Finite Group Databasebeta

← #35315253

Subgroups — #35315253

Order 21248.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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#24918201C211yesmaximal, characteristic
2#24907612C211yesmaximal, characteristic, complemented
3#24905455C212yesmaximal, complemented
4#24905449C212yesmaximal, complemented
5C83×#2165C211yesmaximal, characteristic, complemented
6#19128251—21no—
7#19128099C411yescharacteristic, complemented
8#19128078C411yescharacteristic, complemented
9#19128048—21no—
10#19127954—21no—
11#19127954C2212yes—
12#19127814—21no—
13#19127814C412yescomplemented
14#19127472—21no—
15#19127461—21no—
16C83⋊#338C2211yescharacteristic
17#19127458C2211yescharacteristic
18C4×M16×C83C2211yescharacteristic
19#364×C83—21no—
20#349×C83C2212yes—
21#347×C83—21no—
22C2×#13050590—41no—
23C2×D664—21no—
24C2×D664C2×C412yes—
25C2×#13050574—21no—
26#14581792—21no—
27#14581789—21no—
28#14581780—41no—
29#14581766—22no—
30#14581766—41no—
31C4×D332—22no—
32C4×D332—41no—
33#14581751—21no—
34#14581724—21no—
35#14581724C2×C412yes—
36#14581714—22no—
37#14581714—22no—
38C2×M16×C83D411yescharacteristic
39C2×M16×C83C2311yescharacteristic
40C2×C42×C83D411yescharacteristic
41M32×C83—22no—
42#97×C83C2×C411yescharacteristic
43#97×C83D411yescharacteristic
44C4×C8×C83C2×C411yescharacteristic
45C4×C8×C83D411yescharacteristic
46#13050590—84no—
47C2×Dic166—41no—
48C2×D332—22no—
49C2×D332—41no—
50C2×D332—41no—
51C2×C22⋊D83—42no—
52C2×C4×D83—42no—
53D664—42no—
54D664—42no—
55#13050574—41no—
56#13050574—41no—
57C2×C83⋊C8—41no—
58C4⋊Dic83—22no—
59C4⋊Dic83—41no—
60C4⋊Dic83—41no—
61C22⋊Dic83—41no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—42no—
64C4×Dic83—41no—
65C22×C4×C83—21no—
66C22×C4×C83C2×D411yescharacteristic
67M16×C83—41no—
68M16×C83—41no—
69C2×C8×C83C22⋊C411yescharacteristic
70C2×C8×C83C22⋊C411yescharacteristic
71C2×C8×C83C22×C411yescharacteristic
72C2×C8×C83C2×D411yescharacteristic
73C42×C83—21no—
74C42×C83C22⋊C411yescharacteristic
75C42×C83C22⋊C411yescharacteristic
76C16×C83—42no—
77C23×C83AGL(1,ℤ/8)11yescharacteristic
78C22×D83—42no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—42no—
84C2×C4×C83C2×C22⋊C411yescharacteristic
85C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
86C22⋊D83—84no—
87C2×Dic83—42no—
88C2×Dic83—81no—
89D332—42no—
90D332—42no—
91D332—82no—
92D332—82no—
93C4×D83—84no—
94Dic166—82no—
95C8×C83—21no—
96C8×C83—21no—
97C8×C83—21no—
98C8×C83—21no—
99C83⋊C8—81no—
100C22×C83—21no—
101C22×C83—41no—
102C22×C83#41811yescharacteristic
103C2×D83—82no—
104C2×D83—82no—
105C2×D83—82no—
106C4×C83—22no—
107C4×C83—21no—
108C4×C83—41no—
109C4×C83#41011yescharacteristic
110C4×C83#42111yescharacteristic
111Dic83—82no—
112Dic83—161no—
113#11837—831nomaximal, Sylow
114C2×C83—21no—
115C2×C83—41no—
116C2×C83#151411yescharacteristic
117D83—162no—
118#3184—831no—
