Finite Group Databasebeta

← #36362869

Subgroups — #36362869

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×#19149725C211yesmaximal, characteristic, complemented
2C2×#19149707C211yesmaximal, characteristic, complemented
3C2×#19149699C211yesmaximal, characteristic, complemented
4#26012072C211yesmaximal, characteristic, complemented
5#26011656C212yesmaximal, complemented
6#26011643C212yesmaximal, complemented
7#26011639C212yesmaximal, complemented
8#26011626C212yesmaximal, complemented
9C2×#19149599C211yesmaximal, characteristic, complemented
10#26011459C211yesmaximal, characteristic, complemented
11#417×C173C211yesmaximal, characteristic, complemented
12#19149725—24no—
13#19149725—24no—
14C2×#14583827C2212yes—
15C2×#14583826—21no—
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic, complemented
18C22×D692—21no—
19C22×D692C2211yescharacteristic
20C2×D4×D173—22no—
21C2×D4×D173C2212yes—
22#19149707—24no—
23#19149699—24no—
24C2×#14583816C2211yescharacteristic, complemented
25C2×#14583815C2212yescomplemented
26C2×#14583814C2211yescharacteristic, complemented
27C2×#14583813C2211yescharacteristic, complemented
28C2×D1384C2212yescomplemented
29C2×#14583811C2211yescharacteristic, complemented
30C22×#13063436C2211yescharacteristic, complemented
31#19149659—21no—
32#19149659C2212yescomplemented
33#19149654—21no—
34#19149654C2212yescomplemented
35#19149653—22no—
36#19149599C2212yescomplemented
37#19149599C2212yescomplemented
38#19149591C2212yescomplemented
39#19149591C2212yescomplemented
40C2×#41×C173C2211yescharacteristic
41C2×M16×C173C2211yescharacteristic, complemented
42C2×C4⋊C4×C173C2211yescharacteristic, complemented
43#103×C173C2212yescomplemented
44#102×C173C2212yescomplemented
45C23×D173—22no—
46C23×D173—21no—
47#14583827—24no—
48#14583827—24no—
49#14583826—28no—
50#14583826—22no—
51#14583826—22no—
52#14583826—22no—
53#14583826—22no—
54C2×Dic346D411yescharacteristic
55C2×Dic346C2311yescharacteristic
56C2×D692—28no—
57C2×D692—22no—
58C2×D692—22no—
59C2×D692—21no—
60C2×D692—22no—
61C2×D692D411yescharacteristic
62C2×D692D414yescomplemented
63C2×D692C2311yescharacteristic
64C2×D692C2312yes—
65C2×C22⋊D173—22no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173D412yes—
68D4×D173—216no—
69D4×D173—24no—
70D4×D173—24no—
71C2×C4×D173—22no—
72C2×C4×D173—21no—
73C2×C4×D173C2312yes—
74#14583816—22no—
75#14583815—24no—
76#14583814—22no—
77#14583813—22no—
78D1384—24no—
79#14583811—22no—
80C2×#13063436—22no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173—22no—
84#14583803—22no—
85#14583803—22no—
86#41×C173—22no—
87#41×C173D414yescomplemented
88C2×Q8×C173C2311yescharacteristic
89C2×D4×C173—21no—
90C2×D4×C173C2311yescharacteristic
91C22×C4×C173—21no—
92C22×C4×C173—21no—
93C22×C4×C173C2311yescharacteristic
94M16×C173—22no—
95C2×C8×C173C2312yes—
96C4⋊C4×C173—21no—
97C4⋊C4×C173C2312yes—
98C22×D173—22no—
99C22×D173—28no—
100C22×D173—24no—
101C22×D173—21no—
102C22×D173—82no—
