Finite Group Databasebeta

← #38769075

Subgroups — #38769075

Order 25472.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#27164499C211yesmaximal, characteristic
2#27164058C211yesmaximal, characteristic, complemented
3C2×#20263883C211yesmaximal, characteristic
4#27164017C211yesmaximal, characteristic
5#27163787C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7#384×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9C2×#15691870C2211yescharacteristic
10C2×#15691870C2211yescharacteristic
11#20263920—21no—
12#20263917—21no—
13C4×C22⋊D199—22no—
14#20263914C2211yescharacteristic
15#20263913—21no—
16C4⋊C4×D199—21no—
17#20263888—21no—
18#20263886—21no—
19#20263883C414yescomplemented
20#20263878—21no—
21#20263875—21no—
22C2×C4⋊C4×C199—22no—
23C2×C4⋊C4×C199C2211yescharacteristic
24C2×C4⋊C4×C199C2211yescharacteristic
25C2×C42×C199C2211yescharacteristic
26#15691886—21no—
27#15691886—21no—
28#15691886C2×C412yes—
29#15691886Q812yes—
30C2×Dic398D411yescharacteristic
31C2×D796D411yescharacteristic
32C2×C22⋊D199—21no—
33C2×C4×D199—22no—
34C2×C4×D199—21no—
35#15691870—21no—
36#15691870—21no—
37#15691870—21no—
38#15691870C2×C412yescomplemented
39C2×C199⋊C8—21no—
40C2×C199⋊C8—21no—
41C4⋊Dic199D411yescharacteristic
42C4⋊Dic199D411yescharacteristic
43Dic199⋊C4—22no—
44C22⋊Dic199—21no—
45(C2×D199)⋊C4—22no—
46C4×Dic199—22no—
47C4×Dic199—21no—
48C22×C4×C199—21no—
49C22×C4×C199—21no—
50C22×C4×C199D412yes—
51C22×C4×C199C2311yescharacteristic
52C4⋊C4×C199—21no—
53C4⋊C4×C199—21no—
54C4⋊C4×C199—21no—
55C4⋊C4×C199—21no—
56C4⋊C4×C199—44no—
57C42×C199—21no—
58C42×C199C2×C412yescomplemented
59C22×D199—21no—
60Dic398—21no—
61Dic398C4⋊C412yescomplemented
62D796—21no—
63D796C4⋊C412yescomplemented
64C22⋊D199—22no—
65C22⋊D199—22no—
66C2×Dic199—21no—
67C2×Dic199—21no—
68C2×Dic199—21no—
69C4×D199—22no—
70C4×D199—22no—
71C4×D199—82no—
72C199⋊C8—22no—
73C199⋊C8—41no—
74C23×C199C2×D411yescharacteristic
75C2×C4×C199—22no—
76C2×C4×C199—22no—
77C2×C4×C199—22no—
78C2×C4×C199—22no—
79C2×C4×C199—42no—
80C2×C4×C199—42no—
81C2×C4×C199—41no—
82C2×C4×C199C22×C411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C2×C4×C199C2×D411yescharacteristic
85C2×C4×C199C2×Q811yescharacteristic
86C2×C4×C199#4111yescharacteristic
87C2×C4×C199#4111yescharacteristic
88C22×C199—21no—
89C22×C199—21no—
90C22×C199C4×D411yescharacteristic
91C22×C199#12011yescharacteristic
92C22×C199C22⋊Q811yescharacteristic
93C2×D199—22no—
94C2×D199—41no—
95C2×D199—41no—
96C4×C199—24no—
97C4×C199—42no—
98C4×C199—81no—
99C4×C199C2×C4⋊C411yescharacteristic
100C4×C199C4×D411yescharacteristic
101C4×C199C22⋊Q811yescharacteristic
102C4×C199#12311yescharacteristic
103Dic199—22no—
104Dic199—41no—
105Dic199—41no—
106C2×C199—21no—
107C2×C199—21no—
108C2×C199#38511yescharacteristic
109C2×C199#45311yescharacteristic
110C2×C199#45611yescharacteristic
111D199—42no—
112C199#185911yescharacteristic, complemented, Sylow
113#1859—1991nomaximal, Sylow
