Finite Group Databasebeta

← #35302268

Subgroups — #35302268

Order 21248.

454 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 115 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#24912825C211yesmaximal, characteristic
2#24907459C211yesmaximal, characteristic
3#24905893C212yesmaximal
4#24905322C212yesmaximal
5C83×#2983C211yesmaximal, characteristic
6C2×C4×Dic166C2211yescharacteristic
7#19127725C2212yes—
8C2×#14581711—21no—
9#19127640C412yes—
10#19127621C412yes—
11#19127608—21no—
12#19127597—21no—
13#19127585—22no—
14#19127567C2211yescharacteristic
15#19127470—21no—
16#19127433—24no—
17C2×C4×Q8×C83C2211yescharacteristic
18#446×C83—24no—
19#432×C83—22no—
20#431×C83C2212yes—
21C22×Dic166C2×C411yescharacteristic
22C2×#13050574—22no—
23C4×Dic166—22no—
24C4×Dic166—22no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×Dic83⋊C4—22no—
28C2×Dic83⋊C4—21no—
29C2×C4×Dic83—21no—
30C4⋊(C83⋊C8)—22no—
31C4⋊(C83⋊C8)—22no—
32#14581711—42no—
33C83⋊#95—22no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95—21no—
37C83⋊#95C2×C412yes—
38C22×Q8×C83D411yescharacteristic
39C4×Q8×C83—22no—
40C4×Q8×C83—22no—
41C2×C4⋊C4×C83C2×C412yes—
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C42×C83D412yes—
44C2×C42×C83C2311yescharacteristic
45C4⋊C8×C83—24no—
46C4⋊C8×C83—22no—
47C22⋊C8×C83—22no—
48#97×C83—24no—
49C4×C8×C83D414yes—
50C2×Dic166—22no—
51C2×Dic166—42no—
52C22×Dic83—21no—
53C22×Dic83—22no—
54C22×Dic83—21no—
55#13050574—42no—
56#13050574—42no—
57C2×C83⋊C8—42no—
58C4⋊Dic83—22no—
59C4⋊Dic83—41no—
60C4⋊Dic83—41no—
61Dic83⋊C4—44no—
62Dic83⋊C4—41no—
63Dic83⋊C4—41no—
64C4×Dic83—42no—
65C2×Q8×C83—22no—
66C2×Q8×C83—21no—
67C2×Q8×C83—21no—
68C2×Q8×C83—22no—
69C22×C4×C83C22⋊C412yes—
70C22×C4×C83C22⋊C412yes—
71C22×C4×C83C22×C411yescharacteristic
72C22×C4×C83C2×D411yescharacteristic
73C22×C4×C83C2×D411yescharacteristic
74C2×C8×C83—24no—
75C4⋊C4×C83—24no—
76C4⋊C4×C83—22no—
77C42×C83—22no—
78C42×C83—22no—
79C42×C83C2×D412yes—
80C42×C83#4112yes—
81C23×C83C2×C22⋊C411yescharacteristic
82Q8×C83—42no—
83Q8×C83—42no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—41no—
89C2×C4×C83—41no—
90C2×C4×C83—42no—
91C2×C4×C83C4×D412yes—
92C2×C4×C83#12011yescharacteristic
93C2×C4×C83#12112yes—
94C2×C4×C83#12311yescharacteristic
95C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C83#13712yes—
97C2×C4×C83#13711yescharacteristic
98C2×Dic83—22no—
99C2×Dic83—41no—
100C2×Dic83—41no—
101C2×Dic83—42no—
102C2×Dic83—42no—
103C2×Dic83—42no—
104C2×Dic83—41no—
105C2×Dic83—42no—
106Dic166—24no—
107Dic166—42no—
108C8×C83—44no—
109C83⋊C8—82no—
110C22×C83—21no—
111C22×C83—21no—
112C22×C83#38611yescharacteristic
113C22×C83#41711yescharacteristic
114C22×C83#41911yescharacteristic
115C4×C83—22no—
116C4×C83—21no—
117C4×C83—21no—
118C4×C83—42no—
119C4×C83—42no—
120C4×C83#46812yes—
121C4×C83#47012yes—
122Dic83—42no—
123Dic83—82no—
124Dic83—81no—
125#9488—831nomaximal, Sylow
126C2×C83—21no—
127C2×C83—21no—
128C2×C83#189111yescharacteristic
