Finite Group Databasebeta

← C2×#16810789 #20284037

Subgroups — C2×#16810789 #20284037

Order 6784.

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

normal subgroups only

This group has 844 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×#13133263C211yesmaximal, characteristic, complemented
2C2×#13133243C211yesmaximal, characteristic, complemented
3C2×#13133213C211yesmaximal, characteristic, complemented
4#16810789C218yesmaximal, direct product factor
5C2×#13133144C211yesmaximal, characteristic, complemented
6C2×#13133143C211yesmaximal, characteristic, complemented
7C2×#13133133C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic, complemented
9#13133263—22no—
10#13133263—22no—
11#13133263—22no—
12#13133263C2214yescomplemented
13C2×#11719949—21no—
14C2×#11719945—21no—
15C2×#11719945C2211yescharacteristic
16C22×D212—21no—
17C22×D212C2211yescharacteristic
18C22×C22⋊D53—22no—
19C2×D4×D53—22no—
20C2×D4×D53—21no—
21#13133243—22no—
22#13133243C2214yescomplemented
23C2×#11719923—21no—
24C2×#11719922—21no—
25C2×#11719917C2211yescharacteristic, complemented
26#13133213—22no—
27#13133213C2214yescomplemented
28C2×C4×Dic53C2211yescharacteristic, complemented
29#13133144C2214yescomplemented
30#13133143C2214yescomplemented
31#13133133C2214yescomplemented
32C2×#41×C53C2211yescharacteristic
33C22×D4×C53C2211yescharacteristic, complemented
34AGL(1,ℤ/8)×C53—22no—
35AGL(1,ℤ/8)×C53C2214yescomplemented
36C2×SD16×C53—21no—
37C2×D8×C53—21no—
38C2×M16×C53C2211yescharacteristic, complemented
39C24×C53—21no—
40C23×D53—22no—
41C23×D53D412yes—
42#41×C53—21no—
43#41×C53—21no—
44#41×C53D412yescomplemented
45#41×C53C2312yes—
46C2×Q8×C53D411yescharacteristic, complemented
47C2×D4×C53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—21no—
50C2×D4×C53—21no—
51C2×D4×C53—21no—
52C2×D4×C53—21no—
53C2×D4×C53D411yescharacteristic, complemented
54C2×D4×C53C2312yescomplemented
55C22×C4×C53—21no—
56C22×C4×C53C2311yescharacteristic
57C2×C22⋊D53—22no—
58C2×C22⋊D53—21no—
59C2×C22⋊D53—28no—
60C2×C22⋊D53—48no—
61C22×Dic53—21no—
62#11719949—24no—
63#11719949—24no—
64D4×D53—28no—
65D4×D53—24no—
66D4×D53—28no—
67D4×D53—24no—
68#11719945—24no—
69#11719945—21no—
70#11719945—21no—
71#11719945—24no—
72#11719945D412yescomplemented
73#11719945C2312yes—
74C2×D212—22no—
75C2×D212—24no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—24no—
81C2×D212D412yes—
82C2×D212D411yescharacteristic, complemented
83C2×D212C2312yes—
84C2×C4×D53—22no—
85C2×C4×D53—21no—
86C2×Dic106D411yescharacteristic, complemented
87SD16×C53—24no—
88D8×C53—24no—
89M16×C53—21no—
90M16×C53C2312yescomplemented
91C2×C8×C53—21no—
92#11719923—24no—
93#11719922—24no—
94C4×Dic53—21no—
95C4×Dic53C2312yescomplemented
96#11719917—21no—
97#11719917C2312yescomplemented
98C2×#88322—21no—
99C23×C53—21no—
100C23×C53—22no—
101C23×C53—44no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×D411yescharacteristic
105C22×D53—22no—
106C22×D53—21no—
107C22×D53—28no—
108C22×D53—22no—
109C22×D53—44no—
110C22×D53—84no—
111C22×D53C2×D414yes—
112Q8×C53—21no—