103C22×D173—82no—
104C22×D173—81no—
105C22×D173—81no—
106C22×D173C2×D412yes—
107Dic346—22no—
108Dic346—21no—
109Dic346—21no—
110D692—28no—
111D692—22no—
112D692—24no—
113D692—22no—
114D692—21no—
115D692—21no—
116D692—82no—
117D692C2×D414yes—
118C22⋊D173—28no—
119C22⋊D173—24no—
120C22⋊D173—42no—
121C2×Dic173—21no—
122C2×Dic173C2×D412yes—
123C4×D173—28no—
124C4×D173—22no—
125C4×D173—22no—
126C4×D173—42no—
127#13063436—21no—
128#13063436—21no—
129C23×C173—21no—
130C23×C173C2×D411yescharacteristic
131Q8×C173—21no—
132Q8×C173C2×D412yes—
133D4×C173—24no—
134D4×C173—21no—
135D4×C173C2×D412yes—
136C2×C4×C173—21no—
137C2×C4×C173—24no—
138C2×C4×C173—22no—
139C2×C4×C173—81no—
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—22no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173#12012yes—
151C22×C173C22×D411yescharacteristic
152C2×D173—22no—
153C2×D173—24no—
154C2×D173—22no—
155C2×D173—22no—
156C2×D173—42no—
157C2×D173—42no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—84no—
162C2×D173—88no—
163C4×C173—22no—
164C4×C173—42no—
165C4×C173#12012yes—
166C4×C173C22×D411yescharacteristic
167C4×C173C22×D411yescharacteristic
168Dic173—22no—
169Dic173—41no—
170C2×C173—22no—
171C2×C173—42no—
172C2×C173C2×#12011yescharacteristic
173C2×C173#57611yescharacteristic
174C2×C173#57711yescharacteristic
175D173—42no—
176D173—44no—
177D173—81no—
178C173#301211yescharacteristic, complemented, Sylow
179#3012—1731nomaximal, Sylow
180C2×H(5,𝔽2)—1731no—
181C2×AGL(1,ℤ/8)—1731no—
182C2×#135—1731no—
183#537—1731no—
184#452—1732no—
185#450—1732no—
186#449—1732no—
187#447—1732no—
188#417D17311yescharacteristic, complemented
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
198C22×D4—3461no—
199C22×D4—3462no—
200C22×D4—1732no—
201C22×D4—1731no—
202AGL(1,ℤ/8)—3464no—
203#135—3464no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×SD16—1732no—
208C2×D8—1731no—
209C2×D8—1732no—
210C2×M16C2×D17311yescharacteristic, complemented
211C22×C8—1731no—
212C22⋊Q8—3461no—
213C22⋊Q8—1732no—
214#121—3461no—
215#121—3462no—
216#121—1732no—
217C2×C4⋊C4C2×D17311yescharacteristic, complemented
218#103C2×D17312yescomplemented
219#102—1732no—
220#102—1732no—
221#102C2×D17312yescomplemented
222C22⋊C8—1732no—
223C22⋊C8—1732no—
224C24—3462no—
225C24—3461no—
226#41—22no—
227#41—3468no—
228#41—3464no—
229#41—3464no—
230#41—3462no—
231#41—3462no—
232#41—3462no—
233#41—3462no—
234#41D69214yescomplemented
235C2×Q8—1731no—
236C2×Q8—1731no—
237C2×Q8C22×D17311yescharacteristic
238C2×D4—21no—
239C2×D4—3468no—
240C2×D4—3462no—
241C2×D4—3462no—
242C2×D4—34616no—
243C2×D4—3464no—
244C2×D4—3464no—
245C2×D4—3461no—
246C2×D4—3462no—
247C2×D4—3461no—
248C2×D4—3462no—
249C2×D4—1732no—
250C2×D4—1731no—
251C2×D4—1734no—
252C2×D4—1731no—
253C2×D4—1732no—
254C2×D4C22×D17311yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—3462no—
258C22×C4—3461no—
259C22×C4—1732no—
260C22×C4C22×D17311yescharacteristic
261Q16—3462no—
262SD16—3464no—
