Finite Group Databasebeta

← #38768047

Subgroups — #38768047

Order 25472.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164688C211yesmaximal, characteristic
2#27164300C211yesmaximal, characteristic, complemented
3#27164220C211yesmaximal, characteristic, complemented
4#27163762C211yesmaximal, characteristic, complemented
5#27163750C211yesmaximal, characteristic, complemented
6#27163747C211yesmaximal, characteristic, complemented
7#486×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9C2×#15691873—21no—
10#20263951—21no—
11#20263927—22no—
12#20263926—21no—
13C4×Dic398C2211yescharacteristic
14C4×D796C2211yescharacteristic
15D4×Dic199—21no—
16#20263886—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19#20263879—21no—
20#20263878—21no—
21C4⋊Q8×C199C2211yescharacteristic
22#127×C199C2211yescharacteristic
23#102×C199—22no—
24C4×C8×C199C2211yescharacteristic
25#15691885—42no—
26#15691885—42no—
27C2×Dic398D411yescharacteristic
28C2×D796D411yescharacteristic
29C2×C22⋊D199—22no—
30C2×C22⋊D199—22no—
31C22×Dic199—22no—
32C2×C4×D199—21no—
33#15691873—42no—
34C2×C199⋊C8—21no—
35C2×C199⋊C8—21no—
36C4⋊Dic199D411yescharacteristic
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—21no—
39C22⋊Dic199—22no—
40(C2×D199)⋊C4—21no—
41C4×Dic199—21no—
42C2×Q8×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199—21no—
45C2×D4×C199D412yes—
46C2×C8×C199—21no—
47C4⋊C4×C199—21no—
48C4⋊C4×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51Dic398—21no—
52Dic398SD1612yescomplemented
53D796—21no—
54D796—21no—
55C22⋊D199—44no—
56C22⋊D199—82no—
57C2×Dic199—21no—
58C2×Dic199—21no—
59C2×Dic199—21no—
60C2×Dic199—44no—
61C4×D199—41no—
62C4×D199—41no—
63C199⋊C8—41no—
64C199⋊C8—41no—
65C23×C199—22no—
66Q8×C199—41no—
67Q8×C199—41no—
68D4×C199—42no—
69D4×C199—41no—
70D4×C199—41no—
71D4×C199—41no—
72D4×C199—41no—
73C2×C4×C199—41no—
74C2×C4×C199C2×D411yescharacteristic
75C2×C4×C199C2×D411yescharacteristic
76C2×C4×C199C2×D411yescharacteristic
77C8×C199—41no—
78C22×C199—42no—
79C22×C199—42no—
80C22×C199—42no—
81C22×C199#12011yescharacteristic
82C2×D199—41no—
83C2×D199—41no—
84C2×D199—41no—
85C4×C199—21no—
86C4×C199—81no—
87C4×C199C2×SD1611yescharacteristic
88C4×C199#13511yescharacteristic
89C4×C199AGL(1,ℤ/8)11yescharacteristic
90C4×C199AGL(1,ℤ/8)11yescharacteristic
91Dic199—22no—
92Dic199—41no—
93Dic199—41no—
94C2×C199—82no—
95C2×C199#44911yescharacteristic
96C2×C199#45011yescharacteristic
97C2×C199#45511yescharacteristic
98D199—81no—
99C199#164211yescharacteristic, complemented, Sylow
100#1642—1991nomaximal, Sylow
101#550—1991no—
102#486D19911yescharacteristic, complemented
103#475—1991no—
104#461—1991no—
105#331—1991no—
106Q8⋊C8—1991no—
107D4⋊C8—1991no—
108C2×#41—3981no—
109C2×SD16—3981no—
110C4⋊Q8C2×D19911yescharacteristic
111#127—3981no—
112#127C2×D19911yescharacteristic
113#121—3982no—
114#121—3981no—
115C4×Q8—1991no—
116C4×D4—3981no—
117C4×D4—1991no—
118#106—3981no—
119C4⋊C8—1991no—
120C4⋊C8—1991no—
121#103—3981no—
122#103—3981no—
123#102—22no—
124C4×C8C2×D19911yescharacteristic
125#41—7962no—
126#41—7962no—
127C2×Q8—21no—
128C2×Q8—1991no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3982no—
132C2×D4—3982no—
133C2×D4—1991no—
134C2×D4C22⋊D19912yes—
135C22×C4—3982no—
136C22×C4—3981no—
137SD16—7962no—
138C2×C8—21no—
139C2×C8—3981no—
140C2×C8—3981no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—3981no—
144C4⋊C4—1991no—
145C4⋊C4—1991no—
146C22⋊C4—3982no—
147C22⋊C4—3981no—
148C42—3981no—
149C42C22×D19911yescharacteristic
150C23—22no—
151C23—3981no—
152Q8—41no—
153Q8—41no—
154Q8—3981no—
155Q8—1992no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—7964no—
162D4—15922no—
163D4—3981no—
164D4—3981no—
165C2×C4—41no—
166C2×C4—7964no—
167C2×C4—7961no—
168C2×C4—7961no—
169C2×C4—3981no—
170C2×C4—3981no—
171C2×C4—3981no—
172C2×C4D4×D19911yescharacteristic
173C2×C4D4×D19911yescharacteristic
174C2×C4C2×C22⋊D19911yescharacteristic
175C8—41no—
176C8—7961no—
177C8—7961no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—7961no—
182C22—7961no—
183C22—7961no—
184C22#2026392311yescharacteristic
185C4—21no—
186C4—81no—
187C4—7961no—
188C4—7961no—
189C4—3982no—
190C4SD16×D19911yescharacteristic
191C4#2026397411yescharacteristic
192C4#2026397911yescharacteristic
193C4#2026398211yescharacteristic
194C2—82no—
195C2—15921no—
196C2#2716416011yescharacteristic
197C2#2716416711yescharacteristic
198C2#2716419511yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#486×C199C2
socleC22×C199#120
prime radical#27164688C2
hypercenter#486D199
hypocenterC199#1642
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #27164300C2semidirect
3. #27164220C2semidirect
4. #27163762C2semidirect
5. #27163750C2semidirect
6. #27163747C2semidirect
7. #486×C199C2semidirect
52. Dic398SD16semidirect
99. C199#1642semidirect
102. #486D199semidirect

Maximal subgroups

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

subgroupindex
#271646882
#271643002
#271642202
#271637622
#271637502
#271637472
#486×C1992
#1642199