Finite Group Databasebeta

← #35315250

Subgroups — #35315250

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#24909275C211yesmaximal, characteristic, complemented
3#24905453C212yesmaximal, complemented
4#24905447C212yesmaximal, complemented
5C83×#1841C211yesmaximal, 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#19127522—21no—
15#19127483C2212yes—
16#19127479—21no—
17C4×M16×C83C2211yescharacteristic
18#344×C83—21no—
19#339×C83—21no—
20#338×C83C2211yescharacteristic
21H(3,ℤ/4)×C83C2211yescharacteristic
22C2×#13050590—41no—
23C2×D664—21no—
24C2×D664C2×C412yes—
25#14581792—21no—
26#14581789—21no—
27#14581780—41no—
28#14581766—22no—
29#14581766—41no—
30C4×D332—22no—
31C4×D332—41no—
32#14581732—22no—
33#14581724—21no—
34#14581724C2×C412yes—
35#14581714—22no—
36#14581714—22no—
37C2×M16×C83—21no—
38C2×M16×C83D411yescharacteristic
39C2×M16×C83C2311yescharacteristic
40#117×C83—21no—
41C2×C42×C83D411yescharacteristic
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—
55C4⋊Dic83—22no—
56C4⋊Dic83—41no—
57(C2×D83)⋊C4—42no—
58(C2×D83)⋊C4—42no—
59C83⋊C16—42no—
60C22×C4×C83—21no—
61C22×C4×C83C2×D411yescharacteristic
62M16×C83—41no—
63M16×C83—41no—
64M16×C83—41no—
65M16×C83—41no—
66C2×C8×C83—41no—
67C2×C8×C83C22⋊C411yescharacteristic
68C2×C8×C83C22⋊C411yescharacteristic
69C2×C8×C83C22×C411yescharacteristic
70C2×C8×C83C2×D411yescharacteristic
71C4⋊C4×C83—41no—
72C22⋊C4×C83—41no—
73C42×C83—21no—
74C42×C83—41no—
75C42×C83C22⋊C411yescharacteristic
76C42×C83C22⋊C411yescharacteristic
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×C83—81no—
85C2×C4×C83C2×C22⋊C411yescharacteristic
86C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
87C22⋊D83—84no—
88C2×Dic83—42no—
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—
99C8×C83—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—161no—
110C4×C83#41011yescharacteristic
111C4×C83#42111yescharacteristic
112Dic83—82no—
113#11837—831nomaximal, Sylow
114C2×C83—21no—
115C2×C83—41no—
116C2×C83#151411yescharacteristic
117D83—162no—
118#3184—831no—
119#2165—831no—
120#1841D8311yescharacteristic, complemented
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—1661no—
134#349—832no—
135#347—1661no—
136#344—21no—
137#339—21no—
138#338C2×D8311yescharacteristic
139H(3,ℤ/4)C2×D8311yescharacteristic
140C2×#41—3321no—
141C2×D8—832no—
142C2×D8—1661no—
143C2×M16—21no—
144C2×M16C22⋊D8311yescharacteristic
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—21no—
154C2×C42C22⋊D8311yescharacteristic
155M32—1662no—
156#106—832no—
157#106—1661no—
158#102—1662no—
159#102—1662no—
160#97C4×D8311yescharacteristic
161#97D33211yescharacteristic
162C4×C8C4×D8311yescharacteristic
163C4×C8D33211yescharacteristic
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×C22⋊D8311yescharacteristic
173D8—3322no—
174D8—3322no—
175M16—41no—
176M16—41no—
177M16—41no—
178M16—41no—
179C2×C8—41no—
180C2×C8(C2×D83)⋊C411yescharacteristic
181C2×C8(C2×D83)⋊C411yescharacteristic
182C2×C8C2×C4×D8311yescharacteristic
183C2×C8C2×D33211yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—3321no—
186C4⋊C4—1662no—
187C22⋊C4—41no—
188C22⋊C4—3322no—
189C22⋊C4—3322no—
190C42—21no—
191C42—41no—
192C42(C2×D83)⋊C411yescharacteristic
193C42(C2×D83)⋊C411yescharacteristic
194C16—3322no—
195C23—3322no—
196C23#1458181711yescharacteristic
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—81no—
209C2×C4—3322no—
210C2×C4—6644no—
211C2×C4C2×(C2×D83)⋊C411yescharacteristic
212C2×C4#1458181711yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—81no—
218C22—21no—
219C22—41no—
220C22—6642no—
221C22—6642no—
222C22—6642no—
223C22#1912770511yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—161no—
228C4—6642no—
229C4#1912767411yescharacteristic
230C4#1912771911yescharacteristic
231C2—21no—
232C2—41no—
233C2—13282no—
234C2#2490600911yescharacteristic
subgroupG/H
centerC2#24906009
commutatorC2×C8×C83C22×C4
FrattiniC2×M16C22×D83
FittingC83×#1841C2
socleC2×C83#1514
prime radical#24918201C2
cosocleC2×M16×C83C23
hypercenter#1841D83
hypocenterC83#11837
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #1841

Complements

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

NH ≅ G/Ntype
2. #24909275C2semidirect
3. #24905453C2semidirect
4. #24905447C2semidirect
5. C83×#1841C2semidirect
7. #19128099C4semidirect
8. #19128078C4semidirect
13. #19127814C4semidirect
120. #1841D83semidirect
123. C83#11837semidirect

Maximal subgroups

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

subgroupindex
#249182012
#249092752
#249054532
#249054472
C83×#18412
#1183783