Finite Group Databasebeta

← C2×#16810805 #20284056

Subgroups — C2×#16810805 #20284056

Order 6784.

708 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133265C211yesmaximal, characteristic
2C2×#13133248C212yesmaximal, complemented
3#16810805C218yesmaximal, direct product factor
4C2×#13133145C212yesmaximal, complemented
5C2×#13133136C211yesmaximal, characteristic, complemented
6C2×C4⋊Q8×C53C211yesmaximal, characteristic, complemented
7#13133265—22no—
8#13133265—24no—
9#13133265C2214yes—
10C2×#11719949—22no—
11C2×#11719945—22no—
12C2×#11719945C2212yes—
13C2×Q8×D53—22no—
14#13133248—24no—
15#13133248C2218yescomplemented
16C2×#11719925—22no—
17C2×#11719924—22no—
18C2×#11719917C2212yes—
19#13133145C2218yescomplemented
20#13133136C2214yescomplemented
21C22×Q8×C53C2211yescharacteristic
22C22×Q8×C53C2211yescharacteristic
23C4⋊Q8×C53—22no—
24C4⋊Q8×C53C2214yescomplemented
25C2×C4⋊C4×C53—22no—
26C2×C42×C53C2211yescharacteristic
27C2×Q8×C53—22no—
28C2×Q8×C53—22no—
29C2×Q8×C53—21no—
30C2×Q8×C53—22no—
31C2×Q8×C53—22no—
32C2×Q8×C53D412yes—
33C2×Q8×C53C2312yes—
34C2×Q8×C53C2312yes—
35C22×C4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53D412yes—
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—22no—
40#11719949—28no—
41#11719949—28no—
42Q8×D53—28no—
43Q8×D53—28no—
44#11719945—28no—
45#11719945—28no—
46#11719945—22no—
47#11719945—22no—
48#11719945D414yes—
49#11719945C2314yes—
50C2×D212—22no—
51C2×D212D412yes—
52C2×C4×D53—24no—
53C2×C4×D53—22no—
54C2×Dic106—22no—
55C2×Dic106D412yes—
56C4⋊C4×C53—28no—
57C42×C53—21no—
58C42×C53C2312yes—
59#11719925—28no—
60#11719924—28no—
61#11719917—22no—
62#11719917C2314yes—
63C2×#88322—22no—
64C23×C53C2×D411yescharacteristic
65C22×D53—22no—
66Q8×C53—24no—
67Q8×C53—24no—
68Q8×C53—24no—
69Q8×C53—42no—
70C2×C4×C53—24no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×C4×C53C2×D412yes—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×D414yes—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2411yescharacteristic
82C22⋊D53—24no—
83C22⋊D53—24no—
84C2×Dic53—22no—
85D212—28no—
86D212—22no—
87D212C2×D414yes—
88C4×D53—216no—
89C4×D53—24no—
90C4×D53—24no—
91Dic106—28no—
92Dic106—22no—
93Dic106C2×D414yes—
94#88322—24no—
95C22×C53—22no—
96C22×C53#12011yescharacteristic
97C22×C53#12011yescharacteristic
98C22×C53C22×D411yescharacteristic
99C2×D53—24no—
100C2×D53—44no—
101C4×C53—24no—
102C4×C53—24no—
103C4×C53—42no—
104C4×C53#12012yes—
105C4×C53C22×D412yes—
106Dic53—24no—
107C2×#456—531nomaximal, Sylow
108C2×C53—21no—
109C2×C53—21no—
110C2×C53#45612yescomplemented
111C2×C53C2×#12011yescharacteristic
112D53—44no—
113C2×2−1+4—531no—
114C2×#137—532no—
115C2×C4⋊Q8D5311yescharacteristic, complemented
116#456—538no—
117C2×#104—532no—
118C2×#101—531no—
119C53C2×#45611yescharacteristic, complemented, Sylow
1202−1+4—534no—
1212−1+4—1064no—
1222−1+4—1062no—
123C2×#41—532no—
124C2×#41—1062no—
125C2×#41—1062no—
126C22×Q8—1062no—
127C22×Q8C2×D5311yescharacteristic
128C22×Q8C2×D5311yescharacteristic
129#137—538no—
130#137—1064no—
131C2×Q16—1062no—
132C2×SD16—1062no—
133C2×M16—532no—
134C4⋊Q8—22no—
135C4⋊Q8C2×D5314yescomplemented
136C2×C4⋊C4—22no—
137C2×C42C2×D5311yescharacteristic
138#104—538no—
139#101—534no—
140#41—534no—
141#41—534no—
142#41—1068no—
143#41—1068no—
144#41—1068no—
145#41—1062no—
146#41—1062no—
147#41—1068no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—532no—
154C2×Q8—1062no—
155C2×Q8—1068no—
156C2×Q8—1068no—
157C2×Q8C22⋊D5312yes—
158C2×Q8C22×D5312yes—
159C2×Q8C22×D5312yes—
160C2×D4—532no—
161C2×D4—1062no—
162C2×D4—1062no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—1064no—
166C22×C4—1062no—
167C22×C4C22⋊D5312yes—
168C22×C4C22×D5311yescharacteristic
169Q16—1068no—
170SD16—1068no—
171M16—534no—
172M16—1062no—
173C2×C8—1062no—
174C4⋊C4—28no—
175C42—21no—
176C42C22×D5312yes—
177C23—1062no—
178C23C2×C22⋊D5311yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—24no—
182Q8—42no—
183Q8—534no—
184Q8—1068no—
185Q8—1062no—
186D4—534no—
187D4—1068no—
188D4—1064no—
189D4—1064no—
190D4—1062no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—10616no—
199C2×C4—1062no—
200C2×C4—1064no—
201C2×C4—1064no—
202C2×C4D4×D5312yes—
203C2×C4D4×D5312yes—
204C2×C4C2×C22⋊D5314yes—
205C2×C4C2×C22⋊D5311yescharacteristic
206C2×C4C23×D5311yescharacteristic
207C8—1064no—
208C22—22no—
209C22—1064no—
210C22—2124no—
211C22#1313318511yescharacteristic
212C22#1313318511yescharacteristic
213C22C22×C22⋊D5311yescharacteristic
214C4—24no—
215C4—24no—
216C4—42no—
217C4—1064no—
218C4#1313318512yes—
219C4C2×D4×D5312yes—
220C2—21no—
221C2—21no—
222C2—2124no—
223C2#1681080512yesdirect product factor
224C2C2×#1313318511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×C4⋊Q8×C53C2
socleC22×C53#120
hypercenterC2×C4⋊Q8D53
hypocenterC53C2×#456
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#13133248C2semidirect
3. #16810805C2direct (N × H)
4. C2×#13133145C2semidirect
5. C2×#13133136C2semidirect
6. C2×C4⋊Q8×C53C2semidirect
15. #13133248C22semidirect
19. #13133145C22semidirect
20. #13133136C22semidirect
24. C4⋊Q8×C53C22semidirect
110. C2×C53#456semidirect
115. C2×C4⋊Q8D53semidirect
119. C53C2×#456semidirect
135. C4⋊Q8C2×D53semidirect
223. C2#16810805direct (N × H)