Finite Group Databasebeta

← #38767994

Subgroups — #38767994

Order 25472.

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

normal subgroups only

This group has 218 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#27164397C211yesmaximal, characteristic, complemented
3#27164305C211yesmaximal, characteristic, complemented
4#27163766C211yesmaximal, characteristic, complemented
5D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#476×C199C211yesmaximal, characteristic, complemented
8C2×#15691876—22no—
9#20263955C2211yescharacteristic
10#20263941—22no—
11#20263929—21no—
12#20263926—21no—
13C4×D796C2211yescharacteristic
14#20263914—21no—
15#20263913—21no—
16#20263886—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18#20263875—21no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×D4×C199C2211yescharacteristic
22#106×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25C2×Dic398D412yes—
26C2×D796D411yescharacteristic
27C2×C22⋊D199—22no—
28C2×C4×D199—21no—
29#15691876—44no—
30C2×C199⋊C8—21no—
31C2×C199⋊C8—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199D411yescharacteristic
35C22⋊Dic199—22no—
36(C2×D199)⋊C4—22no—
37(C2×D199)⋊C4—21no—
38C4×Dic199—22no—
39C2×Q8×C199—21no—
40C2×D4×C199D411yescharacteristic
41C22×C4×C199—21no—
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—22no—
58C199⋊C8—41no—
59C23×C199—21no—
60Q8×C199—42no—
61D4×C199—21no—
62D4×C199—21no—
63C2×C4×C199—21no—
64C2×C4×C199—41no—
65C2×C4×C199—41no—
66C2×C4×C199—41no—
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C8×C199—41no—
71C22×C199—41no—
72C22×C199—41no—
73C22×C199—41no—
74C22×C199#12011yescharacteristic
75C2×D199—41no—
76C2×D199—41no—
77C2×D199—41no—
78C4×C199—21no—
79C4×C199—41no—
80C4×C199—81no—
81C4×C199#13511yescharacteristic
82C4×C199#13511yescharacteristic
83C4×C199#13711yescharacteristic
84C4×C199#13711yescharacteristic
85Dic199—41no—
86Dic199—42no—
87C2×C199—81no—
88C2×C199#45211yescharacteristic
89C2×C199#45211yescharacteristic
90C2×C199#45311yescharacteristic
91D199—81no—
92C199#163411yescharacteristic, complemented, Sylow
93#1634—1991nomaximal, Sylow
94#555—1991no—
95#494—1991no—
96#476—1991no—
97#476D19911yescharacteristic, complemented
98#333—1991no—
99D4⋊C8—1991no—
100D4⋊C8—1991no—
101C2×Q16—3982no—
102C4⋊Q8—1991no—
103C4⋊Q8C2×D19911yescharacteristic
104#124—3982no—
105#121—3981no—
106#121—3981no—
107C4×D4—1991no—
108C4×D4C2×D19911yescharacteristic
109#117—3981no—
110#117—3981no—
111#106—21no—
112#106—3981no—
113C4⋊C8—1991no—
114C4⋊C8C2×D19911yescharacteristic
115#102—21no—
116#102—3981no—
117C4×C8—1991no—
118C2×Q8—21no—
119C2×Q8—1992no—
120C2×D4—3982no—
121C2×D4—1991no—
122C2×D4C22⋊D19911yescharacteristic
123C22×C4—21no—
124C22×C4—3981no—
125Q16—7964no—
126C2×C8—21no—
127C2×C8—3981no—
128C2×C8—3981no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3981no—
132C4⋊C4—3981no—
133C4⋊C4—1991no—
134C4⋊C4C22⋊D19911yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—3982no—
137C22⋊C4—3982no—
138C22⋊C4—3981no—
139C42—3982no—
140C42C22×D19911yescharacteristic
141C23—21no—
142C23—3981no—
143Q8—42no—
144Q8—7962no—
145D4—21no—
146D4—21no—
147D4—15922no—
148D4—3981no—
149D4—3981no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—7961no—
155C2×C4—7961no—
156C2×C4—3981no—
157C2×C4—3982no—
158C2×C4D4×D19911yescharacteristic
159C2×C4D4×D19911yescharacteristic
160C2×C4C2×C22⋊D19911yescharacteristic
161C8—41no—
162C8—7961no—
163C8—3982no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—7961no—
168C22—7961no—
169C22—7961no—
170C22#2026392311yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—7961no—
175C4—7962no—
176C4#2026397511yescharacteristic
177C4#2026397611yescharacteristic
178C4#2026398411yescharacteristic
179C4#2026398611yescharacteristic
180C2—81no—
181C2—15921no—
182C2#2716417711yescharacteristic
183C2#2716417811yescharacteristic
184C2#2716418411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#476×C199C2
socleC22×C199#120
prime radical#27164733C2
hypercenter#476D199
hypocenterC199#1634
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #27164397C2semidirect
3. #27164305C2semidirect
4. #27163766C2semidirect
5. D4⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #476×C199C2semidirect
92. C199#1634semidirect
97. #476D199semidirect

Maximal subgroups

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

subgroupindex
#271647332
#271643972
#271643052
#271637662
D4⋊(C199⋊C8)2
#271637452
#476×C1992
#1634199