Finite Group Databasebeta

← #28364680

Subgroups — #28364680

Order 13952.

266 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#20295591C211yesmaximal, characteristic
2#20295007C211yesmaximal, characteristic, complemented
3#20294998C211yesmaximal, characteristic, complemented
4#20294814C211yesmaximal, characteristic, complemented
5#20294813C211yesmaximal, characteristic, complemented
6#20294812C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C109C211yesmaximal, characteristic, complemented
8C2×#13182028C2211yescharacteristic
9C22×C22⋊D109C411yescharacteristic, complemented
10M16×D109—21no—
11#16812695—21no—
12#16812685—21no—
13#16812672—21no—
14#16812671—21no—
15#16812669C411yescharacteristic, complemented
16#16812668C411yescharacteristic, complemented
17#16812667C2211yescharacteristic
18#16812655—21no—
19#16812655C411yescharacteristic, complemented
20#16812617—21no—
21#16812616C2211yescharacteristic
22#16812614C2211yescharacteristic
23#16812613—21no—
24#120×C109C2211yescharacteristic
25#100×C109C2211yescharacteristic
26#99×C109C2211yescharacteristic
27C23×D109C2×C411yescharacteristic
28#13182028—41no—
29#13182028—41no—
30C2×Dic218D411yescharacteristic
31C2×D436—21no—
32C2×C22⋊D109—21no—
33C2×C22⋊D109—21no—
34C2×C22⋊D109—21no—
35C2×C22⋊D109—21no—
36C2×C22⋊D109—41no—
37C2×C22⋊D109—41no—
38C2×C22⋊D109C2×C411yescharacteristic
39C2×C22⋊D109D411yescharacteristic
40C22×Dic109C2×C411yescharacteristic
41C22×Dic109D411yescharacteristic
42C2×C4×D109D411yescharacteristic
43#13182013—21no—
44#13182012—41no—
45C8×D109—41no—
46#13182008—21no—
47#13182008—41no—
48C22⋊Dic109—21no—
49C22⋊Dic109—21no—
50C22⋊Dic109C2×C411yescharacteristic
51#13182006—21no—
52#13182006—21no—
53C4×Dic109—21no—
54C4×Dic109—41no—
55C24×C109C2×C411yescharacteristic
56C2×D4×C109—21no—
57C2×D4×C109C2311yescharacteristic
58M16×C109—21no—
59C22⋊C4×C109—21no—
60C22⋊C4×C109—21no—
61C22⋊C4×C109C2×C411yescharacteristic
62C23×C109—21no—
63C23×C109—41no—
64C23×C109—81no—
65C23×C109C22×C411yescharacteristic
66C23×C109C2×D411yescharacteristic
67C22×D109—21no—
68C22×D109—41no—
69C22×D109—81no—
70C22×D109C22⋊C411yescharacteristic
71D4×C109—41no—
72D4×C109—81no—
73C2×C4×C109—21no—
74C2×C4×C109—41no—
75C2×C4×C109C2×D411yescharacteristic
76C22⋊D109—41no—
77C22⋊D109—41no—
78C22⋊D109—42no—
79C22⋊D109—41no—
80C22⋊D109—41no—
81C22⋊D109—81no—
82C22⋊D109—81no—
83C2×Dic109—21no—
84C2×Dic109—21no—
85C2×Dic109—21no—
86C2×Dic109—21no—
87C2×Dic109—21no—
88C2×Dic109—41no—
89C2×Dic109C22⋊C411yescharacteristic
90C2×Dic109C22⋊C411yescharacteristic
91C2×Dic109C22⋊C411yescharacteristic
92D436—81no—
93C4×D109—21no—
94C4×D109—21no—
95Dic218—41no—
96C8×C109—41no—
97#11724784—41no—
98C22×C109—21no—
99C22×C109—21no—
100C22×C109—41no—
101C22×C109—41no—
102C22×C109—41no—
103C22×C109—41no—
104C22×C109—41no—
105C22×C109—82no—
106C22×C109C2×C22⋊C411yescharacteristic
107C2×D109—21no—
108C2×D109—41no—
109C2×D109—41no—
110C2×D109—41no—
111C2×D109—41no—
112C2×D109—82no—
113C4×C109—21no—
114C4×C109—41no—
115C4×C109—81no—
116Dic109—21no—
117Dic109—21no—
118Dic109—41no—
119Dic109—81no—
120Dic109#9911yescharacteristic
121Dic109#9911yescharacteristic
122C2×C109—21no—
