Finite Group Databasebeta

← #38767957

Subgroups — #38767957

Order 25472.

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

normal subgroups only

This group has 216 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#27164742C211yesmaximal, characteristic, complemented
2#27164215C211yesmaximal, characteristic, complemented
3#27164210C211yesmaximal, characteristic
4#27163764C211yesmaximal, characteristic, complemented
5#27163750C211yesmaximal, characteristic, complemented
6#27163747C211yesmaximal, characteristic, complemented
7#486×C199C211yesmaximal, characteristic, complemented
8C2×#15691874—21no—
9C2×#15691871—21no—
10#20263950—22no—
11#20263944—21no—
12#20263935—21no—
13Q8×Dic199—21no—
14C4×Dic398C2211yescharacteristic
15C4×D796C2211yescharacteristic
16#20263913—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19#20263876—21no—
20#20263876—21no—
21C4⋊Q8×C199C2211yescharacteristic
22#127×C199C2211yescharacteristic
23#102×C199—22no—
24C4×C8×C199C2211yescharacteristic
25C2×Dic398D411yescharacteristic
26C2×D796D411yescharacteristic
27C2×C4×D199—21no—
28#15691874—42no—
29#15691871—42no—
30C2×C199⋊C8—21no—
31C2×C199⋊C8—21no—
32C4⋊Dic199—22no—
33C4⋊Dic199D411yescharacteristic
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—21no—
36(C2×D199)⋊C4—22no—
37(C2×D199)⋊C4—22no—
38(C2×D199)⋊C4—21no—
39C4×Dic199—22no—
40C4×Dic199—21no—
41C2×Q8×C199—21no—
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199—21no—
45C2×C8×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D412yes—
48C42×C199C2311yescharacteristic
49C22×D199—21no—
50Dic398—21no—
51Dic398—21no—
52D796—21no—
53D796—21no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C2×Dic199—21no—
57C4×D199—41no—
58C4×D199—41no—
59C199⋊C8—41no—
60C199⋊C8—41no—
61C23×C199—41no—
62Q8×C199—41no—
63Q8×C199—41no—
64D4×C199—42no—
65D4×C199—41no—
66D4×C199—41no—
67D4×C199—41no—
68D4×C199—41no—
69C2×C4×C199—22no—
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C8×C199—41no—
74C22×C199—81no—
75C22×C199—81no—
76C22×C199—81no—
77C22×C199#12011yescharacteristic
78C2×D199—41no—
79C2×D199—41no—
80C2×D199—41no—
81C4×C199—21no—
82C4×C199—42no—
83C4×C199#13511yescharacteristic
84C4×C199#13511yescharacteristic
85C4×C199AGL(1,ℤ/8)11yescharacteristic
86C4×C199AGL(1,ℤ/8)11yescharacteristic
87Dic199—41no—
88Dic199—41no—
89Dic199—41no—
90C2×C199—161no—
91C2×C199#44911yescharacteristic
92C2×C199#44911yescharacteristic
93C2×C199#45411yescharacteristic
94D199—81no—
95C199#162811yescharacteristic, complemented, Sylow
96#1628—1991nomaximal, Sylow
97#556—1991no—
98#486D19911yescharacteristic, complemented
99#460—1991no—
100#459—1991no—
101#332—1991no—
102Q8⋊C8—1991no—
103D4⋊C8—1991no—
104C2×SD16—3981no—
105C2×D8—3981no—
106C4⋊Q8C2×D19911yescharacteristic
107#127C2×D19911yescharacteristic
108#126—3982no—
109#124—3981no—
110C22⋊Q8—3981no—
111C4×Q8—3981no—
112C4×Q8—1991no—
113C4×D4—1991no—
114#117—3981no—
115C4⋊C8—1991no—
116C4⋊C8—1991no—
117#102—22no—
118#102—3981no—
119#102—3981no—
120C4×C8C2×D19911yescharacteristic
121C2×Q8—21no—
122C2×Q8—1991no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1991no—
127C22×C4—3981no—
128SD16—7962no—
129D8—7962no—
130C2×C8—21no—
131C2×C8—3981no—
132C2×C8—3981no—
133C4⋊C4—21no—
134C4⋊C4—3982no—
135C4⋊C4—3981no—
136C4⋊C4—1991no—
137C4⋊C4—1991no—
138C4⋊C4C22⋊D19912yes—
139C22⋊C4—3982no—
140C22⋊C4—3982no—
141C22⋊C4—3981no—
142C42—3982no—
143C42—3981no—
144C42C22×D19911yescharacteristic
145C23—41no—
146C23—3981no—
147Q8—41no—
148Q8—41no—
149Q8—3981no—
150Q8—3981no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—3981no—
157D4—3981no—
158C2×C4—22no—
159C2×C4—7961no—
160C2×C4—7961no—
161C2×C4—3981no—
162C2×C4—3981no—
163C2×C4—3981no—
164C2×C4D4×D19911yescharacteristic
165C2×C4D4×D19911yescharacteristic
166C2×C4C2×C22⋊D19911yescharacteristic
167C8—41no—
168C8—7961no—
169C8—7961no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—7961no—
174C22—7961no—
175C22—7961no—
176C22#2026392311yescharacteristic
177C4—21no—
178C4—42no—
179C4—7961no—
180C4—7961no—
181C4—7961no—
182C4#2026397411yescharacteristic
183C4#2026397611yescharacteristic
184C4#2026397911yescharacteristic
185C4#2026398211yescharacteristic
186C2—161no—
187C2—15921no—
188C2#2716416011yescharacteristic
189C2#2716416211yescharacteristic
190C2#2716419011yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#486×C199C2
socleC22×C199#120
prime radical#27164210C2
hypercenter#486D199
hypocenterC199#1628
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #27164742C2semidirect
2. #27164215C2semidirect
4. #27163764C2semidirect
5. #27163750C2semidirect
6. #27163747C2semidirect
7. #486×C199C2semidirect
95. C199#1628semidirect
98. #486D199semidirect

Maximal subgroups

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

subgroupindex
#271647422
#271642152
#271642102
#271637642
#271637502
#271637472
#486×C1992
#1628199