Finite Group Databasebeta

← #38766846

Subgroups — #38766846

Order 25472.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 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#27163927C211yesmaximal, characteristic
2#27163742C211yesmaximal, characteristic, complemented
3#342×C199C211yesmaximal, characteristic, complemented
4C22×D796C411yescharacteristic, complemented
5C22×C22⋊D199—21no—
6C2×(C2×D199)⋊C4—21no—
7C22⋊(C199⋊C8)—21no—
8C199⋊#95C411yescharacteristic, complemented
9C2×C22⋊C4×C199—21no—
10C2×C22⋊C4×C199C2211yescharacteristic
11C23×D199—21no—
12C2×D796—41no—
13C2×D796—42no—
14C2×C22⋊D199—42no—
15C2×C22⋊D199—42no—
16C2×C22⋊D199—41no—
17C2×C22⋊D199—41no—
18C22×Dic199—21no—
19C2×C199⋊C8—41no—
20(C2×D199)⋊C4—42no—
21C24×C199D411yescharacteristic
22C22×C4×C199—41no—
23C22×C4×C199C2×C411yescharacteristic
24C22×C4×C199D411yescharacteristic
25C22⋊C4×C199—22no—
26C22⋊C4×C199—41no—
27C22⋊C4×C199—41no—
28C22×D199—22no—
29C22×D199—41no—
30C22×D199—41no—
31C22×D199—41no—
32C22×D199—41no—
33C22×D199—41no—
34C22×D199—41no—
35D796—42no—
36D796—42no—
37C22⋊D199—44no—
38C22⋊D199—44no—
39C2×Dic199—22no—
40C2×Dic199—41no—
41C2×Dic199—41no—
42C199⋊C8—81no—
43C23×C199—21no—
44C23×C199—21no—
45C23×C199—41no—
46C23×C199—41no—
47C23×C199C22⋊C411yescharacteristic
48C23×C199D811yescharacteristic
49C23×C199SD1611yescharacteristic
50C2×C4×C199—21no—
51C2×C4×C199—21no—
52C2×C4×C199—41no—
53C2×C4×C199—41no—
54C2×C4×C199—42no—
55C2×C4×C199—81no—
56C2×C4×C199—81no—
57C22×C199—21no—
58C22×C199—21no—
59C22×C199—21no—
60C22×C199—21no—
61C22×C199—41no—
62C22×C199—41no—
63C22×C199—41no—
64C22×C199—41no—
65C22×C199—81no—
66C22×C199#9911yescharacteristic
67C22×C199#10211yescharacteristic
68C22×C199#10411yescharacteristic
69C2×D199—42no—
70C2×D199—42no—
71C2×D199—42no—
72C2×D199—81no—
73C2×D199—81no—
74C2×D199—81no—
75C2×D199—81no—
76C4×C199—41no—
77C4×C199—41no—
78C4×C199—82no—
79Dic199—42no—
80C2×C199—21no—
81C2×C199—21no—
82C2×C199—41no—
83C2×C199—41no—
84C2×C199#32711yescharacteristic
85C2×C199AΣL(1,𝔽16)11yescharacteristic
86C2×C199#35311yescharacteristic
87D199—82no—
88C199#134311yescharacteristic, complemented, Sylow
89#1343—1991nomaximal, Sylow
90#392—1991no—
91#342D19911yescharacteristic, complemented
92#323—1991no—
93C22×D4—3981no—
94C22×D4—1991no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—3981no—
97C2×C22⋊C4C2×D19911yescharacteristic
98C22⋊C8—3981no—
99#95—1991no—
100C24—3981no—
101C24C22⋊D19911yescharacteristic
102C2×D4—7961no—
103C2×D4—7962no—
104C2×D4—7962no—
105C2×D4—7961no—
106C2×D4—7961no—
107C2×D4—7962no—
108C22×C4—41no—
109C22×C4—3981no—
110C22×C4C4×D19911yescharacteristic
111C22×C4D79611yescharacteristic
112C2×C8—7961no—
113C22⋊C4—22no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—7962no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—7961no—
122C23—7961no—
123C23—7961no—
124C23—7961no—
125C23—7961no—
126C23—7961no—
127C23—3982no—
128C23(C2×D199)⋊C411yescharacteristic
129C23#1569187111yescharacteristic
130C23#1569187311yescharacteristic
131D4—7962no—
132D4—7962no—
133D4—7964no—
134D4—7964no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—7961no—
143C2×C4—7961no—
144C2×C4—3982no—
145C8—15921no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—7962no—
156C22—7962no—
157C22—7962no—
158C22—15921no—
159C22—15921no—
160C22—15921no—
161C22—15921no—
162C22#2026386911yescharacteristic
163C22#2026387511yescharacteristic
164C22#2026388311yescharacteristic
165C4—41no—
166C4—41no—
167C4—82no—
168C4—7962no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—15922no—
174C2#2716375111yescharacteristic
175C2#2716380411yescharacteristic
176C2#2716381011yescharacteristic
subgroupG/H
centerC22#20263869
commutatorC22×C4×C199C2×C4
FrattiniC2×C22⋊C4C2×D199
Fitting#342×C199C2
socleC22×C199#99
prime radical#27163927C2
cosocleC2×C22⋊C4×C199C22
hypercenter#342D199
hypocenterC199#1343
LCSG ▷ C22×C4×C199 ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #27163742C2semidirect
3. #342×C199C2semidirect
4. C22×D796C4semidirect
8. C199⋊#95C4semidirect
88. C199#1343semidirect
91. #342D199semidirect

Maximal subgroups

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

subgroupindex
#271639272
#271637422
#342×C1992
#1343199