Finite Group Databasebeta

← #38769393

Subgroups — #38769393

Order 25472.

380 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 93 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#27164493C211yesmaximal, characteristic, complemented
2C2×#20263870C212yesmaximal, complemented
3C2×#20263870C211yesmaximal, characteristic, complemented
4#27163784C212yesmaximal, complemented
5#534×C199C211yesmaximal, characteristic
6D4×Dic199—24no—
7#20263914—21no—
8C2×C22⋊Dic199—21no—
9C2×C22⋊Dic199C2212yes—
10C2×C22⋊Dic199C2212yes—
11#20263870—24no—
12#20263870—22no—
13C22×D4×C199C2211yescharacteristic
14C22×D4×C199C2211yescharacteristic
15#123×C199—21no—
16#123×C199C412yescomplemented
17#121×C199—22no—
18#120×C199—21no—
19#120×C199C412yescomplemented
20C2×C22⋊C4×C199C2211yescharacteristic
21C22×Dic199—22no—
22C22×Dic199—22no—
23C22×Dic199—22no—
24C4⋊Dic199—22no—
25C22⋊Dic199—22no—
26C22⋊Dic199—22no—
27C22⋊Dic199—22no—
28C22⋊Dic199—22no—
29C22⋊Dic199D414yescomplemented
30C4×Dic199—22no—
31C24×C199C2×C411yescharacteristic, complemented
32C24×C199D411yescharacteristic
33C24×C199C2311yescharacteristic
34C2×D4×C199—22no—
35C2×D4×C199—22no—
36C2×D4×C199—21no—
37C2×D4×C199—21no—
38C2×D4×C199—22no—
39C2×D4×C199—22no—
40C2×D4×C199—22no—
41C2×D4×C199C2×C412yes—
42C2×D4×C199D412yes—
43C22×C4×C199—21no—
44C22×C4×C199C2×C411yescharacteristic, complemented
45C22×C4×C199D411yescharacteristic
46C4⋊C4×C199—22no—
47C22⋊C4×C199—22no—
48C22⋊C4×C199—21no—
49C22⋊C4×C199—22no—
50C22⋊C4×C199C2×C412yescomplemented
51C2×Dic199—24no—
52C2×Dic199—42no—
53C2×Dic199—82no—
54C2×Dic199—82no—
55C2×Dic199—82no—
56C23×C199—21no—
57C23×C199—21no—
58C23×C199—21no—
59C23×C199—22no—
60C23×C199—81no—
61C23×C199—81no—
62C23×C199—81no—
63C23×C199C22⋊C412yes—
64C23×C199C22×C411yescharacteristic
65C23×C199C2×D412yes—
66C23×C199C2×D411yescharacteristic
67C23×C199C2×D411yescharacteristic
68C23×C199#4112yes—
69D4×C199—24no—
70D4×C199—24no—
71D4×C199—41no—
72D4×C199—42no—
73D4×C199—41no—
74D4×C199—82no—
75C2×C4×C199—21no—
76C2×C4×C199—21no—
77C2×C4×C199—22no—
78C2×C4×C199—21no—
79C2×C4×C199—22no—
80C2×C4×C199—81no—
81C2×C4×C199C22⋊C412yes—
82C22×C199—22no—
83C22×C199—41no—
84C22×C199—41no—
85C22×C199—41no—
86C22×C199—42no—
87C22×C199—42no—
88C22×C199—42no—
89C22×C199—42no—
90C22×C199—42no—
91C22×C199—41no—
92C22×C199—81no—
93C22×C199—82no—
94C22×C199C2×C22⋊C411yescharacteristic
95C22×C199C4×D412yes—
96C22×C199#12011yescharacteristic
97C22×C199#12112yes—
98C22×C199#12311yescharacteristic
99C4×C199—22no—
100C4×C199—42no—
101C4×C199—41no—
102Dic199—44no—
103Dic199—82no—
104C2×C199—21no—
105C2×C199—22no—
106C2×C199—21no—
107C2×C199—22no—
108C2×C199—21no—
109C2×C199—42no—
110C2×C199—81no—
111C2×C199#38611yescharacteristic
112C199#191311yescharacteristic, complemented, Sylow
113#1913—1991nomaximal, Sylow
114#534D19911yescharacteristic
