Finite Group Databasebeta

← #35315568

Subgroups — #35315568

Order 21248.

372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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#24912675C211yesmaximal, characteristic
2#24907459C211yesmaximal, characteristic
3#24905491C214yesmaximal
4C83×#1461C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic
6#19127725—21no—
7C2×#14581711—21no—
8#19127640C412yes—
9#19127640C2212yes—
10#19127608—21no—
11#19127607—21no—
12#19127607—21no—
13#19127566—21no—
14C4×C4⋊Dic83C411yescharacteristic
15#19127470C2212yes—
16C2×C4×Q8×C83C2211yescharacteristic
17#431×C83—21no—
18#324×C83—21no—
19#324×C83C2212yes—
20C22×Dic166—21no—
21C2×#13050574—22no—
22#14581792—41no—
23#14581792—41no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83C2×C412yes—
27C2×C4×Dic83—21no—
28C4⋊(C83⋊C8)—41no—
29C4⋊(C83⋊C8)—41no—
30#14581711—42no—
31C83⋊#95—22no—
32C83⋊#95—22no—
33C83⋊#95—22no—
34C83⋊#95—22no—
35C83⋊#95C2×C412yes—
36C22×Q8×C83D411yescharacteristic
37C4×Q8×C83—41no—
38C4×Q8×C83—41no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C4⋊C4×C83C2311yescharacteristic
42C2×C42×C83C2×C411yescharacteristic
43C2×C42×C83D411yescharacteristic
44C2×C42×C83D411yescharacteristic
45C4⋊C8×C83—41no—
46C22⋊C8×C83—22no—
47C4×C8×C83—41no—
48C2×Dic166—44no—
49C2×Dic166—41no—
50C2×Dic166—41no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53#13050574—42no—
54#13050574—42no—
55C2×C83⋊C8—42no—
56C4⋊Dic83—42no—
57C4⋊Dic83—42no—
58C4⋊Dic83—42no—
59C4×Dic83—42no—
60C2×Q8×C83—41no—
61C2×Q8×C83—41no—
62C2×Q8×C83SD1612yes—
63C2×Q8×C83Q1612yes—
64C22×C4×C83C22⋊C411yescharacteristic
65C22×C4×C83C22⋊C411yescharacteristic
66C22×C4×C83C22⋊C411yescharacteristic
67C22×C4×C83C22⋊C411yescharacteristic
68C22×C4×C83C22×C411yescharacteristic
69C22×C4×C83C2×D411yescharacteristic
70C22×C4×C83C2×D411yescharacteristic
71C2×C8×C83—42no—
72C4⋊C4×C83—41no—
73C4⋊C4×C83—41no—
74C4⋊C4×C83—41no—
75C42×C83—21no—
76C42×C83—21no—
77C42×C83—21no—
78C42×C83—21no—
79C42×C83—21no—
80C42×C83—21no—
81C23×C83C2×C22⋊C411yescharacteristic
82Q8×C83—42no—
83Q8×C83—42no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—41no—
95C2×C4×C83—41no—
96C2×C4×C83—41no—
97C2×C4×C83#9911yescharacteristic
98C2×C4×C83#9911yescharacteristic
99C2×C4×C83#10312yes—
100C2×C4×C83#10312yes—
101C2×C4×C83C2×SD1611yescharacteristic
102C2×C4×C83C2×Q1611yescharacteristic
103C2×Dic83—42no—
104C2×Dic83—42no—
105C2×Dic83—44no—
106C2×Dic83—42no—
107C2×Dic83—42no—
108Dic166—82no—
109Dic166—82no—
110C8×C83—82no—
111C83⋊C8—82no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83C2×#9911yescharacteristic
115C22×C83C2×#10311yescharacteristic
116C22×C83#42111yescharacteristic
117C4×C83—21no—
118C4×C83—21no—
119C4×C83—22no—
120C4×C83—22no—
121C4×C83—21no—
122C4×C83—41no—
123C4×C83—41no—
124C4×C83—41no—
125C4×C83#35512yes—
126Dic83—82no—
127Dic83—82no—
128#11907—831nomaximal, Sylow
129C2×C83—21no—
130C2×C83—21no—
131C2×C83#152011yescharacteristic