114#520—1991no—
115#428—1991no—
116C2×#104—1991no—
117#417—1991no—
118#384D19911yescharacteristic, complemented
119#343—1991no—
120#339—1991no—
121C2×#41—1991no—
122C2×M16—1991no—
123C2×M16—1991no—
124C4×Q8—3981no—
125C4×D4—3981no—
126C4×D4—3982no—
127#117—3981no—
128#117—1991no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—3981no—
131C2×C4⋊C4C2×D19911yescharacteristic
132C2×C4⋊C4C2×D19911yescharacteristic
133C2×C42C2×D19911yescharacteristic
134#107—3981no—
135#106—3981no—
136#104—1994no—
137#103—3981no—
138#102—3981no—
139#41—3981no—
140#41—3981no—
141#41—1992no—
142#41—1992no—
143C2×Q8—1991no—
144C2×D4—3981no—
145C2×D4—1991no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3982no—
149C22×C4—3981no—
150C22×C4C22⋊D19912yes—
151C22×C4C22×D19911yescharacteristic
152M16—3981no—
153M16—3981no—
154M16—3981no—
155M16—1992no—
156C2×C8—3981no—
157C2×C8—3981no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—44no—
163C4⋊C4—3982no—
164C4⋊C4—1991no—
165C4⋊C4—1991no—
166C22⋊C4—3982no—
167C22⋊C4—3981no—
168C42—21no—
169C42—3982no—
170C42—3981no—
171C42C4×D19912yescomplemented
172C23—3981no—
173C23C2×C22⋊D19911yescharacteristic
174Q8—3981no—
175Q8—1992no—
176D4—3982no—
177D4—3982no—
178D4—3981no—
179D4—1992no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—15922no—
188C2×C4—3981no—
189C2×C4—3981no—
190C2×C4—3981no—
191C2×C4—3982no—
192C2×C4—3982no—
193C2×C4C2×C4×D19911yescharacteristic
194C2×C4D4×D19911yescharacteristic
195C2×C4D4×D19911yescharacteristic
196C2×C4Q8×D19911yescharacteristic
197C2×C4#1569188611yescharacteristic
198C2×C4#1569188711yescharacteristic
199C8—7961no—
200C8—3982no—
201C22—21no—
202C22—21no—
203C22—7961no—
204C22—7961no—
205C22—3982no—
206C22C4×C22⋊D19911yescharacteristic
207C22#2026392311yescharacteristic
208C22#2026393111yescharacteristic
209C4—24no—
210C4—42no—
211C4—81no—
212C4—7961no—
213C4—7961no—
214C4—3982no—
215C4C4⋊C4×D19911yescharacteristic
216C4#2026391711yescharacteristic
217C4#2026393311yescharacteristic
218C4#2026394011yescharacteristic
219C2—21no—
220C2—21no—
221C2—7962no—
222C2#2716388911yescharacteristic
223C2#2716418511yescharacteristic
224C2#2716420011yescharacteristic
subgroupG/H
centerC22#20263923
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#384×C199C2
socleC22×C199#120
prime radicalC2×#15691886C22
cosocleC22×C4×C199C23
hypercenter#384D199
hypocenterC199#1859
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #27164058C2semidirect
5. #27163787C2semidirect
6. #27163778C2semidirect
7. #384×C199C2semidirect
19. #20263883C4semidirect
38. #15691870C2×C4semidirect
58. C42×C199C2×C4semidirect
61. Dic398C4⋊C4semidirect
63. D796C4⋊C4semidirect
112. C199#1859semidirect
118. #384D199semidirect
171. C42C4×D199semidirect

Maximal subgroups

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

subgroupindex
#271644992
#271640582
C2×#202638832
#271640172
#271637872
#271637782
#384×C1992
#1859199