Finite Group Databasebeta

← C2×#14521227 #16855041

Subgroups — C2×#14521227 #16855041

Order 3968.

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

normal subgroups only

This group has 760 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×#11758528C212yesmaximal, complemented
2C2×#11758513C211yesmaximal, characteristic, complemented
3#14521227C218yesmaximal, direct product factor
4C2×#11758425C212yesmaximal, complemented
5C2×#11758421C211yesmaximal, characteristic, complemented
6C2×C31×H(5,𝔽2)C211yesmaximal, characteristic
7C31×H(5,𝔽2)—24no—
8C31×H(5,𝔽2)—22no—
9C31×H(5,𝔽2)C2214yes—
10C2×#41×C31—22no—
11C2×#41×C31C2212yes—
12C22×D4×C31—22no—
13C22×D4×C31—22no—
14C22×D4×C31C2211yescharacteristic
15C22×Dic62C2211yescharacteristic, complemented
16C2×C22⋊Dic31—22no—
17#11758528—24no—
18#11758528C2218yescomplemented
19C2×#92482—22no—
20#11758513—22no—
21#11758513C2214yescomplemented
22C2×#92480—22no—
23C2×C4×Dic31C2211yescharacteristic, complemented
24C2×#92474C2212yescomplemented
25#11758425C2218yescomplemented
26#11758421C2214yescomplemented
27C24×C31—22no—
28C24×C31D412yes—
29#41×C31—28no—
30#41×C31—28no—
31#41×C31—22no—
32#41×C31—22no—
33#41×C31D414yes—
34#41×C31C2314yes—
35C2×Q8×C31D412yes—
36C2×D4×C31—24no—
37C2×D4×C31—28no—
38C2×D4×C31—28no—
39C2×D4×C31—22no—
40C2×D4×C31—28no—
41C2×D4×C31—28no—
42C2×D4×C31—22no—
43C2×D4×C31—22no—
44C2×D4×C31D412yes—
45C2×D4×C31D412yes—
46C2×D4×C31C2312yes—
47C22×C4×C31—22no—
48C22×C4×C31—22no—
49C22×C4×C31C2311yescharacteristic
50C22×Dic31—21no—
51C22×Dic31—21no—
52C2×Dic62—22no—
53C2×Dic62—22no—
54C2×Dic62—21no—
55C2×Dic62C2312yescomplemented
56C22⋊Dic31—28no—
57#92482—28no—
58#92480—28no—
59C4×Dic31—21no—
60C4×Dic31C2312yescomplemented
61#92474—22no—
62#92474C2314yescomplemented
63C2×C31⋊C8—22no—
64C23×C31—28no—
65C23×C31—22no—
66C23×C31—22no—
67C23×C31—22no—
68C23×C31—44no—
69C23×C31—84no—
70C23×C31C2×D411yescharacteristic
71C23×C31C2×D414yes—
72Q8×C31—22no—
73Q8×C31C2×D414yes—
74D4×C31—216no—
75D4×C31—24no—
76D4×C31—24no—
77D4×C31—24no—
78D4×C31—22no—
79D4×C31—24no—
80D4×C31C2×D414yes—
81C2×C4×C31—28no—
82C2×C4×C31—22no—
83C2×C4×C31—24no—
84C2×C4×C31—24no—
85C2×C4×C31C2×D411yescharacteristic
86C2×C4×C31C2×D412yes—
87C2×C4×C31C2×D412yes—
88C2×C4×C31C2411yescharacteristic
89C2×Dic31—22no—
90C2×Dic31—22no—
91C2×Dic31—42no—
92C2×Dic31—41no—
93C2×Dic31—41no—
94Dic62—24no—
95Dic62—42no—
96C31⋊C8—24no—
97C2×#455—311nomaximal, Sylow
98C22×C31—22no—
99C22×C31—24no—
100C22×C31—24no—
101C22×C31—44no—
102C22×C31—44no—
103C22×C31—44no—
104C22×C31—82no—
105C22×C31—88no—
106C22×C31#12011yescharacteristic
107C22×C31#12011yescharacteristic
108C22×C31C22×D411yescharacteristic
109C4×C31—24no—
110C4×C31#12012yes—
111C4×C31C22×D412yes—
112Dic31—42no—
113Dic31—42no—
114C2×H(5,𝔽2)D3111yescharacteristic
115C2×#137—312no—
116C2×#124—311no—
117#455—318no—
118C2×#104—312no—
119C2×#100—311no—
120C2×C31—21no—
121C2×C31—21no—
122C2×C31—44no—
123C2×C31—44no—