123C2×C109—41no—
124C2×C109—41no—
125C2×C109—81no—
126C2×C109C2×#9911yescharacteristic
127D109—41no—
128D109—81no—
129#2122—1091nomaximal, Sylow
130C109#212211yescharacteristic, complemented, Sylow
131#535—1091no—
132#413—1091no—
133#410—1091no—
134AΣL(1,𝔽16)—1091no—
135AΣL(1,𝔽16)—1091no—
136AΣL(1,𝔽16)—1091no—
137AΣL(1,𝔽16)D10911yescharacteristic, complemented
138C2×#41—1091no—
139C22×D4—1091no—
140C2×M16—2181no—
141#127—2181no—
142#124—2181no—
143#121—2181no—
144#121—2181no—
145#120—1091no—
146#120—1091no—
147#120—1091no—
148#120C2×D10911yescharacteristic
149#117—1091no—
150#117—2181no—
151#101—2181no—
152#100—1091no—
153#100C2×D10911yescharacteristic
154#99—1091no—
155#99—2181no—
156#99C2×D10911yescharacteristic
157C24—1091no—
158C24C4×D10911yescharacteristic
159#41—4361no—
160#41—4361no—
161C2×Q8—1091no—
162C2×D4—21no—
163C2×D4—1091no—
164C2×D4—1091no—
165C2×D4—4361no—
166C2×D4—4361no—
167C2×D4—2181no—
168C2×D4—2181no—
169C2×D4—2181no—
170C2×D4—2181no—
171C2×D4—2181no—
172C2×D4C22×D10911yescharacteristic
173C22×C4—1091no—
174C22×C4—1091no—
175C22×C4—1091no—
176M16—21no—
177M16—4361no—
178M16—2181no—
179C2×C8—4361no—
180C4⋊C4—4361no—
181C4⋊C4—2181no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—1091no—
185C22⋊C4—2181no—
186C22⋊C4—2181no—
187C22⋊C4—2181no—
188C22⋊C4—2181no—
189C22⋊C4C4×D10911yescharacteristic
190C42—4361no—
191C42—2181no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—8721no—
196C23—1091no—
197C23—4361no—
198C23—2181no—
199C23C2×C4×D10911yescharacteristic
200C23D4×D10911yescharacteristic
201Q8—4361no—
202D4—41no—
203D4—81no—
204D4—8721no—
205D4—8721no—
206D4—8721no—
207D4—4361no—
208D4—4361no—
209D4—4361no—
210D4—4361no—
211D4—4362no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—1091no—
215C2×C4—1091no—
216C2×C4—1091no—
217C2×C4—4361no—
218C2×C4—2181no—
219C2×C4—2181no—
220C2×C4—2181no—
221C2×C4—2181no—
222C2×C4—2181no—
223C2×C4—2181no—
224C2×C4—2181no—
225C2×C4D4×D10911yescharacteristic
226C8—41no—
227C8—4361no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—8722no—
237C22—4361no—
238C22—4361no—
239C22—4361no—
240C22—4361no—
241C22—2181no—
242C22C22⋊C4×D10911yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—8721no—
247C4—1091no—
248C4—1091no—
249C4—4361no—
250C4—2181no—
251C4—2181no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—8721no—
257C2—4361no—
258C2#99×D10911yescharacteristic
subgroupG/H
centerC2#99×D109
commutatorC23×C109C22×C4
FrattiniC2×D4C22×D109
FittingAΣL(1,𝔽16)×C109C2
socleC2×C109C2×#99
prime radical#20295591C2
cosocleC2×D4×C109C23
hypercenterAΣL(1,𝔽16)D109
hypocenterC109#2122
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #20295007C2semidirect
3. #20294998C2semidirect
4. #20294814C2semidirect
5. #20294813C2semidirect
6. #20294812C2semidirect
7. AΣL(1,𝔽16)×C109C2semidirect
9. C22×C22⋊D109C4semidirect
15. #16812669C4semidirect
16. #16812668C4semidirect
19. #16812655C4semidirect
130. C109#2122semidirect
137. AΣL(1,𝔽16)D109semidirect

Maximal subgroups

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

subgroupindex
#202955912
#202950072
#202949982
#202948142
#202948132
#202948122
AΣL(1,𝔽16)×C1092
#2122109