Finite Group Databasebeta

← #35309174

Subgroups — #35309174

Order 21248.

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

normal subgroups only

This group has 330 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#24908295C211yesmaximal, characteristic
2#24907617C211yesmaximal, characteristic, complemented
3#24906110C212yesmaximal
4#24905359C212yesmaximal, complemented
5C83×#1274C211yesmaximal, characteristic, complemented
6#19128251C2211yescharacteristic
7#19127725C2211yescharacteristic
8C2×C4⋊(C83⋊C8)—21no—
9C2×C4⋊(C83⋊C8)—21no—
10C2×C4⋊(C83⋊C8)C2212yes—
11C2×#14581720—21no—
12C2×#14581720—21no—
13#19127446—22no—
14#19127444—22no—
15#431×C83C2212yes—
16#376×C83C2211yescharacteristic
17C2×#13050590—21no—
18C2×#13050574—21no—
19C22×C83⋊C8—22no—
20#14581792D411yescharacteristic
21#14581789D411yescharacteristic
22#14581780—21no—
23#14581766—22no—
24C4×D332—22no—
25C4⋊(C83⋊C8)—22no—
26C4⋊(C83⋊C8)—22no—
27C4⋊(C83⋊C8)—22no—
28C4⋊(C83⋊C8)—42no—
29C4⋊(C83⋊C8)—42no—
30#14581720—42no—
31#14581720—42no—
32C2×C42×C83D411yescharacteristic
33C2×C42×C83D411yescharacteristic
34C2×C42×C83C2311yescharacteristic
35C4⋊C8×C83—22no—
36C22⋊C8×C83—22no—
37C4×C8×C83—22no—
38#95×C83—22no—
39#13050590—41no—
40#13050590—41no—
41#13050590—41no—
42#13050590—41no—
43C2×Dic166—21no—
44C2×D332—21no—
45C2×D332—22no—
46C2×C22⋊D83—41no—
47C2×C4×D83—41no—
48#13050574—41no—
49#13050574—41no—
50C2×C83⋊C8—42no—
51C2×C83⋊C8—41no—
52C2×C83⋊C8—42no—
53C2×C83⋊C8—42no—
54C4⋊Dic83—22no—
55(C2×D83)⋊C4—42no—
56C22×C4×C83—21no—
57C22×C4×C83—21no—
58C22×C4×C83C2×D411yescharacteristic
59C22×C4×C83#4112yes—
60C2×C8×C83—42no—
61C42×C83—21no—
62C42×C83—21no—
63C42×C83D812yes—
64C42×C83SD1612yes—
65C42×C83C2×D411yescharacteristic
66C42×C83#4112yes—
67C42×C83#4111yescharacteristic
68C23×C83#12311yescharacteristic
69C22×D83—41no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—42no—
73C2×C4×C83—42no—
74C2×C4×C83—42no—
75C2×C4×C83#12111yescharacteristic
76C2×C4×C83#12312yes—
77C2×C4×C83#12411yescharacteristic
78C2×C4×C83#12612yes—
79C2×C4×C83C2×D811yescharacteristic
80C2×C4×C83C2×SD1611yescharacteristic
81C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C83#13711yescharacteristic
83C22⋊D83—81no—
84C22⋊D83—81no—
85C2×Dic83—41no—
86D332—41no—
87D332—41no—
88D332—82no—
89C4×D83—81no—
90C4×D83—81no—
91Dic166—41no—
92Dic166—41no—
93C8×C83—82no—
94C83⋊C8—82no—
95C83⋊C8—81no—
96C22×C83—22no—
97C22×C83#39711yescharacteristic
98C22×C83#48011yescharacteristic
99C22×C83#48111yescharacteristic
100C2×D83—81no—
101C2×D83—82no—
102C4×C83—22no—
103C4×C83—22no—
104C4×C83—42no—
105C4×C83—42no—
106C4×C83#45911yescharacteristic
107C4×C83#46111yescharacteristic
108C4×C83#48611yescharacteristic
109C4×C83#48911yescharacteristic
110Dic83—81no—
111#10627—831nomaximal, Sylow
112C2×C83—21no—
113C2×C83—21no—
114C2×C83#207111yescharacteristic
115C2×C83#208012yes—
116D83—161no—
