Finite Group Databasebeta

← #38767394

Subgroups — #38767394

Order 25472.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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#27164529C211yesmaximal, characteristic
2#27163878C211yesmaximal, characteristic, complemented
3#27163756C212yesmaximal, complemented
4#27163753C212yesmaximal, complemented
5#407×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C411yescharacteristic, complemented
7#20263938—21no—
8#20263934C411yescharacteristic, complemented
9#20263934C2211yescharacteristic
10#20263929C411yescharacteristic, complemented
11#20263929C2211yescharacteristic
12#20263924—21no—
13C4×C22⋊D199—21no—
14C4×C22⋊D199—21no—
15#20263914C411yescharacteristic, complemented
16C2×C4⋊Dic199—21no—
17C2×C22⋊Dic199—21no—
18C199⋊#95C2212yes—
19C23×C4×C199C2211yescharacteristic
20C2×M16×C199—21no—
21C22⋊C8×C199—21no—
22C22⋊C8×C199C2212yes—
23#15691886—42no—
24C2×Dic398C2×C411yescharacteristic
25C2×D796C2×C411yescharacteristic
26C2×C22⋊D199—21no—
27C2×C22⋊D199—21no—
28C2×C22⋊D199—21no—
29C22×Dic199—22no—
30C2×C4×D199—21no—
31C4⋊Dic199—42no—
32C4⋊Dic199C2×C411yescharacteristic
33C4⋊Dic199C2×C411yescharacteristic
34Dic199⋊C4—21no—
35Dic199⋊C4—21no—
36C22⋊Dic199—21no—
37C22⋊Dic199—21no—
38C22⋊Dic199—21no—
39C22⋊Dic199—42no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42C4×Dic199—21no—
43C24×C199D411yescharacteristic
44C22×C4×C199—42no—
45C22×C4×C199C2×C411yescharacteristic
46C22×C4×C199C2×C411yescharacteristic
47C22×C4×C199D411yescharacteristic
48C22×C4×C199D411yescharacteristic
49C22×C4×C199D411yescharacteristic
50C22×C4×C199C2311yescharacteristic
51M16×C199—42no—
52C2×C8×C199—22no—
53C22×D199—21no—
54Dic398—41no—
55D796—41no—
56C22⋊D199—42no—
57C22⋊D199—42no—
58C22⋊D199—41no—
59C22⋊D199—41no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—42no—
64C2×Dic199—42no—
65C2×Dic199—42no—
66C4×D199—42no—
67C23×C199—21no—
68C23×C199—21no—
69C23×C199—41no—
70C23×C199—41no—
71C23×C199C22⋊C411yescharacteristic
72C23×C199C22⋊C411yescharacteristic
73C23×C199C2×D411yescharacteristic
74C2×C4×C199—22no—
75C2×C4×C199—22no—
76C2×C4×C199—22no—
77C2×C4×C199—41no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199C22⋊C411yescharacteristic
81C2×C4×C199C22⋊C411yescharacteristic
82C2×C4×C199C22×C411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C8×C199—42no—
85C22×C199—21no—
86C22×C199—21no—
87C22×C199—21no—
88C22×C199—21no—
89C22×C199—21no—
90C22×C199—21no—
91C22×C199—21no—
92C22×C199—21no—
93C22×C199—42no—
94C22×C199—42no—
95C22×C199C2×C22⋊C411yescharacteristic
96C22×C199AGL(1,ℤ/8)11yescharacteristic
97C22×C199#13711yescharacteristic
98C2×D199—41no—
99C2×D199—41no—
100C2×D199—41no—
101C4×C199—21no—
102C4×C199—21no—
103C4×C199—21no—
104C4×C199#9912yes—
105Dic199—41no—
106Dic199—41no—
107Dic199—41no—
108Dic199—82no—
109C2×C199—21no—
110C2×C199—21no—
111C2×C199—41no—
112C2×C199—41no—
113C2×C199C2×#9911yescharacteristic
114C2×C199#41711yescharacteristic
