Finite Group Databasebeta

← #36355206

Subgroups — #36355206

Order 22144.

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

normal subgroups only

This group has 280 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#26012051C211yesmaximal, characteristic
2#100×D173C211yesmaximal, characteristic, complemented
3#99×D173C211yesmaximal, characteristic, complemented
4#26011282C211yesmaximal, characteristic, complemented
5#26011280C211yesmaximal, characteristic, complemented
6#26011278C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C411yescharacteristic, complemented
9C2×D4×D173C2211yescharacteristic
10M16×D173—21no—
11#19149663C411yescharacteristic, complemented
12#19149661—21no—
13#19149660C2211yescharacteristic
14#19149652—21no—
15#19149650—21no—
16#19149648C411yescharacteristic, complemented
17#19149647—21no—
18C22⋊C4×D173—21no—
19C22⋊C4×D173C411yescharacteristic, complemented
20#19149596C2211yescharacteristic
21#19149595—21no—
22#19149594C2211yescharacteristic
23#19149593—21no—
24#120×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic
26#99×C173C2211yescharacteristic
27C23×D173C2×C411yescharacteristic
28C23×D173D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—41no—
34C2×C22⋊D173—41no—
35C2×C22⋊D173C2×C411yescharacteristic
36C2×C22⋊D173D411yescharacteristic
37D4×D173—41no—
38D4×D173—41no—
39C22×Dic173C2×C411yescharacteristic, complemented
40C2×C4×D173—21no—
41C2×C4×D173—41no—
42C2×C4×D173D411yescharacteristic
43#14583810—21no—
44#14583809—41no—
45C8×D173—41no—
46C4⋊Dic173—21no—
47#14583805—21no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173—21no—
51C22⋊Dic173C2×C411yescharacteristic, complemented
52#14583803—21no—
53#14583803—21no—
54#14583803—41no—
55C24×C173C2×C411yescharacteristic, complemented
56C2×D4×C173—21no—
57C2×D4×C173C2311yescharacteristic
58M16×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173C2×C411yescharacteristic, complemented
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173—21no—
66C22×D173—81no—
67C22×D173—81no—
68C22×D173C22⋊C411yescharacteristic
69C22×D173C22⋊C411yescharacteristic
70C22×D173C22⋊C411yescharacteristic
71D692—41no—
72C22⋊D173—41no—
73C22⋊D173—41no—
74C22⋊D173—42no—
75C22⋊D173—41no—
76C22⋊D173—41no—
77C22⋊D173—81no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—41no—
81C2×Dic173—41no—
82C2×Dic173C22⋊C411yescharacteristic
83C4×D173—21no—
84C4×D173—21no—
85C4×D173—81no—
86C4×D173—82no—
87#13063436—41no—
88C23×C173—21no—
89C23×C173—41no—
90C23×C173—81no—
91C23×C173C22×C411yescharacteristic
92C23×C173C2×D411yescharacteristic
93D4×C173—41no—
94D4×C173—81no—
95C2×C4×C173—21no—
96C2×C4×C173—41no—
97C2×C4×C173C2×D411yescharacteristic
98C8×C173—41no—
99C22×C173—21no—
100C22×C173—21no—
101C22×C173—41no—
102C22×C173—41no—
103C22×C173—41no—
104C22×C173—41no—
105C22×C173—41no—
106C22×C173—82no—
107C22×C173C2×C22⋊C411yescharacteristic
108C2×D173—21no—
109C2×D173—21no—
110C2×D173—41no—
111C2×D173—42no—
112C2×D173—41no—
113C2×D173—41no—
114C2×D173—81no—
115C2×D173—81no—
116C2×D173—81no—
117C2×D173#9911yescharacteristic
118C2×D173#9911yescharacteristic
119C4×C173—21no—
120C4×C173—41no—
121C4×C173—81no—
122Dic173—21no—
123Dic173—41no—
124Dic173—41no—
125Dic173—81no—
126C2×C173—21no—
127C2×C173—41no—
128C2×C173—41no—
129C2×C173—81no—
130C2×C173C2×#9911yescharacteristic
