Finite Group Databasebeta

← #35404153

Subgroups — #35404153

Order 21248.

1266 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 275 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
1C2×#19128178C211yesmaximal, characteristic
2#24910048C212yesmaximal
3#24907540C211yesmaximal, characteristic
4#24907539C214yesmaximal
5#24907538C212yesmaximal
6#24907194C214yesmaximal
7C83×#1820C211yesmaximal, characteristic
8C23×Dic166C2211yescharacteristic
9#19128178—28no—
10C2×C4×Dic166C414yescomplemented
11C2×#14581757C414yescomplemented
12C2×#14581751C2211yescharacteristic
13C2×#14581748C2214yes—
14C2×#14581748C2212yes—
15C2×#14581748C2212yes—
16#19127605—28no—
17#19127564C2212yes—
18#19127562C2214yes—
19#19127468C2214yes—
20#19127467C2218yes—
21C2×#117×C83C2211yescharacteristic
22#381×C83C2212yes—
23#342×C83C2214yes—
24C22×Dic166—28no—
25C22×Dic166C2×C412yes—
26C22×Dic166C2×C412yes—
27C22×Dic166D418yes—
28C23×Dic83C2311yescharacteristic
29C23×Dic83C2311yescharacteristic
30C4×Dic166—216no—
31#14581757—216no—
32#14581751—24no—
33#14581748—28no—
34#14581748—28no—
35#14581748—216no—
36C2×C4⋊Dic83C2×C412yes—
37C2×Dic83⋊C4—24no—
38C2×Dic83⋊C4C2×C414yes—
39C2×Dic83⋊C4C2×C414yes—
40C2×C22⋊Dic83C2314yes—
41C2×C22⋊Dic83C2312yes—
42C2×C4×Dic83—24no—
43C2×C4×Dic83C2×C414yes—
44C2×C4×Dic83C2×C414yes—
45C2×C4×Dic83C2×C412yes—
46C83⋊#95—28no—
47C83⋊#95—24no—
48C23×C4×C83C2311yescharacteristic
49#117×C83—24no—
50C2×C4⋊C4×C83C2×C412yes—
51C2×C22⋊C4×C83C2314yes—
52C2×C22⋊C4×C83C2312yes—
53C2×C42×C83C2×C412yes—
54#95×C83—24no—
55C2×Dic166—28no—
56C2×Dic166—28no—
57C2×Dic166—216no—
58C2×Dic166—216no—
59C2×Dic166C22⋊C418yes—
60C2×Dic166C22⋊C418yes—
61C22×Dic83—24no—
62C22×Dic83—24no—
63C22×Dic83C22×C412yes—
64C22×Dic83C22×C414yes—
65C22×Dic83C22×C412yes—
66C22×Dic83C2×D414yes—
67C22×Dic83C2×D414yes—
68C22×Dic83#4118yes—
69C4⋊Dic83—24no—
70Dic83⋊C4—28no—
71Dic83⋊C4—28no—
72Dic83⋊C4—48no—
73C22⋊Dic83—24no—
74C22⋊Dic83—28no—
75C4×Dic83—24no—
76C4×Dic83—28no—
77C4×Dic83—28no—
78C4×Dic83—48no—
79C24×C83C2411yescharacteristic
80C22×C4×C83—24no—
81C22×C4×C83—24no—
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C22×C411yescharacteristic
84C22×C4×C83C22×C414yes—
85C22×C4×C83C2×D414yes—
86C4⋊C4×C83—24no—
87C22⋊C4×C83—24no—
88C22⋊C4×C83—28no—
89C42×C83—24no—
90C23×C83—42no—
91C23×C83C23×C411yescharacteristic
92C23×C83C22×D412yes—
93C23×C83C2×#4114yes—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—28no—
97C2×C4×C83—24no—
98C2×C4×C83—44no—
99C2×C4×C83—48no—
100C2×C4×C83—44no—
101C2×C4×C83C2×C22⋊C412yes—
102C2×C4×C83C2×C22⋊C412yes—
103C2×Dic83—24no—
104C2×Dic83—28no—
105C2×Dic83—24no—
106C2×Dic83—44no—
107C2×Dic83—48no—
108C2×Dic83—44no—
109C2×Dic83C2×C22⋊C414yes—
110C2×Dic83C2×C22⋊C414yes—
111C2×Dic83#11718yes—
112C2×Dic83#12318yes—
113C2×Dic83#12318yes—
114Dic166—216no—
115Dic166—216no—
116C22×C83—22no—
117C22×C83—24no—
118C22×C83—44no—
119C22×C83C22×C22⋊C411yescharacteristic
120C22×C83C2×#11712yes—
121C22×C83C2×#12314yes—
122C4×C83—22no—
123C4×C83—22no—
124C4×C83—44no—
125C4×C83—84no—
126Dic83—24no—
127Dic83—44no—
128Dic83—84no—
129Dic83#38118yes—
130#20691—831nomaximal, Sylow
