Finite Group Databasebeta

← #35315905

Subgroups — #35315905

Order 21248.

352 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 85 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#24918043C211yesmaximal, characteristic
2#24909217C211yesmaximal, characteristic, complemented
3#24906220C211yesmaximal, characteristic, complemented
4#24905521C211yesmaximal, characteristic, complemented
5#24905510C211yesmaximal, characteristic, complemented
6#24905508C211yesmaximal, characteristic, complemented
7C83×#1365C211yesmaximal, characteristic, complemented
8#19128252—21no—
9C2×#14581731—21no—
10#19128078C2211yescharacteristic
11#19128049—21no—
12#19127938—21no—
13#19127938C412yes—
14#19127783—21no—
15#19127783C412yes—
16#19127723—21no—
17C4×C8×D83C2211yescharacteristic
18C4⋊(C83⋊C16)C2211yescharacteristic
19Dic83⋊C16—21no—
20#19127454C2211yescharacteristic
21#19127453—21no—
22#500×C83C2211yescharacteristic
23C4⋊C16×C83C2211yescharacteristic
24#335×C83C2211yescharacteristic
25C2×#13050591—41no—
26C2×D664—21no—
27C2×D664C2×C412yes—
28C2×#13050573—41no—
29C2×C8×D83C2×C411yescharacteristic
30C2×C8×D83D411yescharacteristic
31#14581789—21no—
32#14581781—41no—
33#14581767—22no—
34#14581767—41no—
35#14581754—22no—
36#14581754—41no—
37C42×D83D411yescharacteristic
38#14581731—42no—
39C2×C83⋊C16—21no—
40C4⋊(C83⋊C8)—21no—
41Dic83⋊C8—41no—
42#14581712—22no—
43#14581712—22no—
44C4×C83⋊C8D411yescharacteristic
45C8×Dic83C2×C411yescharacteristic
46C8×Dic83D411yescharacteristic
47C4⋊Q8×C83—21no—
48C2×C16×C83—21no—
49#107×C83—21no—
50#107×C83C2×C412yes—
51C4⋊C8×C83—21no—
52C4×C8×C83C2311yescharacteristic
53#13050591—84no—
54C2×D332—22no—
55C2×D332—42no—
56C2×D332—41no—
57C2×D332—41no—
58C2×C4×D83—42no—
59C2×C4×D83C22⋊C411yescharacteristic
60C2×C4×D83D811yescharacteristic
61C2×C4×D83SD1611yescharacteristic
62D664—42no—
63D664—41no—
64D664—41no—
65#13050573—82no—
66C2×C83⋊C8—41no—
67C2×C83⋊C8C22⋊C411yescharacteristic
68C2×C83⋊C8C22⋊C411yescharacteristic
69C8×D83—21no—
70C8×D83—21no—
71C8×D83—41no—
72(C2×D83)⋊C4—42no—
73(C2×D83)⋊C4—42no—
74C4×Dic83C22⋊C411yescharacteristic
75C4×Dic83D811yescharacteristic
76C4×Dic83SD1611yescharacteristic
77C83⋊C16—41no—
78C2×Q8×C83—41no—
79C2×C8×C83—41no—
80C2×C8×C83C22×C411yescharacteristic
81C2×C8×C83C2×D411yescharacteristic
82C4⋊C4×C83—22no—
83C4⋊C4×C83—41no—
84C42×C83C2×D411yescharacteristic
85C16×C83—41no—
86C22×D83—42no—
87C22×D83#10211yescharacteristic
88Q8×C83—82no—
89C2×C4×C83—42no—
90C2×C4×C83C2×C22⋊C411yescharacteristic
91C2×C4×C83C2×D811yescharacteristic
92C2×C4×C83C2×SD1611yescharacteristic
93C2×Dic83#10211yescharacteristic
94C2×Dic83#10211yescharacteristic
95C2×Dic83#10211yescharacteristic
96D332—42no—
97D332—42no—
98D332—82no—
99D332—84no—
100D332—81no—
101D332—81no—
102C4×D83—21no—
103C4×D83—21no—
104C4×D83—21no—
105C4×D83—21no—
106C4×D83—22no—
107C4×D83—84no—
108C8×C83—21no—
109C8×C83—81no—
110C8×C83AGL(1,ℤ/8)11yescharacteristic
111C8×C83#13711yescharacteristic
112C83⋊C8—21no—
113C83⋊C8—21no—
114C83⋊C8—81no—
115C22×C83C2×#10211yescharacteristic
116C2×D83—21no—
117C2×D83—21no—
118C2×D83—21no—
119C2×D83—82no—
120C2×D83—82no—
121C2×D83—82no—
122C4×C83—21no—
123C4×C83—82no—
124C4×C83#41311yescharacteristic
125C4×C83#41711yescharacteristic
126C4×C83#50911yescharacteristic
127C4×C83#51011yescharacteristic
128Dic83—21no—
129Dic83—21no—
130Dic83#36112yes—
131#11986—831nomaximal, Sylow
