Finite Group Databasebeta

← #38768803

Subgroups — #38768803

Order 25472.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#20264000C211yesmaximal, characteristic
2C2×#20263883C212yesmaximal, complemented
3#27164017C212yesmaximal, complemented
4C2×#20263872C211yesmaximal, characteristic, complemented
5#390×C199C211yesmaximal, characteristic, complemented
6#20264000—24no—
7#20264000—22no—
8#20264000C414yescomplemented
9C2×#15691886C2212yes—
10C2×#15691885—22no—
11C22×C22⋊D199—22no—
12C2×D4×D199—22no—
13C2×#15691870C2212yescomplemented
14#20263883—24no—
15#20263878—22no—
16#20263875—22no—
17#20263872—22no—
18C22×D4×C199C2211yescharacteristic
19C2×C4⋊C4×C199C2211yescharacteristic, complemented
20C2×C22⋊C4×C199—22no—
21C2×C42×C199C2211yescharacteristic, complemented
22C23×D199—22no—
23#15691886—22no—
24#15691886—22no—
25#15691886C2×C414yes—
26#15691886D414yes—
27#15691885—28no—
28#15691885—28no—
29C2×Dic398D412yes—
30C2×D796D412yes—
31C2×C22⋊D199—24no—
32C2×C22⋊D199—28no—
33C2×C22⋊D199—28no—
34C2×C22⋊D199—22no—
35D4×D199—28no—
36D4×D199—28no—
37C22×Dic199—22no—
38C2×C4×D199—22no—
39#15691870—22no—
40#15691870—22no—
41C2×C199⋊C8—22no—
42C24×C199D412yes—
43C2×D4×C199—22no—
44C2×D4×C199—22no—
45C2×D4×C199C2×C412yes—
46C2×D4×C199D412yes—
47C22×C4×C199—21no—
48C22×C4×C199—21no—
49C22×C4×C199C2311yescharacteristic
50C4⋊C4×C199—22no—
51C22⋊C4×C199—44no—
52C42×C199—22no—
53C22×D199—22no—
54C22×D199—28no—
55C22×D199—82no—
56C22×D199—82no—
57Dic398—22no—
58Dic398C22⋊C414yescomplemented
59D796—22no—
60D796C22⋊C414yescomplemented
61C22⋊D199—216no—
62C22⋊D199—24no—
63C22⋊D199—24no—
64C2×Dic199—28no—
65C2×Dic199—22no—
66C4×D199—24no—
67C4×D199—24no—
68C199⋊C8—42no—
69C23×C199—22no—
70C23×C199—22no—
71C23×C199—42no—
72C23×C199—42no—
73C23×C199C22⋊C414yes—
74C23×C199C2×D411yescharacteristic
75D4×C199—24no—
76D4×C199—24no—
77C2×C4×C199—22no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199—42no—
81C2×C4×C199—41no—
82C2×C4×C199C22×C411yescharacteristic
83C2×C4×C199C2×D412yes—
84C2×C4×C199C2×D412yes—
85C2×C4×C199C2×D411yescharacteristic
86C22×C199—21no—
87C22×C199—21no—
88C22×C199—24no—
89C22×C199—44no—
90C22×C199—42no—
91C22×C199—42no—
92C22×C199—82no—
93C22×C199C2×C22⋊C411yescharacteristic
94C22×C199#12011yescharacteristic
95C22×C199#12011yescharacteristic
96C2×D199—24no—
97C2×D199—42no—
98C2×D199—42no—
99C2×D199—88no—
100C4×C199—42no—
101C4×C199—81no—
102C4×C199C2×C22⋊C412yes—
103C4×C199#12012yes—
104Dic199—24no—
105C2×C199—21no—
106C2×C199—21no—
107C2×C199—44no—
108C2×C199#37911yescharacteristic
109C2×C199#45311yescharacteristic
110C2×C199#45511yescharacteristic
111D199—44no—
112C199#179211yescharacteristic, complemented, Sylow
113#1792—1991nomaximal, Sylow
114C2×H(5,𝔽2)—1991no—
115C2×#104—1992no—
116#417—1992no—
117C2×#100—1991no—
118#390D19911yescharacteristic, complemented
119H(5,𝔽2)—3984no—
120H(5,𝔽2)—3982no—
121H(5,𝔽2)—1994no—
122C2×#41—3982no—
123C2×#41—1992no—
124C22×D4—3982no—
125C22×D4—3982no—
126C22×D4C2×D19911yescharacteristic
127C2×M16—1992no—
128C2×C4⋊C4C2×D19911yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C2×D19911yescharacteristic, complemented
131#104—3984no—
132#103—3982no—
133#102—3982no—
134#100—3982no—
135C24—3982no—
136C24C22⋊D19912yes—
137#41—3988no—
138#41—3988no—
139#41—3982no—
140#41—3982no—
141#41—1994no—
142#41—1994no—
143C2×Q8—1992no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—3988no—
147C2×D4—3988no—
148C2×D4—3984no—
149C2×D4—3988no—
150C2×D4—3988no—
151C2×D4—3982no—
152C2×D4—1992no—
153C2×D4C4×D19912yes—
154C2×D4D79612yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—3982no—
158C22×C4—3982no—
159C22×C4C22×D19911yescharacteristic
160M16—3982no—
161M16—3982no—
162C2×C8—3982no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23—15922no—
171C23—15922no—
172C23—3988no—
173C23—3982no—
174C23(C2×D199)⋊C414yes—
175C23C2×C22⋊D19911yescharacteristic
176Q8—3982no—
177Q8—1994no—
178D4—24no—
179D4—24no—
180D4—39816no—
181D4—3984no—
182D4—3984no—
183D4—3982no—
184D4—1994no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—3988no—
191C2×C4—3982no—
192C2×C4—3984no—
193C2×C4—3984no—
194C2×C4C2×C4×D19911yescharacteristic
195C2×C4D4×D19912yes—
196C2×C4D4×D19912yes—
197C2×C4C2×D79611yescharacteristic
198C8—7962no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—82no—
206C22—7962no—
207C22—7962no—
208C22—15928no—
209C22—3984no—
210C22C2×(C2×D199)⋊C411yescharacteristic
211C22#2026392311yescharacteristic
212C22#2026392311yescharacteristic
213C4—42no—
214C4—81no—
215C4—3984no—
216C4C22⋊C4×D19912yes—
217C4#2026392512yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—7964no—
222C2#2716387311yescharacteristic
223C2#2716418511yescharacteristic
224C2#2716419511yescharacteristic
subgroupG/H
centerC22#20263923
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#390×C199C2
socleC22×C199#120
prime radicalC2×#20264000C2
cosocleC22×C4×C199C23
hypercenter#390D199
hypocenterC199#1792
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#20263883C2semidirect
3. #27164017C2semidirect
4. C2×#20263872C2semidirect
5. #390×C199C2semidirect
8. #20264000C4semidirect
13. C2×#15691870C22semidirect
19. C2×C4⋊C4×C199C22semidirect
21. C2×C42×C199C22semidirect
58. Dic398C22⋊C4semidirect
60. D796C22⋊C4semidirect
112. C199#1792semidirect
118. #390D199semidirect
128. C2×C4⋊C4C2×D199semidirect
130. C2×C42C2×D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202640002
C2×#202638832
#271640172
C2×#202638722
#390×C1992
#1792199