129C2×C83#189411yescharacteristic
130C2×C83#191211yescharacteristic
131#2983D8311yescharacteristic
132#2581—831no—
133#1801—831no—
134#1487—832no—
135#1286—832no—
136C83#948811yescharacteristic, complemented, Sylow
137C2×C4×Q8—831no—
138C2×C4×Q8C2×D8311yescharacteristic
139#446—24no—
140#432—22no—
141#431C2×D8312yes—
142#423—832no—
143C2×#101—1661no—
144#400—832no—
145#395—832no—
146#391—1661no—
147#389—1661no—
148#387—1662no—
149#382—831no—
150#343—1661no—
151Q8⋊C8—1664no—
152C22×Q8—831no—
153C22×Q8D33211yescharacteristic
154C2×M16—1662no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1662no—
158C4×Q8—1662no—
159C2×C4⋊C4—831no—
160C2×C4⋊C4—1662no—
161C2×C4⋊C4—1661no—
162C2×C4⋊C4—1661no—
163C2×C4⋊C4C4×D8312yes—
164C2×C4⋊C4D33211yescharacteristic
165C2×C42—1661no—
166C2×C42C22⋊D8312yes—
167C2×C42C22×D8311yescharacteristic
168C4⋊C8—24no—
169C4⋊C8—22no—
170C4⋊C8—1662no—
171C4⋊C8—1662no—
172#101—3322no—
173C22⋊C8—22no—
174#97—24no—
175C4×C8C22⋊D8314yes—
176#95—832no—
177#95—1662no—
178#95—1662no—
179#95—1662no—
180#95—1661no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—3322no—
186C2×Q8—1662no—
187C22×C4—1662no—
188C22×C4—1661no—
189C22×C4—1661no—
190C22×C4(C2×D83)⋊C412yes—
191C22×C4(C2×D83)⋊C412yes—
192C22×C4C2×C4×D8311yescharacteristic
193C22×C4C2×C22⋊D8311yescharacteristic
194C22×C4C2×D33211yescharacteristic
195M16—3322no—
196M16—3322no—
197C2×C8—24no—
198C2×C8—3322no—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—3324no—
202C4⋊C4—3321no—
203C4⋊C4—3321no—
204C4⋊C4—3321no—
205C4⋊C4—3321no—
206C4⋊C4—1662no—
207C42—22no—
208C42—22no—
209C42—3322no—
210C42C2×C22⋊D8312yes—
211C42#1305059012yes—
212C23C2×(C2×D83)⋊C411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—3322no—
216Q8—1664no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—3321no—
225C2×C4—3321no—
226C2×C4—3322no—
227C2×C4—3322no—
228C2×C4—3322no—
229C2×C4—3321no—
230C2×C4—3322no—
231C2×C4—1662no—
232C2×C4C4×C22⋊D8312yes—
233C2×C4#1458176111yescharacteristic
234C2×C4#1458176612yes—
235C2×C4#1458177511yescharacteristic
236C2×C4#1458182011yescharacteristic
237C2×C4#1458182512yes—
238C2×C4#1458182511yescharacteristic
239C8—44no—
240C8—6642no—
241C22—21no—
242C22—21no—
243C22#1912758011yescharacteristic
244C22#1912770311yescharacteristic
245C22#1912771111yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—3322no—
252C4—6642no—
253C4—6641no—
254C4#1912794712yes—
255C4#1912795912yes—
256C2—21no—
257C2—21no—
258C2#2490788111yescharacteristic
259C2#2490789611yescharacteristic
260C2#2490800011yescharacteristic
subgroupG/H
centerC22#19127580
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2983C2
socleC22×C83#386
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenter#2983D83
hypocenterC83#9488
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
136. C83#9488semidirect

Maximal subgroups

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

subgroupindex
#249128252
#249074592
#249058932
#249053222
C83×#29832
#948883