Finite Group Databasebeta

← C2×#19127869 #24916787

Subgroups — C2×#19127869 #24916787

Order 10624.

844 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581817C212yesmaximal, complemented
2C2×#14581788C211yesmaximal, characteristic, complemented
3#19127869C218yesmaximal, direct product factor
4C2×#14581719C212yesmaximal, complemented
5C2×#14581709C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C83C211yesmaximal, characteristic, complemented
7C22×D332C2211yescharacteristic, complemented
8C22×C22⋊D83—22no—
9#14581817—24no—
10#14581817C2218yescomplemented
11C2×#13050577—22no—
12C2×#13050575—22no—
13C2×#13050574C2212yescomplemented
14#14581788—22no—
15#14581788C2214yescomplemented
16C2×C4×Dic83C2211yescharacteristic, complemented
17#14581719C2218yescomplemented
18#14581709C2214yescomplemented
19H(5,𝔽2)×C83—24no—
20H(5,𝔽2)×C83—22no—
21H(5,𝔽2)×C83C2214yescomplemented
22C2×#41×C83—22no—
23C2×#41×C83C2212yes—
24C22×D4×C83—22no—
25C22×D4×C83—22no—
26C22×D4×C83C2211yescharacteristic
27C23×D83—21no—
28C2×D332—22no—
29C2×D332—22no—
30C2×D332—21no—
31C2×D332C2312yescomplemented
32C2×C22⋊D83—28no—
33C2×C22⋊D83—48no—
34C22×Dic83—21no—
35#13050577—28no—
36#13050575—28no—
37#13050574—22no—
38#13050574C2314yescomplemented
39C2×C83⋊C8—22no—
40C4×Dic83—21no—
41C4×Dic83C2312yescomplemented
42C24×C83—22no—
43C24×C83D412yes—
44#41×C83—28no—
45#41×C83—28no—
46#41×C83—22no—
47#41×C83—22no—
48#41×C83D414yescomplemented
49#41×C83C2314yes—
50C2×Q8×C83D412yescomplemented
51C2×D4×C83—24no—
52C2×D4×C83—28no—
53C2×D4×C83—28no—
54C2×D4×C83—22no—
55C2×D4×C83—28no—
56C2×D4×C83—28no—
57C2×D4×C83—22no—
58C2×D4×C83—22no—
59C2×D4×C83D412yes—
60C2×D4×C83D412yescomplemented
61C2×D4×C83C2312yes—
62C22×C4×C83—22no—
63C22×C4×C83—22no—
64C22×C4×C83C2311yescharacteristic
65C23×C83—28no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—44no—
70C23×C83—84no—
71C23×C83C2×D411yescharacteristic
72C23×C83C2×D414yes—
73C22×D83—22no—
74C22×D83—44no—
75C22×D83—41no—
76C22×D83—41no—
77Q8×C83—22no—
78Q8×C83C2×D414yescomplemented
79D4×C83—216no—
80D4×C83—24no—
81D4×C83—24no—
82D4×C83—24no—
83D4×C83—22no—
84D4×C83—24no—
85D4×C83C2×D414yescomplemented
86C2×C4×C83—28no—
87C2×C4×C83—22no—
88C2×C4×C83—24no—
89C2×C4×C83—24no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D412yes—
92C2×C4×C83C2×D412yes—
93C2×C4×C83C2411yescharacteristic
94C22⋊D83—416no—
95C2×Dic83—24no—
96C2×Dic83—22no—
97D332—24no—
98D332—42no—
99C83⋊C8—24no—
100C22×C83—22no—
101C22×C83—24no—
102C22×C83—24no—
103C22×C83—44no—
104C22×C83—44no—
105C22×C83—44no—
106C22×C83—82no—
107C22×C83—88no—
108C22×C83#12011yescharacteristic
109C22×C83#12011yescharacteristic
110C22×C83C22×D411yescharacteristic
111C2×D83—44no—
112C2×D83—42no—
113C2×D83—82no—
114C2×D83—82no—
115C4×C83—24no—
116C4×C83#12012yes—
117C4×C83C22×D412yes—
118Dic83—24no—
119C2×C83—21no—
120C2×C83—21no—
121C2×C83—44no—
122C2×C83—44no—
123C2×C83#45312yescomplemented
124C2×C83C2×#12011yescharacteristic
125D83—82no—
126C2×#453—831nomaximal, Sylow
127C83C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)D8311yescharacteristic, complemented
129C2×AGL(1,ℤ/8)—832no—
130C2×#127—831no—
131#453—838no—
132C2×#104—832no—
133C2×#100—831no—
134H(5,𝔽2)—24no—
135H(5,𝔽2)—22no—