132C2×C83#154711yescharacteristic
133C2×C83#213911yescharacteristic
134C2×C83C2×#36111yescharacteristic
135D83—41no—
136D83—162no—
137#3169—831no—
138#2150—831no—
139#1567—831no—
140#1365—831no—
141#1365D8311yescharacteristic, complemented
142#1361—831no—
143#1361—831no—
144C83#1198611yescharacteristic, complemented, Sylow
145#532—1661no—
146C2×M32—1661no—
147#500C2×D8311yescharacteristic
148#493—831no—
149#486—1661no—
150#466—832no—
151#466—1661no—
152C4×D8—832no—
153C4×D8—1661no—
154#423—1661no—
155C2×C4×C8—831no—
156C4⋊C16—831no—
157C4⋊C16—1661no—
158C4⋊C16C2×D8311yescharacteristic
159#335—831no—
160#335—1661no—
161#335C2×D8311yescharacteristic
162C2×#41—3321no—
163C2×D8—832no—
164C2×D8—1661no—
165C2×M16—3321no—
166C22×C8—831no—
167C22×C8—831no—
168C4⋊Q8—21no—
169#127—1661no—
170#124—3321no—
171#121—3321no—
172#121—1662no—
173C4×D4—3321no—
174C4×D4—1662no—
175C2×C42—831no—
176M32—3322no—
177C2×C16—21no—
178C2×C16—1661no—
179#107—21no—
180#107C4×D8312yes—
181C4⋊C8—21no—
182C4⋊C8—3321no—
183C4⋊C8—1661no—
184#102—1662no—
185#102—1662no—
186C4×C8—831no—
187C4×C8—831no—
188C4×C8—831no—
189C4×C8C22×D8311yescharacteristic
190#41—6644no—
191C2×Q8—41no—
192C2×D4—3322no—
193C2×D4—3321no—
194C2×D4—3321no—
195C2×D4—1662no—
196C22×C4—3322no—
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—831no—
200D8—3322no—
201D8—3321no—
202D8—3321no—
203M16—6642no—
204C2×C8—41no—
205C2×C8—3321no—
206C2×C8—3321no—
207C2×C8—831no—
208C2×C8—831no—
209C2×C8—1661no—
210C2×C8—1661no—
211C2×C8C2×C4×D8311yescharacteristic
212C2×C8D4×D8311yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—41no—
215C22⋊C4—3322no—
216C22⋊C4—3322no—
217C42—831no—
218C42—831no—
219C42—831no—
220C42D4×D8311yescharacteristic
221C16—41no—
222C16—3321no—
223C23—3322no—
224C23—831no—
225Q8—82no—
226D4—3322no—
227D4—3322no—
228D4—6642no—
229D4—6644no—
230D4—6641no—
231D4—6641no—
232C2×C4—42no—
233C2×C4—831no—
234C2×C4—831no—
235C2×C4—831no—
236C2×C4—6644no—
237C2×C4—1662no—
238C2×C4—1661no—
239C2×C4—1661no—
240C2×C4—1661no—
241C2×C4—1661no—
242C2×C4C22⋊C4×D8311yescharacteristic
243C2×C4D8×D8311yescharacteristic
244C2×C4SD16×D8311yescharacteristic
245C8—21no—
246C8—81no—
247C8—6641no—
248C8—1661no—
249C8—1661no—
250C8#1458181911yescharacteristic
251C8#1458182411yescharacteristic
252C22—6642no—
253C22—6642no—
254C22—6642no—
255C22—1661no—
256C22—1661no—
257C22—1661no—
258C22#102×D8311yescharacteristic
259C4—21no—
260C4—82no—
261C4—832no—
262C4—1661no—
263C4—1661no—
264C4#1912768411yescharacteristic
265C4#1912770111yescharacteristic
266C4#1912813811yescharacteristic
267C4#1912814311yescharacteristic
268C2—13282no—
269C2—3321no—
270C2#2490613911yescharacteristic
271C2#2490917411yescharacteristic
272C2#361×D8311yescharacteristic
subgroupG/H
centerC22#102×D83
commutatorC2×C8×C83C22×C4
FrattiniC4×C8C22×D83
FittingC83×#1365C2
socleC22×C83C2×#102
prime radical#24918043C2
cosocleC4×C8×C83C23
hypercenter#1365D83
hypocenterC83#11986
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
2. #24909217C2semidirect
3. #24906220C2semidirect
4. #24905521C2semidirect
5. #24905510C2semidirect
6. #24905508C2semidirect
7. C83×#1365C2semidirect
141. #1365D83semidirect
144. C83#11986semidirect

Maximal subgroups

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

subgroupindex
#249180432
#249092172
#249062202
#249055212
#249055102
#249055082
C83×#13652
#1198683