Finite Group Databasebeta

← C2×#16810796 #20284051

Subgroups — C2×#16810796 #20284051

Order 6784.

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

normal subgroups only

This group has 728 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×#13133242C212yesmaximal, complemented
2C2×#13133206C211yesmaximal, characteristic, complemented
3#16810796C218yesmaximal, direct product factor
4C2×#13133144C212yesmaximal, complemented
5C2×#13133136C211yesmaximal, characteristic, complemented
6C2×2−1+4×C53C211yesmaximal, characteristic, complemented
7C22×D212C2211yescharacteristic
8#13133242—24no—
9#13133242C2218yescomplemented
10C2×#11719924—22no—
11C2×#11719922—22no—
12C2×#11719917C2212yes—
13#13133206—22no—
14#13133206C2214yescomplemented
15C2×#11719921—22no—
16C2×C4×Dic53C2211yescharacteristic
17#13133144C2218yescomplemented
18#13133136C2214yescomplemented
192−1+4×C53—24no—
202−1+4×C53—22no—
212−1+4×C53C2214yescomplemented
22C2×#41×C53—22no—
23C2×#41×C53—22no—
24C2×#41×C53C2212yes—
25C22×Q8×C53—22no—
26C22×Q8×C53C2211yescharacteristic
27C23×D53—21no—
28#41×C53—28no—
29#41×C53—28no—
30#41×C53—28no—
31#41×C53—28no—
32#41×C53—22no—
33#41×C53—22no—
34#41×C53D414yes—
35#41×C53C2314yes—
36C2×Q8×C53—22no—
37C2×Q8×C53—28no—
38C2×Q8×C53—28no—
39C2×Q8×C53—22no—
40C2×Q8×C53—22no—
41C2×Q8×C53D412yes—
42C2×Q8×C53D412yes—
43C2×Q8×C53C2312yes—
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53D412yes—
47C22×C4×C53—24no—
48C22×C4×C53—22no—
49C22×C4×C53D412yes—
50C22×C4×C53C2311yescharacteristic
51C22×Dic53—21no—
52C2×D212—22no—
53C2×D212—22no—
54C2×D212—21no—
55C2×D212C2312yes—
56#11719924—28no—
57#11719922—28no—
58#11719921—28no—
59C4×Dic53—21no—
60C4×Dic53C2312yes—
61#11719917—22no—
62#11719917C2314yes—
63C2×#88322—22no—
64C23×C53—22no—
65C23×C53C2×D411yescharacteristic
66C22×D53—22no—
67C22×D53—44no—
68C22×D53—41no—
69C22×D53—41no—
70Q8×C53—28no—
71Q8×C53—22no—
72Q8×C53—24no—
73Q8×C53—24no—
74Q8×C53C2×D414yes—
75D4×C53—28no—
76D4×C53—24no—
77D4×C53—24no—
78D4×C53—22no—
79D4×C53C2×D414yes—
80C2×C4×C53—216no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—24no—
85C2×C4×C53—24no—
86C2×C4×C53C2×D414yes—
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D412yes—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2411yescharacteristic
91C2×Dic53—22no—
92C2×Dic53—42no—
93D212—24no—
94D212—42no—
95#88322—24no—
96C22×C53—22no—
97C22×C53—24no—
98C22×C53—44no—
99C22×C53#12011yescharacteristic
100C22×C53#12011yescharacteristic
101C22×C53C22×D411yescharacteristic
102C2×D53—44no—
103C2×D53—42no—
104C2×D53—82no—
105C2×D53—82no—
106C4×C53—24no—
107C4×C53—24no—
108C4×C53#12012yes—
109C4×C53C22×D412yes—
110Dic53—42no—
111C2×#454—531nomaximal, Sylow
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—44no—
115C2×C53#45412yescomplemented
116C2×C53C2×#12011yescharacteristic
