4361no—
243C4—2181no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2—8721no—
248C2—4361no—
249C2—4361no—
250C2—2181no—
251C2—2181no—
252C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
Fitting#123×C109C4
socleC2×C109C2×#99
prime radical#20295587C2
cosocleC2×C22⋊D109C23
hypercenter#123#89137
hypocenterC109#2121
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
1. D109⋊#100C2semidirect
2. D109⋊#99C2semidirect
3. #20295928C2semidirect
4. #20295927C2semidirect
12. C22×D436C4semidirect
20. #16812667C4semidirect
21. C22⋊C4×D109C4semidirect
22. #123×C109C4semidirect
30. C23×D109C2×C4semidirect
46. C22⋊Dic109C2×C4semidirect
52. C22×C4×C109C2×C4semidirect
55. C22⋊C4×C109C2×C4semidirect
127. C109#2121semidirect
138. #123#89137semidirect
168. C22×C4C2×#89137semidirect
182. 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
#202959282
#202959272
#202955872
#2121109