117#1963—831no—
118#1843—831no—
119#1540—832no—
120#1301—832no—
121#1274D8311yescharacteristic, complemented
122C83#1062711yescharacteristic, complemented, Sylow
123#532—831no—
124#431C2×D8312yes—
125#423—831no—
126C2×C4⋊C8—832no—
127C2×C4⋊C8—1661no—
128C2×C4⋊C8—1661no—
129C2×#104—1661no—
130C2×#104—1661no—
131#376C2×D8311yescharacteristic
132#332—1662no—
133#331—1662no—
134C2×#41—1661no—
135C2×M16—1661no—
136C22×C8—1662no—
137C4⋊Q8—831no—
138#127—831no—
139#124—1661no—
140#121—1662no—
141C4×D4—1662no—
142C2×C42D33211yescharacteristic
143C2×C42D33211yescharacteristic
144C2×C42C22×D8311yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—3322no—
147C4⋊C8—3322no—
148C4⋊C8—1662no—
149C4⋊C8—1662no—
150C4⋊C8—1662no—
151#104—3322no—
152#104—3322no—
153C22⋊C8—22no—
154C4×C8—22no—
155#95—22no—
156#41—3321no—
157#41—3321no—
158#41—3321no—
159#41—3321no—
160C2×Q8—1661no—
161C2×D4—3321no—
162C2×D4—1661no—
163C2×D4—1662no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—3321no—
167C22×C4C2×D33211yescharacteristic
168C22×C4#1305058912yes—
169M16—3321no—
170M16—3321no—
171C2×C8—42no—
172C2×C8—3322no—
173C2×C8—3321no—
174C2×C8—3322no—
175C2×C8—3322no—
176C4⋊C4—1662no—
177C22⋊C4—3322no—
178C42—21no—
179C42—21no—
180C42D66412yes—
181C42#1305057912yes—
182C42D4×D8311yescharacteristic
183C42#1305059012yes—
184C42#1305059111yescharacteristic
185C23—3321no—
186C23#1458177611yescharacteristic
187Q8—3321no—
188Q8—3321no—
189D4—3321no—
190D4—3321no—
191D4—6642no—
192D4—6641no—
193D4—6641no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—3321no—
200C2×C4—6641no—
201C2×C4—6641no—
202C2×C4#1458176711yescharacteristic
203C2×C4#1458177412yes—
204C2×C4#1458178011yescharacteristic
205C2×C4#1458178712yes—
206C2×C4C2×D66411yescharacteristic
207C2×C4C2×#1305057911yescharacteristic
208C2×C4#1458182011yescharacteristic
209C2×C4#1458182511yescharacteristic
210C8—82no—
211C8—6642no—
212C8—6641no—
213C22—22no—
214C22—6641no—
215C22—6642no—
216C22#1912762811yescharacteristic
217C22#1912801511yescharacteristic
218C22#1912802011yescharacteristic
219C4—22no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—6641no—
224C4#1912789711yescharacteristic
225C4#1912790711yescharacteristic
226C4#1912804811yescharacteristic
227C4#1912806111yescharacteristic
228C2—21no—
229C2—21no—
230C2—13281no—
231C2#2490886811yescharacteristic
232C2#2490891512yes—
subgroupG/H
centerC2×C4#14581780
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1274C2
socleC22×C83#397
prime radical#19128251C22
hypercenter#1274D83
hypocenterC83#10627
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
2. #24907617C2semidirect
4. #24905359C2semidirect
5. C83×#1274C2semidirect
121. #1274D83semidirect
122. C83#10627semidirect

Maximal subgroups

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

subgroupindex
#249082952
#249076172
#249061102
#249053592
C83×#12742
#1062783