Finite Group Databasebeta

← C2×#26011657 #36362885

Subgroups — C2×#26011657 #36362885

Order 22144.

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×#19149725C211yesmaximal, characteristic, complemented
2C2×#19149705C211yesmaximal, characteristic, complemented
3C2×#19149675C211yesmaximal, characteristic, complemented
4#26011657C218yesmaximal, direct product factor
5C2×#19149606C211yesmaximal, characteristic, complemented
6C2×#19149605C211yesmaximal, characteristic, complemented
7C2×#19149595C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
9#19149725—22no—
10#19149725—22no—
11#19149725—22no—
12#19149725C2214yescomplemented
13C2×#14583827—21no—
14C2×#14583826—21no—
15C2×#14583826C2211yescharacteristic
16C22×D692—21no—
17C22×D692C2211yescharacteristic
18C22×C22⋊D173—22no—
19C2×D4×D173—21no—
20C2×D4×D173—22no—
21#19149705—22no—
22#19149705C2214yescomplemented
23C2×#14583814—21no—
24C2×#14583811—21no—
25C2×#14583810C2211yescharacteristic, complemented
26#19149675—22no—
27#19149675C2214yescomplemented
28C2×C4×Dic173C2211yescharacteristic, complemented
29#19149606C2214yescomplemented
30#19149605C2214yescomplemented
31#19149595C2214yescomplemented
32C2×#41×C173C2211yescharacteristic
33C22×D4×C173C2211yescharacteristic, complemented
34AGL(1,ℤ/8)×C173—22no—
35AGL(1,ℤ/8)×C173C2214yescomplemented
36C2×SD16×C173—21no—
37C2×D8×C173—21no—
38C2×M16×C173C2211yescharacteristic, complemented
39C23×D173—22no—
40C23×D173D412yes—
41#14583827—24no—
42#14583827—24no—
43#14583826—24no—
44#14583826—24no—
45#14583826—21no—
46#14583826—21no—
47#14583826D412yescomplemented
48#14583826C2312yes—
49C2×Dic346D411yescharacteristic, complemented
50C2×D692—22no—
51C2×D692—24no—
52C2×D692—24no—
53C2×D692—21no—
54C2×D692—21no—
55C2×D692—21no—
56C2×D692—21no—
57C2×D692D412yes—
58C2×D692D411yescharacteristic, complemented
59C2×D692C2312yes—
60C2×C22⋊D173—22no—
61C2×C22⋊D173—28no—
62C2×C22⋊D173—21no—
63C2×C22⋊D173—48no—
64D4×D173—28no—
65D4×D173—24no—
66D4×D173—28no—
67D4×D173—24no—
68C22×Dic173—21no—
69C2×C4×D173—22no—
70C2×C4×D173—21no—
71#14583814—24no—
72#14583811—24no—
73#14583810—21no—
74#14583810C2312yescomplemented
75C2×#13063436—21no—
76C4×Dic173—21no—
77C4×Dic173C2312yescomplemented
78C24×C173—21no—
79#41×C173—21no—
80#41×C173—21no—
81#41×C173D412yescomplemented
82#41×C173C2312yes—
83C2×Q8×C173D411yescharacteristic, complemented
84C2×D4×C173—21no—
85C2×D4×C173—21no—
86C2×D4×C173—21no—
87C2×D4×C173—21no—
88C2×D4×C173—21no—
89C2×D4×C173—21no—
90C2×D4×C173D411yescharacteristic, complemented
91C2×D4×C173C2312yescomplemented
92C22×C4×C173—21no—
93C22×C4×C173C2311yescharacteristic
94SD16×C173—24no—
95D8×C173—24no—
96M16×C173—21no—
97M16×C173C2312yescomplemented
98C2×C8×C173—21no—
99C22×D173—22no—
100C22×D173—21no—
101C22×D173—28no—
102C22×D173—22no—
103C22×D173—44no—
104C22×D173—84no—
105C22×D173C2×D414yes—
106Dic346—21no—
107Dic346C2×D412yescomplemented
108D692—28no—
109D692—22no—
110D692—22no—
111D692—21no—
112D692—22no—
113D692—22no—
114D692C2×D412yescomplemented
115C22⋊D173—28no—
116C22⋊D173—22no—
117C22⋊D173—22no—
118C22⋊D173—416no—
119C2×Dic173—24no—
120C2×Dic173—21no—
121C2×Dic173—22no—
122C4×D173—28no—
123C4×D173—22no—
