Finite Group Databasebeta

← #38770518

Subgroups — #38770518

Order 25472.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164586C211yesmaximal, characteristic
2#99×D199C212yesmaximal, complemented
3#27163813C211yesmaximal, characteristic, complemented
4#27163810C212yesmaximal, complemented
5#354×C199C211yesmaximal, characteristic, complemented
6C2×#15691887C411yescharacteristic, complemented
7C2×D4×D199C2211yescharacteristic
8#20263952C411yescharacteristic, complemented
9#20263951C2211yescharacteristic
10#20263944C411yescharacteristic, complemented
11#20263928—22no—
12#20263925—22no—
13#20263912C411yescharacteristic, complemented
14C22⋊C4×D199—22no—
15#20263870C2212yes—
16#20263869—22no—
17#124×C199C2211yescharacteristic
18#99×C199C2212yes—
19C23×D199D412yes—
20#15691887—42no—
21C2×D796—21no—
22C2×D796—22no—
23C2×D796C2×C411yescharacteristic
24C2×D796C2×C411yescharacteristic, complemented
25C2×C22⋊D199—22no—
26C2×C22⋊D199D412yes—
27D4×D199—42no—
28C2×C4×D199—21no—
29C2×C4×D199—42no—
30C2×C4×D199C2×C411yescharacteristic
31Dic199⋊C4—21no—
32C22⋊Dic199—22no—
33(C2×D199)⋊C4—22no—
34(C2×D199)⋊C4—21no—
35(C2×D199)⋊C4—42no—
36C4×Dic199C2×C411yescharacteristic, complemented
37C2×Q8×C199C2×C411yescharacteristic, complemented
38C2×D4×C199C2311yescharacteristic
39C22⋊C4×C199—22no—
40C22⋊C4×C199—22no—
41C42×C199C2×C411yescharacteristic, complemented
42C22×D199—22no—
43C22×D199—22no—
44C22×D199—21no—
45C22×D199—82no—
46C22×D199C22⋊C412yes—
47C22×D199C22⋊C411yescharacteristic
48D796—41no—
49D796—41no—
50D796—42no—
51D796—82no—
52C22⋊D199—42no—
53C22⋊D199—82no—
54C2×Dic199—21no—
55C2×Dic199—42no—
56C2×Dic199C22⋊C411yescharacteristic
57C4×D199—22no—
58C4×D199—42no—
59C4×D199—84no—
60C23×C199C2×D412yes—
61Q8×C199—41no—
62D4×C199—41no—
63C2×C4×C199—21no—
64C2×C4×C199—21no—
65C2×C4×C199—42no—
66C2×C4×C199C22×C411yescharacteristic
67C22×C199—22no—
68C22×C199—42no—
69C22×C199C2×C22⋊C411yescharacteristic
70C2×D199—22no—
71C2×D199—41no—
72C2×D199—44no—
73C2×D199—42no—
74C2×D199—42no—
75C2×D199—41no—
76C2×D199—82no—
77C2×D199#9912yes—
78C4×C199—21no—
79C4×C199—41no—
80C4×C199—41no—
81C4×C199—82no—
82Dic199—21no—
83Dic199—41no—
84Dic199—82no—
85C2×C199—21no—
86C2×C199—42no—
87C2×C199C2×#9911yescharacteristic
88D199—22no—
89D199—42no—
90D199—81no—
91C199#212711yescharacteristic, complemented, Sylow
92#2127—1991nomaximal, Sylow
93#535—1991no—
94C2×#99—1992no—
95#354—1991no—
96#354D19911yescharacteristic, complemented
97#353—1992no—
98C2×#41—1991no—
99C22×D4—1991no—
100#127—1991no—
101#127—1991no—
102#124—1991no—
103#124C2×D19911yescharacteristic
104#121—3982no—
105#120—3982no—
106#117—1991no—
107C2×C22⋊C4—3982no—
108#99—3982no—
109#99—1992no—
110#99C2×D19912yes—
111C24—1992no—
112#41—7962no—
113C2×Q8C4×D19911yescharacteristic, complemented
114C2×D4—7962no—
115C2×D4—3981no—
116C2×D4—3982no—
117C2×D4—3982no—
118C2×D4—1992no—
119C2×D4—1991no—
120C2×D4—1991no—
121C2×D4C22×D19911yescharacteristic
122C22×C4—7962no—
123C22×C4—3981no—
124C22×C4—1991no—
125C4⋊C4—3981no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—7962no—
129C22⋊C4—3981no—
130C22⋊C4—3982no—
131C22⋊C4—3982no—
132C42—1991no—
133C42C4×D19911yescharacteristic, complemented
134C23—15922no—
135C23—3982no—
136C23—3982no—
137C23—3981no—
138C23—1992no—
139C23—1991no—
140C23D4×D19912yes—
141Q8—41no—
142D4—41no—
143D4—7961no—
144D4—7961no—
145D4—7962no—
146D4—7962no—
147D4—15922no—
148D4—15922no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—7962no—
153C2×C4—7962no—
154C2×C4—15924no—
155C2×C4—3982no—
156C2×C4—3981no—
157C2×C4—1991no—
158C2×C4C2×C4×D19911yescharacteristic
159C22—22no—
160C22—42no—
161C22—7962no—
162C22—7962no—
163C22—7961no—
164C22—7964no—
165C22—7961no—
166C22—15922no—
167C22—3982no—
168C22—1992no—
169C22C22⋊C4×D19911yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—7961no—
175C4—15922no—
176C4—3981no—
177C2—21no—
178C2—42no—
179C2—7962no—
180C2—15921no—
181C2—3982no—
182C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#354×C199C2
socleC2×C199C2×#99
prime radical#27164586C2
cosocleC2×D4×C199C23
hypercenter#354D199
hypocenterC199#2127
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D199C2semidirect
3. #27163813C2semidirect
4. #27163810C2semidirect
5. #354×C199C2semidirect
6. C2×#15691887C4semidirect
8. #20263952C4semidirect
10. #20263944C4semidirect
13. #20263912C4semidirect
24. C2×D796C2×C4semidirect
36. C4×Dic199C2×C4semidirect
37. C2×Q8×C199C2×C4semidirect
41. C42×C199C2×C4semidirect
91. C199#2127semidirect
96. #354D199semidirect
113. C2×Q8C4×D199semidirect
133. C42C4×D199semidirect

Maximal subgroups

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

subgroupindex
#271645862
#99×D1992
#271638132
#271638102
#354×C1992
#2127199