Finite Group Databasebeta

← #35313813

Subgroups — #35313813

Order 21248.

418 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 77 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#24910088C211yesmaximal, characteristic
2#24905796C212yesmaximal
3#24905407C211yesmaximal, characteristic, complemented
4#24905406C212yesmaximal, complemented
5C83×#1324C211yesmaximal, characteristic, complemented
6C2×#14581750C2211yescharacteristic
7C2×#14581749C411yescharacteristic
8C2×C4⋊C4×D83C411yescharacteristic
9#19127755—22no—
10#19127598—22no—
11#19127575—22no—
12#19127571C411yescharacteristic
13#19127570C411yescharacteristic
14#19127566C2211yescharacteristic
15#19127427C2212yes—
16#19127424—22no—
17#382×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×C4×D83—21no—
20C22×C4×D83C2×C411yescharacteristic
21#14581750—42no—
22#14581749—22no—
23#14581749—22no—
24C2×C4⋊Dic83—22no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—22no—
30C4⋊C4×D83—42no—
31C2×(C2×D83)⋊C4—21no—
32C2×(C2×D83)⋊C4D412yes—
33C2×C4×Dic83C2×C411yescharacteristic
34C2×C4×Dic83D412yes—
35Dic83⋊C8—42no—
36C22⋊(C83⋊C8)—22no—
37#14581702—42no—
38C83⋊#95—22no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83C2311yescharacteristic
41C2×C42×C83C2×C411yescharacteristic
42C22⋊C8×C83—22no—
43#95×C83—22no—
44C23×D83C22⋊C411yescharacteristic
45C22×Dic83—21no—
46C22×Dic83—21no—
47C22×Dic83C22⋊C411yescharacteristic
48C22×Dic83C22⋊C412yes—
49C2×C4×D83—22no—
50C2×C4×D83—21no—
51C2×C4×D83—21no—
52C2×C4×D83—22no—
53C2×C4×D83—42no—
54C2×C4×D83—82no—
55C2×C83⋊C8—42no—
56C4⋊Dic83—41no—
57C4⋊Dic83—41no—
58C4⋊Dic83—82no—
59Dic83⋊C4—24no—
60Dic83⋊C4—42no—
61Dic83⋊C4—82no—
62(C2×D83)⋊C4—24no—
63(C2×D83)⋊C4—42no—
64C4×Dic83—21no—
65C4×Dic83—22no—
66C4×Dic83—22no—
67C4×Dic83M1612yes—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83C22×C411yescharacteristic
71C22×C4×C83C2×D412yes—
72C2×C8×C83—42no—
73C4⋊C4×C83—41no—
74C4⋊C4×C83—41no—
75C42×C83—21no—
76C42×C83M1612yes—
77C23×C83C2×C22⋊C411yescharacteristic
78C22×D83—21no—
79C22×D83—21no—
80C22×D83—41no—
81C22×D83—41no—
82C22×D83#10112yes—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83—42no—
90C2×C4×C83—42no—
91C2×C4×C83—41no—
92C2×C4×C83C2×M1611yescharacteristic
93C2×C4×C83C2×M1611yescharacteristic
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—41no—
99C2×Dic83—41no—
100C2×Dic83—42no—
101C2×Dic83—42no—
102C2×Dic83—41no—
103C2×Dic83#9912yes—
104C4×D83—24no—
105C4×D83—24no—
106C4×D83—84no—
107C8×C83—82no—
108C83⋊C8—82no—
109C22×C83—21no—
110C22×C83—21no—
111C22×C83#40711yescharacteristic
112C22×C83C2×#9911yescharacteristic
113C22×C83C2×#10111yescharacteristic
114C2×D83—22no—
115C2×D83—42no—
116C2×D83—41no—
117C2×D83—41no—
118C2×D83—42no—
119C4×C83—21no—
120C4×C83—21no—
121C4×C83—42no—
122C4×C83—42no—
123C4×C83—81no—
124Dic83—22no—
125Dic83—42no—
126Dic83—42no—
127Dic83—81no—
128#11515—831nomaximal, Sylow
129C2×C83—21no—
130C2×C83—21no—
131C2×C83#146311yescharacteristic