115C2×C199#42111yescharacteristic
116D199—81no—
117C199#151311yescharacteristic, complemented, Sylow
118#1513—1991nomaximal, Sylow
119#525—1991no—
120#407D19911yescharacteristic, complemented
121#380—1991no—
122#328—1992no—
123#327—1992no—
124C2×#41—1991no—
125C23×C4C2×D19911yescharacteristic
126C2×M16—21no—
127#123—3981no—
128C22⋊Q8—1991no—
129C22⋊Q8—1991no—
130#121—1991no—
131#121—1991no—
132#120—3981no—
133C4×D4—3981no—
134C4×D4—3981no—
135#117—1991no—
136C2×C4⋊C4—3981no—
137C2×C22⋊C4—3981no—
138C22⋊C8—21no—
139C22⋊C8C2×D19912yes—
140#95—1992no—
141C24D79611yescharacteristic
142#41—7962no—
143C2×Q8—1991no—
144C2×D4—3981no—
145C2×D4—3981no—
146C2×D4—3981no—
147C2×D4—1991no—
148C22×C4—42no—
149C22×C4—3982no—
150C22×C4—3981no—
151C22×C4C4×D19911yescharacteristic
152C22×C4C4×D19911yescharacteristic
153C22×C4C22⋊D19911yescharacteristic
154C22×C4C22⋊D19911yescharacteristic
155C22×C4D79611yescharacteristic
156C22×C4C22×D19911yescharacteristic
157M16—42no—
158C2×C8—22no—
159C4⋊C4—7962no—
160C4⋊C4—3981no—
161C4⋊C4—3981no—
162C4⋊C4—1991no—
163C4⋊C4—1991no—
164C22⋊C4—7962no—
165C22⋊C4—3981no—
166C22⋊C4—3981no—
167C22⋊C4—3981no—
168C22⋊C4—3981no—
169C22⋊C4—3981no—
170C42—3981no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—3981no—
176C23(C2×D199)⋊C411yescharacteristic
177C23(C2×D199)⋊C411yescharacteristic
178C23C2×D79611yescharacteristic
179Q8—7961no—
180D4—7961no—
181D4—7962no—
182D4—7962no—
183D4—7961no—
184D4—7961no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—7962no—
192C2×C4—7962no—
193C2×C4—7962no—
194C2×C4—7962no—
195C2×C4—3981no—
196C2×C4—3981no—
197C2×C4—3981no—
198C2×C4(C2×D199)⋊C411yescharacteristic
199C2×C4(C2×D199)⋊C411yescharacteristic
200C2×C4C2×C4×D19911yescharacteristic
201C2×C4C2×C22⋊D19911yescharacteristic
202C8—42no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—7961no—
214C22—7961no—
215C22—7961no—
216C22C2×(C2×D199)⋊C411yescharacteristic
217C22#2026398311yescharacteristic
218C22#2026398811yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—7961no—
223C4—7961no—
224C4—7961no—
225C4—15922no—
226C4#2026386912yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—15921no—
232C2C2×#2026386911yescharacteristic
233C2#2716402011yescharacteristic
234C2#2716403411yescharacteristic
subgroupG/H
centerC22C2×(C2×D199)⋊C4
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#407×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164529C2
cosocleC22×C4×C199C23
hypercenter#407D199
hypocenterC199#1513
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #27163878C2semidirect
3. #27163756C2semidirect
4. #27163753C2semidirect
5. #407×C199C2semidirect
6. C2×#15691886C4semidirect
8. #20263934C4semidirect
10. #20263929C4semidirect
15. #20263914C4semidirect
117. C199#1513semidirect
120. #407D199semidirect

Maximal subgroups

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

subgroupindex
#271645292
#271638782
#271637562
#271637532
#407×C1992
#1513199