Finite Group Databasebeta

← #38770495

Subgroups — #38770495

Order 25472.

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#27164579C211yesmaximal, characteristic
2#100×D199C211yesmaximal, characteristic, complemented
3#99×D199C211yesmaximal, characteristic, complemented
4#27163808C211yesmaximal, characteristic, complemented
5#27163806C211yesmaximal, characteristic, complemented
6#27163804C211yesmaximal, characteristic, complemented
7#352×C199C211yesmaximal, characteristic, complemented
8C22×D796C411yescharacteristic, complemented
9C2×D4×D199C2211yescharacteristic
10M16×D199—21no—
11#20263940—21no—
12#20263938C411yescharacteristic, complemented
13#20263930—21no—
14#20263928—21no—
15#20263925—21no—
16#20263925C411yescharacteristic, complemented
17#20263923C2211yescharacteristic
18C22⋊C4×D199—21no—
19C22⋊C4×D199C411yescharacteristic, complemented
20#20263872C2211yescharacteristic
21#20263871—21no—
22#20263870C2211yescharacteristic
23#20263869—21no—
24#123×C199C2211yescharacteristic
25#100×C199C2211yescharacteristic
26#99×C199C2211yescharacteristic
27C23×D199C2×C411yescharacteristic
28C23×D199C2×C411yescharacteristic, complemented
29C23×D199D411yescharacteristic
30C2×D796—21no—
31C2×D796—21no—
32C2×D796—21no—
33C2×D796—41no—
34C2×D796—41no—
35C2×D796D411yescharacteristic
36C2×C22⋊D199—21no—
37C2×C22⋊D199C2×C411yescharacteristic
38C2×C22⋊D199D411yescharacteristic
39D4×D199—41no—
40D4×D199—41no—
41C2×C4×D199—21no—
42C2×C4×D199—41no—
43C2×C4×D199D411yescharacteristic
44#15691870—21no—
45#15691869—41no—
46C8×D199—41no—
47Dic199⋊C4—21no—
48C22⋊Dic199—21no—
49C22⋊Dic199C2×C411yescharacteristic, complemented
50(C2×D199)⋊C4—21no—
51(C2×D199)⋊C4—21no—
52(C2×D199)⋊C4—21no—
53(C2×D199)⋊C4—21no—
54(C2×D199)⋊C4—41no—
55C2×D4×C199C2311yescharacteristic
56C22×C4×C199C2×C411yescharacteristic, complemented
57M16×C199—21no—
58C4⋊C4×C199—21no—
59C22⋊C4×C199—21no—
60C22⋊C4×C199—21no—
61C22⋊C4×C199C2×C411yescharacteristic, complemented
62C22×D199—21no—
63C22×D199—21no—
64C22×D199—21no—
65C22×D199—21no—
66C22×D199—21no—
67C22×D199—41no—
68C22×D199—81no—
69C22×D199—81no—
70C22×D199—81no—
71C22×D199C22⋊C411yescharacteristic
72C22×D199C22⋊C411yescharacteristic
73C22×D199C22⋊C411yescharacteristic
74D796—41no—
75D796—42no—
76D796—41no—
77D796—41no—
78D796—81no—
79C22⋊D199—41no—
80C22⋊D199—41no—
81C22⋊D199—81no—
82C2×Dic199—21no—
83C2×Dic199—41no—
84C2×Dic199C22⋊C411yescharacteristic
85C4×D199—21no—
86C4×D199—21no—
87C4×D199—81no—
88C4×D199—82no—
89C199⋊C8—41no—
90C23×C199C22×C411yescharacteristic
91C23×C199C2×D411yescharacteristic
92D4×C199—41no—
93C2×C4×C199—21no—
94C2×C4×C199—21no—
95C2×C4×C199—41no—
96C2×C4×C199—41no—
97C2×C4×C199C2×D411yescharacteristic
98C8×C199—41no—
99C22×C199—21no—
100C22×C199—21no—
101C22×C199—41no—
102C22×C199—41no—
103C22×C199C2×C22⋊C411yescharacteristic
104C2×D199—21no—
105C2×D199—21no—
106C2×D199—41no—
107C2×D199—42no—
108C2×D199—41no—
109C2×D199—41no—
110C2×D199—41no—
111C2×D199—41no—
112C2×D199—41no—
113C2×D199—81no—
114C2×D199—81no—
115C2×D199—81no—
116C2×D199—82no—
117C2×D199#9911yescharacteristic
118C2×D199#9911yescharacteristic
119C4×C199—21no—
120C4×C199—41no—
121C4×C199—41no—
122C4×C199—81no—
123Dic199—21no—
124Dic199—41no—
125Dic199—81no—
126C2×C199—21no—
127C2×C199—41no—
128C2×C199—41no—
129C2×C199C2×#9911yescharacteristic
130D199—21no—
131D199—21no—
132D199—41no—
133D199—41no—
