Finite Group Databasebeta

← #36362872

Subgroups — #36362872

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×#19149705C211yesmaximal, characteristic, complemented
3C2×#19149701C211yesmaximal, characteristic, complemented
4#26012072C211yesmaximal, characteristic, complemented
5#26011654C212yesmaximal, complemented
6#26011643C212yesmaximal, complemented
7#26011637C212yesmaximal, complemented
8#26011626C212yesmaximal, complemented
9#26011490C211yesmaximal, characteristic, complemented
10#26011457C211yesmaximal, characteristic, complemented
11C2×#102×C173C211yesmaximal, characteristic, complemented
12#19149724—24no—
13#19149724—24no—
14C2×#14583826C2211yescharacteristic
15C2×#14583826C2211yescharacteristic
16C2×#14583826C2211yescharacteristic, complemented
17C2×#14583825—21no—
18C2×#14583825C2212yes—
19C22×C22⋊D173—22no—
20C2×D4×D173—21no—
21C2×D4×D173C2212yes—
22#19149705—24no—
23#19149701—24no—
24C2×Dic692C2211yescharacteristic, complemented
25C2×#14583815C2211yescharacteristic, complemented
26C2×#14583815C2211yescharacteristic, complemented
27C2×#14583814C2212yescomplemented
28C2×D1384C2211yescharacteristic, complemented
29C2×#14583811C2212yescomplemented
30C2×#14583810C2211yescharacteristic, complemented
31#19149659—21no—
32#19149659C2212yescomplemented
33#19149654—21no—
34#19149654C2212yescomplemented
35#19149653—22no—
36#19149602C2212yescomplemented
37#19149599C2212yescomplemented
38#19149591C2212yescomplemented
39#19149591C2212yescomplemented
40C22×D4×C173C2211yescharacteristic
41C22×C8×C173C2211yescharacteristic, complemented
42C2×C4⋊C4×C173C2211yescharacteristic, complemented
43#102×C173C2212yescomplemented
44#102×C173C2212yescomplemented
45C23×D173—22no—
46#14583826—22no—
47#14583826—22no—
48#14583826—22no—
49#14583826—22no—
50#14583826—22no—
51#14583826D414yescomplemented
52#14583825—28no—
53#14583825—24no—
54#14583825—24no—
55C2×Dic346D411yescharacteristic
56C2×Dic346C2311yescharacteristic
57C2×Dic346C2311yescharacteristic
58C2×D692—22no—
59C2×D692D411yescharacteristic
60C2×D692C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—216no—
63C2×C22⋊D173—22no—
64C2×C22⋊D173—22no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173D412yes—
68D4×D173—28no—
69D4×D173—24no—
70D4×D173—24no—
71C22×Dic173—22no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2312yes—
75Dic692—22no—
76#14583815—22no—
77#14583815—22no—
78#14583814—24no—
79D1384—22no—
80#14583811—24no—
81#14583810—22no—
82C2×#13063436C2312yes—
83C4⋊Dic173—22no—
84#14583803—22no—
85#14583803—22no—
86C24×C173—21no—
87C2×D4×C173—21no—
88C2×D4×C173—22no—
89C2×D4×C173D414yescomplemented
90C2×D4×C173C2312yes—
91C22×C4×C173—21no—
92C22×C4×C173C2311yescharacteristic
93C2×C8×C173—22no—
94C2×C8×C173C2311yescharacteristic
95C2×C8×C173C2311yescharacteristic
96C4⋊C4×C173—21no—
97C4⋊C4×C173C2312yes—
98C22×D173—21no—
99C22×D173—28no—
100C22×D173—21no—
101C22×D173—82no—
102C22×D173—82no—
103C22×D173C2×D412yes—
104Dic346—22no—
105Dic346—21no—
106Dic346—21no—
107Dic346—21no—
108Dic346C2×D412yes—
109D692—22no—
110D692—21no—
111D692—21no—
112D692—21no—
113D692—82no—
114D692C2×D412yes—
115C22⋊D173—28no—
116C22⋊D173—28no—
117C22⋊D173—24no—
118C22⋊D173—24no—
119C22⋊D173—42no—
120C2×Dic173—28no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173C2×D412yes—
124C4×D173—24no—
125C4×D173—22no—
126C4×D173—22no—
127C4×D173—42no—
128#13063436—22no—
129C23×C173—22no—
130C23×C173—24no—
131C23×C173—81no—
132C23×C173—81no—
133C23×C173C2×D411yescharacteristic
134D4×C173—24no—
135D4×C173—22no—
136D4×C173C2×D414yes—
137C2×C4×C173—22no—
138C2×C4×C173—81no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D412yes—
141C2×C4×C173C2×D412yes—
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—24no—
148C22×C173—42no—
149C22×C173—42no—
150C22×C173—84no—
151C22×C173#12012yes—
152C22×C173C22×D411yescharacteristic
153C2×D173—22no—
154C2×D173—22no—
155C2×D173—22no—
156C2×D173—22no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—88no—
163C4×C173—42no—
164C4×C173#12012yes—
165C4×C173C22×D411yescharacteristic
166C4×C173C22×D411yescharacteristic
167Dic173—22no—
168Dic173—22no—
169Dic173—41no—
170C2×C173—22no—
171C2×C173—44no—
172C2×C173C2×#12011yescharacteristic
173C2×C173#57611yescharacteristic
174C2×C173#57711yescharacteristic
175D173—42no—
176D173—42no—