124C4×D173—22no—
125#13063436—22no—
126C23×C173—21no—
127C23×C173—22no—
128C23×C173—44no—
129C23×C173—41no—
130C23×C173—41no—
131C23×C173C2×D411yescharacteristic
132Q8×C173—21no—
133Q8×C173C2×D412yescomplemented
134D4×C173—22no—
135D4×C173—22no—
136D4×C173—21no—
137D4×C173—22no—
138D4×C173—22no—
139D4×C173—41no—
140D4×C173—41no—
141D4×C173C2×D412yescomplemented
142C2×C4×C173—21no—
143C2×C4×C173—22no—
144C2×C4×C173—22no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2411yescharacteristic
151C8×C173—22no—
152C22×C173—22no—
153C22×C173—22no—
154C22×C173—44no—
155C22×C173—42no—
156C22×C173—42no—
157C22×C173—82no—
158C22×C173—82no—
159C22×C173#12011yescharacteristic
160C22×C173#12011yescharacteristic
161C22×C173C22×D411yescharacteristic
162C2×D173—24no—
163C2×D173—22no—
164C2×D173—44no—
165C2×D173—42no—
166C2×D173—44no—
167C2×D173—82no—
168C2×D173—88no—
169C4×C173—22no—
170C4×C173#12011yescharacteristic
171C4×C173#12011yescharacteristic
172C4×C173C22×D411yescharacteristic
173C4×C173C22×D411yescharacteristic
174Dic173—24no—
175Dic173—22no—
176C2×C173—21no—
177C2×C173—21no—
178C2×C173—42no—
179C2×C173—82no—
180C2×C173#45312yescomplemented
181C2×C173C2×#12011yescharacteristic
182D173—44no—
183D173—42no—
184C173C2×#45311yescharacteristic, complemented, Sylow
185C2×#453—1731nomaximal, Sylow
186C2×H(5,𝔽2)—1731no—
187C2×AGL(1,ℤ/8)—1731no—
188C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
189C2×#127—1731no—
190#453—1738no—
191C2×#104—1731no—
192C2×#104—1731no—
193C2×#100—1731no—
194H(5,𝔽2)—3462no—
195H(5,𝔽2)—3462no—
196H(5,𝔽2)—3462no—
197H(5,𝔽2)—1734no—
198C2×#41—3461no—
199C2×#41—3461no—
200C2×#41—1731no—
201C2×#41C2×D17311yescharacteristic
202C22×D4—3461no—
203C22×D4—3462no—
204C22×D4—3461no—
205C22×D4—3462no—
206C22×D4—1731no—
207C22×D4C2×D17311yescharacteristic, complemented
208AGL(1,ℤ/8)—22no—
209AGL(1,ℤ/8)—3462no—
210AGL(1,ℤ/8)—1734no—
211AGL(1,ℤ/8)C2×D17314yescomplemented
212C2×SD16—21no—
213C2×SD16—3461no—
214C2×D8—21no—
215C2×D8—3461no—
216C2×M16—1731no—
217C2×M16C2×D17311yescharacteristic, complemented
218#127—3462no—
219#127—1734no—
220C2×C42—1731no—
221#104—1734no—
222#104—1734no—
223#100—1734no—
224C24—21no—
225C24—3462no—
226C24—1732no—
227#41—21no—
228#41—21no—
229#41—3464no—
230#41—3464no—
231#41—3464no—
232#41—3464no—
233#41—3461no—
234#41—3461no—
235#41—1732no—
236#41—1732no—
237#41C22⋊D17312yescomplemented
238#41C22×D17312yes—
239C2×Q8—1731no—
240C2×Q8C22⋊D17311yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—3462no—
248C2×D4—3464no—
249C2×D4—3464no—
250C2×D4—3468no—
251C2×D4—3464no—
252C2×D4—3468no—
253C2×D4—3462no—
254C2×D4—3461no—
255C2×D4—3464no—
256C2×D4—3468no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1732no—
262C2×D4—1731no—
263C2×D4—1732no—
264C2×D4—6928no—
265C2×D4C22⋊D17311yescharacteristic, complemented
266C2×D4C22×D17312yescomplemented
267C22×C4—21no—
268C22×C4—3462no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4C22×D17311yescharacteristic
272SD16—24no—
273SD16—3464no—
274D8—24no—
275D8—3464no—
276M16—21no—
277M16—3461no—
278M16—1732no—
279M16C22×D17312yescomplemented
280C2×C8—21no—
281C2×C8—3461no—
282C42—3461no—
