Finite Group Databasebeta

← #38769390

Subgroups — #38769390

Order 25472.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 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#27164577C211yesmaximal, characteristic
2#27164493C211yesmaximal, characteristic, complemented
3C2×#20263869C212yesmaximal, complemented
4#27163785C212yesmaximal, complemented
5C2×#99×C199C211yesmaximal, characteristic, complemented
6C22×C22⋊D199C2211yescharacteristic
7#20263938—21no—
8#20263936C412yescomplemented
9#20263927—22no—
10#20263924—21no—
11#20263923C412yescomplemented
12D4×Dic199—24no—
13#20263914—21no—
14C2×C22⋊Dic199—21no—
15C2×C22⋊Dic199C2211yescharacteristic
16C2×C22⋊Dic199C2212yes—
17#20263869—24no—
18C22×D4×C199C2211yescharacteristic
19C2×C22⋊C4×C199C2212yes—
20#99×C199—22no—
21C23×D199C2×C411yescharacteristic, complemented
22C2×C22⋊D199—22no—
23C2×C22⋊D199—21no—
24C2×C22⋊D199—22no—
25C2×C22⋊D199—21no—
26C2×C22⋊D199—22no—
27C22×Dic199—22no—
28C22×Dic199—22no—
29C22×Dic199—21no—
30C22×Dic199C2×C411yescharacteristic, complemented
31C4⋊Dic199—22no—
32Dic199⋊C4—22no—
33C22⋊Dic199—22no—
34C22⋊Dic199—22no—
35C22⋊Dic199—21no—
36C22⋊Dic199—22no—
37C22⋊Dic199C2×C412yescomplemented
38C22⋊Dic199D414yescomplemented
39(C2×D199)⋊C4—22no—
40C4×Dic199—22no—
41C24×C199D411yescharacteristic
42C24×C199C2311yescharacteristic
43C2×D4×C199—22no—
44C2×D4×C199—22no—
45C2×D4×C199C2×C412yes—
46C2×D4×C199D412yes—
47C22×C4×C199—22no—
48C22×C4×C199D411yescharacteristic
49C22⋊C4×C199—22no—
50C22⋊C4×C199—22no—
51C22×D199—21no—
52C22×D199—22no—
53C22×D199—81no—
54C22⋊D199—41no—
55C22⋊D199—42no—
56C22⋊D199—41no—
57C22⋊D199—82no—
58C2×Dic199—22no—
59C2×Dic199—21no—
60C2×Dic199—22no—
61C2×Dic199—24no—
62C2×Dic199—82no—
63C2×Dic199—82no—
64C2×Dic199—81no—
65C23×C199—21no—
66C23×C199—21no—
67C23×C199—81no—
68C23×C199—81no—
69C23×C199C22⋊C412yes—
70C23×C199C22×C411yescharacteristic
71C23×C199C2×D412yes—
72C23×C199C2×D411yescharacteristic
73C23×C199C2×D411yescharacteristic
74C23×C199#4112yes—
75D4×C199—24no—
76D4×C199—24no—
77C2×C4×C199—21no—
78C2×C4×C199—21no—
79C2×C4×C199—42no—
80C2×C4×C199—82no—
81C2×C4×C199C22⋊C412yes—
82C22×C199—22no—
83C22×C199—42no—
84C22×C199—41no—
85C22×C199—41no—
86C22×C199—42no—
87C22×C199—41no—
88C22×C199—42no—
89C22×C199—42no—
90C22×C199—81no—
91C22×C199C2×C22⋊C411yescharacteristic
92C22×C199C4×D412yes—
93C22×C199#12011yescharacteristic
94C22×C199#12112yes—
95C22×C199#12311yescharacteristic
96C2×D199—42no—
97C2×D199—41no—
98C2×D199—82no—
99C4×C199—22no—
100C4×C199—82no—
101Dic199—44no—
102Dic199—42no—
103Dic199—41no—
104C2×C199—21no—
105C2×C199—21no—
106C2×C199—22no—
107C2×C199—22no—
108C2×C199—21no—
109C2×C199—42no—
110C2×C199#38611yescharacteristic
111D199—81no—
112C199#191311yescharacteristic, complemented, Sylow
113#1913—1991nomaximal, Sylow
114#534—1991no—
115#518—1991no—
116C2×#99—1992no—
117C2×#99D19911yescharacteristic, complemented
118#342—1992no—
119C22×D4—1991no—
120C22×D4C2×D19911yescharacteristic
121#123—3981no—
122#123—1992no—
123#121—3982no—
124#120—3981no—
125#120—1992no—
126C4×D4—3984no—
127#117—3981no—
128C2×C22⋊C4—3981no—
129C2×C22⋊C4—1992no—
130C2×C22⋊C4—1991no—
131C2×C22⋊C4C2×D19912yes—
132#99—22no—
133#99—3984no—
134C24—1991no—
135C24D79611yescharacteristic
136C24C22×D19911yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—3982no—
140C2×D4—3981no—
141C2×D4—3981no—
142C2×D4—3982no—
143C2×D4—3982no—
144C2×D4C4×D19912yes—
145C2×D4C22⋊D19912yes—
146C22×C4—22no—
147C22×C4—3982no—
148C22×C4—3982no—
149C22×C4—3981no—
150C22×C4—1991no—
151C22×C4D79611yescharacteristic
152C4⋊C4—3982no—
153C4⋊C4—3982no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—3982no—
157C22⋊C4—3982no—
158C22⋊C4—3982no—
159C22⋊C4—3981no—
160C22⋊C4—3982no—
161C22⋊C4—1992no—
162C22⋊C4—1994no—
163C42—3982no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—15921no—
169C23—3981no—
170C23—3982no—
171C23(C2×D199)⋊C412yes—
172C23C2×C4×D19911yescharacteristic
173C23D4×D19912yes—
174C23C2×C22⋊D19911yescharacteristic
175C23C2×D79611yescharacteristic
176C23#1569188512yes—
177D4—24no—
178D4—24no—
179D4—7961no—
180D4—7961no—
181D4—7962no—
182D4—15922no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—15922no—
188C2×C4—15922no—
189C2×C4—15921no—
190C2×C4—3982no—
191C2×C4—3981no—
192C2×C4—3982no—
193C2×C4—3984no—
194C2×C4(C2×D199)⋊C412yes—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—7962no—
205C22—7961no—
206C22—15922no—
207C22C2×(C2×D199)⋊C411yescharacteristic
208C22(C22⋊D199)⋊C412yes—
209C22#2026392511yescharacteristic
210C22#2026392712yes—
211C22#2026393911yescharacteristic
212C4—22no—
213C4—82no—
214C4—7964no—
215C4—7962no—
216C4—7961no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—15921no—
224C2#2716389511yescharacteristic
subgroupG/H
centerC2#27163895
commutatorC23×C199C22×C4
FrattiniC24C22×D199
FittingC2×#99×C199C2
socleC2×C199#386
prime radical#27164577C2
cosocleC24×C199C23
hypercenterC2×#99D199
hypocenterC199#1913
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #27164493C2semidirect
3. C2×#20263869C2semidirect
4. #27163785C2semidirect
5. C2×#99×C199C2semidirect
8. #20263936C4semidirect
11. #20263923C4semidirect
21. C23×D199C2×C4semidirect
30. C22×Dic199C2×C4semidirect
37. C22⋊Dic199C2×C4semidirect
38. C22⋊Dic199D4semidirect
112. C199#1913semidirect
117. C2×#99D199semidirect

Maximal subgroups

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

subgroupindex
#271645772
#271644932
C2×#202638692
#271637852
C2×#99×C1992
#1913199