Finite Group Databasebeta

← #38767908

Subgroups — #38767908

Order 25472.

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

normal subgroups only

This group has 240 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#27164753C211yesmaximal, characteristic, complemented
2#27164394C211yesmaximal, characteristic
3#27164215C211yesmaximal, characteristic, complemented
4#27163761C211yesmaximal, characteristic, complemented
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163748C211yesmaximal, characteristic, complemented
7#460×C199C211yesmaximal, characteristic, complemented
8C2×#15691874—21no—
9C2×#15691871—22no—
10#20263952C2211yescharacteristic
11#20263944—21no—
12#20263943—21no—
13#20263940—22no—
14C4×Dic398C2211yescharacteristic
15C4×D796—21no—
16#20263917—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18#20263876—21no—
19C4×C199⋊C8C2211yescharacteristic
20C2×SD16×C199—21no—
21#127×C199C2211yescharacteristic
22C4×Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25C2×Dic398D411yescharacteristic
26C2×D796—21no—
27C2×D796—21no—
28C2×D796D412yes—
29C2×C4×D199—22no—
30#15691874—42no—
31#15691871—44no—
32C2×C199⋊C8—21no—
33C2×C199⋊C8—21no—
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—21no—
36(C2×D199)⋊C4—22no—
37(C2×D199)⋊C4—22no—
38(C2×D199)⋊C4—22no—
39C4×Dic199—21no—
40C2×Q8×C199D411yescharacteristic
41C2×D4×C199—21no—
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44SD16×C199—42no—
45C2×C8×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D411yescharacteristic
48C42×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—22no—
51Dic398—21no—
52Dic398—21no—
53D796—42no—
54D796—41no—
55D796—41no—
56D796—41no—
57D796—41no—
58C2×Dic199—21no—
59C2×Dic199—21no—
60C4×D199—82no—
61C199⋊C8—22no—
62C199⋊C8—41no—
63C23×C199—41no—
64Q8×C199—21no—
65Q8×C199—21no—
66D4×C199—42no—
67D4×C199—41no—
68D4×C199—41no—
69D4×C199—41no—
70D4×C199—41no—
71C2×C4×C199—21no—
72C2×C4×C199—21no—
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C2×C4×C199C2×D411yescharacteristic
76C8×C199—41no—
77C22×C199—81no—
78C22×C199—81no—
79C22×C199—81no—
80C22×C199#12011yescharacteristic
81C2×D199—42no—
82C2×D199—42no—
83C2×D199—42no—
84C4×C199—21no—
85C4×C199—41no—
86C4×C199—41no—
87C4×C199#13511yescharacteristic
88C4×C199#13511yescharacteristic
89C4×C199AGL(1,ℤ/8)11yescharacteristic
90C4×C199AGL(1,ℤ/8)11yescharacteristic
91Dic199—41no—
92Dic199—41no—
93C2×C199—161no—
94C2×C199#44911yescharacteristic
95C2×C199#44911yescharacteristic
96C2×C199#45311yescharacteristic
97D199—82no—
98C199#162011yescharacteristic, complemented, Sylow
99#1620—1991nomaximal, Sylow
100#559—1991no—
101#493—1991no—
102#460—1991no—
103#460D19911yescharacteristic, complemented
104#331—1991no—
105Q8⋊C8—1991no—
106Q8⋊C8—1991no—
107C2×SD16—21no—
108C2×SD16—3981no—
109C2×D8—3982no—
110#127—1991no—
111#127C2×D19911yescharacteristic
112#124—3981no—
113#124—3981no—
114#123—3982no—
115C4×Q8—1991no—
116C4×Q8C2×D19911yescharacteristic
117C4×D4—3981no—
118C4×D4—3981no—
119C4⋊C8—1991no—
120C4⋊C8C2×D19911yescharacteristic
121#102—21no—
122#102—3981no—
123C4×C8—1991no—
124C2×Q8—1991no—
125C2×Q8C22⋊D19911yescharacteristic
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—3981no—
130C2×D4—3981no—
131C2×D4—1992no—
132C22×C4—3982no—
133SD16—42no—
134SD16—7962no—
135D8—7964no—
136C2×C8—21no—
137C2×C8—3981no—
138C2×C8—3981no—
139C4⋊C4—21no—
140C4⋊C4—3981no—
141C4⋊C4—1991no—
142C4⋊C4C22⋊D19911yescharacteristic
143C22⋊C4—3982no—
144C22⋊C4—3982no—
145C22⋊C4—3982no—
146C42—21no—
147C42—3981no—
148C42C22×D19911yescharacteristic
149C23—41no—
150C23—3982no—
151Q8—21no—
152Q8—21no—
153Q8—3981no—
154Q8—3981no—
155D4—42no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—7962no—
161D4—7961no—
162D4—7961no—
163D4—7961no—
164D4—7961no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—15922no—
168C2×C4—3981no—
169C2×C4—3981no—
170C2×C4D4×D19911yescharacteristic
171C2×C4D4×D19911yescharacteristic
172C2×C4C2×C22⋊D19911yescharacteristic
173C8—41no—
174C8—7961no—
175C8—3982no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—7962no—
180C22—7962no—
181C22—7962no—
182C22#2026392311yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—7961no—
187C4—7961no—
188C4#2026397511yescharacteristic
189C4#2026397611yescharacteristic
190C4#2026398111yescharacteristic
191C4#2026398211yescharacteristic
192C2—161no—
193C2—15922no—
194C2#2716416111yescharacteristic
195C2#2716416211yescharacteristic
196C2#2716418411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#460×C199C2
socleC22×C199#120
prime radical#27164394C2
hypercenter#460D199
hypocenterC199#1620
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #27164753C2semidirect
3. #27164215C2semidirect
4. #27163761C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
6. #27163748C2semidirect
7. #460×C199C2semidirect
98. C199#1620semidirect
103. #460D199semidirect

Maximal subgroups

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

subgroupindex
#271647532
#271643942
#271642152
#271637612
Q8⋊(C199⋊C8)2
#271637482
#460×C1992
#1620199