124C2×C31#45512yescomplemented
125C2×C31C2×#12011yescharacteristic
126H(5,𝔽2)—24no—
127H(5,𝔽2)—22no—
128H(5,𝔽2)C2×D3114yes—
129C2×#41—22no—
130C2×#41C2×D3112yes—
131C22×Q8—311no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C2×D3111yescharacteristic
135#137—318no—
136#137—624no—
137C2×Q16—622no—
138C2×SD16—622no—
139C2×M16—312no—
140#124—314no—
141#124—622no—
142C2×C22⋊C4—622no—
143C2×C42—311no—
144#104—318no—
145#100—314no—
146C31C2×#45511yescharacteristic, complemented, Sylow
147C24—22no—
148C24C22⋊D3112yes—
149#41—28no—
150#41—28no—
151#41—22no—
152#41—22no—
153#41C22⋊D3114yes—
154#41C22×D3114yes—
155C2×Q8—312no—
156C2×Q8—621no—
157C2×Q8—622no—
158C2×Q8—622no—
159C2×Q8C22⋊D3112yes—
160C2×D4—24no—
161C2×D4—28no—
162C2×D4—28no—
163C2×D4—22no—
164C2×D4—28no—
165C2×D4—28no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4C22⋊D3112yes—
169C2×D4C22⋊D3112yes—
170C2×D4C22×D3112yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—621no—
174C22×C4—621no—
175C22×C4C22×D3111yescharacteristic
176Q16—628no—
177SD16—628no—
178M16—314no—
179M16—622no—
180C2×C8—622no—
181C22⋊C4—628no—
182C42—312no—
183C42—621no—
184C23—28no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—44no—
189C23—84no—
190C23C2×C22⋊D3111yescharacteristic
191C23C2×C22⋊D3114yes—
192Q8—22no—
193Q8—624no—
194Q8—1242no—
195Q8C2×C22⋊D3114yes—
196D4—216no—
197D4—24no—
198D4—24no—
199D4—24no—
200D4—22no—
201D4—24no—
202D4C2×C22⋊D3114yes—
203C2×C4—28no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—622no—
208C2×C4—622no—
209C2×C4—1242no—
210C2×C4—1241no—
211C2×C4—1241no—
212C2×C4C2×C22⋊D3111yescharacteristic
213C2×C4C2×C22⋊D3112yes—
214C2×C4C2×C22⋊D3112yes—
215C2×C4C23×D3111yescharacteristic
216C8—624no—
217C22—22no—
218C22—24no—
219C22—24no—
220C22—44no—
221C22—44no—
222C22—44no—
223C22—82no—
224C22—88no—
225C22#1175853011yescharacteristic
226C22#1175853011yescharacteristic
227C22C22×C22⋊D3111yescharacteristic
228C4—24no—
229C4—1242no—
230C4—1242no—
231C4#1175853012yes—
232C4C22×C22⋊D3112yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—44no—
237C2#1452122712yesdirect product factor
238C2C2×#1175853011yescharacteristic
subgroupG/H
centerC22#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC2×C31×H(5,𝔽2)C2
socleC22×C31#120
hypercenterC2×H(5,𝔽2)D31
hypocenterC31C2×#455
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. C2×#11758513C2semidirect
3. #14521227C2direct (N × H)
4. C2×#11758425C2semidirect
5. C2×#11758421C2semidirect
15. C22×Dic62C22semidirect
18. #11758528C22semidirect
21. #11758513C22semidirect
23. C2×C4×Dic31C22semidirect
24. C2×#92474C22semidirect
25. #11758425C22semidirect
26. #11758421C22semidirect
55. C2×Dic62C23semidirect
60. C4×Dic31C23semidirect
62. #92474C23semidirect
124. C2×C31#455semidirect
146. C31C2×#455semidirect
237. C2#14521227direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585282
C2×#117585132
#145212272
C2×#117584252
C2×#117584212
C2×C31×H(5,𝔽2)2
C2×#45531