117D53—82no—
118C2×2−1+4D5311yescharacteristic, complemented
119C2×AGL(1,ℤ/8)—532no—
120C2×#124—531no—
121#454—538no—
122C2×#104—532no—
123C2×#101—531no—
124C53C2×#45411yescharacteristic, complemented, Sylow
1252−1+4—24no—
1262−1+4—22no—
1272−1+4C2×D5314yescomplemented
128C2×#41—22no—
129C2×#41—22no—
130C2×#41C2×D5312yes—
131C22×Q8—22no—
132C22×Q8C2×D5311yescharacteristic
133C22×D4—531no—
134AGL(1,ℤ/8)—538no—
135AGL(1,ℤ/8)—1064no—
136C2×SD16—1062no—
137C2×D8—1062no—
138C2×M16—532no—
139#124—534no—
140#124—1062no—
141C2×C22⋊C4—1062no—
142C2×C42—531no—
143#104—538no—
144#101—534no—
145C24—1061no—
146#41—28no—
147#41—28no—
148#41—28no—
149#41—28no—
150#41—22no—
151#41—22no—
152#41C22⋊D5314yes—
153#41C22×D5314yes—
154C2×Q8—22no—
155C2×Q8—28no—
156C2×Q8—28no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8C22⋊D5312yes—
160C2×Q8C22⋊D5312yes—
161C2×Q8C22×D5312yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—532no—
165C2×D4—1062no—
166C2×D4—1062no—
167C2×D4—1061no—
168C2×D4C22⋊D5312yes—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—1061no—
172C22×C4C22⋊D5312yes—
173C22×C4C22×D5311yescharacteristic
174SD16—1068no—
175D8—1068no—
176M16—534no—
177M16—1062no—
178C2×C8—1062no—
179C22⋊C4—1068no—
180C42—532no—
181C42—1061no—
182C23—22no—
183C23—1062no—
184C23—2124no—
185C23—2121no—
186C23—2121no—
187C23C2×C22⋊D5311yescharacteristic
188Q8—28no—
189Q8—22no—
190Q8—24no—
191Q8—24no—
192Q8C2×C22⋊D5314yes—
193D4—28no—
194D4—24no—
195D4—24no—
196D4—22no—
197D4—1064no—
198D4—2122no—
199D4C2×C22⋊D5314yes—
200C2×C4—216no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—1062no—
207C2×C4—2122no—
208C2×C4C2×C22⋊D5314yes—
209C2×C4C2×C22⋊D5311yescharacteristic
210C2×C4C2×C22⋊D5312yes—
211C2×C4C2×C22⋊D5312yes—
212C2×C4C23×D5311yescharacteristic
213C8—1064no—
214C22—22no—
215C22—24no—
216C22—44no—
217C22—4242no—
218C22—4242no—
219C22—2124no—
220C22—2122no—
221C22#1313318611yescharacteristic
222C22#1313318611yescharacteristic
223C22C22×C22⋊D5311yescharacteristic
224C4—24no—
225C4—24no—
226C4—2122no—
227C4#1313318612yes—
228C4C22×C22⋊D5312yes—
229C2—21no—
230C2—21no—
231C2—44no—
232C2—4242no—
233C2#1681079612yesdirect product factor
234C2C2×#1313318611yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×2−1+4×C53C2
socleC22×C53#120
hypercenterC2×2−1+4D53
hypocenterC53C2×#454
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#13133242C2semidirect
2. C2×#13133206C2semidirect
3. #16810796C2direct (N × H)
4. C2×#13133144C2semidirect
5. C2×#13133136C2semidirect
6. C2×2−1+4×C53C2semidirect
9. #13133242C22semidirect
14. #13133206C22semidirect
17. #13133144C22semidirect
18. #13133136C22semidirect
21. 2−1+4×C53C22semidirect
115. C2×C53#454semidirect
118. C2×2−1+4D53semidirect
124. C53C2×#454semidirect
127. 2−1+4C2×D53semidirect
233. C2#16810796direct (N × H)