Finite Group Databasebeta

← #35307543

Subgroups — #35307543

Order 21248.

402 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 89 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#24910768C211yesmaximal, characteristic, complemented
2#24908295C211yesmaximal, characteristic
3#24905959C212yesmaximal
4#24905365C212yesmaximal, complemented
5C83×#1274C211yesmaximal, characteristic, complemented
6#19128251C2211yescharacteristic
7#19127725C2211yescharacteristic
8C2×#14581720—22no—
9#19127643—22no—
10#19127635—22no—
11C4×Dic83⋊C4—22no—
12C4×C4⋊Dic83C2212yes—
13#19127436—22no—
14#19127436—22no—
15#431×C83C2212yes—
16#376×C83C2211yescharacteristic
17C2×#13050590—21no—
18C2×#13050574—21no—
19#14581792Q811yescharacteristic
20#14581789Q811yescharacteristic
21#14581780—21no—
22#14581766—22no—
23C4×D332—22no—
24C2×C4⋊Dic83—22no—
25C2×Dic83⋊C4—22no—
26C2×Dic83⋊C4—22no—
27C2×C4×Dic83—22no—
28C4⋊(C83⋊C8)—22no—
29#14581720—44no—
30C83⋊#95—22no—
31C83⋊#95—22no—
32C83⋊#95D412yes—
33C83⋊#95D412yes—
34C2×C42×C83D412yes—
35C2×C42×C83C2311yescharacteristic
36C4⋊C8×C83—22no—
37C22⋊C8×C83—22no—
38C4×C8×C83—22no—
39#95×C83—22no—
40#13050590—42no—
41#13050590—42no—
42C2×Dic166—21no—
43C2×D332—21no—
44C2×D332—22no—
45C2×C22⋊D83—41no—
46C22×Dic83—22no—
47C22×Dic83—22no—
48C2×C4×D83—41no—
49#13050574—42no—
50C2×C83⋊C8—41no—
51C4⋊Dic83—22no—
52C4⋊Dic83—44no—
53Dic83⋊C4—44no—
54Dic83⋊C4—44no—
55(C2×D83)⋊C4—42no—
56C4×Dic83—42no—
57C4×Dic83—42no—
58C22×C4×C83—22no—
59C22×C4×C83C2×D412yes—
60C22×C4×C83C2×D411yescharacteristic
61C2×C8×C83—42no—
62C42×C83—22no—
63C42×C83—22no—
64C42×C83C2×Q811yescharacteristic
65C42×C83#4111yescharacteristic
66C42×C83#4112yes—
67C23×C83#12011yescharacteristic
68C22×D83—41no—
69C2×C4×C83—42no—
70C2×C4×C83—42no—
71C2×C4×C83—42no—
72C2×C4×C83—42no—
73C2×C4×C83C22⋊Q812yes—
74C2×C4×C83C22⋊Q811yescharacteristic
75C2×C4×C83#12412yes—
76C2×C4×C83#12411yescharacteristic
77C2×C4×C83#13512yes—
78C2×C4×C83AGL(1,ℤ/8)12yes—
79C22⋊D83—82no—
80C2×Dic83—24no—
81C2×Dic83—42no—
82C2×Dic83—42no—
83C2×Dic83—42no—
84C2×Dic83—42no—
85C2×Dic83—41no—
86C2×Dic83—42no—
87D332—42no—
88D332—82no—
89C4×D83—81no—
90C4×D83—81no—
91Dic166—42no—
92C8×C83—82no—
93C83⋊C8—81no—
94C22×C83—22no—
95C22×C83#39311yescharacteristic
96C22×C83#44912yes—
97C2×D83—81no—
98C2×D83—82no—
99C4×C83—22no—
100C4×C83—22no—
101C4×C83—82no—
102C4×C83#47812yes—
103C4×C83#48811yescharacteristic
104C4×C83#49011yescharacteristic
105Dic83—44no—
106Dic83—82no—
107Dic83—81no—
108#10372—831nomaximal, Sylow
109C2×C83—22no—
110C2×C83#202511yescharacteristic
111C2×C83#203112yes—
112D83—161no—
113#2402—831no—
114#1963—831no—
