Finite Group Databasebeta

← #35309384

Subgroups — #35309384

Order 21248.

372 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 93 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#24908260C211yesmaximal, characteristic
2#24907637C211yesmaximal, characteristic, complemented
3#24906233C211yesmaximal, characteristic, complemented
4#24905889C211yesmaximal, characteristic
5#24905887C211yesmaximal, characteristic
6#24905293C211yesmaximal, characteristic, complemented
7C83×#1570C211yesmaximal, characteristic, complemented
8C2×C4×D332C2211yescharacteristic
9#19127754C2211yescharacteristic
10C2×#14581727—22no—
11#19127725C2211yescharacteristic
12C2×(C2×D83)⋊C8—22no—
13C2×#14581703C2211yescharacteristic
14#19127451—22no—
15#19127430—22no—
16#19127428—22no—
17#431×C83C2211yescharacteristic
18#423×C83C2211yescharacteristic
19C2×C4⋊C8×C83C2211yescharacteristic
20#334×C83—22no—
21C22×D332Q811yescharacteristic
22C22×C4×D83—21no—
23C2×#13050574—21no—
24C22×C83⋊C8D412yes—
25C4×D332—21no—
26C4×D332—21no—
27C4×D332—21no—
28C4×D332—21no—
29C2×C4⋊Dic83Q811yescharacteristic
30C2×(C2×D83)⋊C4—21no—
31#14581727—44no—
32C4⋊(C83⋊C8)—21no—
33C4⋊(C83⋊C8)—21no—
34C4⋊(C83⋊C8)—21no—
35C22⋊(C83⋊C8)—21no—
36(C2×D83)⋊C8—44no—
37#14581703—21no—
38#14581703—21no—
39C4×C83⋊C8D412yes—
40C2×M16×C83—21no—
41C22×C8×C83—21no—
42C2×C42×C83C2311yescharacteristic
43C4⋊C8×C83—21no—
44C4⋊C8×C83—21no—
45C4⋊C8×C83—21no—
46C4⋊C8×C83—21no—
47C4⋊C8×C83—21no—
48C22⋊C8×C83—21no—
49C4×C8×C83D412yes—
50C23×D83—21no—
51C2×D332—21no—
52C2×D332—21no—
53C2×D332—42no—
54C22×Dic83—21no—
55C2×C4×D83—41no—
56C2×C4×D83—41no—
57C2×C4×D83—42no—
58C2×C4×D83—42no—
59#13050574—42no—
60C2×C83⋊C8—22no—
61C2×C83⋊C8—22no—
62C2×C83⋊C8—22no—
63C2×C83⋊C8—41no—
64C2×C83⋊C8SD1614yes—
65C4⋊Dic83—21no—
66C4⋊Dic83—21no—
67(C2×D83)⋊C4—41no—
68(C2×D83)⋊C4—41no—
69C22×C4×C83C2×D411yescharacteristic
70C22×C4×C83C2×Q811yescharacteristic
71C22×C4×C83#4111yescharacteristic
72M16×C83—42no—
73C2×C8×C83—22no—
74C2×C8×C83—41no—
75C2×C8×C83—41no—
76C2×C8×C83—42no—
77C42×C83C2×D411yescharacteristic
78C42×C83C2×D411yescharacteristic
79C42×C83#4111yescharacteristic
80C42×C83#4111yescharacteristic
81C23×C83C22⋊Q811yescharacteristic
82C22×D83—22no—
83C22×D83—41no—
84C22×D83—42no—
85C22×D83—41no—
86C22×D83—42no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83C22⋊Q811yescharacteristic
90C2×C4×C83C22⋊Q811yescharacteristic
91C2×C4×C83#12311yescharacteristic
92C2×C4×C83#12311yescharacteristic
93C2×C4×C83#12311yescharacteristic
94C2×C4×C83#12711yescharacteristic
95C2×C4×C83C2×SD1612yes—
96C2×C4×C83AGL(1,ℤ/8)12yes—
97C2×Dic83—41no—
98C2×Dic83—42no—
99D332—22no—
100D332—22no—
101D332—41no—
102D332—41no—
103C4×D83—82no—
104C4×D83—81no—
105C4×D83—81no—
106C8×C83—42no—
107C8×C83—81no—
108C8×C83—81no—
109C83⋊C8—24no—
110C83⋊C8—42no—
111C83⋊C8—81no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83#39911yescharacteristic
115C22×C83#47612yes—
116C2×D83—44no—
117C2×D83—42no—
118C2×D83—82no—
119C2×D83—81no—
120C2×D83—81no—
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—21no—
124C4×C83—21no—
125C4×C83#48112yes—
126C4×C83#49211yescharacteristic
127C4×C83#49611yescharacteristic
128Dic83—81no—
129#10661—831nomaximal, Sylow
130C2×C83—22no—
131C2×C83#209111yescharacteristic
132C2×C83#209411yescharacteristic
