Finite Group Databasebeta

← #38770799

Subgroups — #38770799

Order 25472.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#27164815C211yesmaximal, characteristic
2#27164459C211yesmaximal, characteristic, complemented
3#27164273C211yesmaximal, characteristic, complemented
4#27163829C211yesmaximal, characteristic, complemented
5#27163828C211yesmaximal, characteristic, complemented
6#27163799C211yesmaximal, characteristic, complemented
7#510×C199C211yesmaximal, characteristic, complemented
8#41×D199—21no—
9#20263987—21no—
10#20263984—21no—
11#20263982—21no—
12#20263979—21no—
13#20263977C2211yescharacteristic
14#20263975—21no—
15C2×#15691869C2211yescharacteristic
16#20263931—21no—
17#20263901—21no—
18#20263898—21no—
19#20263895C2211yescharacteristic, complemented
20#20263887C2211yescharacteristic, complemented
21#20263879—21no—
22#135×C199C2211yescharacteristic
23C2×Q16×C199C2211yescharacteristic, complemented
24Q32×C199—21no—
25SD32×C199—21no—
26M32×C199C2211yescharacteristic, complemented
27#15691887—21no—
28#15691886—21no—
29#15691886—41no—
30#15691885—21no—
31#15691885—41no—
32Q8×D199—21no—
33D4×D199—21no—
34D4×D199—41no—
35C2×C4×D199—41no—
36C2×C4×D199D411yescharacteristic
37Dic796D411yescharacteristic
38#15691876—21no—
39#15691875—21no—
40#15691874—21no—
41#15691873—21no—
42D1592D411yescharacteristic
43#15691871—21no—
44#15691869—21no—
45#15691869—21no—
46C2×C199⋊C8D411yescharacteristic
47C4⋊Dic199—21no—
48Dic199⋊C4—41no—
49(C2×D199)⋊C4—41no—
50C199⋊C16—21no—
51#41×C199—21no—
52C2×Q8×C199—21no—
53Q16×C199—21no—
54Q16×C199—21no—
55Q16×C199D411yescharacteristic
56SD16×C199—21no—
57D8×C199D411yescharacteristic
58C2×C8×C199C2311yescharacteristic
59C16×C199—21no—
60C22×D199—41no—
61C22×D199D811yescharacteristic
62Dic398—21no—
63Dic398—41no—
64D796—21no—
65D796—41no—
66C22⋊D199—41no—
67C22⋊D199—41no—
68C22⋊D199—41no—
69C2×Dic199—41no—
70C2×Dic199—41no—
71C2×Dic199D811yescharacteristic
72C4×D199—21no—
73C4×D199—21no—
74C4×D199—41no—
75C4×D199—41no—
76C4×D199—41no—
77C199⋊C8—21no—
78Q8×C199—21no—
79Q8×C199—41no—
80Q8×C199—41no—
81D4×C199—21no—
82D4×C199—41no—
83C2×C4×C199—41no—
84C2×C4×C199—41no—
85C2×C4×C199C2×D411yescharacteristic
86C8×C199C2×D411yescharacteristic
87C8×C199C2×D411yescharacteristic
88C22×C199—41no—
89C22×C199C2×D811yescharacteristic
90C2×D199—21no—
91C2×D199—41no—
92C2×D199—41no—
93C2×D199—82no—
94C4×C199—41no—
95C4×C199—81no—
96C4×C199#12011yescharacteristic
97C4×C199AGL(1,ℤ/8)11yescharacteristic
98Dic199—21no—
99Dic199—41no—
100Dic199—81no—
101C2×C199—21no—
102C2×C199—81no—
103C2×C199#44711yescharacteristic
104D199—41no—
105D199—81no—
106C199#219111yescharacteristic, complemented, Sylow
107#2191—1991nomaximal, Sylow
108#575—1991no—
109#510—1991no—
110#510D19911yescharacteristic, complemented
111#470—1991no—
112#360—1991no—
113#360—1991no—
114#349—1991no—
115C2×#41—3981no—
116#137—3981no—
117#137—3981no—
118AGL(1,ℤ/8)—3981no—
119AGL(1,ℤ/8)—3981no—
120#135—3981no—
121#135—1991no—
122#135C2×D19911yescharacteristic
123C2×Q16C2×D19911yescharacteristic, complemented
124C2×M16—1991no—
125C22⋊Q8—3981no—
126Q32—21no—
127Q32—3981no—
128SD32—21no—
129SD32—3981no—
130M32—1991no—
131M32C2×D19911yescharacteristic, complemented
132#106—1991no—
133#103—3981no—
134#41—21no—
135#41—7961no—
136#41—7961no—
137#41—3981no—
138#41—3981no—
139#41—3981no—
140C2×Q8—21no—
141C2×Q8—3981no—
142C2×D4—7961no—
143C2×D4—3981no—
144C22×C4—7961no—
145C22×C4—1991no—
146Q16—21no—
147Q16—21no—
148Q16—3981no—
149Q16—1991no—
150Q16C22⋊D19911yescharacteristic
151SD16—21no—
152SD16—3981no—
153SD16—3981no—
154SD16—3981no—
155D8—3981no—
156D8—1991no—
157D8C22⋊D19911yescharacteristic
158M16—3981no—
159M16—3981no—
160C2×C8—1991no—
161C2×C8C22×D19911yescharacteristic
162C4⋊C4—7961no—
163C4⋊C4—3981no—
164C22⋊C4—7961no—
165C16—21no—
166C16—3981no—
167C23—7961no—
168C23—1991no—
169Q8—21no—
170Q8—41no—
171Q8—41no—
172Q8—7961no—
173Q8—3981no—
174D4—21no—
175D4—41no—
176D4—7961no—
177D4—7961no—
178D4—7961no—
179D4—7961no—
180D4—3981no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—7961no—
184C2×C4—7961no—
185C2×C4—7961no—
186C2×C4—7961no—
187C2×C4—7961no—
188C2×C4—3981no—
189C2×C4—3981no—
190C2×C4—1991no—
191C2×C4D4×D19911yescharacteristic
192C8—3981no—
193C8D4×D19911yescharacteristic
194C8C2×C22⋊D19911yescharacteristic
195C22—41no—
196C22—7961no—
197C22—7961no—
198C22—15922no—
199C22—3981no—
200C22D8×D19911yescharacteristic
201C4—41no—
202C4—81no—
203C4—7961no—
204C4—15921no—
205C4—3981no—
206C4#2026392311yescharacteristic
207C4#2026397911yescharacteristic
208C2—21no—
209C2—81no—
210C2—7961no—
211C2—15921no—
212C2#2716415011yescharacteristic
subgroupG/H
centerC2#27164150
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
Fitting#510×C199C2
socleC2×C199#447
prime radical#27164815C2
hypercenter#510D199
hypocenterC199#2191
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #27164459C2semidirect
3. #27164273C2semidirect
4. #27163829C2semidirect
5. #27163828C2semidirect
6. #27163799C2semidirect
7. #510×C199C2semidirect
19. #20263895C22semidirect
20. #20263887C22semidirect
23. C2×Q16×C199C22semidirect
26. M32×C199C22semidirect
106. C199#2191semidirect
110. #510D199semidirect
123. C2×Q16C2×D199semidirect
131. M32C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648152
#271644592
#271642732
#271638292
#271638282
#271637992
#510×C1992
#2191199