Finite Group Databasebeta

← C83×#3022 #24904707

Subgroups — C83×#3022 #24904707

Order 10624.

704 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 212 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#585×C83C211yesmaximal, characteristic
2C2×#137×C83C212yesmaximal, complemented
3#536×C83C211yesmaximal, characteristic, complemented
4#456×C83C214yesmaximal, complemented
5#455×C83C214yesmaximal, complemented
6#421×C83C212yesmaximal, complemented
7#413×C83C211yesmaximal, characteristic, complemented
82−1+4×C83—22no—
92−1+4×C83C2212yes—
10H(5,𝔽2)×C83—22no—
11H(5,𝔽2)×C83—22no—
12H(5,𝔽2)×C83C2212yes—
13C2×#41×C83—24no—
14C2×#41×C83—22no—
15C2×#41×C83C2212yes—
16C2×#41×C83C2211yescharacteristic
17C22×Q8×C83C2211yescharacteristic, complemented
18#137×C83—24no—
19#137×C83C2218yescomplemented
20C2×Q16×C83—22no—
21C2×SD16×C83—22no—
22C2×M16×C83C2212yescomplemented
23C4⋊Q8×C83C2212yescomplemented
24#124×C83C2212yescomplemented
25#123×C83—22no—
26C22⋊Q8×C83—22no—
27#117×C83C2211yescharacteristic, complemented
28#104×C83C2218yescomplemented
29#101×C83C2212yescomplemented
30#100×C83C2212yescomplemented
31#41×C83—24no—
32#41×C83—28no—
33#41×C83—24no—
34#41×C83—24no—
35#41×C83—24no—
36#41×C83—22no—
37#41×C83—22no—
38#41×C83—24no—
39#41×C83—22no—
40#41×C83—22no—
41#41×C83D414yes—
42#41×C83C2314yes—
43C2×Q8×C83—22no—
44C2×Q8×C83—24no—
45C2×Q8×C83—21no—
46C2×Q8×C83—22no—
47C2×Q8×C83—22no—
48C2×Q8×C83D412yes—
49C2×Q8×C83C2312yes—
50C2×Q8×C83C2311yescharacteristic
51C2×D4×C83—28no—
52C2×D4×C83—22no—
53C2×D4×C83—24no—
54C2×D4×C83—22no—
55C2×D4×C83—24no—
56C2×D4×C83D412yes—
57C2×D4×C83D412yes—
58C2×D4×C83C2311yescharacteristic
59C22×C4×C83—24no—
60C22×C4×C83—22no—
61C22×C4×C83—21no—
62C22×C4×C83D412yes—
63C22×C4×C83C2311yescharacteristic
64Q16×C83—28no—
65SD16×C83—28no—
66M16×C83—22no—
67M16×C83C2314yes—
68C2×C8×C83—22no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83—24no—
71C22⋊C4×C83—21no—
72C22⋊C4×C83—24no—
73C42×C83C2312yes—
74C23×C83—24no—
75C23×C83—22no—
76C23×C83C2×D412yes—
77C23×C83C2×D411yescharacteristic
78Q8×C83—24no—
79Q8×C83—22no—
80Q8×C83—22no—
81Q8×C83—24no—
82Q8×C83—41no—
83Q8×C83—41no—
84Q8×C83C2×D414yes—
85D4×C83—28no—
86D4×C83—24no—
87D4×C83—24no—
88D4×C83—24no—
89D4×C83—24no—
90D4×C83—22no—
91D4×C83—22no—
92D4×C83C2×D414yes—
93C2×C4×C83—24no—
94C2×C4×C83—28no—
95C2×C4×C83—22no—
96C2×C4×C83—24no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—24no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—42no—
103C2×C4×C83C2×D412yes—
104C2×C4×C83C2×D411yescharacteristic
105C2×C4×C83C2×D412yes—
106C2×C4×C83C2×D412yes—
107C2×C4×C83C2411yescharacteristic
108C8×C83—24no—
109C22×C83—22no—
110C22×C83—24no—
111C22×C83—41no—
112C22×C83—42no—
113C22×C83—82no—
114C22×C83—84no—
115C22×C83#12012yes—
116C22×C83C22×D411yescharacteristic
117C4×C83—24no—
118C4×C83—22no—
119C4×C83—42no—
120C4×C83—42no—
121C4×C83#12011yescharacteristic
122C4×C83#12011yescharacteristic
123C4×C83C22×D412yes—
124C2×C83—21no—
125C2×C83—22no—
126C2×C83—42no—
127C2×C83—44no—
128C2×C83C2×#12011yescharacteristic
129#3022C8311yesmaximal, characteristic, direct product factor, Sylow
130C83#302211yescharacteristic, direct product factor, Sylow
131#585C2×C8311yescharacteristic
132C2×#137C2×C8312yescomplemented
133#536C2×C8311yescharacteristic, complemented
