Finite Group Databasebeta

← #38767809

Subgroups — #38767809

Order 25472.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164644C211yesmaximal, characteristic
2#27164176C212yesmaximal, complemented
3#27163930C211yesmaximal, characteristic
4#27163756C212yesmaximal, complemented
5#323×C199C211yesmaximal, characteristic
6C22×Dic398C2211yescharacteristic
7C2×Dic796—22no—
8#20263934—21no—
9#20263934C2212yes—
10#20263932—22no—
11#20263932—22no—
12C2×C22⋊Dic199—21no—
13C2×C22⋊Dic199—21no—
14#20263880—22no—
15C199⋊#95—21no—
16C199⋊#95C2211yescharacteristic
17C2×C22⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C2×Dic398—21no—
20C2×Dic398—22no—
21C2×Dic398—42no—
22C2×Dic398D412yes—
23C22×Dic199—21no—
24C22×Dic199—21no—
25C22×Dic199—21no—
26C22×Dic199—21no—
27Dic796—44no—
28C4⋊Dic199—21no—
29C4⋊Dic199D412yes—
30Dic199⋊C4—22no—
31Dic199⋊C4—22no—
32C22⋊Dic199—22no—
33C22⋊Dic199—22no—
34C22⋊Dic199—42no—
35C24×C199D411yescharacteristic
36C22×C4×C199D411yescharacteristic
37C22×C4×C199C2311yescharacteristic
38C2×C8×C199—22no—
39C22⋊C4×C199—22no—
40Dic398—24no—
41Dic398—42no—
42Dic398—42no—
43Dic398—81no—
44C2×Dic199—22no—
45C2×Dic199—22no—
46C2×Dic199—22no—
47C2×Dic199—41no—
48C2×Dic199—41no—
49C2×Dic199—41no—
50C2×Dic199—41no—
51C2×Dic199—41no—
52C2×Dic199—81no—
53C2×Dic199—81no—
54C23×C199—21no—
55C23×C199—21no—
56C23×C199—41no—
57C23×C199—41no—
58C23×C199Q1612yescomplemented
59C23×C199C2×D411yescharacteristic
60C2×C4×C199—21no—
61C2×C4×C199—41no—
62C2×C4×C199—41no—
63C2×C4×C199C2×D412yes—
64C8×C199—42no—
65C22×C199—21no—
66C22×C199—21no—
67C22×C199—22no—
68C22×C199—42no—
69C22×C199—41no—
70C22×C199—41no—
71C22×C199—81no—
72C22×C199#12011yescharacteristic
73C22×C199C2×Q1611yescharacteristic
74C22×C199#13711yescharacteristic
75C4×C199—22no—
76C4×C199—41no—
77Dic199—42no—
78Dic199—42no—
79Dic199—42no—
80Dic199—81no—
81C2×C199—21no—
82C2×C199—21no—
83C2×C199—42no—
84C2×C199#45111yescharacteristic
85C2×C199#45511yescharacteristic
86C2×C199B(4,𝔽2)11yescharacteristic
87C199#160211yescharacteristic, complemented, Sylow
88#1602—1991nomaximal, Sylow
89#543—1991no—
90#451—1992no—
91#393—1991no—
92#328—1992no—
93#323D19911yescharacteristic
94C22×Q8—1991no—
95C2×Q16—3982no—
96C22⋊Q8—3982no—
97C22⋊Q8—3982no—
98C22⋊Q8—3981no—
99C22⋊Q8—1992no—
100C2×C22⋊C4—3981no—
101C2×C22⋊C4—3981no—
102C2×C22⋊C4C2×D19911yescharacteristic
103#103—3982no—
104C22⋊C8C2×D19912yes—
105#95—3981no—
106#95—1991no—
107C24D79611yescharacteristic
108C2×Q8—7962no—
109C2×Q8—3981no—
110C2×Q8—3982no—
111C2×Q8—1992no—
112C22×C4—3981no—
113C22×C4—3981no—
114C22×C4—3981no—
115C22×C4—3981no—
116C22×C4D79611yescharacteristic
117C22×C4C22×D19911yescharacteristic
118Q16—7964no—
119C2×C8—22no—
120C4⋊C4—3982no—
121C4⋊C4—3981no—
122C4⋊C4—3982no—
123C4⋊C4—1992no—
124C22⋊C4—22no—
125C22⋊C4—7962no—
126C22⋊C4—3982no—
127C22⋊C4—3982no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic79612yescomplemented
133C23C2×D79611yescharacteristic
134Q8—7962no—
135Q8—7962no—
136Q8—15921no—
137Q8—3984no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—7961no—
142C2×C4—7961no—
143C2×C4—7961no—
144C2×C4—7961no—
145C2×C4—7961no—
146C2×C4—15921no—
147C2×C4—15921no—
148C2×C4—3982no—
149C2×C4—3982no—
150C2×C4—3982no—
151C2×C4D4×D19912yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2026392511yescharacteristic
161C22C2×Dic79611yescharacteristic
162C22#2026398811yescharacteristic
163C4—22no—
164C4—41no—
165C4—7962no—
166C4—7962no—
167C4—7962no—
168C4—15921no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2716417611yescharacteristic
173C2#2716419811yescharacteristic
174C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#323×C199C2
socleC22×C199#120
prime radicalC24×C199D4
hypercenter#323D199
hypocenterC199#1602
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27164176C2semidirect
4. #27163756C2semidirect
58. C23×C199Q16semidirect
87. C199#1602semidirect
132. C23Dic796semidirect

Maximal subgroups

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

subgroupindex
#271646442
#271641762
#271639302
#271637562
#323×C1992
#1602199