Finite Group Databasebeta

← C83×#34177 #35262001

Subgroups — C83×#34177 #35262001

Order 21248.

2446 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C83C211yesmaximal, characteristic
2C83×#2343C2116yesmaximal, complemented
3C83×#2315C212yesmaximal
4C83×#2313C214yesmaximal, complemented
5C83×#2283C214yesmaximal, complemented
6C2×#381×C83C212yesmaximal
7C83×#2273C212yesmaximal
8C24×C4×C83C2211yescharacteristic
9C2×#117×C83C4116yescomplemented
10C2×#117×C83C2218yes—
11C22×C4⋊C4×C83C2212yes—
12C22×C22⋊C4×C83C2214yes—
13C22×C22⋊C4×C83C2211yescharacteristic
14C22×C22⋊C4×C83C2211yescharacteristic
15C22×C42×C83C2212yes—
16#387×C83C22132yescomplemented
17#385×C83C22116yes—
18#383×C83C2218yescomplemented
19#382×C83C2218yescomplemented
20#381×C83C2214yes—
21#381×C83C2214yes—
22#381×C83C2218yes—
23#380×C83C2218yes—
24C4×C4⋊C4×C83C2218yescomplemented
25C4×C22⋊C4×C83C2218yes—
26C4×C22⋊C4×C83C22116yes—
27#376×C83C2218yescomplemented
28C2×#95×C83C2218yes—
29C25×C83C2311yescharacteristic
30C23×C4×C83—28no—
31C23×C4×C83C2×C418yes—
32C23×C4×C83C2311yescharacteristic
33C23×C4×C83C2311yescharacteristic
34C23×C4×C83C2314yes—
35C23×C4×C83C2318yes—
36C23×C4×C83C2312yes—
37C23×C4×C83C2312yes—
38#117×C83C2×C4164yescomplemented
39C2×C4⋊C4×C83C2×C4116yescomplemented
40C2×C4⋊C4×C83C23116yes—
41C2×C4⋊C4×C83C2318yes—
42C2×C22⋊C4×C83—216no—
43C2×C22⋊C4×C83C2×C418yes—
44C2×C22⋊C4×C83C2×C418yes—
45C2×C22⋊C4×C83C2318yes—
46C2×C22⋊C4×C83C2318yes—
47C2×C22⋊C4×C83C2314yes—
48C2×C22⋊C4×C83C2314yes—
49C2×C42×C83C2×C4116yescomplemented
50C2×C42×C83C23116yes—
51C2×C42×C83C2314yes—
52C2×C42×C83C2314yes—
53#95×C83C23116yes—
54#95×C83C23116yes—
55#95×C83C23116yes—
56#95×C83C23116yes—
57C24×C83—44no—
58C24×C83C22×C414yes—
59C24×C83#4118yes—
60C24×C83C2412yes—
61C24×C83C2411yescharacteristic
62C22×C4×C83—232no—
63C22×C4×C83—24no—
64C22×C4×C83—24no—
65C22×C4×C83—216no—
66C22×C4×C83—216no—
67C22×C4×C83—44no—
68C22×C4×C83—44no—
69C22×C4×C83—416no—
70C22×C4×C83C22×C414yes—
71C22×C4×C83C22×C414yes—
72C22×C4×C83C22×C4116yes—
73C22×C4×C83C22×C418yes—
74C22×C4×C83C22×C418yes—
75C22×C4×C83C24116yes—
76C22×C4×C83C2412yes—
77C22×C4×C83C2412yes—
78C22×C4×C83C2414yes—
79C22×C4×C83C2414yes—
80C4⋊C4×C83—416no—
81C4⋊C4×C83C22×C4132yes—
82C22⋊C4×C83—216no—
83C22⋊C4×C83—216no—
84C22⋊C4×C83C22×C4116yes—
85C22⋊C4×C83C22×C4116yes—
86C42×C83—416no—
87C42×C83C22×C4116yes—
88C42×C83C22×C4116yes—
89C23×C83—24no—
90C23×C83—28no—
91C23×C83—48no—
92C23×C83—416no—
93C23×C83#117116yes—
94C23×C83C23×C414yes—
95C23×C83C23×C412yes—
96C23×C83C2×#4114yes—