115#1502—832no—
116#1303—832no—
117#1274D8311yescharacteristic, complemented
118C83#1037211yescharacteristic, complemented, Sylow
119#532—831no—
120#431C2×D8312yes—
121#423—831no—
122C2×#104—1662no—
123#400—1662no—
124#398—1662no—
125C4×C4⋊C4—832no—
126C4×C4⋊C4—1662no—
127#376C2×D8311yescharacteristic
128#327—1662no—
129#327—1662no—
130C2×#41—1661no—
131C2×M16—1661no—
132C4⋊Q8—831no—
133#127—831no—
134#124—1661no—
135#121—1662no—
136C4×D4—1662no—
137C2×C4⋊C4—1662no—
138C2×C4⋊C4—1662no—
139C2×C4⋊C4—1662no—
140C2×C42—1662no—
141C2×C42D33212yes—
142C2×C42C22×D8311yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—1662no—
145#104—3324no—
146C22⋊C8—22no—
147C4×C8—22no—
148#95—22no—
149#95—832no—
150#95—832no—
151#95—1662no—
152#95—1662no—
153#41—3322no—
154#41—3322no—
155C2×Q8—1661no—
156C2×D4—3321no—
157C2×D4—1662no—
158C2×D4—1661no—
159C22×C4—22no—
160C22×C4—3321no—
161C22×C4—1662no—
162C22×C4—1662no—
163C22×C4D4×D8312yes—
164C22×C4C2×D33211yescharacteristic
165M16—3322no—
166C2×C8—42no—
167C2×C8—3321no—
168C4⋊C4—3324no—
169C4⋊C4—3324no—
170C4⋊C4—3324no—
171C4⋊C4—1662no—
172C22⋊C4—3322no—
173C42—22no—
174C42—22no—
175C42—3322no—
176C42—3322no—
177C42Q8×D8311yescharacteristic
178C42#1305058911yescharacteristic
179C42#1305059012yes—
180C23—3321no—
181C23#1458176211yescharacteristic
182Q8—3322no—
183D4—3322no—
184D4—6642no—
185D4—6642no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—3322no—
191C2×C4—3322no—
192C2×C4—3322no—
193C2×C4—3322no—
194C2×C4—3322no—
195C2×C4—3321no—
196C2×C4—6641no—
197C2×C4—6641no—
198C2×C4—1664no—
199C2×C4#1458177012yes—
200C2×C4#1458177211yescharacteristic
201C2×C4#1458177812yes—
202C2×C4#1458178011yescharacteristic
203C2×C4#1458181412yes—
204C2×C4#1458182012yes—
205C8—82no—
206C8—6641no—
207C22—22no—
208C22—6641no—
209C22—6642no—
210C22#1912761411yescharacteristic
211C22#1912784812yes—
212C4—22no—
213C4—22no—
214C4—82no—
215C4—3324no—
216C4—6642no—
217C4—6641no—
218C4#1912800212yes—
219C4#1912805611yescharacteristic
220C4#1912806611yescharacteristic
221C2—22no—
222C2—13281no—
223C2#2490861011yescharacteristic
224C2#2490863412yes—
subgroupG/H
centerC2×C4#14581780
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1274C2
socleC22×C83#393
prime radical#19128251C22
hypercenter#1274D83
hypocenterC83#10372
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
1. #24910768C2semidirect
4. #24905365C2semidirect
5. C83×#1274C2semidirect
117. #1274D83semidirect
118. C83#10372semidirect

Maximal subgroups

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

subgroupindex
#249107682
#249082952
#249059592
#249053652
C83×#12742
#1037283