Finite Group Databasebeta

← #38770498

Subgroups — #38770498

Order 25472.

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#27164583C211yesmaximal, characteristic
2#27163999C211yesmaximal, characteristic, complemented
3#27163990C211yesmaximal, characteristic, complemented
4#27163806C211yesmaximal, characteristic, complemented
5#27163805C211yesmaximal, characteristic, complemented
6#27163804C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C199C211yesmaximal, characteristic, complemented
8C2×#15691885C2211yescharacteristic
9C22×C22⋊D199C411yescharacteristic, complemented
10M16×D199—21no—
11#20263951—21no—
12#20263941—21no—
13#20263928—21no—
14#20263927—21no—
15#20263925C411yescharacteristic, complemented
16#20263924C411yescharacteristic, complemented
17#20263923C2211yescharacteristic
18#20263911—21no—
19#20263911C411yescharacteristic, complemented
20#20263873—21no—
21#20263872C2211yescharacteristic
22#20263870C2211yescharacteristic
23#20263869—21no—
24#120×C199C2211yescharacteristic
25#100×C199C2211yescharacteristic
26#99×C199C2211yescharacteristic
27C23×D199C2×C411yescharacteristic
28#15691885—41no—
29#15691885—41no—
30C2×Dic398D411yescharacteristic
31C2×D796—21no—
32C2×C22⋊D199—21no—
33C2×C22⋊D199—21no—
34C2×C22⋊D199—21no—
35C2×C22⋊D199—21no—
36C2×C22⋊D199—41no—
37C2×C22⋊D199—41no—
38C2×C22⋊D199C2×C411yescharacteristic
39C2×C22⋊D199D411yescharacteristic
40C22×Dic199C2×C411yescharacteristic
41C22×Dic199D411yescharacteristic
42C2×C4×D199D411yescharacteristic
43#15691870—21no—
44#15691869—41no—
45C8×D199—41no—
46Dic199⋊C4—21no—
47Dic199⋊C4—41no—
48C22⋊Dic199—21no—
49C22⋊Dic199—21no—
50C22⋊Dic199C2×C411yescharacteristic
51(C2×D199)⋊C4—21no—
52(C2×D199)⋊C4—21no—
53C4×Dic199—21no—
54C4×Dic199—41no—
55C24×C199C2×C411yescharacteristic
56C2×D4×C199—21no—
57C2×D4×C199C2311yescharacteristic
58M16×C199—21no—
59C22⋊C4×C199—21no—
60C22⋊C4×C199—21no—
61C22⋊C4×C199C2×C411yescharacteristic
62C22×D199—21no—
63C22×D199—41no—
64C22×D199—81no—
65C22×D199C22⋊C411yescharacteristic
66Dic398—41no—
67D796—81no—
68C22⋊D199—41no—
69C22⋊D199—41no—
70C22⋊D199—42no—
71C22⋊D199—41no—
72C22⋊D199—41no—
73C22⋊D199—81no—
74C22⋊D199—81no—
75C2×Dic199—21no—
76C2×Dic199—21no—
77C2×Dic199—21no—
78C2×Dic199—21no—
79C2×Dic199—21no—
80C2×Dic199—41no—
81C2×Dic199C22⋊C411yescharacteristic
82C2×Dic199C22⋊C411yescharacteristic
83C2×Dic199C22⋊C411yescharacteristic
84C4×D199—21no—
85C4×D199—21no—
86C199⋊C8—41no—
87C23×C199—21no—
88C23×C199—41no—
89C23×C199—81no—
90C23×C199C22×C411yescharacteristic
91C23×C199C2×D411yescharacteristic
92D4×C199—41no—
93D4×C199—81no—
94C2×C4×C199—21no—
95C2×C4×C199—41no—
96C2×C4×C199C2×D411yescharacteristic
97C8×C199—41no—
98C22×C199—21no—
99C22×C199—21no—
100C22×C199—41no—
101C22×C199—41no—
102C22×C199—41no—
103C22×C199—41no—
104C22×C199—41no—
105C22×C199—82no—
106C22×C199C2×C22⋊C411yescharacteristic
107C2×D199—21no—
108C2×D199—41no—
109C2×D199—41no—
110C2×D199—41no—
111C2×D199—41no—
112C2×D199—82no—
113C4×C199—21no—
114C4×C199—41no—
115C4×C199—81no—
116Dic199—21no—
117Dic199—21no—
118Dic199—41no—
119Dic199—81no—
120Dic199#9911yescharacteristic
121Dic199#9911yescharacteristic
122C2×C199—21no—