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#417—1731no—
189C2×#102D17311yescharacteristic, complemented
190#408—1731no—
191H(5,𝔽2)—3464no—
192H(5,𝔽2)—3464no—
193C2×#41—3461no—
194C2×#41—1732no—
195C2×#41—1731no—
196C2×#41—1731no—
197C2×#41—1731no—
198C22×D4—3461no—
199C22×D4—3462no—
200C22×D4—1732no—
201C22×D4C2×D17311yescharacteristic
202AGL(1,ℤ/8)—3464no—
203#135—3464no—
204C2×Q16—1731no—
205C2×SD16—1732no—
206C2×SD16—1731no—
207C2×SD16—1731no—
208C2×D8—1732no—
209C2×D8—1731no—
210C2×M16—1731no—
211C22×C8C2×D17311yescharacteristic, complemented
212C22⋊Q8—3461no—
213C22⋊Q8—1732no—
214#121—3461no—
215#121—3462no—
216#121—1732no—
217C2×C4⋊C4C2×D17311yescharacteristic, complemented
218#103—1732no—
219#102—1732no—
220#102C2×D17312yescomplemented
221#102C2×D17312yescomplemented
222C22⋊C8—1732no—
223C22⋊C8—1732no—
224C24—21no—
225C24—3462no—
226#41—3468no—
227#41—3464no—
228#41—3464no—
229#41—3462no—
230#41—3462no—
231#41—3462no—
232#41—3462no—
233#41—3462no—
234#41—1734no—
235C2×Q8—1731no—
236C2×Q8—1731no—
237C2×Q8—1731no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—3462no—
241C2×D4—3468no—
242C2×D4—3464no—
243C2×D4—34616no—
244C2×D4—3462no—
245C2×D4—3461no—
246C2×D4—3464no—
247C2×D4—3461no—
248C2×D4—3462no—
249C2×D4—1732no—
250C2×D4—1731no—
251C2×D4—1731no—
252C2×D4—1731no—
253C2×D4D69214yescomplemented
254C2×D4C22×D17312yes—
255C22×C4—21no—
256C22×C4—3462no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—1732no—
260C22×C4C22×D17311yescharacteristic
261Q16—3462no—
262SD16—3462no—
263SD16—3462no—
264SD16—3464no—
265D8—3462no—
266D8—3464no—
267M16—3462no—
268C2×C8—22no—
269C2×C8—1732no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3462no—
274C4⋊C4C22×D17312yes—
275C22⋊C4—3462no—
276C22⋊C4—3462no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—3468no—
282C23—3461no—
283C23—3461no—
284C23—13842no—
285C23—13842no—
286C23—1732no—
287C23D4×D17311yescharacteristic
288Q8—3462no—
289Q8—3461no—
290Q8—3461no—
291Q8—3461no—
292Q8—1732no—
293D4—24no—
294D4—22no—
295D4—3468no—
296D4—3468no—
297D4—3464no—
298D4—3464no—
299D4—3462no—
300D4—3461no—
301D4—3461no—
302D4—3461no—
303D4—13842no—
304D4—1732no—
305D4—6922no—
306D4C2×D69214yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—3468no—
310C2×C4—3461no—
311C2×C4—3464no—
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—21no—
322C8—21no—
323C8—3462no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—3462no—
331C22—3462no—
332C22—3462no—
333C22—3462no—
334C22—13848no—
335C22—6921no—
336C22—6921no—
337C22—6921no—
338C22—6921no—
339C22—6921no—
340C22#1914964912yes—
341C22C2×D4×D17311yescharacteristic
342C4—42no—
343C4—3462no—
344C4—3462no—
345C4—6921no—
346C4#1914964912yes—
347C4C2×D4×D17311yescharacteristic
348C4C22×D69211yescharacteristic
349C2—22no—
350C2—44no—
351C2—13841no—
352C2—6922no—
353C2—6922no—
354C2C2×#1914964911yescharacteristic
355C2#2601229211yescharacteristic
356C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#102×C173C2
socleC22×C173C22×D4
hypercenterC2×#102D173
hypocenterC173#3012
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#19149724C2semidirect
2. C2×#19149705C2semidirect
3. C2×#19149701C2semidirect
4. #26012072C2semidirect
5. #26011654C2semidirect
6. #26011643C2semidirect
7. #26011637C2semidirect
8. #26011626C2semidirect
9. #26011490C2semidirect
10. #26011457C2semidirect
11. C2×#102×C173C2semidirect
16. C2×#14583826C22semidirect
24. C2×Dic692C22semidirect
25. C2×#14583815C22semidirect
26. C2×#14583815C22semidirect
27. C2×#14583814C22semidirect
28. C2×D1384C22semidirect
29. C2×#14583811C22semidirect
30. C2×#14583810C22semidirect
32. #19149659C22semidirect
34. #19149654C22semidirect
36. #19149602C22semidirect
37. #19149599C22semidirect
38. #19149591C22semidirect
39. #19149591C22semidirect
41. C22×C8×C173C22semidirect
42. C2×C4⋊C4×C173C22semidirect
43. #102×C173C22semidirect
44. #102×C173C22semidirect
51. #14583826D4semidirect
89. C2×D4×C173D4semidirect
178. C173#3012semidirect
189. C2×#102D173semidirect
211. C22×C8C2×D173semidirect
217. C2×C4⋊C4C2×D173semidirect
220. #102C2×D173semidirect
221. #102C2×D173semidirect
253. C2×D4D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497242
C2×#191497052
C2×#191497012
#260120722
#260116542
#260116432
#260116372
#260116262
#260114902
#260114572
C2×#102×C1732
#3012173