113Q8×C53C2×D412yescomplemented
114D4×C53—22no—
115D4×C53—22no—
116D4×C53—21no—
117D4×C53—22no—
118D4×C53—22no—
119D4×C53—41no—
120D4×C53—41no—
121D4×C53C2×D412yescomplemented
122C2×C4×C53—21no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—28no—
132C22⋊D53—22no—
133C22⋊D53—22no—
134C22⋊D53—416no—
135C2×Dic53—24no—
136C2×Dic53—21no—
137C2×Dic53—22no—
138D212—28no—
139D212—22no—
140D212—21no—
141D212—22no—
142D212—22no—
143D212—22no—
144D212C2×D412yescomplemented
145C4×D53—28no—
146C4×D53—22no—
147C4×D53—22no—
148Dic106—21no—
149Dic106C2×D412yescomplemented
150C8×C53—22no—
151#88322—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—44no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—82no—
158C22×C53—82no—
159C22×C53#12011yescharacteristic
160C22×C53#12011yescharacteristic
161C22×C53C22×D411yescharacteristic
162C2×D53—24no—
163C2×D53—22no—
164C2×D53—44no—
165C2×D53—42no—
166C2×D53—44no—
167C2×D53—82no—
168C2×D53—88no—
169C4×C53—22no—
170C4×C53#12011yescharacteristic
171C4×C53#12011yescharacteristic
172C4×C53C22×D411yescharacteristic
173C4×C53C22×D411yescharacteristic
174Dic53—24no—
175Dic53—22no—
176C2×#453—531nomaximal, Sylow
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—42no—
180C2×C53—82no—
181C2×C53#45312yescomplemented
182C2×C53C2×#12011yescharacteristic
183D53—44no—
184D53—42no—
185C2×H(5,𝔽2)—531no—
186C2×AGL(1,ℤ/8)—531no—
187C2×AGL(1,ℤ/8)D5311yescharacteristic, complemented
188C2×#127—531no—
189#453—538no—
190C2×#104—531no—
191C2×#104—531no—
192C2×#100—531no—
193C53C2×#45311yescharacteristic, complemented, Sylow
194H(5,𝔽2)—534no—
195H(5,𝔽2)—1062no—
196H(5,𝔽2)—1062no—
197H(5,𝔽2)—1062no—
198C2×#41—531no—
199C2×#41—1061no—
200C2×#41—1061no—
201C2×#41C2×D5311yescharacteristic
202C22×D4—531no—
203C22×D4—1061no—
204C22×D4—1062no—
205C22×D4—1061no—
206C22×D4—1062no—
207C22×D4C2×D5311yescharacteristic, complemented
208AGL(1,ℤ/8)—22no—
209AGL(1,ℤ/8)—534no—
210AGL(1,ℤ/8)—1062no—
211AGL(1,ℤ/8)C2×D5314yescomplemented
212C2×SD16—21no—
213C2×SD16—1061no—
214C2×D8—21no—
215C2×D8—1061no—
216C2×M16—531no—
217C2×M16C2×D5311yescharacteristic, complemented
218#127—534no—
219#127—1062no—
220C2×C42—531no—
221#104—534no—
222#104—534no—
223#100—534no—
224C24—21no—
225C24—532no—
226C24—1062no—
227#41—21no—
228#41—21no—
229#41—532no—
230#41—532no—
231#41—1064no—
232#41—1064no—
233#41—1064no—
234#41—1064no—
235#41—1061no—
236#41—1061no—
237#41C22⋊D5312yescomplemented
238#41C22×D5312yes—
239C2×Q8—531no—
240C2×Q8C22⋊D5311yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—531no—
248C2×D4—532no—
249C2×D4—532no—
250C2×D4—1064no—
251C2×D4—1062no—
252C2×D4—1064no—
253C2×D4—1068no—
254C2×D4—1068no—
255C2×D4—1064no—
256C2×D4—1062no—
257C2×D4—1061no—
258C2×D4—1064no—
259C2×D4—1068no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—2128no—
265C2×D4C22⋊D5311yescharacteristic, complemented
266C2×D4C22×D5312yescomplemented
267C22×C4—21no—
268C22×C4—1062no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272SD16—24no—
273SD16—1064no—
274D8—24no—
275D8—1064no—
276M16—21no—
277M16—532no—