97C23×C83C2×#4118yes—
98C23×C83C2511yescharacteristic
99C2×C4×C83—28no—
100C2×C4×C83—28no—
101C2×C4×C83—232no—
102C2×C4×C83—24no—
103C2×C4×C83—24no—
104C2×C4×C83—28no—
105C2×C4×C83—28no—
106C2×C4×C83—216no—
107C2×C4×C83—232no—
108C2×C4×C83—48no—
109C2×C4×C83—48no—
110C2×C4×C83—816no—
111C2×C4×C83C23×C414yes—
112C2×C4×C83C23×C418yes—
113C2×C4×C83C23×C418yes—
114C2×C4×C83C23×C414yes—
115C22×C83—24no—
116C22×C83—28no—
117C22×C83—216no—
118C22×C83—28no—
119C22×C83—416no—
120C22×C83C2×#11718yes—
121C22×C83C2×#11714yes—
122C22×C83C24×C411yescharacteristic
123C22×C83C22×#4112yes—
124C22×C83#58514yes—
125C4×C83—24no—
126C4×C83—24no—
127C4×C83—28no—
128C4×C83—28no—
129C4×C83—416no—
130#34177C8311yesmaximal, characteristic, direct product factor, Sylow
131C2×C83—24no—
132C2×C83—28no—
133C2×C83C22×#11711yescharacteristic
134C2×C83#342812yes—
135C2×C83#349114yes—
136C22×#117C2×C8311yescharacteristic
137#2343C2×C83116yescomplemented
138#2315C2×C8312yes—
139#2313C2×C8314yescomplemented
140#2283C2×C8314yescomplemented
141C2×#381C2×C8312yes—
142#2273C2×C8312yes—
143C83#3417711yescharacteristic, direct product factor, Sylow
144C24×C4C22×C8311yescharacteristic
145C2×#117C4×C83116yescomplemented
146C2×#117C22×C8318yes—
147C22×C4⋊C4C22×C8312yes—
148C22×C22⋊C4C22×C8314yes—
149C22×C22⋊C4C22×C8311yescharacteristic
150C22×C22⋊C4C22×C8311yescharacteristic
151C22×C42C22×C8312yes—
152#387C22×C83132yescomplemented
153#385C22×C83116yes—
154#383C22×C8318yescomplemented
155#382C22×C8318yescomplemented
156#381C22×C8314yes—
157#381C22×C8318yes—
158#381C22×C8314yes—
159#380C22×C8318yes—
160C4×C4⋊C4C22×C8318yescomplemented
161C4×C22⋊C4C22×C8318yes—
162C4×C22⋊C4C22×C83116yes—
163#376C22×C8318yescomplemented
164C2×#95C22×C8318yes—
165C25C23×C8311yescharacteristic
166C23×C4—28no—
167C23×C4C2×C4×C8318yes—
168C23×C4C23×C8311yescharacteristic
169C23×C4C23×C8314yes—
170C23×C4C23×C8311yescharacteristic
171C23×C4C23×C8318yes—
172C23×C4C23×C8312yes—
173C23×C4C23×C8312yes—
174#117C2×C4×C83164yescomplemented
175C2×C4⋊C4C2×C4×C83116yescomplemented
176C2×C4⋊C4C23×C83116yes—
177C2×C4⋊C4C23×C8318yes—
178C2×C22⋊C4—216no—
179C2×C22⋊C4C2×C4×C8318yes—
180C2×C22⋊C4C2×C4×C8318yes—
181C2×C22⋊C4C23×C8318yes—
182C2×C22⋊C4C23×C8318yes—
183C2×C22⋊C4C23×C8314yes—
184C2×C22⋊C4C23×C8314yes—
185C2×C42C2×C4×C83116yescomplemented
186C2×C42C23×C83116yes—
187C2×C42C23×C8314yes—
188C2×C42C23×C8314yes—
189#95C23×C83116yes—
190#95C23×C83116yes—
191#95C23×C83116yes—
192#95C23×C83116yes—
193C24—44no—
194C24C22×C4×C8314yes—
195C24#41×C8318yes—
