Finite Group Databasebeta

← C2×#16810814 #20284104

Subgroups — C2×#16810814 #20284104

Order 6784.

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

normal subgroups only

This group has 794 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×#13133262C211yesmaximal, characteristic
2C2×#13133201C212yesmaximal, complemented
3C2×#13133198C211yesmaximal, characteristic, complemented
4#16810814C218yesmaximal, direct product factor
5C2×#13133131C212yesmaximal, complemented
6C2×#99×C53C211yesmaximal, characteristic, complemented
7#13133262—22no—
8#13133262—24no—
9#13133262C2214yes—
10C2×#11719945—21no—
11C2×#11719947—22no—
12C22×C22⋊D53—21no—
13C22×C22⋊D53C2212yes—
14C2×D4×D53—22no—
15C23×Dic53C2211yescharacteristic, complemented
16#13133201—24no—
17#13133201C2218yescomplemented
18#13133198—22no—
19#13133198C2214yescomplemented
20C2×C4⋊Dic53—22no—
21C2×#11719919—21no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53C2211yescharacteristic, complemented
24C2×#11719921—22no—
25#13133131C2218yescomplemented
26C22×D4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2212yescomplemented
28#99×C53C2214yescomplemented
29C24×C53D411yescharacteristic
30C24×C53C2311yescharacteristic
31C23×D53—21no—
32C23×D53D412yes—
33C2×D4×C53—22no—
34C2×D4×C53—22no—
35C2×D4×C53D412yes—
36C2×D4×C53C2312yes—
37C22×C4×C53—22no—
38C22×C4×C53D411yescharacteristic
39C2×C22⋊D53—22no—
40C2×C22⋊D53—24no—
41C2×C22⋊D53—22no—
42C2×C22⋊D53—22no—
43C2×C22⋊D53—24no—
44C2×C22⋊D53—22no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53D414yes—
47C2×C22⋊D53C2314yes—
48C22×Dic53—21no—
49C22×Dic53—22no—
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53—22no—
54C22×Dic53D412yes—
55C22×Dic53C2312yescomplemented
56#11719947—28no—
57#11719947—28no—
58D4×D53—28no—
59D4×D53—28no—
60#11719945—24no—
61#11719945—24no—
62C2×D212—21no—
63C2×C4×D53—22no—
64C2×Dic106—21no—
65C22⋊C4×C53—22no—
66C22⋊C4×C53C2314yescomplemented
67C22⋊Dic53—24no—
68C22⋊Dic53—21no—
69C22⋊Dic53C2312yescomplemented
70#11719921—28no—
71C4⋊Dic53—28no—
72#11719919—24no—
73C23×C53—21no—
74C23×C53—21no—
75C23×C53—42no—
76C23×C53—42no—
77C23×C53C2×D411yescharacteristic
78C23×C53C2×D412yes—
79C23×C53C2×D411yescharacteristic
80C23×C53C2×D412yes—
81C23×C53C2×D412yes—
82C23×C53C2411yescharacteristic
83C22×D53—22no—
84C22×D53—24no—
85C22×D53—22no—
86C22×D53—44no—
87C22×D53—82no—
88C22×D53C2×D414yes—
89D4×C53—24no—
90D4×C53—24no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—24no—
94C2×C4×C53—44no—
95C2×C4×C53C2×D412yes—
96C22⋊D53—28no—
97C22⋊D53—24no—
98C22⋊D53—24no—
99C22⋊D53—24no—
100C22⋊D53—24no—
101C2×Dic53—24no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—24no—
106C2×Dic53—22no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53C2×D414yes—
112D212—24no—
113C4×D53—28no—
114Dic106—24no—
115C22×C53—22no—
116C22×C53—24no—
117C22×C53—22no—
118C22×C53—41no—
119C22×C53—42no—
120C22×C53—41no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—81no—
124C22×C53#12012yes—
125C22×C53#12011yescharacteristic
126C22×C53#12011yescharacteristic
127C22×C53C22×D412yes—
128C22×C53C22×D411yescharacteristic
129C2×D53—24no—
130C2×D53—44no—
131C2×D53—44no—
132C2×D53—82no—
133C2×D53—84no—
134C4×C53—22no—
135C4×C53—44no—
136Dic53—24no—
137Dic53—42no—
138Dic53—42no—
139C2×#458—531nomaximal, Sylow
140C2×C53—21no—
141C2×C53—22no—
142C2×C53—22no—
143C2×C53—21no—
144C2×C53—42no—
145C2×C53#45812yescomplemented
146C2×C53C2×#12011yescharacteristic
