Finite Group Databasebeta

← #38769265

Subgroups — #38769265

Order 25472.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#27164784C211yesmaximal, characteristic, complemented
2#27164529C211yesmaximal, characteristic
3#27164000C211yesmaximal, characteristic, complemented
4#27163982C211yesmaximal, characteristic, complemented
5#27163790C211yesmaximal, characteristic, complemented
6#27163789C211yesmaximal, characteristic, complemented
7#407×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9#20263963—22no—
10#20263961—22no—
11C2×#15691870C2211yescharacteristic
12C2×#15691870C2211yescharacteristic
13M16×D199—22no—
14#20263938C412yes—
15#20263934—21no—
16#20263929—21no—
17#20263924C412yes—
18C4×C22⋊D199—22no—
19#20263914C2211yescharacteristic
20#20263873—21no—
21#20263871—21no—
22C22⋊(C199⋊C8)—22no—
23C23×C4×C199C2211yescharacteristic
24C2×M16×C199C2211yescharacteristic
25C2×M16×C199C2211yescharacteristic
26C22⋊C8×C199—22no—
27#15691886—21no—
28#15691886—21no—
29#15691886—21no—
30#15691886—21no—
31C2×Dic398D411yescharacteristic
32C2×D796D411yescharacteristic
33C2×C22⋊D199—22no—
34C2×C22⋊D199C2×C412yes—
35C2×C4×D199D412yes—
36#15691870—21no—
37#15691870—21no—
38#15691870—21no—
39#15691870—21no—
40#15691869—24no—
41C2×C199⋊C8—21no—
42C2×C199⋊C8D412yescomplemented
43C8×D199—24no—
44C4⋊Dic199—21no—
45Dic199⋊C4—22no—
46C22⋊Dic199—22no—
47C22⋊Dic199C2×C412yes—
48(C2×D199)⋊C4—22no—
49C4×Dic199—21no—
50C24×C199C2×C411yescharacteristic
51C22×C4×C199—21no—
52C22×C4×C199—21no—
53C22×C4×C199—21no—
54C22×C4×C199—21no—
55C22×C4×C199—21no—
56C22×C4×C199—21no—
57C22×C4×C199C2×C411yescharacteristic
58C22×C4×C199C2311yescharacteristic
59M16×C199—21no—
60M16×C199—21no—
61M16×C199—21no—
62M16×C199—21no—
63C2×C8×C199—21no—
64C2×C8×C199D412yescomplemented
65C22×D199C22⋊C412yes—
66Dic398—21no—
67Dic398—21no—
68D796—21no—
69D796—21no—
70C22⋊D199—22no—
71C22⋊D199—22no—
72C22⋊D199—44no—
73C2×Dic199—22no—
74C2×Dic199C22⋊C412yes—
75C4×D199—22no—
76C4×D199—22no—
77C199⋊C8—22no—
78C199⋊C8—41no—
79C23×C199—21no—
80C23×C199—21no—
81C23×C199—21no—
82C23×C199—42no—
83C23×C199C22×C411yescharacteristic
84C2×C4×C199—21no—
85C2×C4×C199—21no—
86C2×C4×C199—21no—
87C2×C4×C199—41no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199C2×D411yescharacteristic
92C2×C4×C199C2×D411yescharacteristic
93C2×C4×C199C2×D411yescharacteristic
94C2×C4×C199C2×D411yescharacteristic
95C2×C4×C199#4111yescharacteristic
96C2×C4×C199#4111yescharacteristic
97C8×C199—22no—
98C8×C199—41no—
99C22×C199—22no—
100C22×C199—22no—
101C22×C199—22no—
102C22×C199—41no—
103C22×C199—41no—
104C22×C199—41no—
105C22×C199—41no—
106C22×C199—41no—
107C22×C199C2×C22⋊C411yescharacteristic
108C22×C199C4×D411yescharacteristic
109C22×C199C4×D411yescharacteristic
110C2×D199—22no—
111C2×D199—42no—
112C4×C199—41no—
113C4×C199#12011yescharacteristic
114C4×C199#12111yescharacteristic
115C4×C199#12111yescharacteristic
116C4×C199#12311yescharacteristic
117Dic199—22no—
118Dic199—42no—
119C2×C199—21no—