119#2165D8311yescharacteristic, complemented
120#1841—831no—
121#1341—832no—
122#1339—832no—
123C83#1183711yescharacteristic, complemented, Sylow
124#532—1661no—
125#498—831no—
126#493—831no—
127#486—1661no—
128#469—832no—
129#469—1661no—
130#442—832no—
131#442—1661no—
132C4×M16C2×D8311yescharacteristic
133#364—21no—
134#349C2×D8312yes—
135#347—21no—
136#344—1661no—
137#339—1661no—
138#338—831no—
139H(3,ℤ/4)—831no—
140C2×#41—3321no—
141C2×D8—832no—
142C2×D8—1661no—
143C2×M16—1661no—
144C2×M16D33211yescharacteristic
145C2×M16C22×D8311yescharacteristic
146C4⋊Q8—1661no—
147#127—1661no—
148#124—3321no—
149#121—3321no—
150#121—1662no—
151C4×D4—3321no—
152C4×D4—1662no—
153#117—1661no—
154C2×C42D33211yescharacteristic
155M32—22no—
156#106—832no—
157#106—1661no—
158#102—1662no—
159#102—1662no—
160#97C4×D8311yescharacteristic
161#97C22⋊D8311yescharacteristic
162C4×C8C4×D8311yescharacteristic
163C4×C8C22⋊D8311yescharacteristic
164#41—6644no—
165C2×Q8—3321no—
166C2×D4—3321no—
167C2×D4—3321no—
168C2×D4—3322no—
169C2×D4—1662no—
170C22×C4—21no—
171C22×C4—3322no—
172C22×C4C2×D33211yescharacteristic
173D8—3322no—
174D8—3322no—
175M16—41no—
176M16—41no—
177M16—3321no—
178M16—3321no—
179C2×C8—3321no—
180C2×C8(C2×D83)⋊C411yescharacteristic
181C2×C8(C2×D83)⋊C411yescharacteristic
182C2×C8C2×C4×D8311yescharacteristic
183C2×C8C2×C22⋊D8311yescharacteristic
184C4⋊C4—3321no—
185C4⋊C4—3321no—
186C4⋊C4—1662no—
187C22⋊C4—3322no—
188C22⋊C4—3322no—
189C22⋊C4—3321no—
190C42—21no—
191C42—3321no—
192C42(C2×D83)⋊C411yescharacteristic
193C42(C2×D83)⋊C411yescharacteristic
194C16—42no—
195C23—3322no—
196C23#1458182011yescharacteristic
197Q8—6642no—
198D4—3322no—
199D4—3322no—
200D4—6642no—
201D4—6642no—
202D4—6644no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—3322no—
209C2×C4—6644no—
210C2×C4—6641no—
211C2×C4C2×(C2×D83)⋊C411yescharacteristic
212C2×C4#1458182011yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—6641no—
218C22—21no—
219C22—41no—
220C22—6642no—
221C22—6642no—
222C22—6642no—
223C22#1912770711yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—13281no—
228C4—6642no—
229C4#1912767411yescharacteristic
230C4#1912771711yescharacteristic
231C2—21no—
232C2—41no—
233C2—13282no—
234C2#2490601111yescharacteristic
subgroupG/H
centerC2#24906011
commutatorC2×C8×C83C22×C4
FrattiniC2×M16C22×D83
FittingC83×#2165C2
socleC2×C83#1514
prime radical#24918201C2
cosocleC2×M16×C83C23
hypercenter#2165D83
hypocenterC83#11837
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #24907612C2semidirect
3. #24905455C2semidirect
4. #24905449C2semidirect
5. C83×#2165C2semidirect
7. #19128099C4semidirect
8. #19128078C4semidirect
13. #19127814C4semidirect
119. #2165D83semidirect
123. C83#11837semidirect

Maximal subgroups

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

subgroupindex
#249182012
#249076122
#249054552
#249054492
C83×#21652
#1183783