147D53—44no—
148C2×H(5,𝔽2)—531no—
149C2×#123—531no—
150C2×#123—532no—
151#458—538no—
152C2×#99—532no—
153C2×#99D5311yescharacteristic, complemented
154C53C2×#45811yescharacteristic, complemented, Sylow
155H(5,𝔽2)—534no—
156H(5,𝔽2)—1064no—
157H(5,𝔽2)—1062no—
158C2×#41—1062no—
159C2×#41—1061no—
160C22×D4—532no—
161C22×D4—1062no—
162C22×D4—1061no—
163C22×D4C2×D5311yescharacteristic
164C23×C4—531no—
165#123—534no—
166#123—538no—
167#123—1064no—
168#123—1062no—
169C2×C4⋊C4—1062no—
170C2×C4⋊C4—1061no—
171C2×C22⋊C4—531no—
172C2×C22⋊C4—1062no—
173C2×C22⋊C4—1061no—
174C2×C22⋊C4C2×D5312yescomplemented
175#99—538no—
176#99C2×D5314yescomplemented
177C24—532no—
178C24—1061no—
179C24D21211yescharacteristic
180C24C22×D5311yescharacteristic
181#41—1064no—
182#41—1068no—
183#41—1068no—
184#41—1064no—
185C2×Q8—1061no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—534no—
189C2×D4—534no—
190C2×D4—1061no—
191C2×D4—1068no—
192C2×D4—1068no—
193C2×D4—1062no—
194C2×D4—1064no—
195C2×D4—1062no—
196C2×D4—1062no—
197C2×D4—1064no—
198C2×D4—1062no—
199C2×D4—1062no—
200C2×D4D21212yes—
201C2×D4C22×D5312yes—
202C22×C4—22no—
203C22×C4—532no—
204C22×C4—532no—
205C22×C4—1062no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—1062no—
211C22×C4—1062no—
212C22×C4D21211yescharacteristic
213C4⋊C4—1068no—
214C4⋊C4—1064no—
215C22⋊C4—22no—
216C22⋊C4—532no—
217C22⋊C4—1068no—
218C22⋊C4—1061no—
219C22⋊C4—1064no—
220C22⋊C4C22×D5314yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—534no—
226C23—1064no—
227C23—1062no—
228C23—1062no—
229C23—4242no—
230C23—2124no—
231C23C2×D21211yescharacteristic
232C23C2×D21212yes—
233C23C2×D21211yescharacteristic
234C23D4×D5312yes—
235C23D4×D5312yes—
236C23C23×D5311yescharacteristic
237Q8—1064no—
238D4—24no—
239D4—24no—
240D4—1064no—
241D4—1068no—
242D4—1064no—
243D4—1064no—
244D4—1064no—
245D4—1064no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—44no—
250C2×C4—534no—
251C2×C4—1068no—
252C2×C4—1064no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1064no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—2122no—
259C2×C4—2121no—
260C2×C4—2122no—
261C2×C4—2121no—
262C2×C4C2×D21212yes—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—1064no—
273C22—4242no—
274C22—4244no—
275C22—2124no—
276C22—2124no—
277C22#1313318711yescharacteristic
278C22#1313318712yes—
279C22#1313318711yescharacteristic
280C22C2×D4×D5312yes—
281C22C22×D21211yescharacteristic
282C4—22no—
283C4—44no—
284C4—1064no—
285C4—2122no—
286C4—2122no—
287C2—22no—
288C2—21no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—2124no—
293C2#1681081412yesdirect product factor
294C2C2×#1313318711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC23×C53C24
FrattiniC23C23×D53
FittingC2×#99×C53C2
socleC22×C53#120
hypercenterC2×#99D53
hypocenterC53C2×#458
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#13133201C2semidirect
3. C2×#13133198C2semidirect
4. #16810814C2direct (N × H)
5. C2×#13133131C2semidirect
6. C2×#99×C53C2semidirect
15. C23×Dic53C22semidirect
17. #13133201C22semidirect
19. #13133198C22semidirect
23. C2×C22⋊Dic53C22semidirect
25. #13133131C22semidirect
27. C2×C22⋊C4×C53C22semidirect
28. #99×C53C22semidirect
55. C22×Dic53C23semidirect
66. C22⋊C4×C53C23semidirect
69. C22⋊Dic53C23semidirect
145. C2×C53#458semidirect
153. C2×#99D53semidirect
154. C53C2×#458semidirect
174. C2×C22⋊C4C2×D53semidirect
176. #99C2×D53semidirect
220. C22⋊C4C22×D53semidirect
293. C2#16810814direct (N × H)