Finite Group Databasebeta

← #38767790

Subgroups — #38767790

Order 25472.

266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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#27164168C212yesmaximal
2#27163935C211yesmaximal, characteristic
3#27163755C212yesmaximal, complemented
4#27163744C211yesmaximal, characteristic, complemented
5#544×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7C2×#15691873—22no—
8C2×(C2×D199)⋊C4—21no—
9C2×(C2×D199)⋊C4—21no—
10#20263875—22no—
11C22⋊(C199⋊C8)C2212yes—
12C199⋊#95C2211yescharacteristic
13C4⋊Q8×C199—22no—
14#125×C199—22no—
15C22⋊Q8×C199—21no—
16C22⋊Q8×C199C2212yes—
17#117×C199—21no—
18C2×C4⋊C4×C199C2211yescharacteristic
19C23×D199—21no—
20C2×D796—22no—
21C2×D796—42no—
22C22×Dic199—21no—
23#15691873—44no—
24C2×C199⋊C8—22no—
25(C2×D199)⋊C4—42no—
26(C2×D199)⋊C4—42no—
27C2×Q8×C199—21no—
28C2×Q8×C199D412yes—
29C22×C4×C199D411yescharacteristic
30C22×C4×C199D411yescharacteristic
31C22×C4×C199C2311yescharacteristic
32C4⋊C4×C199—21no—
33C4⋊C4×C199—21no—
34C4⋊C4×C199—21no—
35C4⋊C4×C199—22no—
36C4⋊C4×C199—22no—
37C4⋊C4×C199—22no—
38C4⋊C4×C199D412yes—
39C22⋊C4×C199—22no—
40C42×C199—22no—
41C22×D199—22no—
42C22×D199—41no—
43C22×D199—41no—
44C22×D199—41no—
45C22×D199—41no—
46C22×D199—41no—
47C22×D199—41no—
48D796—24no—
49D796—42no—
50C2×Dic199—41no—
51C2×Dic199—41no—
52C2×Dic199—41no—
53C199⋊C8—42no—
54C23×C199C2×D411yescharacteristic
55Q8×C199—42no—
56Q8×C199—81no—
57C2×C4×C199—21no—
58C2×C4×C199—21no—
59C2×C4×C199—22no—
60C2×C4×C199—22no—
61C2×C4×C199—21no—
62C2×C4×C199—41no—
63C2×C4×C199—41no—
64C2×C4×C199SD1612yescomplemented
65C2×C4×C199C2×D412yes—
66C22×C199—21no—
67C22×C199—21no—
68C22×C199#12011yescharacteristic
69C22×C199C2×SD1611yescharacteristic
70C22×C199#13711yescharacteristic
71C2×D199—42no—
72C2×D199—42no—
73C2×D199—42no—
74C2×D199—81no—
75C2×D199—81no—
76C2×D199—81no—
77C2×D199—81no—
78C4×C199—22no—
79C4×C199—22no—
80C4×C199—42no—
81C4×C199—41no—
82C4×C199—42no—
83Dic199—81no—
84C2×C199—21no—
85C2×C199—21no—
86C2×C199#44811yescharacteristic
87C2×C199#45411yescharacteristic
88C2×C199B(4,𝔽2)11yescharacteristic
89D199—82no—
90C199#159911yescharacteristic, complemented, Sylow
91#1599—1991nomaximal, Sylow
92#544D19911yescharacteristic, complemented
93#450—1992no—
94#394—1991no—
95#328—1992no—
96#324—1991no—
97C22×D4—1991no—
98C2×SD16—3982no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C2×D19912yes—
103#117—21no—
104C2×C4⋊C4C2×D19911yescharacteristic
105C2×C22⋊C4—3981no—
106C2×C22⋊C4—3981no—
107#102—3982no—
108C22⋊C8—1992no—
109#95—1991no—
110C24—3981no—
111C2×Q8—21no—
112C2×Q8C22⋊D19912yes—
113C2×D4—7962no—
114C2×D4—3982no—
115C22×C4—3981no—
116C22×C4C22⋊D19911yescharacteristic
117C22×C4C22⋊D19911yescharacteristic
118C22×C4C22×D19911yescharacteristic
119SD16—7964no—
120C2×C8—3982no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4C22⋊D19912yes—
128C22⋊C4—22no—
129C22⋊C4—7962no—
130C22⋊C4—7962no—
131C42—22no—
132C23—7961no—
133C23—7961no—
134C23—7961no—
135C23—7961no—
136C23—7961no—
137C23—7961no—
138C23—3982no—
139C23C2×C22⋊D19911yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—7962no—
143D4—3984no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—7961no—
152C2×C4—7961no—
153C2×C4—7961no—
154C2×C4#1569187312yescomplemented
155C2×C4C2×C22⋊D19912yes—
156C8—7962no—
157C22—21no—
158C22—21no—
159C22—7962no—
160C22—7962no—
161C22—7962no—
162C22—15921no—
163C22—15921no—
164C22—15921no—
165C22—15921no—
166C22#2026392411yescharacteristic
167C22C2×#1569187311yescharacteristic
168C22#2026398611yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—15921no—
175C2—21no—
176C2—21no—
177C2—15922no—
178C2#2716415711yescharacteristic
179C2#2716419111yescharacteristic
180C2#2716420511yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#544×C199C2
socleC22×C199#120
prime radicalC22×D796C22
hypercenter#544D199
hypocenterC199#1599
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #27163755C2semidirect
4. #27163744C2semidirect
5. #544×C199C2semidirect
64. C2×C4×C199SD16semidirect
90. C199#1599semidirect
92. #544D199semidirect
154. C2×C4#15691873semidirect

Maximal subgroups

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

subgroupindex
#271641682
#271639352
#271637552
#271637442
#544×C1992
#1599199