Finite Group Databasebeta

← #38769288

Subgroups — #38769288

Order 25472.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#20263929C211yesmaximal, characteristic
2C2×#20263877C212yesmaximal, complemented
3#27163878C211yesmaximal, characteristic, complemented
4#27163781C212yesmaximal, complemented
5C22×M16×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7C22×C22⋊D199—21no—
8#20263929—22no—
9#20263929C414yescomplemented
10C2×C4⋊Dic199C2212yes—
11C2×C4⋊Dic199C2211yescharacteristic
12C2×C22⋊Dic199—21no—
13C2×(C2×D199)⋊C4—21no—
14#20263877—24no—
15C199⋊#95—21no—
16C23×C4×C199C2211yescharacteristic
17C2×M16×C199—22no—
18C2×M16×C199—24no—
19C22×C8×C199C2212yes—
20C23×D199—21no—
21C2×D796—22no—
22C2×D796—21no—
23C2×D796—22no—
24C2×D796C2×C412yes—
25C2×C22⋊D199—24no—
26C2×C22⋊D199—44no—
27C22×Dic199—21no—
28C22×Dic199—22no—
29C4⋊Dic199—21no—
30C4⋊Dic199—22no—
31C4⋊Dic199—22no—
32C4⋊Dic199C2×C412yes—
33C22⋊Dic199—42no—
34(C2×D199)⋊C4—24no—
35C24×C199D411yescharacteristic
36C22×C4×C199—22no—
37C22×C4×C199—22no—
38C22×C4×C199C2×C412yes—
39C22×C4×C199D412yes—
40C22×C4×C199D411yescharacteristic
41C22×C4×C199C2311yescharacteristic
42M16×C199—28no—
43C2×C8×C199—22no—
44C2×C8×C199—22no—
45C2×C8×C199D414yescomplemented
46C22×D199—22no—
47C22×D199—42no—
48C22×D199—42no—
49C22×D199—41no—
50C22×D199—41no—
51D796—24no—
52D796—41no—
53D796—41no—
54C22⋊D199—48no—
55C2×Dic199—22no—
56C2×Dic199—42no—
57C2×Dic199—42no—
58C2×Dic199—42no—
59C2×Dic199—42no—
60C23×C199—22no—
61C23×C199—21no—
62C23×C199—22no—
63C23×C199—21no—
64C23×C199C22⋊C412yes—
65C23×C199C2×D411yescharacteristic
66C2×C4×C199—22no—
67C2×C4×C199—24no—
68C2×C4×C199—22no—
69C2×C4×C199—22no—
70C2×C4×C199C22⋊C412yes—
71C2×C4×C199C22×C411yescharacteristic
72C2×C4×C199C2×D412yes—
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199#4112yes—
75C8×C199—24no—
76C22×C199—22no—
77C22×C199—22no—
78C22×C199—22no—
79C22×C199—42no—
80C22×C199—42no—
81C22×C199C2×C22⋊C411yescharacteristic
82C22×C199#12112yes—
83C22×C199AGL(1,ℤ/8)11yescharacteristic
84C22×C199AGL(1,ℤ/8)12yes—
85C22×C199#13711yescharacteristic
86C2×D199—42no—
87C2×D199—42no—
88C2×D199—42no—
89C2×D199—82no—
90C2×D199—82no—
91C4×C199—22no—
92C4×C199C4×D412yes—
93C4×C199#12011yescharacteristic
94C4×C199#12311yescharacteristic
95Dic199—42no—
96Dic199—82no—
97C2×C199—42no—
98C2×C199#38611yescharacteristic
99C2×C199#41711yescharacteristic
100C2×C199#41811yescharacteristic
101C2×C199#46812yes—
102C2×C199#46912yes—
103D199—82no—
104C199#189511yescharacteristic, complemented, Sylow
105#1895—1991nomaximal, Sylow
106C22×M16D19911yescharacteristic, complemented
107C2×#121—1991no—
108C2×#102—1992no—
109#380—1991no—
110#340—1992no—
111C22×D4—3981no—
112C22×D4—1991no—
113C23×C4C2×D19911yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D19912yes—
117#121—3982no—
118#121—1994no—
119C2×C4⋊C4—1992no—
120C2×C4⋊C4—1991no—
121C2×C22⋊C4—3981no—
122C2×C22⋊C4—3981no—
123#102—3984no—
124#95—3981no—
125C24—3981no—
126C24D79611yescharacteristic
127C2×D4—7964no—
128C2×D4—3984no—
129C2×D4—3981no—
130C2×D4—3982no—
131C2×D4—3982no—
132C2×D4—1992no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—3982no—
136C22×C4—3981no—
137C22×C4C4×D19912yes—
138C22×C4C22⋊D19912yes—
139C22×C4D79611yescharacteristic
140C22×C4C22×D19911yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D19914yescomplemented
145C4⋊C4—3982no—
146C4⋊C4—3982no—
147C4⋊C4—3981no—
148C4⋊C4—1992no—
149C22⋊C4—7962no—
150C22⋊C4—3984no—
151C23—21no—
152C23—22no—
153C23—21no—
154C23—22no—
155C23—7962no—
156C23—7961no—
157C23—7961no—
158C23—7962no—
159C23—3982no—
160C23(C2×D199)⋊C412yes—
161C23C2×D79611yescharacteristic
162D4—7968no—
163D4—7961no—
164D4—7961no—
165D4—3984no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—7962no—
171C2×C4—7962no—
172C2×C4—7962no—
173C2×C4—7962no—
174C2×C4—3982no—
175C2×C4(C2×D199)⋊C412yes—
176C2×C4C2×C4×D19911yescharacteristic
177C2×C4C2×C22⋊D19911yescharacteristic
178C2×C4C2×C22⋊D19912yes—
179C2×C4#1569188612yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—7962no—
187C22—7962no—
188C22—7962no—
189C22—15922no—
190C22—15922no—
191C22C2×(C2×D199)⋊C411yescharacteristic
192C22#2026392912yes—
193C22#2026398312yes—
194C22#2026398311yescharacteristic
195C22#2026398811yescharacteristic
196C4—22no—
197C4—7962no—
198C4—15922no—
199C4C4×C22⋊D19912yes—
200C4#2026392411yescharacteristic
201C4#2026393811yescharacteristic
202C2—42no—
203C2—15922no—
204C2#2716389711yescharacteristic
205C2#2716402011yescharacteristic
206C2#2716402411yescharacteristic
207C2#2716426412yes—
208C2#2716427112yes—
subgroupG/H
centerC23C2×D796
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC22×M16×C199C2
socleC23×C199C2×D4
prime radicalC2×#20263929C2
cosocleC22×C4×C199C23
hypercenterC22×M16D199
hypocenterC199#1895
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#20263877C2semidirect
3. #27163878C2semidirect
4. #27163781C2semidirect
5. C22×M16×C199C2semidirect
9. #20263929C4semidirect
45. C2×C8×C199D4semidirect
104. C199#1895semidirect
106. C22×M16D199semidirect
144. C2×C8C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639292
C2×#202638772
#271638782
#271637812
C22×M16×C1992
#1895199