132C2×C83#212911yescharacteristic
133C2×C83#213411yescharacteristic
134D83—42no—
135#2317—831no—
136#1464—832no—
137#1324—831no—
138#1324—832no—
139#1324D8311yescharacteristic, complemented
140C83#1151511yescharacteristic, complemented, Sylow
141C2×#117—831no—
142C2×#117—831no—
143C22×C4⋊C4—831no—
144#432—1662no—
145#389—1662no—
146#385—1662no—
147#384—831no—
148#384—831no—
149#382—831no—
150#382C2×D8311yescharacteristic
151#324—832no—
152#324C2×D8312yes—
153#323—1662no—
154C23×C4—831no—
155C23×C4—1661no—
156#117—3322no—
157#117—1662no—
158#117—1662no—
159C2×C4⋊C4—3322no—
160C2×C4⋊C4—831no—
161C2×C4⋊C4—831no—
162C2×C4⋊C4—1661no—
163C2×C4⋊C4—1662no—
164C2×C4⋊C4—1662no—
165C2×C4⋊C4—1661no—
166C2×C4⋊C4C4×D8311yescharacteristic
167C2×C4⋊C4C22×D8311yescharacteristic
168C2×C22⋊C4—832no—
169C2×C22⋊C4—1661no—
170C2×C42—831no—
171C2×C42—832no—
172C2×C42C4×D8311yescharacteristic
173C4⋊C8—3322no—
174C22⋊C8—22no—
175C22⋊C8—1662no—
176#97—3322no—
177#95—22no—
178#95—1662no—
179C24—831no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—3322no—
183C22×C4—832no—
184C22×C4—831no—
185C22×C4—6642no—
186C22×C4—1662no—
187C22×C4—1662no—
188C22×C4—1661no—
189C22×C4—1661no—
190C22×C4—1661no—
191C22×C4—1661no—
192C22×C4C2×C4×D8311yescharacteristic
193C22×C4D4×D8312yes—
194C2×C8—42no—
195C2×C8—3322no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—3321no—
199C4⋊C4—3321no—
200C4⋊C4—3322no—
201C4⋊C4—6642no—
202C4⋊C4—6642no—
203C4⋊C4—1664no—
204C22⋊C4—3322no—
205C22⋊C4—1664no—
206C42—21no—
207C42—832no—
208C42—1662no—
209C42—1662no—
210C42—1661no—
211C42#1305057312yes—
212C23—3321no—
213C23—3321no—
214C23—832no—
215C23—1661no—
216C23—1661no—
217C23C22⋊C4×D8311yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—3321no—
228C2×C4—3321no—
229C2×C4—3322no—
230C2×C4—3322no—
231C2×C4—3321no—
232C2×C4—832no—
233C2×C4—6644no—
234C2×C4—1664no—
235C2×C4—1664no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1661no—
239C2×C4—1661no—
240C2×C4M16×D8311yescharacteristic
241C2×C4C2×#1305057311yescharacteristic
242C8—82no—
243C8—6642no—
244C22—21no—
245C22—21no—
246C22—3322no—
247C22—3322no—
248C22—3321no—
249C22—3321no—
250C22—1662no—
251C22#1912766311yescharacteristic
252C22#99×D8311yescharacteristic
253C22#101×D8311yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—3322no—
260C4—3322no—
261C4—6641no—
262C4—1662no—
263C2—21no—
264C2—21no—
265C2—3322no—
266C2#2490579111yescharacteristic
267C2#2490913611yescharacteristic
268C2#2490915311yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1324C2
socleC22×C83C2×#99
prime radicalC23×D83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1324D83
hypocenterC83#11515
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
3. #24905407C2semidirect
4. #24905406C2semidirect
5. C83×#1324C2semidirect
139. #1324D83semidirect
140. C83#11515semidirect

Maximal subgroups

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

subgroupindex
#249100882
#249057962
#249054072
#249054062
C83×#13242
#1151583