278M16—1061no—
279M16C22×D5312yescomplemented
280C2×C8—21no—
281C2×C8—1061no—
282C42—532no—
283C42—1061no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—41no—
289C23—534no—
290C23—1068no—
291C23—1062no—
292C23—1061no—
293C23—1062no—
294C23—4244no—
295C23—2124no—
296C23D4×D5311yescharacteristic
297Q8—21no—
298Q8—532no—
299Q8—1061no—
300Q8C2×C22⋊D5312yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308D4—532no—
309D4—1068no—
310D4—1068no—
311D4—1062no—
312D4—1062no—
313D4—1061no—
314D4—1062no—
315D4—1062no—
316D4—1062no—
317D4—1062no—
318D4—21216no—
319D4C2×C22⋊D5312yescomplemented
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—1068no—
324C2×C4—1064no—
325C2×C4—1061no—
326C2×C4—1062no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4D4×D5311yescharacteristic
330C2×C4D4×D5311yescharacteristic
331C2×C4D4×D5311yescharacteristic
332C2×C4C2×C22⋊D5311yescharacteristic
333C2×C4C2×C22⋊D5311yescharacteristic
334C2×C4C23×D5311yescharacteristic
335C8—22no—
336C8—1062no—
337C22—22no—
338C22—22no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—1062no—
345C22—1064no—
346C22—4242no—
347C22—4248no—
348C22—2124no—
349C22—2122no—
350C22—2124no—
351C22#1313318511yescharacteristic
352C22#1313318511yescharacteristic
353C22C2×D4×D5311yescharacteristic
354C4—22no—
355C4—1064no—
356C4—1062no—
357C4#1313318511yescharacteristic
358C4#1313318511yescharacteristic
359C4C2×D4×D5311yescharacteristic
360C4C22×C22⋊D5311yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—82no—
365C2—2122no—
366C2—2124no—
367C2#1681078912yesdirect product factor
368C2C2×#1313318511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×AGL(1,ℤ/8)×C53C2
socleC22×C53#120
hypercenterC2×AGL(1,ℤ/8)D53
hypocenterC53C2×#453
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. C2×#13133243C2semidirect
3. C2×#13133213C2semidirect
4. #16810789C2direct (N × H)
5. C2×#13133144C2semidirect
6. C2×#13133143C2semidirect
7. C2×#13133133C2semidirect
8. C2×AGL(1,ℤ/8)×C53C2semidirect
12. #13133263C22semidirect
22. #13133243C22semidirect
25. C2×#11719917C22semidirect
27. #13133213C22semidirect
28. C2×C4×Dic53C22semidirect
29. #13133144C22semidirect
30. #13133143C22semidirect
31. #13133133C22semidirect
33. C22×D4×C53C22semidirect
35. AGL(1,ℤ/8)×C53C22semidirect
38. C2×M16×C53C22semidirect
44. #41×C53D4semidirect
46. C2×Q8×C53D4semidirect
53. C2×D4×C53D4semidirect
54. C2×D4×C53C23semidirect
72. #11719945D4semidirect
82. C2×D212D4semidirect
86. C2×Dic106D4semidirect
90. M16×C53C23semidirect
95. C4×Dic53C23semidirect
97. #11719917C23semidirect
113. Q8×C53C2×D4semidirect
121. D4×C53C2×D4semidirect
144. D212C2×D4semidirect
149. Dic106C2×D4semidirect
181. C2×C53#453semidirect
187. C2×AGL(1,ℤ/8)D53semidirect
193. C53C2×#453semidirect
207. C22×D4C2×D53semidirect
211. AGL(1,ℤ/8)C2×D53semidirect
217. C2×M16C2×D53semidirect
237. #41C22⋊D53semidirect
240. C2×Q8C22⋊D53semidirect
265. C2×D4C22⋊D53semidirect
266. C2×D4C22×D53semidirect
279. M16C22×D53semidirect
300. Q8C2×C22⋊D53semidirect
319. D4C2×C22⋊D53semidirect
367. C2#16810789direct (N × H)