Finite Group Databasebeta

← #36362866

Subgroups — #36362866

Order 22144.

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

normal subgroups only

This group has 764 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×#19149724C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3#26012069C211yesmaximal, characteristic, complemented
4#26011650C212yesmaximal, complemented
5#26011640C212yesmaximal, complemented
6#26011633C212yesmaximal, complemented
7#26011623C212yesmaximal, complemented
8#26011493C211yesmaximal, characteristic, complemented
9C2×#19149600C211yesmaximal, characteristic, complemented
10#26011459C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
12#19149724—24no—
13#19149724—24no—
14C2×#14583826C2211yescharacteristic
15C2×#14583826C2211yescharacteristic
16C2×#14583825—21no—
17C2×#14583825C2212yes—
18C22×C22⋊D173—22no—
19C2×D4×D173—21no—
20C2×D4×D173C2212yes—
21#19149699—24no—
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583814C2211yescharacteristic, complemented
24C2×#14583813C2211yescharacteristic, complemented
25C2×#14583811C2211yescharacteristic, complemented
26C22×#13063436C2211yescharacteristic, complemented
27#19149658—21no—
28#19149655C2212yescomplemented
29#19149653—21no—
30#19149651—22no—
31#19149650C2212yescomplemented
32C2×C4⋊Dic173C2211yescharacteristic, complemented
33#19149604C2212yescomplemented
34#19149601C2212yescomplemented
35#19149600C2212yescomplemented
36#19149600C2212yescomplemented
37#19149591C2212yescomplemented
38#19149591C2212yescomplemented
39C2×#41×C173C2211yescharacteristic, complemented
40C22×D4×C173C2211yescharacteristic
41AGL(1,ℤ/8)×C173—24no—
42C2×SD16×C173C2212yescomplemented
43C2×D8×C173C2212yescomplemented
44C2×M16×C173C2211yescharacteristic, complemented
45C23×D173—22no—
46#14583826—22no—
47#14583826—22no—
48#14583826—22no—
49#14583826D414yescomplemented
50#14583825—28no—
51#14583825—24no—
52#14583825—24no—
53C2×Dic346D411yescharacteristic
54C2×Dic346C2311yescharacteristic
55C2×D692D411yescharacteristic
56C2×D692C2311yescharacteristic
57C2×C22⋊D173—216no—
58C2×C22⋊D173—22no—
59C2×C22⋊D173—22no—
60C2×C22⋊D173—22no—
61C2×C22⋊D173—21no—
62C2×C22⋊D173D412yes—
63D4×D173—28no—
64D4×D173—24no—
65D4×D173—24no—
66C22×Dic173—22no—
67C22×Dic173—21no—
68C2×C4×D173—21no—
69C2×C4×D173C2312yes—
70#14583816—22no—
71#14583814—22no—
72#14583813—22no—
73#14583811—22no—
74C2×#13063436—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173—21no—
78C4⋊Dic173C2312yes—
79#14583805—22no—
80C22⋊Dic173—22no—
81#14583803—22no—
82C24×C173—21no—
83#41×C173—22no—
84#41×C173—22no—
85C2×Q8×C173C2311yescharacteristic
86C2×D4×C173—21no—
87C2×D4×C173—21no—
88C2×D4×C173—22no—
89C2×D4×C173D414yescomplemented
90C2×D4×C173C2311yescharacteristic
91C2×D4×C173C2312yes—
92C22×C4×C173—21no—
93C22×C4×C173C2311yescharacteristic
94SD16×C173—24no—
95D8×C173—24no—
96M16×C173—22no—
97C2×C8×C173C2312yes—
98C22×D173—21no—
99C22×D173—28no—
100C22×D173—82no—
101C22×D173—82no—
102C22×D173C2×D412yes—
103Dic346—22no—
104Dic346—21no—
105Dic346C2×D412yes—
106D692—22no—
107D692—21no—
108D692C2×D412yes—
109C22⋊D173—28no—
110C22⋊D173—28no—
111C22⋊D173—24no—
112C22⋊D173—24no—
113C22⋊D173—82no—
114C2×Dic173—28no—
115C2×Dic173—21no—
116C2×Dic173—22no—
117C2×Dic173—81no—
118C2×Dic173C2×D412yes—
119C4×D173—24no—
120C4×D173—22no—
121C4×D173—22no—
122#13063436—21no—
123#13063436—21no—
124C23×C173—22no—
125C23×C173—24no—
126C23×C173—21no—
127C23×C173—81no—
128C23×C173—81no—
129C23×C173C2×D411yescharacteristic
130Q8×C173—21no—
131Q8×C173—21no—
132D4×C173—24no—
133D4×C173—22no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—42no—
137D4×C173C2×D414yes—
138C2×C4×C173—21no—
139C2×C4×C173—42no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D412yes—
142C2×C4×C173C2×D412yes—
143C2×C4×C173C2411yescharacteristic
144C8×C173—22no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—24no—
148C22×C173—42no—
149C22×C173—42no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—84no—
154C22×C173#12012yes—
155C22×C173C22×D411yescharacteristic
156C2×D173—22no—
157C2×D173—22no—
158C2×D173—22no—
159C2×D173—22no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—88no—
163C4×C173—41no—
164C4×C173#12012yes—
165C4×C173C22×D411yescharacteristic
166C4×C173C22×D411yescharacteristic
167Dic173—22no—
168Dic173—22no—
169Dic173—42no—
170C2×C173—22no—
171C2×C173—44no—
172C2×C173—81no—
173C2×C173C2×#12011yescharacteristic
174C2×C173#57611yescharacteristic
175C2×C173#57711yescharacteristic
176D173—42no—
177D173—42no—