131C2×C83—24no—
132C2×C83—22no—
133C2×C83C2×#38111yescharacteristic
134C2×C83#288412yes—
135C2×#519—831no—
136#2310—832no—
137#1820—832no—
138#1820—834no—
139#1820—831no—
140#1820D8311yescharacteristic
141#1740—834no—
142C83#2069111yescharacteristic, complemented, Sylow
143C23×Q8—831no—
144#519—1668no—
145C2×C4×Q8—834no—
146C2×C4×Q8—834no—
147C2×#117—831no—
148C2×#117—832no—
149C2×#117—832no—
150C2×#117—834no—
151C2×#117C2×D8311yescharacteristic
152#391—1668no—
153#381—832no—
154#381—834no—
155#381C2×D8312yes—
156#342—838no—
157#342—834no—
158#342C2×D8314yes—
159C22×Q8—832no—
160C22×Q8—832no—
161C22×Q8—838no—
162C22×Q8—1668no—
163C23×C4—831no—
164C23×C4—831no—
165C23×C4C22×D8311yescharacteristic
166C4×Q8—16616no—
167C4×Q8—16616no—
168#117—24no—
169#117—16616no—
170#117—1668no—
171#117—1668no—
172#117—1664no—
173C2×C4⋊C4—834no—
174C2×C4⋊C4—834no—
175C2×C4⋊C4—832no—
176C2×C4⋊C4—1664no—
177C2×C4⋊C4C4×D8312yes—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—834no—
180C2×C22⋊C4C22×D8314yes—
181C2×C22⋊C4C22×D8312yes—
182C2×C42—834no—
183C2×C42—834no—
184C2×C42—832no—
185C2×C42—1664no—
186C2×C42C4×D8312yes—
187#95—24no—
188#95—1668no—
189#95—1664no—
190C24C23×D8311yescharacteristic
191C2×Q8—838no—
192C2×Q8—838no—
193C2×Q8—1668no—
194C2×Q8—1668no—
195C2×Q8—16616no—
196C2×Q8—16616no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—834no—
200C22×C4—832no—
201C22×C4—834no—
202C22×C4—838no—
203C22×C4—832no—
204C22×C4—834no—
205C22×C4—1664no—
206C22×C4—1664no—
207C22×C4C2×C4×D8311yescharacteristic
208C22×C4C2×C4×D8311yescharacteristic
209C22×C4C2×C4×D8314yes—
210C22×C4D4×D8314yes—
211C4⋊C4—24no—
212C4⋊C4—3328no—
213C4⋊C4—1668no—
214C4⋊C4—1668no—
215C4⋊C4—1664no—
216C22⋊C4—24no—
217C22⋊C4—28no—
218C22⋊C4—1668no—
219C22⋊C4—1664no—
220C42—24no—
221C42—3328no—
222C42—1668no—
223C42—1668no—
224C42—1664no—
225C23—42no—
226C23C22×C4×D8311yescharacteristic
227C23C2×D4×D8312yes—
228C23#41×D8314yes—
229Q8—16616no—
230Q8—16616no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—28no—
234C2×C4—24no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—44no—
238C2×C4—3324no—
239C2×C4—3328no—
240C2×C4—3324no—
241C2×C4—834no—
242C2×C4—838no—
243C2×C4—838no—
244C2×C4—838no—
245C2×C4—834no—
246C2×C4—1664no—
247C2×C4—1664no—
248C2×C4—1668no—
249C2×C4C22⋊C4×D8312yes—
250C2×C4C22⋊C4×D8312yes—
251C22—22no—
252C22—24no—
253C22—44no—
254C22C2×C22⋊C4×D8311yescharacteristic
255C22#117×D8312yes—
256C22#123×D8314yes—
257C4—22no—
258C4—22no—
259C4—44no—
260C4—84no—
261C4—3324no—
262C4—838no—
263C4—6644no—
264C4—1664no—
265C2—24no—
266C2—22no—
267C2#381×D8311yescharacteristic
268C2#2491592512yes—
subgroupG/H
centerC22C2×C22⋊C4×D83
commutatorC23×C83C23×C4
FrattiniC24C23×D83
FittingC83×#1820C2
socleC22×C83C22×C22⋊C4
prime radical, cosocleC24×C83C24
hypercenter#1820D83
hypocenterC83#20691
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
10. C2×C4×Dic166C4semidirect
11. C2×#14581757C4semidirect
142. C83#20691semidirect

Maximal subgroups

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

subgroupindex
C2×#191281782
#249100482
#249075402
#249075392
#249075382
#249071942
C83×#18202
#2069183