136H(5,𝔽2)C2×D8314yescomplemented
137C2×#41—22no—
138C2×#41C2×D8312yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—831no—
142C22×D4—1662no—
143C22×D4C2×D8311yescharacteristic
144AGL(1,ℤ/8)—838no—
145AGL(1,ℤ/8)—1664no—
146C2×SD16—1662no—
147C2×D8—1662no—
148C2×M16—832no—
149#127—834no—
150#127—1662no—
151C2×C42—831no—
152#104—838no—
153#100—834no—
154C24—22no—
155C24—1661no—
156C24C22⋊D8312yes—
157#41—28no—
158#41—28no—
159#41—22no—
160#41—22no—
161#41C22⋊D8314yescomplemented
162#41C22×D8314yes—
163C2×Q8C22⋊D8312yescomplemented
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—3328no—
173C2×D4—832no—
174C2×D4—1668no—
175C2×D4—1661no—
176C2×D4—1662no—
177C2×D4—1662no—
178C2×D4C22⋊D8312yes—
179C2×D4C22⋊D8312yescomplemented
180C2×D4C22×D8312yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—1661no—
184C22×C4C22×D8311yescharacteristic
185SD16—1668no—
186D8—1668no—
187M16—834no—
188M16—1662no—
189C2×C8—1662no—
190C42—832no—
191C42—1661no—
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—44no—
197C23—84no—
198C23—3324no—
199C23—3321no—
200C23—3321no—
201C23—1662no—
202C23C2×C22⋊D8311yescharacteristic
203C23C2×C22⋊D8314yes—
204Q8—22no—
205Q8C2×C22⋊D8314yescomplemented
206D4—216no—
207D4—24no—
208D4—24no—
209D4—24no—
210D4—22no—
211D4—24no—
212D4—33216no—
213D4—3322no—
214D4—1664no—
215D4C2×C22⋊D8314yescomplemented
216C2×C4—28no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—1664no—
221C2×C4—1662no—
222C2×C4C2×C22⋊D8311yescharacteristic
223C2×C4C2×C22⋊D8312yes—
224C2×C4C2×C22⋊D8312yes—
225C2×C4C23×D8311yescharacteristic
226C8—1664no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—82no—
234C22—88no—
235C22—3324no—
236C22—3322no—
237C22—6642no—
238C22—6642no—
239C22#1458176111yescharacteristic
240C22#1458176111yescharacteristic
241C22C22×C22⋊D8311yescharacteristic
242C4—24no—
243C4—1664no—
244C4#1458176112yes—
245C4C22×C22⋊D8312yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—44no—
250C2—6642no—
251C2#1912786912yesdirect product factor
252C2C2×#1458176111yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×H(5,𝔽2)×C83C2
socleC22×C83#120
hypercenterC2×H(5,𝔽2)D83
hypocenterC83C2×#453
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#14581817C2semidirect
2. C2×#14581788C2semidirect
3. #19127869C2direct (N × H)
4. C2×#14581719C2semidirect
5. C2×#14581709C2semidirect
6. C2×H(5,𝔽2)×C83C2semidirect
7. C22×D332C22semidirect
10. #14581817C22semidirect
13. C2×#13050574C22semidirect
15. #14581788C22semidirect
16. C2×C4×Dic83C22semidirect
17. #14581719C22semidirect
18. #14581709C22semidirect
21. H(5,𝔽2)×C83C22semidirect
31. C2×D332C23semidirect
38. #13050574C23semidirect
41. C4×Dic83C23semidirect
48. #41×C83D4semidirect
50. C2×Q8×C83D4semidirect
60. C2×D4×C83D4semidirect
78. Q8×C83C2×D4semidirect
85. D4×C83C2×D4semidirect
123. C2×C83#453semidirect
127. C83C2×#453semidirect
128. C2×H(5,𝔽2)D83semidirect
136. H(5,𝔽2)C2×D83semidirect
161. #41C22⋊D83semidirect
163. C2×Q8C22⋊D83semidirect
179. C2×D4C22⋊D83semidirect
205. Q8C2×C22⋊D83semidirect
215. D4C2×C22⋊D83semidirect
251. C2#19127869direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145818172
C2×#145817882
#191278692
C2×#145817192
C2×#145817092
C2×H(5,𝔽2)×C832
C2×#45383