115#518—1991no—
116C2×#99—1991no—
117C2×#99—1992no—
118#342—1992no—
119C22×D4C2×D19911yescharacteristic
120C22×D4C2×D19911yescharacteristic
121#123—21no—
122#123Dic19912yescomplemented
123#121—22no—
124#120—21no—
125#120Dic19912yescomplemented
126C4×D4—3984no—
127#117—3981no—
128C2×C22⋊C4—3981no—
129C2×C22⋊C4—1992no—
130C2×C22⋊C4—1992no—
131C2×C22⋊C4C2×D19911yescharacteristic
132#99—3982no—
133#99—3984no—
134C24C2×Dic19911yescharacteristic, complemented
135C24C22⋊D19911yescharacteristic
136C24C22×D19911yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4C2×Dic19912yes—
145C2×D4C22⋊D19912yes—
146C22×C4—21no—
147C22×C4—3982no—
148C22×C4—3982no—
149C22×C4—3982no—
150C22×C4C2×Dic19911yescharacteristic, complemented
151C22×C4C22⋊D19911yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—3982no—
154C22⋊C4—22no—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C22⋊C4—3982no—
158C22⋊C4—3982no—
159C22⋊C4—3982no—
160C22⋊C4—3982no—
161C22⋊C4—1994no—
162C22⋊C4C2×Dic19912yescomplemented
163C42—3982no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—81no—
169C23—81no—
170C23—81no—
171C23C22⋊Dic19912yes—
172C23C22×Dic19911yescharacteristic
173C23D4×D19912yes—
174C23C2×C22⋊D19911yescharacteristic
175C23C2×C22⋊D19911yescharacteristic
176C23#1569188512yes—
177D4—24no—
178D4—24no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—82no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—81no—
189C2×C4—7962no—
190C2×C4—15922no—
191C2×C4—15922no—
192C2×C4—15922no—
193C2×C4—3984no—
194C2×C4C22⋊Dic19912yes—
195C22—22no—
196C22—41no—
197C22—42no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—81no—
206C22—82no—
207C22C2×C22⋊Dic19911yescharacteristic
208C22D4×Dic19912yes—
209C22#2026392311yescharacteristic
210C22#2026392712yes—
211C22#2026393611yescharacteristic
212C4—22no—
213C4—42no—
214C4—41no—
215C4—7964no—
216C4—15922no—
217C2—21no—
218C2—21no—
219C2—22no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—81no—
224C2#2716389811yescharacteristic
subgroupG/H
centerC2#27163898
commutatorC23×C199C22×C4
FrattiniC24C22×D199
Fitting, prime radical#534×C199C2
socleC2×C199#386
cosocleC24×C199C23
hypercenter#534D199
hypocenterC199#1913
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #27164493C2semidirect
2. C2×#20263870C2semidirect
3. C2×#20263870C2semidirect
4. #27163784C2semidirect
16. #123×C199C4semidirect
19. #120×C199C4semidirect
29. C22⋊Dic199D4semidirect
31. C24×C199C2×C4semidirect
44. C22×C4×C199C2×C4semidirect
50. C22⋊C4×C199C2×C4semidirect
112. C199#1913semidirect
122. #123Dic199semidirect
125. #120Dic199semidirect
134. C24C2×Dic199semidirect
150. C22×C4C2×Dic199semidirect
162. C22⋊C4C2×Dic199semidirect

Maximal subgroups

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

subgroupindex
#271644932
C2×#202638702
#271637842
#534×C1992
#1913199