133C2×C83#209811yescharacteristic
134D83—82no—
135#1955—831no—
136#1849—831no—
137#1570—831no—
138#1570D8311yescharacteristic, complemented
139#1486—831no—
140#1486—831no—
141#1275—831no—
142C83#1066111yescharacteristic, complemented, Sylow
143C2×C4×D4—831no—
144#431—831no—
145#431C2×D8311yescharacteristic
146C2×#108—1662no—
147#423—831no—
148#423C2×D8311yescharacteristic
149C2×C4⋊C8C2×D8311yescharacteristic
150C2×C22⋊C8—1662no—
151C2×#97—831no—
152#334—22no—
153#334—1662no—
154D4⋊C8—1662no—
155D4⋊C8—1662no—
156C22×D4—831no—
157C23×C4—1661no—
158C2×M16—21no—
159C2×M16—1661no—
160C22×C8—21no—
161C22×C8—832no—
162C4×D4—1661no—
163C4×D4—1661no—
164C4×D4—1661no—
165C4×D4—1661no—
166C2×C4⋊C4—831no—
167C2×C22⋊C4—1661no—
168C2×C42C22×D8311yescharacteristic
169#108—3324no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—1661no—
176C4⋊C8—1661no—
177C4⋊C8—1661no—
178C22⋊C8—21no—
179C22⋊C8—3324no—
180C22⋊C8—1661no—
181#97—1661no—
182#97—1661no—
183C4×C8—832no—
184C4×C8C22⋊D8312yes—
185C24—1661no—
186C2×D4—3322no—
187C2×D4—1661no—
188C2×D4—1661no—
189C22×C4—3321no—
190C22×C4—3321no—
191C22×C4—3322no—
192C22×C4—3322no—
193C22×C4—1661no—
194C22×C4D4×D8311yescharacteristic
195C22×C4Q8×D8311yescharacteristic
196C22×C4#1305059011yescharacteristic
197M16—42no—
198M16—3322no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—3321no—
204C2×C8—834no—
205C2×C8—1662no—
206C2×C8—1662no—
207C2×C8—1662no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C22⋊C4—3321no—
211C22⋊C4—3321no—
212C42D4×D8311yescharacteristic
213C42C2×C22⋊D8311yescharacteristic
214C42#1305059011yescharacteristic
215C42#1305059111yescharacteristic
216C23—3321no—
217C23—3322no—
218C23—3321no—
219C23—3322no—
220C23—1662no—
221C23#1458177011yescharacteristic
222D4—3321no—
223D4—3321no—
224D4—1662no—
225D4—1662no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3321no—
229C2×C4—3322no—
230C2×C4—6641no—
231C2×C4—6641no—
232C2×C4—6642no—
233C2×C4#1458176811yescharacteristic
234C2×C4#1458177011yescharacteristic
235C2×C4#1458177511yescharacteristic
236C2×C4#1458177711yescharacteristic
237C2×C4#1458177711yescharacteristic
238C2×C4#1458178811yescharacteristic
239C2×C4SD16×D8312yes—
240C2×C4#1458181912yes—
241C8—42no—
242C8—81no—
243C8—81no—
244C8—3322no—
245C8—6641no—
246C8—1664no—
247C22—21no—
248C22—21no—
249C22—3324no—
250C22—3322no—
251C22—6642no—
252C22—6641no—
253C22—6641no—
254C22#1912763611yescharacteristic
255C22#1912799012yes—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—21no—
260C4—6641no—
261C4#1912802112yes—
262C4#1912807211yescharacteristic
263C4#1912808711yescharacteristic
264C2—22no—
265C2—6642no—
266C2#2490897811yescharacteristic
267C2#2490899311yescharacteristic
268C2#2490901911yescharacteristic
subgroupG/H
centerC2×C4#14581788
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1570C2
socleC22×C83#399
prime radicalC22×D332Q8
hypercenter#1570D83
hypocenterC83#10661
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
2. #24907637C2semidirect
3. #24906233C2semidirect
6. #24905293C2semidirect
7. C83×#1570C2semidirect
138. #1570D83semidirect
142. C83#10661semidirect

Maximal subgroups

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

subgroupindex
#249082602
#249076372
#249062332
#249058892
#249058872
#249052932
C83×#15702
#1066183