134D199—81no—
135C199#212111yescharacteristic, complemented, Sylow
136#2121—1991nomaximal, Sylow
137#534—1991no—
138C2×#100—1991no—
139C2×#99—1991no—
140#352—1991no—
141#352D19911yescharacteristic, complemented
142AΣL(1,𝔽16)—1991no—
143AΣL(1,𝔽16)—1991no—
144C22×D4—1991no—
145C22×D4—1991no—
146C2×M16—3981no—
147#123—3981no—
148#123—1991no—
149#123C2×D19911yescharacteristic
150#121—3981no—
151#121—3981no—
152#120—3981no—
153#120—1991no—
154#120—1991no—
155C2×C22⋊C4—3981no—
156C2×C22⋊C4—1991no—
157#100—3981no—
158#100—1991no—
159#100C2×D19911yescharacteristic
160#99—3981no—
161#99—1991no—
162#99C2×D19911yescharacteristic
163C24—1991no—
164C24—1991no—
165C24—1991no—
166C2×D4—7961no—
167C2×D4—7961no—
168C2×D4—7961no—
169C2×D4—7961no—
170C2×D4—3981no—
171C2×D4—3981no—
172C2×D4—3981no—
173C2×D4—3981no—
174C2×D4—1991no—
175C2×D4—1991no—
176C2×D4—1991no—
177C2×D4C22×D19911yescharacteristic
178C22×C4—7961no—
179C22×C4—3981no—
180C22×C4—1991no—
181C22×C4C4×D19911yescharacteristic, complemented
182M16—21no—
183M16—7961no—
184M16—3981no—
185C2×C8—7961no—
186C4⋊C4—21no—
187C4⋊C4—3981no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—7961no—
191C22⋊C4—3981no—
192C22⋊C4—3981no—
193C22⋊C4—3981no—
194C22⋊C4—3981no—
195C22⋊C4—3981no—
196C22⋊C4—1991no—
197C22⋊C4C4×D19911yescharacteristic, complemented
198C23—7961no—
199C23—15921no—
200C23—15921no—
201C23—15921no—
202C23—3981no—
203C23—3981no—
204C23—3981no—
205C23—3981no—
206C23—3981no—
207C23—1991no—
208C23—1991no—
209C23—1991no—
210C23C2×C4×D19911yescharacteristic
211C23D4×D19911yescharacteristic
212D4—41no—
213D4—7961no—
214D4—7962no—
215D4—7961no—
216D4—7961no—
217D4—7961no—
218D4—7961no—
219D4—15921no—
220D4—15921no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—7961no—
226C2×C4—15921no—
227C2×C4—15922no—
228C2×C4—3981no—
229C2×C4—3981no—
230C2×C4—3981no—
231C2×C4—1991no—
232C2×C4D4×D19911yescharacteristic
233C8—41no—
234C8—7961no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—7961no—
240C22—7961no—
241C22—7961no—
242C22—7961no—
243C22—7961no—
244C22—7962no—
245C22—7961no—
246C22—15921no—
247C22—15921no—
248C22—15921no—
249C22—15922no—
250C22—3981no—
251C22—3981no—
252C22—1991no—
253C22—1991no—
254C22C22⋊C4×D19911yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—7961no—
260C4—15921no—
261C4—3981no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—7961no—
266C2—7961no—
267C2—15921no—
268C2—3981no—
269C2—3981no—
270C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC23×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#352×C199C2
socleC2×C199C2×#99
prime radical#27164579C2
cosocleC2×D4×C199C23
hypercenter#352D199
hypocenterC199#2121
LCSG ▷ C23×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #100×D199C2semidirect
3. #99×D199C2semidirect
4. #27163808C2semidirect
5. #27163806C2semidirect
6. #27163804C2semidirect
7. #352×C199C2semidirect
8. C22×D796C4semidirect
12. #20263938C4semidirect
16. #20263925C4semidirect
19. C22⋊C4×D199C4semidirect
28. C23×D199C2×C4semidirect
49. C22⋊Dic199C2×C4semidirect
56. C22×C4×C199C2×C4semidirect
61. C22⋊C4×C199C2×C4semidirect
135. C199#2121semidirect
141. #352D199semidirect
181. C22×C4C4×D199semidirect
197. C22⋊C4C4×D199semidirect

Maximal subgroups

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

subgroupindex
#271645792
#100×D1992
#99×D1992
#271638082
#271638062
#271638042
#352×C1992
#2121199