Finite Group Databasebeta

← #38768105

Subgroups — #38768105

Order 25472.

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

normal subgroups only

This group has 200 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#27164741C211yesmaximal, characteristic
2#27164421C211yesmaximal, characteristic, complemented
3#27164220C211yesmaximal, characteristic
4#27163763C211yesmaximal, characteristic, complemented
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic
7#475×C199C211yesmaximal, characteristic, complemented
8C2×#15691873—21no—
9#20263955C2211yescharacteristic
10#20263950—22no—
11#20263943—21no—
12#20263931—21no—
13C4×Dic398—21no—
14C4×D796C2211yescharacteristic
15#20263913—21no—
16#20263888—22no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18#20263879—21no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×Q8×C199C2211yescharacteristic
22#106×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#103×C199—21no—
25C2×Dic398—21no—
26C2×D796D411yescharacteristic
27C2×C4×D199—21no—
28#15691873—42no—
29C2×C199⋊C8—21no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—21no—
32C4⋊Dic199D412yes—
33C4⋊Dic199D411yescharacteristic
34Dic199⋊C4—22no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—22no—
37(C2×D199)⋊C4—21no—
38C4×Dic199—22no—
39C2×Q8×C199—21no—
40C2×Q8×C199D411yescharacteristic
41C2×C8×C199—21no—
42C4⋊C4×C199—21no—
43C4⋊C4×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C42×C199—21no—
47C42×C199C2311yescharacteristic
48C22×D199—21no—
49Dic398—41no—
50Dic398—41no—
51D796—21no—
52D796—21no—
53C2×Dic199—21no—
54C2×Dic199—22no—
55C4×D199—41no—
56C4×D199—41no—
57C199⋊C8—22no—
58C199⋊C8—41no—
59Q8×C199—21no—
60Q8×C199—21no—
61Q8×C199—41no—
62Q8×C199—41no—
63C2×C4×C199—21no—
64C2×C4×C199—21no—
65C2×C4×C199—41no—
66C2×C4×C199C2×D411yescharacteristic
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C8×C199—41no—
70C22×C199#12011yescharacteristic
71C2×D199—41no—
72C2×D199—41no—
73C2×D199—41no—
74C4×C199—21no—
75C4×C199—41no—
76C4×C199—41no—
77C4×C199—81no—
78C4×C199#13511yescharacteristic
79C4×C199#13511yescharacteristic
80C4×C199AGL(1,ℤ/8)11yescharacteristic
81C4×C199#13711yescharacteristic
82Dic199—42no—
83Dic199—41no—
84C2×C199#44911yescharacteristic
85C2×C199#45211yescharacteristic
86C2×C199#45611yescharacteristic
87D199—81no—
88C199#165011yescharacteristic, complemented, Sylow
89#1650—1991nomaximal, Sylow
90#556—1991no—
91#500—1991no—
92#475D19911yescharacteristic, complemented
93#461—1991no—
94#332—1991no—
95Q8⋊C8—1991no—
96D4⋊C8—1991no—
97C2×SD16—3981no—
98C4⋊Q8—1991no—
99C4⋊Q8C2×D19911yescharacteristic
100#126—3982no—
101#124—3981no—
102C22⋊Q8—3981no—
103C4×Q8—3981no—
104C4×Q8C2×D19911yescharacteristic
105C4×D4—1991no—
106#117—3981no—
107#107—3982no—
108#106—21no—
109C4⋊C8—1991no—
110C4⋊C8C2×D19911yescharacteristic
111#103—21no—
112#103—3981no—
113C4×C8—1991no—
114C2×Q8—21no—
115C2×Q8—3981no—
116C2×Q8C22⋊D19911yescharacteristic
117C2×D4—1991no—
118C22×C4—3981no—
119SD16—7962no—
120C2×C8—21no—
121C2×C8—3981no—
122C2×C8—3981no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3982no—
127C4⋊C4—3981no—
128C4⋊C4—1991no—
129C4⋊C4—1992no—
130C4⋊C4C22⋊D19911yescharacteristic
131C22⋊C4—3982no—
132C22⋊C4—3982no—
133C22⋊C4—3981no—
134C42—21no—
135C42—3982no—
136C42C22×D19911yescharacteristic
137C23—3981no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—7961no—
143Q8—7961no—
144D4—3981no—
145D4—3981no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—7961no—
150C2×C4—7961no—
151C2×C4—3981no—
152C2×C4—3982no—
153C2×C4D4×D19911yescharacteristic
154C2×C4D4×D19911yescharacteristic
155C2×C4C2×C22⋊D19911yescharacteristic
156C8—41no—
157C8—7961no—
158C8—3982no—
159C22—7961no—
160C22—7961no—
161C22—7961no—
162C22#2026392311yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—7962no—
168C4—7961no—
169C4#2026397511yescharacteristic
170C4#2026397611yescharacteristic
171C4#2026398211yescharacteristic
172C4#2026398611yescharacteristic
173C2—15921no—
174C2#2716416211yescharacteristic
175C2#2716417811yescharacteristic
176C2#2716419911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#475×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#475D199
hypocenterC199#1650
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #27164421C2semidirect
4. #27163763C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
7. #475×C199C2semidirect
88. C199#1650semidirect
92. #475D199semidirect

Maximal subgroups

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

subgroupindex
#271647412
#271644212
#271642202
#271637632
Q8⋊(C199⋊C8)2
#271637452
#475×C1992
#1650199