Finite Group Databasebeta

← #38744661

Subgroups — #38744661

Order 25216.

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#27161876C211yesmaximal, characteristic
2#27161520C211yesmaximal, characteristic, complemented
3#27161334C211yesmaximal, characteristic, complemented
4#27160890C211yesmaximal, characteristic, complemented
5#27160889C211yesmaximal, characteristic, complemented
6#27160860C211yesmaximal, characteristic, complemented
7#510×C197C211yesmaximal, characteristic, complemented
8#41×D197—21no—
9#20263358—21no—
10#20263355—21no—
11#20263353—21no—
12#20263350—21no—
13#20263348C2211yescharacteristic
14#20263346—21no—
15C2×#15691341C2211yescharacteristic
16#20263302—21no—
17#20263272—21no—
18#20263269—21no—
19#20263266C2211yescharacteristic, complemented
20#20263258C2211yescharacteristic, complemented
21#20263250—21no—
22#135×C197C2211yescharacteristic
23C2×Q16×C197C2211yescharacteristic, complemented
24Q32×C197—21no—
25SD32×C197—21no—
26M32×C197C2211yescharacteristic, complemented
27#15691359—21no—
28#15691358—21no—
29#15691358—41no—
30#15691357—21no—
31#15691357—41no—
32Q8×D197—21no—
33D4×D197—21no—
34D4×D197—41no—
35C2×C4×D197—41no—
36C2×C4×D197D411yescharacteristic
37Dic788D411yescharacteristic
38#15691348—21no—
39#15691347—21no—
40#15691346—21no—
41#15691345—21no—
42D1576D411yescharacteristic
43#15691343—21no—
44#15691341—21no—
45#15691341—21no—
46C2×#13096802D411yescharacteristic
47C4⋊Dic197—21no—
48#15691337—41no—
49#15691335—41no—
50#15691333—21no—
51#41×C197—21no—
52C2×Q8×C197—21no—
53Q16×C197—21no—
54Q16×C197—21no—
55Q16×C197D411yescharacteristic
56SD16×C197—21no—
57D8×C197D411yescharacteristic
58C2×C8×C197C2311yescharacteristic
59C16×C197—21no—
60C22×D197—41no—
61C22×D197D811yescharacteristic
62Dic394—21no—
63Dic394—41no—
64D788—21no—
65D788—41no—
66C22⋊D197—41no—
67C22⋊D197—41no—
68C22⋊D197—41no—
69C2×Dic197—41no—
70C2×Dic197—41no—
71C2×Dic197D811yescharacteristic
72C4×D197—21no—
73C4×D197—21no—
74C4×D197—41no—
75C4×D197—41no—
76C4×D197—41no—
77#13096802—21no—
78Q8×C197—21no—
79Q8×C197—41no—
80Q8×C197—41no—
81D4×C197—21no—
82D4×C197—41no—
83C2×C4×C197—41no—
84C2×C4×C197—41no—
85C2×C4×C197C2×D411yescharacteristic
86C8×C197C2×D411yescharacteristic
87C8×C197C2×D411yescharacteristic
88C22×C197—41no—
89C22×C197C2×D811yescharacteristic
90C2×D197—21no—
91C2×D197—41no—
92C2×D197—41no—
93C2×D197—82no—
94C4×C197—41no—
95C4×C197—81no—
96C4×C197#12011yescharacteristic
97C4×C197AGL(1,ℤ/8)11yescharacteristic
98Dic197—21no—
99Dic197—41no—
100Dic197—81no—
101C2×C197—21no—
102C2×C197—81no—
103C2×C197#44711yescharacteristic
104D197—41no—
105D197—81no—
106C197#219111yescharacteristic, complemented, Sylow
107#2191—1971nomaximal, Sylow
108#575—1971no—
109#510—1971no—
110#510D19711yescharacteristic, complemented
111#470—1971no—
112#360—1971no—
113#360—1971no—
114#349—1971no—
115C2×#41—3941no—
116#137—3941no—
117#137—3941no—
118AGL(1,ℤ/8)—3941no—
119AGL(1,ℤ/8)—3941no—
120#135—3941no—
121#135—1971no—
122#135C2×D19711yescharacteristic
123C2×Q16C2×D19711yescharacteristic, complemented
124C2×M16—1971no—
125C22⋊Q8—3941no—
126Q32—21no—
127Q32—3941no—
128SD32—21no—
129SD32—3941no—
130M32—1971no—
131M32C2×D19711yescharacteristic, complemented
132#106—1971no—
133#103—3941no—
134#41—21no—
135#41—7881no—
136#41—7881no—
137#41—3941no—
138#41—3941no—
139#41—3941no—
140C2×Q8—21no—
141C2×Q8—3941no—
142C2×D4—7881no—
143C2×D4—3941no—
144C22×C4—7881no—
145C22×C4—1971no—
146Q16—21no—
147Q16—21no—
148Q16—3941no—
149Q16—1971no—
150Q16C22⋊D19711yescharacteristic
151SD16—21no—
152SD16—3941no—
153SD16—3941no—
154SD16—3941no—
155D8—3941no—
156D8—1971no—
157D8C22⋊D19711yescharacteristic
158M16—3941no—
159M16—3941no—
160C2×C8—1971no—
161C2×C8C22×D19711yescharacteristic
162C4⋊C4—7881no—
163C4⋊C4—3941no—
164C22⋊C4—7881no—
165C16—21no—
166C16—3941no—
167C23—7881no—
168C23—1971no—
169Q8—21no—
170Q8—41no—
171Q8—41no—
172Q8—7881no—
173Q8—3941no—
174D4—21no—
175D4—41no—
176D4—7881no—
177D4—7881no—
178D4—7881no—
179D4—7881no—
180D4—3941no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—7881no—
184C2×C4—7881no—
185C2×C4—7881no—
186C2×C4—7881no—
187C2×C4—7881no—
188C2×C4—3941no—
189C2×C4—3941no—
190C2×C4—1971no—
191C2×C4D4×D19711yescharacteristic
192C8—3941no—
193C8D4×D19711yescharacteristic
194C8C2×C22⋊D19711yescharacteristic
195C22—41no—
196C22—7881no—
197C22—7881no—
198C22—15762no—
199C22—3941no—
200C22D8×D19711yescharacteristic
201C4—41no—
202C4—81no—
203C4—7881no—
204C4—15761no—
205C4—3941no—
206C4#2026329411yescharacteristic
207C4#2026335011yescharacteristic
208C2—21no—
209C2—81no—
210C2—7881no—
211C2—15761no—
212C2#2716121111yescharacteristic
subgroupG/H
centerC2#27161211
commutator, cosocleC2×C8×C197C23
FrattiniC2×C8C22×D197
Fitting#510×C197C2
socleC2×C197#447
prime radical#27161876C2
hypercenter#510D197
hypocenterC197#2191
LCSG ▷ C2×C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #27161520C2semidirect
3. #27161334C2semidirect
4. #27160890C2semidirect
5. #27160889C2semidirect
6. #27160860C2semidirect
7. #510×C197C2semidirect
19. #20263266C22semidirect
20. #20263258C22semidirect
23. C2×Q16×C197C22semidirect
26. M32×C197C22semidirect
106. C197#2191semidirect
110. #510D197semidirect
123. C2×Q16C2×D197semidirect
131. M32C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618762
#271615202
#271613342
#271608902
#271608892
#271608602
#510×C1972
#2191197