Finite Group Databasebeta

← #35301807

Subgroups — #35301807

Order 21248.

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

normal subgroups only

This group has 534 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#24916504C211yesmaximal, characteristic
2#24912455C211yesmaximal, characteristic
3#24906136C212yesmaximal, complemented
4#24905364C212yesmaximal, complemented
5C83×#1800C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7C2×#14581779—21no—
8#19127779—24no—
9#19127754C2212yes—
10#19127665—22no—
11#19127626C414yes—
12#19127606—21no—
13#19127561—22no—
14C4×C4⋊Dic83C2211yescharacteristic
15#19127448—24no—
16C2×C4×D4×C83C2211yescharacteristic
17#423×C83C2212yes—
18C2×#100×C83—21no—
19#343×C83—21no—
20C22×Dic166—21no—
21C2×#13050574—24no—
22#14581792—22no—
23#14581792—22no—
24#14581779—44no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83C2×C412yes—
28C2×C22⋊Dic83—24no—
29C2×C4×Dic83—21no—
30C4⋊(C83⋊C8)D414yes—
31C22⋊(C83⋊C8)—24no—
32C22⋊(C83⋊C8)—22no—
33C4×C83⋊C8—22no—
34C83⋊#95—24no—
35C83⋊#95C2×C412yes—
36C22×D4×C83D411yescharacteristic
37C23×C4×C83D412yes—
38C2×M16×C83—22no—
39C4×D4×C83—22no—
40C4×D4×C83—22no—
41C2×C4⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83C2×C412yes—
43C2×C42×C83C2311yescharacteristic
44C4⋊C8×C83—22no—
45C4⋊C8×C83—22no—
46#100×C83—42no—
47C2×Dic166—42no—
48C2×Dic166—42no—
49C2×Dic166—41no—
50C2×Dic166—41no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53#13050574—84no—
54C2×C83⋊C8—24no—
55C4⋊Dic83—24no—
56C4⋊Dic83—42no—
57C4⋊Dic83—41no—
58C4⋊Dic83—41no—
59C22⋊Dic83—48no—
60C4×Dic83—42no—
61C24×C83C22⋊C412yes—
62C2×D4×C83—22no—
63C2×D4×C83—22no—
64C2×D4×C83—21no—
65C2×D4×C83—21no—
66C22×C4×C83—22no—
67C22×C4×C83—22no—
68C22×C4×C83—44no—
69C22×C4×C83C22⋊C412yes—
70C22×C4×C83C22×C411yescharacteristic
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83C2×D411yescharacteristic
73M16×C83—42no—
74M16×C83—42no—
75C2×C8×C83—42no—
76C4⋊C4×C83—22no—
77C22⋊C4×C83—24no—
78C42×C83C2×D412yes—
79C42×C83#4112yes—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—42no—
84C23×C83—42no—
85C23×C83C2×C22⋊C411yescharacteristic
86D4×C83—42no—
87D4×C83—42no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—41no—
91C2×C4×C83—44no—
92C2×C4×C83—42no—
93C2×C4×C83—82no—
94C2×C4×C83C4×D412yes—
95C2×C4×C83#12011yescharacteristic
96C2×C4×C83#12112yes—
97C2×C4×C83#12311yescharacteristic
98C2×C4×C83AGL(1,ℤ/8)12yes—
99C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C83#13711yescharacteristic
101C2×Dic83—42no—
102C2×Dic83—42no—
103C2×Dic83—42no—
104C2×Dic83—42no—
105C2×Dic83—42no—
106C2×Dic83—42no—
107Dic166—82no—
108Dic166—82no—
109C8×C83—82no—
110C83⋊C8—44no—
111C22×C83—21no—
112C22×C83—21no—
113C22×C83—44no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83—82no—
118C22×C83#38611yescharacteristic
119C22×C83#41711yescharacteristic
120C22×C83#41811yescharacteristic
121C4×C83—21no—
122C4×C83—22no—
123C4×C83—42no—
124C4×C83#45512yes—
125C4×C83#46812yes—
126C4×C83#46912yes—
127Dic83—82no—
128Dic83—82no—