132C2×C83#212311yescharacteristic
133C2×C83C2×#35511yescharacteristic
134#2568—831no—
135#1801—831no—
136#1461D8311yescharacteristic
137#1353—834no—
138C83#1190711yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—831no—
140C2×C4×Q8C2×D8311yescharacteristic
141#431—21no—
142#423—1661no—
143C2×#101—1661no—
144#400—832no—
145#400—832no—
146#391—1661no—
147#391—1661no—
148#391—1661no—
149#382—1661no—
150C4×C4⋊C4—831no—
151#343—832no—
152#324—21no—
153#324C2×D8312yes—
154C22×Q8—1661no—
155C22×Q8D33211yescharacteristic
156C2×M16—1662no—
157C4⋊Q8—3321no—
158C4⋊Q8—3321no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—1661no—
163C2×C4⋊C4—1661no—
164C2×C4⋊C4C4×D8311yescharacteristic
165C2×C4⋊C4C22⋊D8311yescharacteristic
166C2×C4⋊C4C22×D8311yescharacteristic
167C2×C42—1661no—
168C2×C42C4×D8311yescharacteristic
169C2×C42D33211yescharacteristic
170C2×C42C22⋊D8311yescharacteristic
171C4⋊C8—41no—
172C4⋊C8—3321no—
173C4⋊C8—3321no—
174#101—3322no—
175C22⋊C8—22no—
176C4×C8—41no—
177#95—832no—
178#95—1662no—
179#95—1662no—
180#95—1662no—
181#95—1662no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—3324no—
185C2×Q8—3321no—
186C2×Q8—3321no—
187C2×Q8#1305057912yes—
188C2×Q8Dic33212yes—
189C22×C4—1662no—
190C22×C4—1662no—
191C22×C4(C2×D83)⋊C411yescharacteristic
192C22×C4(C2×D83)⋊C411yescharacteristic
193C22×C4(C2×D83)⋊C411yescharacteristic
194C22×C4(C2×D83)⋊C411yescharacteristic
195C22×C4C2×C4×D8311yescharacteristic
196C22×C4C2×C22⋊D8311yescharacteristic
197C22×C4C2×D33211yescharacteristic
198M16—3322no—
199M16—3322no—
200C2×C8—42no—
201C2×C8—3322no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—3322no—
206C4⋊C4—3322no—
207C4⋊C4—3322no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—3322no—
215C23C2×(C2×D83)⋊C411yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—6642no—
219Q8—6642no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—3322no—
234C2×C4—3324no—
235C2×C4—3322no—
236C2×C4—3322no—
237C2×C4—3322no—
238C2×C4#1458170611yescharacteristic
239C2×C4#1458170611yescharacteristic
240C2×C4#1458171712yes—
241C2×C4#1458171712yes—
242C2×C4C2×#1305057911yescharacteristic
243C2×C4C2×Dic33211yescharacteristic
244C8—82no—
245C8—6642no—
246C22—21no—
247C22—21no—
248C22C2×#1458170611yescharacteristic
249C22C2×#1458171711yescharacteristic
250C22#1912771711yescharacteristic
251C4—21no—
252C4—21no—
253C4—22no—
254C4—22no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—6642no—
260C4—6642no—
261C4#1912749712yes—
262C2—21no—
263C2—21no—
264C2#2490603611yescharacteristic
265C2#2490911611yescharacteristic
266C2C2×#1912749711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1461C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1461D83
hypocenterC83#11907
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
138. C83#11907semidirect

Maximal subgroups

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

subgroupindex
#249126752
#249074592
#249054912
C83×#14612
#1190783