178C173#301211yescharacteristic, complemented, Sylow
179#3012—1731nomaximal, Sylow
180C2×H(5,𝔽2)—1731no—
181C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
182C2×#135—1731no—
183#537—1731no—
184#452—1732no—
185#450—1732no—
186#449—1732no—
187#447—1732no—
188#417—1731no—
189C2×#102—1731no—
190#408—1731no—
191H(5,𝔽2)—3464no—
192H(5,𝔽2)—3464no—
193C2×#41—3461no—
194C2×#41—1732no—
195C2×#41—1731no—
196C2×#41—1731no—
197C2×#41C2×D17311yescharacteristic, complemented
198C22×D4—3461no—
199C22×D4—3462no—
200C22×D4—1732no—
201C22×D4C2×D17311yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—3464no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×SD16C2×D17312yescomplemented
208C2×D8—1731no—
209C2×D8C2×D17312yescomplemented
210C2×M16C2×D17311yescharacteristic, complemented
211C22×C8—1731no—
212C22⋊Q8—3461no—
213C22⋊Q8—1732no—
214#121—3461no—
215#121—3462no—
216#121—1732no—
217C2×C4⋊C4—1731no—
218#103—1732no—
219#102—1732no—
220#102—1732no—
221#102—1732no—
222C22⋊C8—1732no—
223C22⋊C8—1732no—
224C24—21no—
225C24—3462no—
226#41—22no—
227#41—22no—
228#41—3468no—
229#41—3464no—
230#41—3464no—
231#41—3462no—
232#41—3462no—
233#41—3462no—
234#41—1734no—
235C2×Q8—1731no—
236C2×Q8—1731no—
237C2×Q8C22×D17311yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—3468no—
242C2×D4—3464no—
243C2×D4—34616no—
244C2×D4—3462no—
245C2×D4—3461no—
246C2×D4—3464no—
247C2×D4—3462no—
248C2×D4—3462no—
249C2×D4—1732no—
250C2×D4—1731no—
251C2×D4—1731no—
252C2×D4C22⋊D17314yescomplemented
253C2×D4C22×D17311yescharacteristic
254C2×D4C22×D17312yes—
255C22×C4—21no—
256C22×C4—3462no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—1732no—
260C22×C4C22×D17311yescharacteristic
261Q16—3462no—
262SD16—24no—
263SD16—3462no—
264SD16—3462no—
265D8—24no—
266D8—3462no—
267M16—22no—
268C2×C8—3462no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8C22×D17312yes—
272C4⋊C4—3462no—
273C4⋊C4—3461no—
274C4⋊C4—1732no—
275C22⋊C4—3462no—
276C22⋊C4—3462no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—3468no—
283C23—3461no—
284C23—13842no—
285C23—13842no—
286C23—1732no—
287C23D4×D17311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—3462no—
291Q8—3461no—
292Q8—1732no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—3468no—
299D4—3468no—
300D4—3464no—
301D4—3464no—
302D4—3462no—
303D4—3461no—
304D4—13842no—
305D4—1732no—
306D4C2×C22⋊D17314yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—3468no—
310C2×C4—3461no—
311C2×C4—3464no—
312C2×C4—3462no—
313C2×C4—3462no—
314C2×C4—3462no—
315C2×C4—13841no—
316C2×C4—1732no—
317C2×C4D4×D17311yescharacteristic
318C2×C4D4×D17312yes—
319C2×C4C2×C22⋊D17312yes—
320C2×C4C23×D17311yescharacteristic
321C8—22no—
322C8—3461no—
323C8—3461no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—3462no—
334C22—3462no—
335C22—3462no—
336C22—3462no—
337C22—13848no—
338C22—6921no—
339C22—6921no—
340C22#1914964712yes—
341C22C2×D4×D17311yescharacteristic
342C4—41no—
343C4—3462no—
344C4—3462no—
345C4—6922no—
346C4#1914964712yes—
347C4C2×D4×D17311yescharacteristic
348C4C22×C22⋊D17311yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—6922no—
353C2—6922no—
354C2C2×#1914964711yescharacteristic
355C2#2601229511yescharacteristic
356C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3012
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×#19149724C2semidirect
2. C2×#19149699C2semidirect
3. #26012069C2semidirect
4. #26011650C2semidirect
5. #26011640C2semidirect
6. #26011633C2semidirect
7. #26011623C2semidirect
8. #26011493C2semidirect
9. C2×#19149600C2semidirect
10. #26011459C2semidirect
11. C2×AGL(1,ℤ/8)×C173C2semidirect
22. C2×#14583816C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583813C22semidirect
25. C2×#14583811C22semidirect
26. C22×#13063436C22semidirect
28. #19149655C22semidirect
31. #19149650C22semidirect
32. C2×C4⋊Dic173C22semidirect
33. #19149604C22semidirect
34. #19149601C22semidirect
35. #19149600C22semidirect
36. #19149600C22semidirect
37. #19149591C22semidirect
38. #19149591C22semidirect
39. C2×#41×C173C22semidirect
42. C2×SD16×C173C22semidirect
43. C2×D8×C173C22semidirect
44. C2×M16×C173C22semidirect
49. #14583826D4semidirect
89. C2×D4×C173D4semidirect
178. C173#3012semidirect
181. C2×AGL(1,ℤ/8)D173semidirect
197. C2×#41C2×D173semidirect
207. C2×SD16C2×D173semidirect
209. C2×D8C2×D173semidirect
210. C2×M16C2×D173semidirect
252. C2×D4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497242
C2×#191496992
#260120692
#260116502
#260116402
#260116332
#260116232
#260114932
C2×#191496002
#260114592
C2×AGL(1,ℤ/8)×C1732
#3012173