131D173—21no—
132D173—21no—
133D173—41no—
134D173—41no—
135C173#212111yescharacteristic, complemented, Sylow
136#2121—1731nomaximal, Sylow
137#534—1731no—
138C2×#100—1731no—
139C2×#99—1731no—
140#352—1731no—
141#352—1731no—
142AΣL(1,𝔽16)—1731no—
143AΣL(1,𝔽16)D17311yescharacteristic, complemented
144C22×D4—1731no—
145C22×D4—1731no—
146C2×M16—3461no—
147#123—3461no—
148#123—1731no—
149#123—1731no—
150#121—3461no—
151#121—3461no—
152#120—3461no—
153#120—1731no—
154#120C2×D17311yescharacteristic
155C2×C22⋊C4—3461no—
156C2×C22⋊C4—1731no—
157#100—3461no—
158#100—1731no—
159#100C2×D17311yescharacteristic
160#99—3461no—
161#99—1731no—
162#99C2×D17311yescharacteristic
163C24—1731no—
164C24—1731no—
165C24C4×D17311yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—3461no—
168C2×D4—3461no—
169C2×D4—3461no—
170C2×D4—1731no—
171C2×D4—1731no—
172C2×D4—1731no—
173C2×D4—6921no—
174C2×D4—6921no—
175C2×D4—6921no—
176C2×D4—6921no—
177C2×D4C22×D17311yescharacteristic
178C22×C4—3461no—
179C22×C4—1731no—
180C22×C4—1731no—
181C22×C4—6921no—
182M16—21no—
183M16—3461no—
184M16—6921no—
185C2×C8—6921no—
186C4⋊C4—3461no—
187C4⋊C4—3461no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—3461no—
191C22⋊C4—3461no—
192C22⋊C4—3461no—
193C22⋊C4—3461no—
194C22⋊C4—3461no—
195C22⋊C4—1731no—
196C22⋊C4—6921no—
197C22⋊C4C4×D17311yescharacteristic, complemented
198C23—21no—
199C23—41no—
200C23—81no—
201C23—3461no—
202C23—3461no—
203C23—3461no—
204C23—3461no—
205C23—13841no—
206C23—13841no—
207C23—1731no—
208C23—1731no—
209C23—1731no—
210C23C2×C4×D17311yescharacteristic
211C23D4×D17311yescharacteristic
212D4—41no—
213D4—81no—
214D4—13841no—
215D4—6921no—
216D4—6921no—
217D4—6921no—
218D4—6922no—
219D4—6921no—
220D4—6921no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—3461no—
224C2×C4—3461no—
225C2×C4—3461no—
226C2×C4—3461no—
227C2×C4—13841no—
228C2×C4—13842no—
229C2×C4—1731no—
230C2×C4—6921no—
231C2×C4—6921no—
232C2×C4D4×D17311yescharacteristic
233C8—41no—
234C8—6921no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—82no—
243C22—3461no—
244C22—3461no—
245C22—13841no—
246C22—13841no—
247C22—13841no—
248C22—1731no—
249C22—1731no—
250C22—6921no—
251C22—6921no—
252C22—6921no—
253C22—6922no—
254C22C22⋊C4×D17311yescharacteristic
255C4—21no—
256C4—41no—
257C4—81no—
258C4—3461no—
259C4—13841no—
260C4—6921no—
261C4—6921no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—81no—
266C2—3461no—
267C2—3461no—
268C2—6921no—
269C2—6921no—
270C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
FittingAΣL(1,𝔽16)×C173C2
socleC2×C173C2×#99
prime radical#26012051C2
cosocleC2×D4×C173C23
hypercenterAΣL(1,𝔽16)D173
hypocenterC173#2121
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D173C2semidirect
3. #99×D173C2semidirect
4. #26011282C2semidirect
5. #26011280C2semidirect
6. #26011278C2semidirect
7. AΣL(1,𝔽16)×C173C2semidirect
8. C22×C22⋊D173C4semidirect
11. #19149663C4semidirect
16. #19149648C4semidirect
19. C22⋊C4×D173C4semidirect
39. C22×Dic173C2×C4semidirect
51. C22⋊Dic173C2×C4semidirect
55. C24×C173C2×C4semidirect
61. C22⋊C4×C173C2×C4semidirect
135. C173#2121semidirect
143. AΣL(1,𝔽16)D173semidirect
165. C24C4×D173semidirect
197. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
#260120512
#100×D1732
#99×D1732
#260112822
#260112802
#260112782
AΣL(1,𝔽16)×C1732
#2121173