196C24C24×C8312yes—
197C24C24×C8311yescharacteristic
198C22×C4—232no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—216no—
202C22×C4—216no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—416no—
206C22×C4C22×C4×C8314yes—
207C22×C4C22×C4×C8314yes—
208C22×C4C22×C4×C83116yes—
209C22×C4C22×C4×C8318yes—
210C22×C4C22×C4×C8318yes—
211C22×C4C24×C83116yes—
212C22×C4C24×C8312yes—
213C22×C4C24×C8312yes—
214C22×C4C24×C8314yes—
215C22×C4C24×C8314yes—
216C4⋊C4—416no—
217C4⋊C4C22×C4×C83132yes—
218C22⋊C4—216no—
219C22⋊C4—216no—
220C22⋊C4C22×C4×C83116yes—
221C22⋊C4C22×C4×C83116yes—
222C42—416no—
223C42C22×C4×C83116yes—
224C42C22×C4×C83116yes—
225C23—24no—
226C23—28no—
227C23—48no—
228C23—416no—
229C23#117×C83116yes—
230C23C23×C4×C8314yes—
231C23C23×C4×C8312yes—
232C23C2×#41×C8314yes—
233C23C2×#41×C8318yes—
234C23C25×C8311yescharacteristic
235C2×C4—28no—
236C2×C4—28no—
237C2×C4—232no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—28no—
241C2×C4—28no—
242C2×C4—216no—
243C2×C4—232no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—816no—
247C2×C4C23×C4×C8314yes—
248C2×C4C23×C4×C8318yes—
249C2×C4C23×C4×C8318yes—
250C2×C4C23×C4×C8314yes—
251C22—24no—
252C22—28no—
253C22—216no—
254C22—28no—
255C22—416no—
256C22C2×#117×C8318yes—
257C22C2×#117×C8314yes—
258C22C24×C4×C8311yescharacteristic
259C22C22×#41×C8312yes—
260C22#585×C8314yes—
261C4—24no—
262C4—24no—
263C4—28no—
264C4—28no—
265C4—416no—
266C2—24no—
267C2—28no—
268C2C22×#117×C8311yescharacteristic
269C2C83×#342812yes—
270C2C83×#349114yes—
subgroupG/H
center, socleC23×C83C25
commutatorC22C24×C4×C83
Frattini, cosocleC23C25×C83
prime radicalC25×C83C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2343C2semidirect
4. C83×#2313C2semidirect
5. C83×#2283C2semidirect
9. C2×#117×C83C4semidirect
16. #387×C83C22semidirect
18. #383×C83C22semidirect
19. #382×C83C22semidirect
24. C4×C4⋊C4×C83C22semidirect
27. #376×C83C22semidirect
38. #117×C83C2×C4semidirect
39. C2×C4⋊C4×C83C2×C4semidirect
49. C2×C42×C83C2×C4semidirect
130. #34177C83direct (N × H)
137. #2343C2×C83semidirect
139. #2313C2×C83semidirect
140. #2283C2×C83semidirect
143. C83#34177direct (N × H)
145. C2×#117C4×C83semidirect
152. #387C22×C83semidirect
154. #383C22×C83semidirect
155. #382C22×C83semidirect
160. C4×C4⋊C4C22×C83semidirect
163. #376C22×C83semidirect
174. #117C2×C4×C83semidirect
175. C2×C4⋊C4C2×C4×C83semidirect
185. C2×C42C2×C4×C83semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C832
C83×#23432
C83×#23152
C83×#23132
C83×#22832
C2×#381×C832
C83×#22732
#3417783