134#456C2×C8314yescomplemented
135#455C2×C8314yescomplemented
136#421C2×C8312yescomplemented
137#413C2×C8311yescharacteristic, complemented
1382−1+4—22no—
1392−1+4C22×C8312yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C22×C8312yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C22×C8312yes—
146C2×#41C22×C8311yescharacteristic
147C22×Q8C22×C8311yescharacteristic, complemented
148#137—24no—
149#137C22×C8318yescomplemented
150C2×Q16—22no—
151C2×SD16—22no—
152C2×M16C22×C8312yescomplemented
153C4⋊Q8C22×C8312yescomplemented
154#124C22×C8312yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C22×C8311yescharacteristic, complemented
158#104C22×C8318yescomplemented
159#101C22×C8312yescomplemented
160#100C22×C8312yescomplemented
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41D4×C8314yes—
172#41C23×C8314yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8D4×C8312yes—
179C2×Q8C23×C8312yes—
180C2×Q8C23×C8311yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4D4×C8312yes—
187C2×D4D4×C8312yes—
188C2×D4C23×C8311yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4D4×C8312yes—
193C22×C4C23×C8311yescharacteristic
194Q16—28no—
195SD16—28no—
196M16—22no—
197M16C23×C8314yes—
198C2×C8—22no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C23×C8312yes—
204C23—24no—
205C23—22no—
206C23C2×D4×C8312yes—
207C23C2×D4×C8311yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—24no—
212Q8—41no—
213Q8—41no—
214Q8C2×D4×C8314yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×D4×C8314yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4C2×D4×C8312yes—
234C2×C4C2×D4×C8311yescharacteristic
235C2×C4C2×D4×C8312yes—
236C2×C4C2×D4×C8312yes—
237C2×C4C24×C8311yescharacteristic
238C8—24no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#120×C8312yes—
246C22C22×D4×C8311yescharacteristic
247C4—24no—
248C4—22no—
249C4—42no—
250C4—42no—
251C4#120×C8311yescharacteristic
252C4#120×C8311yescharacteristic
253C4C22×D4×C8312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#120×C8311yescharacteristic
subgroupG/H
center, socleC2×C83C2×#120
commutator, Frattini, cosocleC2×C4C24×C83
prime radical#585×C83C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C83 ◁ C22×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#137×C83C2semidirect
3. #536×C83C2semidirect
4. #456×C83C2semidirect
5. #455×C83C2semidirect
6. #421×C83C2semidirect
7. #413×C83C2semidirect
17. C22×Q8×C83C22semidirect
19. #137×C83C22semidirect
22. C2×M16×C83C22semidirect
23. C4⋊Q8×C83C22semidirect
24. #124×C83C22semidirect
27. #117×C83C22semidirect
28. #104×C83C22semidirect
29. #101×C83C22semidirect
30. #100×C83C22semidirect
129. #3022C83direct (N × H)
130. C83#3022direct (N × H)
132. C2×#137C2×C83semidirect
133. #536C2×C83semidirect
134. #456C2×C83semidirect
135. #455C2×C83semidirect
136. #421C2×C83semidirect
137. #413C2×C83semidirect
147. C22×Q8C22×C83semidirect
149. #137C22×C83semidirect
152. C2×M16C22×C83semidirect
153. C4⋊Q8C22×C83semidirect
154. #124C22×C83semidirect
157. #117C22×C83semidirect
158. #104C22×C83semidirect
159. #101C22×C83semidirect
160. #100C22×C83semidirect

Maximal subgroups

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

subgroupindex
#585×C832
C2×#137×C832
#536×C832
#456×C832
#455×C832
#421×C832
#413×C832
#302283