263SD16—3462no—
264SD16—3462no—
265D8—3464no—
266D8—3462no—
267M16—22no—
268C2×C8—3462no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8C22×D17312yes—
272C4⋊C4—21no—
273C4⋊C4—3462no—
274C4⋊C4C22×D17312yes—
275C22⋊C4—3462no—
276C22⋊C4—3462no—
277C23—21no—
278C23—3468no—
279C23—3464no—
280C23—3462no—
281C23—3461no—
282C23—13841no—
283C23—13841no—
284C23—13842no—
285C23—13842no—
286C23—1732no—
287C23D4×D17311yescharacteristic
288Q8—21no—
289Q8—3462no—
290Q8—3461no—
291Q8—3461no—
292Q8C2×D69212yes—
293D4—24no—
294D4—21no—
295D4—3468no—
296D4—3468no—
297D4—3464no—
298D4—3462no—
299D4—3464no—
300D4—3461no—
301D4—3462no—
302D4—3461no—
303D4—13842no—
304D4—1734no—
305D4—6922no—
306D4C2×D69212yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—3468no—
312C2×C4—3462no—
313C2×C4—3462no—
314C2×C4—3461no—
315C2×C4—1732no—
316C2×C4—6922no—
317C2×C4D4×D17311yescharacteristic
318C2×C4D4×D17312yes—
319C2×C4C2×D69212yes—
320C2×C4C23×D17311yescharacteristic
321C8—22no—
322C8—3461no—
323C8—3461no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—3462no—
330C22—3464no—
331C22—3462no—
332C22—3462no—
333C22—13844no—
334C22—13848no—
335C22—6921no—
336C22—6921no—
337C22—6921no—
338C22—6922no—
339C22—6922no—
340C22#1914964912yes—
341C22C2×D4×D17311yescharacteristic
342C4—22no—
343C4—42no—
344C4—3462no—
345C4—6921no—
346C4#1914964912yes—
347C4C2×D4×D17311yescharacteristic
348C4C22×D69211yescharacteristic
349C2—22no—
350C2—42no—
351C2—13841no—
352C2—6922no—
353C2—6924no—
354C2C2×#1914964911yescharacteristic
355C2#2601229511yescharacteristic
356C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#417×C173C2
socleC22×C173C22×D4
hypercenter#417D173
hypocenterC173#3012
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19149725C2semidirect
2. C2×#19149707C2semidirect
3. C2×#19149699C2semidirect
4. #26012072C2semidirect
5. #26011656C2semidirect
6. #26011643C2semidirect
7. #26011639C2semidirect
8. #26011626C2semidirect
9. C2×#19149599C2semidirect
10. #26011459C2semidirect
11. #417×C173C2semidirect
17. C2×#14583826C22semidirect
24. C2×#14583816C22semidirect
25. C2×#14583815C22semidirect
26. C2×#14583814C22semidirect
27. C2×#14583813C22semidirect
28. C2×D1384C22semidirect
29. C2×#14583811C22semidirect
30. C22×#13063436C22semidirect
32. #19149659C22semidirect
34. #19149654C22semidirect
36. #19149599C22semidirect
37. #19149599C22semidirect
38. #19149591C22semidirect
39. #19149591C22semidirect
41. C2×M16×C173C22semidirect
42. C2×C4⋊C4×C173C22semidirect
43. #103×C173C22semidirect
44. #102×C173C22semidirect
62. C2×D692D4semidirect
87. #41×C173D4semidirect
178. C173#3012semidirect
188. #417D173semidirect
210. C2×M16C2×D173semidirect
217. C2×C4⋊C4C2×D173semidirect
218. #103C2×D173semidirect
221. #102C2×D173semidirect
234. #41D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497072
C2×#191496992
#260120722
#260116562
#260116432
#260116392
#260116262
C2×#191495992
#260114592
#417×C1732
#3012173