Finite Group Databasebeta

← #38768091

Subgroups — #38768091

Order 25472.

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

normal subgroups only

This group has 194 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#27164747C211yesmaximal, characteristic
2#27164421C211yesmaximal, characteristic
3#27164225C211yesmaximal, characteristic
4#27163763C211yesmaximal, characteristic
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic
6#27163748C211yesmaximal, characteristic
7#475×C199C211yesmaximal, characteristic
8C2×#15691876—21no—
9#20263955—21no—
10#20263955C2211yescharacteristic
11#20263946—22no—
12#20263946—21no—
13Q8×Dic199—21no—
14C4×Dic398—21no—
15C4×Dic398C2211yescharacteristic
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×Dic398D411yescharacteristic
27#15691876—42no—
28C2×C199⋊C8—21no—
29C2×C199⋊C8—21no—
30C4⋊Dic199—22no—
31C4⋊Dic199—22no—
32C4⋊Dic199—21no—
33C4⋊Dic199D412yes—
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—22no—
36Dic199⋊C4—21no—
37C4×Dic199—22no—
38C4×Dic199—21no—
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
48Dic398—21no—
49Dic398—21no—
50Dic398—41no—
51Dic398—41no—
52C2×Dic199—21no—
53C2×Dic199—21no—
54C2×Dic199—22no—
55C199⋊C8—22no—
56C199⋊C8—41no—
57Q8×C199—21no—
58Q8×C199—41no—
59Q8×C199—41no—
60Q8×C199Q1612yescomplemented
61C2×C4×C199—21no—
62C2×C4×C199—21no—
63C2×C4×C199—41no—
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C8×C199—41no—
68C22×C199#12011yescharacteristic
69C4×C199—22no—
70C4×C199—21no—
71C4×C199—41no—
72C4×C199—81no—
73C4×C199C2×Q1611yescharacteristic
74C4×C199#13511yescharacteristic
75C4×C199AGL(1,ℤ/8)11yescharacteristic
76C4×C199#13711yescharacteristic
77Dic199—41no—
78Dic199—42no—
79Dic199—41no—
80C2×C199#44911yescharacteristic
81C2×C199#45111yescharacteristic
82C2×C199#45611yescharacteristic
83C199#164811yescharacteristic, complemented, Sylow
84#1648—1991nomaximal, Sylow
85#557—1991no—
86#500—1991no—
87#475D19911yescharacteristic
88#462—1991no—
89#332—1991no—
90Q8⋊C8—1991no—
91Q8⋊C8—1991no—
92C2×Q16—3981no—
93C4⋊Q8—3981no—
94C4⋊Q8—1991no—
95C4⋊Q8C2×D19911yescharacteristic
96#125—3982no—
97#125—3981no—
98C4×Q8—3981no—
99C4×Q8—3981no—
100C4×Q8—1991no—
101C4×Q8C2×D19911yescharacteristic
102#107—3982no—
103#106—21no—
104C4⋊C8—1991no—
105C4⋊C8C2×D19911yescharacteristic
106#103—21no—
107#103—3981no—
108C4×C8—1991no—
109C2×Q8—21no—
110C2×Q8—3981no—
111C2×Q8—1991no—
112C2×Q8C22⋊D19911yescharacteristic
113Q16—7962no—
114C2×C8—21no—
115C2×C8—3981no—
116C2×C8—3981no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—3982no—
121C4⋊C4—3982no—
122C4⋊C4—3982no—
123C4⋊C4—3981no—
124C4⋊C4—3981no—
125C4⋊C4—1991no—
126C4⋊C4—1992no—
127C4⋊C4C22⋊D19911yescharacteristic
128C42—21no—
129C42—3982no—
130C42—3981no—
131C42C22×D19911yescharacteristic
132Q8—21no—
133Q8—41no—
134Q8—41no—
135Q8—7961no—
136Q8—7961no—
137Q8—3981no—
138Q8—3981no—
139Q8#1569187612yescomplemented
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—3981no—
144C2×C4—3981no—
145C2×C4—3982no—
146C2×C4D4×D19911yescharacteristic
147C2×C4D4×D19911yescharacteristic
148C2×C4C2×C22⋊D19911yescharacteristic
149C8—41no—
150C8—7961no—
151C8—3982no—
152C22#2026392311yescharacteristic
153C4—22no—
154C4—21no—
155C4—41no—
156C4—81no—
157C4—7962no—
158C4—7961no—
159C4—7961no—
160C4C2×#1569187611yescharacteristic
161C4#2026397611yescharacteristic
162C4#2026398211yescharacteristic
163C4#2026398511yescharacteristic
164C2#2716416211yescharacteristic
165C2#2716417311yescharacteristic
166C2#2716419911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#475×C199C2
socle, prime radicalC22×C199#120
hypercenter#475D199
hypocenterC199#1648
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
60. Q8×C199Q16semidirect
83. C199#1648semidirect
139. Q8#15691876semidirect

Maximal subgroups

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

subgroupindex
#271647472
#271644212
#271642252
#271637632
Q8⋊(C199⋊C8)2
#271637482
#475×C1992
#1648199