Finite Group Databasebeta

← #38767870

Subgroups — #38767870

Order 25472.

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

normal subgroups only

This group has 288 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#27164334C212yesmaximal, complemented
2#27163954C211yesmaximal, characteristic, complemented
3#27163752C212yesmaximal, complemented
4#27163742C211yesmaximal, characteristic, complemented
5#534×C199C211yesmaximal, characteristic
6C2×C4⋊Dic199C2211yescharacteristic, complemented
7C2×Dic199⋊C4—21no—
8C2×C22⋊Dic199—21no—
9#20263886—22no—
10#20263876—22no—
11C22⋊(C199⋊C8)C2212yescomplemented
12C199⋊#95C2211yescharacteristic, complemented
13C22×D4×C199—21no—
14#123×C199—22no—
15#121×C199—21no—
16#121×C199C2212yes—
17#120×C199—22no—
18C2×C22⋊C4×C199C2211yescharacteristic
19C22×Dic199—21no—
20C22×Dic199—21no—
21C2×C199⋊C8—22no—
22C4⋊Dic199—22no—
23Dic199⋊C4—42no—
24C22⋊Dic199—42no—
25C24×C199—21no—
26C24×C199D411yescharacteristic
27C2×D4×C199—22no—
28C2×D4×C199—22no—
29C2×D4×C199—21no—
30C2×D4×C199—42no—
31C2×D4×C199—42no—
32C2×D4×C199D412yes—
33C22×C4×C199—21no—
34C22×C4×C199D411yescharacteristic
35C22×C4×C199C2311yescharacteristic
36C4⋊C4×C199—21no—
37C4⋊C4×C199D412yes—
38C22⋊C4×C199—22no—
39C22⋊C4×C199—22no—
40C22⋊C4×C199—22no—
41C2×Dic199—22no—
42C2×Dic199—41no—
43C2×Dic199—41no—
44C2×Dic199—41no—
45C2×Dic199—41no—
46C2×Dic199—41no—
47C199⋊C8—42no—
48C23×C199—21no—
49C23×C199—21no—
50C23×C199—22no—
51C23×C199—41no—
52C23×C199—41no—
53C23×C199—42no—
54C23×C199—81no—
55C23×C199—81no—
56C23×C199SD1612yes—
57C23×C199C2×D411yescharacteristic
58D4×C199—42no—
59D4×C199—44no—
60D4×C199—42no—
61D4×C199—42no—
62D4×C199—81no—
63C2×C4×C199—21no—
64C2×C4×C199—22no—
65C2×C4×C199—41no—
66C2×C4×C199—42no—
67C2×C4×C199—41no—
68C2×C4×C199C2×D412yes—
69C22×C199—21no—
70C22×C199—21no—
71C22×C199—22no—
72C22×C199—42no—
73C22×C199—41no—
74C22×C199—41no—
75C22×C199—42no—
76C22×C199—42no—
77C22×C199—42no—
78C22×C199—81no—
79C22×C199—84no—
80C22×C199#12011yescharacteristic
81C22×C199C2×SD1611yescharacteristic
82C22×C199AGL(1,ℤ/8)11yescharacteristic
83C4×C199—22no—
84C4×C199—42no—
85C4×C199—41no—
86Dic199—42no—
87Dic199—81no—
88C2×C199—21no—
89C2×C199—21no—
90C2×C199—42no—
91C2×C199—82no—
92C2×C199#45011yescharacteristic
93C2×C199#45311yescharacteristic
94C2×C199#45811yescharacteristic
95C199#161311yescharacteristic, complemented, Sylow
96#1613—1991nomaximal, Sylow
97#534D19911yescharacteristic
98#481—1992no—
99#399—1991no—
100#327—1992no—
101#323—1991no—
102C22×D4—21no—
103#123—22no—
104#121—21no—
105#121C2×D19912yes—
106#120—22no—
107C2×C4⋊C4—3981no—
108C2×C4⋊C4—1991no—
109C2×C22⋊C4—3981no—
110C2×C22⋊C4C2×D19911yescharacteristic
111#106—3982no—
112#102—3982no—
113C22⋊C8—1992no—
114#95—1991no—
115C24—21no—
116C24C22⋊D19911yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C22⋊D19912yes—
123C22×C4—21no—
124C22×C4—3981no—
125C22×C4—3981no—
126C22×C4C22⋊D19911yescharacteristic
127C22×C4C22×D19911yescharacteristic
128C2×C8—3982no—
129C4⋊C4—21no—
130C4⋊C4—7962no—
131C4⋊C4—3982no—
132C4⋊C4C22⋊D19912yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—7962no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#1569187412yes—
146C23C2×C22⋊D19911yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—7961no—
158C2×C4—7961no—
159C2×C4—7961no—
160C2×C4—7961no—
161C2×C4—7961no—
162C2×C4—3982no—
163C2×C4C2×C22⋊D19912yes—
164C8—7962no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#2026392411yescharacteristic
177C22C2×#1569187411yescharacteristic
178C22#2026398111yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—7962no—
183C4—15921no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#2716416911yescharacteristic
189C2#2716418611yescharacteristic
190C2#2716420811yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting, prime radical#534×C199C2
socleC22×C199#120
hypercenter#534D199
hypocenterC199#1613
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #27164334C2semidirect
2. #27163954C2semidirect
3. #27163752C2semidirect
4. #27163742C2semidirect
6. C2×C4⋊Dic199C22semidirect
11. C22⋊(C199⋊C8)C22semidirect
12. C199⋊#95C22semidirect
95. C199#1613semidirect

Maximal subgroups

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

subgroupindex
#271643342
#271639542
#271637522
#271637422
#534×C1992
#1613199