Finite Group Databasebeta

← #38768076

Subgroups — #38768076

Order 25472.

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

normal subgroups only

This group has 212 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#27164733C211yesmaximal, characteristic
2#27164370C211yesmaximal, characteristic, complemented
3#27164295C211yesmaximal, characteristic, complemented
4#27163766C211yesmaximal, characteristic, complemented
5D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#461×C199C211yesmaximal, characteristic, complemented
8#20263955C2211yescharacteristic
9#20263941—22no—
10#20263929—21no—
11#20263926—21no—
12C4×D796C2211yescharacteristic
13#20263914—21no—
14#20263913—21no—
15#20263888—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263878—22no—
18#20263875—21no—
19C4×C199⋊C8C2211yescharacteristic
20C2×SD16×C199—21no—
21C4⋊Q8×C199C2211yescharacteristic
22C4×D4×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24#103×C199—21no—
25C2×Dic398D412yes—
26C2×D796D411yescharacteristic
27C2×C22⋊D199—22no—
28C2×C4×D199—21no—
29C2×C199⋊C8—21no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199D411yescharacteristic
34C22⋊Dic199—22no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—21no—
37C4×Dic199—22no—
38C2×Q8×C199—21no—
39C2×D4×C199D411yescharacteristic
40C22×C4×C199—21no—
41SD16×C199—42no—
42C2×C8×C199—21no—
43C4⋊C4×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C22⋊C4×C199—21no—
47C42×C199C2311yescharacteristic
48C22×D199—21no—
49Dic398—42no—
50D796—21no—
51D796—21no—
52C22⋊D199—82no—
53C2×Dic199—21no—
54C2×Dic199—22no—
55C4×D199—41no—
56C4×D199—41no—
57C199⋊C8—41no—
58C199⋊C8—41no—
59C23×C199—21no—
60Q8×C199—41no—
61Q8×C199—41no—
62D4×C199—21no—
63D4×C199—21no—
64C2×C4×C199—21no—
65C2×C4×C199—41no—
66C2×C4×C199—41no—
67C2×C4×C199—41no—
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C8×C199—41no—
72C22×C199—41no—
73C22×C199—41no—
74C22×C199—41no—
75C22×C199#12011yescharacteristic
76C2×D199—41no—
77C2×D199—41no—
78C2×D199—41no—
79C4×C199—21no—
80C4×C199—41no—
81C4×C199—81no—
82C4×C199#13511yescharacteristic
83C4×C199#13511yescharacteristic
84C4×C199AGL(1,ℤ/8)11yescharacteristic
85C4×C199#13711yescharacteristic
86Dic199—41no—
87Dic199—42no—
88C2×C199—81no—
89C2×C199#44911yescharacteristic
90C2×C199#45211yescharacteristic
91C2×C199#45511yescharacteristic
92D199—81no—
93C199#164611yescharacteristic, complemented, Sylow
94#1646—1991nomaximal, Sylow
95#555—1991no—
96#487—1991no—
97#474—1991no—
98#461D19911yescharacteristic, complemented
99#333—1991no—
100D4⋊C8—1991no—
101D4⋊C8—1991no—
102C2×SD16—21no—
103C4⋊Q8—1991no—
104C4⋊Q8C2×D19911yescharacteristic
105#124—3982no—
106#121—3981no—
107#121—3981no—
108C4×D4—1991no—
109C4×D4C2×D19911yescharacteristic
110#117—3981no—
111#117—3981no—
112#107—3981no—
113C4⋊C8—1991no—
114C4⋊C8C2×D19911yescharacteristic
115#103—21no—
116#103—3982no—
117#102—3981no—
118C4×C8—1991no—
119C2×Q8—21no—
120C2×Q8—1992no—
121C2×D4—3982no—
122C2×D4—1991no—
123C2×D4C22⋊D19911yescharacteristic
124C22×C4—21no—
125C22×C4—3981no—
126SD16—42no—
127C2×C8—21no—
128C2×C8—3981no—
129C2×C8—3981no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3981no—
133C4⋊C4—3981no—
134C4⋊C4—1991no—
135C4⋊C4C22⋊D19911yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3982no—
138C22⋊C4—3982no—
139C22⋊C4—3981no—
140C42—3982no—
141C42C22×D19911yescharacteristic
142C23—21no—
143C23—3981no—
144Q8—41no—
145Q8—41no—
146Q8—7962no—
147D4—21no—
148D4—21no—
149D4—15922no—
150D4—3981no—
151D4—3981no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—7961no—
157C2×C4—7961no—
158C2×C4—3981no—
159C2×C4—3982no—
160C2×C4D4×D19911yescharacteristic
161C2×C4D4×D19911yescharacteristic
162C2×C4C2×C22⋊D19911yescharacteristic
163C8—41no—
164C8—7961no—
165C8—7961no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—7961no—
170C22—7961no—
171C22—7961no—
172C22#2026392311yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—7961no—
177C4—7962no—
178C4#2026397511yescharacteristic
179C4#2026397611yescharacteristic
180C4#2026398111yescharacteristic
181C4#2026398411yescharacteristic
182C2—81no—
183C2—15921no—
184C2#2716416111yescharacteristic
185C2#2716417711yescharacteristic
186C2#2716419411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#461×C199C2
socleC22×C199#120
prime radical#27164733C2
hypercenter#461D199
hypocenterC199#1646
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #27164370C2semidirect
3. #27164295C2semidirect
4. #27163766C2semidirect
5. D4⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #461×C199C2semidirect
93. C199#1646semidirect
98. #461D199semidirect

Maximal subgroups

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

subgroupindex
#271647332
#271643702
#271642952
#271637662
D4⋊(C199⋊C8)2
#271637452
#461×C1992
#1646199