120C2×C199—21no—
121C2×C199—21no—
122C2×C199—42no—
123C2×C199#38611yescharacteristic
124D199—42no—
125C199#189111yescharacteristic, complemented, Sylow
126#1891—1991nomaximal, Sylow
127#568—1991no—
128#525—1991no—
129#413—1991no—
130#407—1991no—
131#407D19911yescharacteristic, complemented
132#344—1991no—
133#344—1991no—
134C2×#41—1991no—
135C23×C4C2×D19911yescharacteristic
136#131—3982no—
137#131—3982no—
138C2×M16—3982no—
139C2×M16—1991no—
140C2×M16—1991no—
141C2×M16C2×D19911yescharacteristic
142C2×M16C2×D19911yescharacteristic
143#123—1992no—
144C22⋊Q8—3981no—
145#121—3981no—
146#120—1992no—
147C4×D4—3982no—
148#117—1991no—
149#101—3981no—
150#100—3981no—
151C22⋊C8—22no—
152C22⋊C8—3982no—
153C24C4×D19911yescharacteristic
154#41—3981no—
155#41—3981no—
156#41—3981no—
157#41—3981no—
158C2×Q8—1991no—
159C2×D4—3982no—
160C2×D4—1992no—
161C2×D4—1991no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1992no—
169C22×C4C4×D19911yescharacteristic
170C22×C4C22×D19911yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—3984no—
176M16—3981no—
177M16—3981no—
178M16—3981no—
179M16—3981no—
180C2×C8—21no—
181C2×C8—3984no—
182C2×C8—3981no—
183C2×C8—1992no—
184C2×C8C22⋊D19912yescomplemented
185C4⋊C4—3982no—
186C4⋊C4—3981no—
187C22⋊C4—3982no—
188C22⋊C4—3982no—
189C22⋊C4—1992no—
190C42—3981no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—1992no—
196C23C2×C4×D19911yescharacteristic
197Q8—3981no—
198Q8—3981no—
199D4—7964no—
200D4—3982no—
201D4—3982no—
202D4—3981no—
203D4—3981no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—3982no—
212C2×C4—3982no—
213C2×C4—3982no—
214C2×C4—1992no—
215C2×C4D4×D19911yescharacteristic
216C2×C4D4×D19911yescharacteristic
217C2×C4D4×D19911yescharacteristic
218C2×C4C2×C22⋊D19911yescharacteristic
219C2×C4#1569188511yescharacteristic
220C2×C4#1569188611yescharacteristic
221C8—22no—
222C8—41no—
223C8—7961no—
224C8—3982no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—7962no—
234C22—3982no—
235C22C22⋊C4×D19911yescharacteristic
236C22(C22⋊D199)⋊C411yescharacteristic
237C22C4×C22⋊D19911yescharacteristic
238C4—41no—
239C4—7962no—
240C4—3982no—
241C4#2026392311yescharacteristic
242C4#2026392611yescharacteristic
243C4#2026392811yescharacteristic
244C4#2026393711yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—7962no—
250C2#2716389411yescharacteristic
subgroupG/H
centerC4#20263923
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#407×C199C2
socleC2×C199#386
prime radical#27164529C2
cosocleC22×C4×C199C23
hypercenter#407D199
hypocenterC199#1891
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #27164784C2semidirect
3. #27164000C2semidirect
4. #27163982C2semidirect
5. #27163790C2semidirect
6. #27163789C2semidirect
7. #407×C199C2semidirect
42. C2×C199⋊C8D4semidirect
64. C2×C8×C199D4semidirect
125. C199#1891semidirect
131. #407D199semidirect
184. C2×C8C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271647842
#271645292
#271640002
#271639822
#271637902
#271637892
#407×C1992
#1891199