283C42—1732no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—41no—
289C23—3468no—
290C23—3462no—
291C23—3462no—
292C23—3461no—
293C23—13844no—
294C23—1734no—
295C23—6924no—
296C23D4×D17311yescharacteristic
297Q8—21no—
298Q8—3461no—
299Q8—1732no—
300Q8C2×C22⋊D17312yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308D4—3468no—
309D4—3468no—
310D4—3462no—
311D4—3462no—
312D4—3462no—
313D4—3461no—
314D4—3462no—
315D4—3462no—
316D4—3462no—
317D4—1732no—
318D4—69216no—
319D4C2×C22⋊D17312yescomplemented
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—3468no—
324C2×C4—3464no—
325C2×C4—3461no—
326C2×C4—3462no—
327C2×C4—3462no—
328C2×C4—3462no—
329C2×C4D4×D17311yescharacteristic
330C2×C4D4×D17311yescharacteristic
331C2×C4D4×D17311yescharacteristic
332C2×C4C2×C22⋊D17311yescharacteristic
333C2×C4C2×C22⋊D17311yescharacteristic
334C2×C4C23×D17311yescharacteristic
335C8—22no—
336C8—3462no—
337C22—22no—
338C22—22no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—3464no—
345C22—3462no—
346C22—13842no—
347C22—13848no—
348C22—6924no—
349C22—6922no—
350C22—6924no—
351C22#1914964711yescharacteristic
352C22#1914964711yescharacteristic
353C22C2×D4×D17311yescharacteristic
354C4—22no—
355C4—3464no—
356C4—3462no—
357C4#1914964711yescharacteristic
358C4#1914964711yescharacteristic
359C4C2×D4×D17311yescharacteristic
360C4C22×C22⋊D17311yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—82no—
365C2—6922no—
366C2—6924no—
367C2#2601165712yesdirect product factor
368C2C2×#1914964711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173#120
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173C2×#453
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
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×#19149725C2semidirect
2. C2×#19149705C2semidirect
3. C2×#19149675C2semidirect
4. #26011657C2direct (N × H)
5. C2×#19149606C2semidirect
6. C2×#19149605C2semidirect
7. C2×#19149595C2semidirect
8. C2×AGL(1,ℤ/8)×C173C2semidirect
12. #19149725C22semidirect
22. #19149705C22semidirect
25. C2×#14583810C22semidirect
27. #19149675C22semidirect
28. C2×C4×Dic173C22semidirect
29. #19149606C22semidirect
30. #19149605C22semidirect
31. #19149595C22semidirect
33. C22×D4×C173C22semidirect
35. AGL(1,ℤ/8)×C173C22semidirect
38. C2×M16×C173C22semidirect
47. #14583826D4semidirect
49. C2×Dic346D4semidirect
58. C2×D692D4semidirect
74. #14583810C23semidirect
77. C4×Dic173C23semidirect
81. #41×C173D4semidirect
83. C2×Q8×C173D4semidirect
90. C2×D4×C173D4semidirect
91. C2×D4×C173C23semidirect
97. M16×C173C23semidirect
107. Dic346C2×D4semidirect
114. D692C2×D4semidirect
133. Q8×C173C2×D4semidirect
141. D4×C173C2×D4semidirect
180. C2×C173#453semidirect
184. C173C2×#453semidirect
188. C2×AGL(1,ℤ/8)D173semidirect
207. C22×D4C2×D173semidirect
211. AGL(1,ℤ/8)C2×D173semidirect
217. C2×M16C2×D173semidirect
237. #41C22⋊D173semidirect
240. C2×Q8C22⋊D173semidirect
265. C2×D4C22⋊D173semidirect
266. C2×D4C22×D173semidirect
279. M16C22×D173semidirect
300. Q8C2×C22⋊D173semidirect
319. D4C2×C22⋊D173semidirect
367. C2#26011657direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497052
C2×#191496752
#260116572
C2×#191496062
C2×#191496052
C2×#191495952
C2×AGL(1,ℤ/8)×C1732
C2×#453173