129#9401—831nomaximal, Sylow
130C2×C83—21no—
131C2×C83—21no—
132C2×C83—82no—
133C2×C83#188511yescharacteristic
134C2×C83#189511yescharacteristic
135C2×C83#191511yescharacteristic
136#2975—831no—
137#2547—831no—
138#1800D8311yescharacteristic
139#1546—832no—
140#1303—832no—
141C83#940111yescharacteristic, complemented, Sylow
142C2×C4⋊Q8—831no—
143C2×#124—1661no—
144C2×C4×D4C2×D8311yescharacteristic
145#436—1664no—
146#431—832no—
147#423C2×D8312yes—
148C2×#100—21no—
149#407—1662no—
150#397—834no—
151#391—1661no—
152#380—1662no—
153C4×C4⋊C4—831no—
154#343—21no—
155#332—1664no—
156C22×Q8—1661no—
157C22×D4C22⋊D8311yescharacteristic
158C23×C4D33212yes—
159C2×M16—22no—
160C2×M16—1664no—
161C4⋊Q8—1662no—
162C4⋊Q8—1662no—
163#124—3324no—
164C4×D4—22no—
165C4×D4—22no—
166C2×C4⋊C4—832no—
167C2×C4⋊C4—1661no—
168C2×C4⋊C4—1661no—
169C2×C4⋊C4C22⋊D8311yescharacteristic
170C2×C22⋊C4—1664no—
171C2×C22⋊C4C4×D8312yes—
172C2×C42—1661no—
173C2×C42C22×D8311yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—834no—
177#100—42no—
178C22⋊C8—1664no—
179C22⋊C8—1662no—
180C4×C8—1662no—
181#95—832no—
182#95—1664no—
183C24(C2×D83)⋊C412yes—
184C2×Q8—3322no—
185C2×Q8—3322no—
186C2×Q8—3321no—
187C2×Q8—3321no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—1662no—
196C22×C4—1662no—
197C22×C4(C2×D83)⋊C412yes—
198C22×C4C2×C4×D8311yescharacteristic
199C22×C4C2×C22⋊D8311yescharacteristic
200C22×C4C2×D33211yescharacteristic
201M16—42no—
202M16—42no—
203M16—6644no—
204C2×C8—42no—
205C2×C8—1664no—
206C4⋊C4—22no—
207C4⋊C4—3322no—
208C4⋊C4—3321no—
209C4⋊C4—3321no—
210C4⋊C4—1664no—
211C22⋊C4—24no—
212C22⋊C4—3328no—
213C42—3322no—
214C42D4×D8312yes—
215C42#1305058912yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×(C2×D83)⋊C411yescharacteristic
222Q8—6642no—
223Q8—6642no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—3322no—
233C2×C4—3322no—
234C2×C4—3322no—
235C2×C4—3322no—
236C2×C4—3322no—
237C2×C4—3322no—
238C2×C4(C22⋊D83)⋊C412yes—
239C2×C4#1458176211yescharacteristic
240C2×C4#1458176412yes—
241C2×C4#1458177611yescharacteristic
242C2×C4#1458181712yes—
243C2×C4#1458181811yescharacteristic
244C2×C4#1458182311yescharacteristic
245C8—82no—
246C8—3324no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22#1912757811yescharacteristic
255C22#1912770011yescharacteristic
256C22#1912770511yescharacteristic
257C4—21no—
258C4—22no—
259C4—42no—
260C4—6642no—
261C4—6642no—
262C4#1912788112yes—
263C4#1912794912yes—
264C4#1912795212yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#2490784911yescharacteristic
269C2#2490789911yescharacteristic
270C2#2490801311yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1800C2
socleC22×C83#386
prime radicalC22×D4×C83D4
cosocleC2×C42×C83C23
hypercenter#1800D83
hypocenterC83#9401
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #24906136C2semidirect
4. #24905364C2semidirect
141. C83#9401semidirect

Maximal subgroups

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

subgroupindex
#249165042
#249124552
#249061362
#249053642
C83×#18002
#940183