123C2×C199—41no—
124C2×C199—41no—
125C2×C199—81no—
126C2×C199C2×#9911yescharacteristic
127D199—41no—
128D199—81no—
129C199#212211yescharacteristic, complemented, Sylow
130#2122—1991nomaximal, Sylow
131#535—1991no—
132#413—1991no—
133#410—1991no—
134AΣL(1,𝔽16)—1991no—
135AΣL(1,𝔽16)—1991no—
136AΣL(1,𝔽16)—1991no—
137AΣL(1,𝔽16)D19911yescharacteristic, complemented
138C2×#41—1991no—
139C22×D4—1991no—
140C2×M16—3981no—
141#127—3981no—
142#124—3981no—
143#121—3981no—
144#121—3981no—
145#120—1991no—
146#120—1991no—
147#120—1991no—
148#120C2×D19911yescharacteristic
149#117—3981no—
150#117—1991no—
151#101—3981no—
152#100—1991no—
153#100C2×D19911yescharacteristic
154#99—3981no—
155#99—1991no—
156#99C2×D19911yescharacteristic
157C24—1991no—
158C24C4×D19911yescharacteristic
159#41—7961no—
160#41—7961no—
161C2×Q8—1991no—
162C2×D4—21no—
163C2×D4—7961no—
164C2×D4—7961no—
165C2×D4—3981no—
166C2×D4—3981no—
167C2×D4—3981no—
168C2×D4—3981no—
169C2×D4—3981no—
170C2×D4—1991no—
171C2×D4—1991no—
172C2×D4C22×D19911yescharacteristic
173C22×C4—1991no—
174C22×C4—1991no—
175C22×C4—1991no—
176M16—21no—
177M16—7961no—
178M16—3981no—
179C2×C8—7961no—
180C4⋊C4—7961no—
181C4⋊C4—3981no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—3981no—
185C22⋊C4—3981no—
186C22⋊C4—3981no—
187C22⋊C4—3981no—
188C22⋊C4—1991no—
189C22⋊C4C4×D19911yescharacteristic
190C42—7961no—
191C42—3981no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—7961no—
196C23—15921no—
197C23—3981no—
198C23—1991no—
199C23C2×C4×D19911yescharacteristic
200C23D4×D19911yescharacteristic
201Q8—7961no—
202D4—41no—
203D4—81no—
204D4—7961no—
205D4—7961no—
206D4—7962no—
207D4—7961no—
208D4—7961no—
209D4—15921no—
210D4—15921no—
211D4—15921no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—7961no—
215C2×C4—3981no—
216C2×C4—3981no—
217C2×C4—3981no—
218C2×C4—3981no—
219C2×C4—3981no—
220C2×C4—3981no—
221C2×C4—3981no—
222C2×C4—1991no—
223C2×C4—1991no—
224C2×C4—1991no—
225C2×C4D4×D19911yescharacteristic
226C8—41no—
227C8—7961no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—7961no—
237C22—7961no—
238C22—7961no—
239C22—7961no—
240C22—15922no—
241C22—3981no—
242C22C22⋊C4×D19911yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—7961no—
247C4—15921no—
248C4—3981no—
249C4—3981no—
250C4—1991no—
251C4—1991no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—7961no—
257C2—15921no—
258C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC23×C199C22×C4
FrattiniC2×D4C22×D199
FittingAΣL(1,𝔽16)×C199C2
socleC2×C199C2×#99
prime radical#27164583C2
cosocleC2×D4×C199C23
hypercenterAΣL(1,𝔽16)D199
hypocenterC199#2122
LCSG ▷ C23×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #27163999C2semidirect
3. #27163990C2semidirect
4. #27163806C2semidirect
5. #27163805C2semidirect
6. #27163804C2semidirect
7. AΣL(1,𝔽16)×C199C2semidirect
9. C22×C22⋊D199C4semidirect
15. #20263925C4semidirect
16. #20263924C4semidirect
19. #20263911C4semidirect
129. C199#2122semidirect
137. AΣL(1,𝔽16)D199semidirect

Maximal subgroups

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

subgroupindex
#271645832
#271639992
#271639902
#271638062
#271638052
#271638042
AΣL(1,𝔽16)×C1992
#2122199