Finite Group Databasebeta

← #448×D173 #36362748

Subgroups — #448×D173 #36362748

Order 22144.

792 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×Q8×D173C211yesmaximal, characteristic, complemented
2C2×SD16×D173C212yesmaximal, complemented
3#121×D173C211yesmaximal, characteristic
4#26011632C211yesmaximal, characteristic, complemented
5#26011631C212yesmaximal, complemented
6#26011630C211yesmaximal, characteristic, complemented
7#26011629C211yesmaximal, characteristic, complemented
8#26011628C212yesmaximal, complemented
9#103×D173C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#448×C173C211yesmaximal, characteristic, complemented
12C22×Dic346—21no—
13C22×Dic346C2211yescharacteristic, complemented
14C2×Q8×D173—28no—
15C2×Q8×D173—21no—
16C2×Q8×D173—21no—
17C2×Q8×D173C2212yes—
18C2×Q8×D173C2212yescomplemented
19C2×D4×D173—21no—
20C2×D4×D173C2211yescharacteristic
21C22×C4×D173—21no—
22C22×C4×D173C2211yescharacteristic
23C2×#14583815C2212yescomplemented
24C2×#14583814C2212yescomplemented
25C2×#14583813C2212yescomplemented
26SD16×D173—28no—
27C2×C8×D173C2212yes—
28#19149654C2211yescharacteristic
29#19149653C2211yescharacteristic
30#19149652—21no—
31#19149651—21no—
32#19149650C2211yescharacteristic
33C4⋊C4×D173C2211yescharacteristic
34C22⋊C4×D173—21no—
35#19149604C2212yescomplemented
36#19149603C2212yescomplemented
37#19149602C2212yescomplemented
38C22⋊#13063436C2211yescharacteristic, complemented
39#19149591C2212yescomplemented
40C22×Q8×C173C2211yescharacteristic, complemented
41C2×SD16×C173C2212yescomplemented
42#121×C173C2211yescharacteristic
43#103×C173C2212yescomplemented
44C22⋊C8×C173C2211yescharacteristic, complemented
45C23×D173—21no—
46C23×D173D411yescharacteristic
47C2×Dic346—28no—
48C2×Dic346—24no—
49C2×Dic346—22no—
50C2×Dic346—22no—
51C2×Dic346—21no—
52C2×Dic346C2312yes—
53Q8×D173—28no—
54Q8×D173—28no—
55Q8×D173—24no—
56Q8×D173—24no—
57Q8×D173—24no—
58Q8×D173D418yescomplemented
59C2×D692—21no—
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64D4×D173—24no—
65D4×D173—44no—
66C22×Dic173—21no—
67C22×Dic173D411yescharacteristic
68C2×C4×D173—28no—
69C2×C4×D173—22no—
70C2×C4×D173—22no—
71C2×C4×D173—24no—
72C2×C4×D173—21no—
73C2×C4×D173D411yescharacteristic
74C2×C4×D173D411yescharacteristic
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2311yescharacteristic
77#14583815—24no—
78#14583814—24no—
79#14583813—24no—
80C2×#13063436C2312yes—
81C8×D173—24no—
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84C22⋊Dic173—21no—
85#14583803—21no—
86#14583803—21no—
87C2×Q8×C173—24no—
88C2×Q8×C173—21no—
89C2×Q8×C173C2312yes—
90C2×D4×C173—21no—
91C2×D4×C173C2311yescharacteristic
92C22×C4×C173—21no—
93C22×C4×C173C2311yescharacteristic
94SD16×C173—24no—
95C2×C8×C173C2312yes—
96C4⋊C4×C173C2311yescharacteristic
97C22⋊C4×C173—21no—
98C22×D173—22no—
99C22×D173—22no—
100C22×D173—21no—
101C22×D173—42no—
102C22×D173—42no—
103C22×D173—42no—
104C22×D173SD1614yescomplemented
105C22×D173C2×D411yescharacteristic
106C22×D173C2×D411yescharacteristic
107Dic346—24no—
108Dic346—28no—
109Dic346—22no—
110Dic346—28no—
111Dic346C2×D414yescomplemented
112D692—22no—
113D692—42no—
114C22⋊D173—42no—
115C22⋊D173—42no—
116C22⋊D173—42no—
117C2×Dic173—24no—
118C2×Dic173—22no—
119C2×Dic173—22no—
120C2×Dic173—21no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—28no—
124C4×D173—28no—
125C4×D173—22no—
126C4×D173—22no—
127C4×D173—22no—
128C4×D173—42no—
129C4×D173C2×D414yes—
130#13063436—22no—
131C23×C173—21no—
132C23×C173C2×D411yescharacteristic
133Q8×C173—24no—
134Q8×C173—22no—
135Q8×C173C2×D414yescomplemented
136D4×C173—22no—
137D4×C173—42no—
138C2×C4×C173—24no—
139C2×C4×C173—22no—
140C2×C4×C173—22no—
141C2×C4×C173—21no—
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173—22no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173C2×SD1612yescomplemented
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—28no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—84no—
159C2×D173#12012yes—
160C2×D173C2×SD1612yes—
161C2×D173C2×SD1612yescomplemented
162C2×D173#13712yes—
163C4×C173—24no—
164C4×C173—21no—
165C4×C173—41no—
166C4×C173C22×D412yes—
167Dic173—24no—
168Dic173—21no—
169Dic173—41no—
170Dic173#12012yes—
171C2×C173—22no—
172C2×C173—81no—
173C2×C173C2×#12011yescharacteristic
174C2×C173C22×SD1611yescharacteristic
175C2×C173C2×#13711yescharacteristic
176D173—22no—
177D173—81no—
178D173#44814yesdirect product factor
179C173C2×#44811yescharacteristic, complemented, Sylow
180C2×#448—1731nomaximal, Sylow
181C23×Q8—1731no—
182C22×SD16—1732no—
183C2×#121—1731no—
184#448—1731no—
185#448—1732no—
186#448—1731no—
187#448—1732no—
188#448—1731no—
189#448D17311yescharacteristic, direct product factor
190C2×#103—1732no—
191C2×C22⋊C8—1731no—
192C22×Q8—3461no—
193C22×Q8—3468no—
194C22×Q8—3461no—
195C22×Q8—3461no—
196C22×Q8—1732no—
197C22×Q8—1732no—
198C22×Q8—1731no—
199C22×Q8C2×D17311yescharacteristic, complemented
200C22×D4—3461no—
201C22×D4—1731no—
202C23×C4—3461no—
203C23×C4—1731no—
204C2×SD16—3468no—
205C2×SD16—1732no—
206C2×SD16—1732no—
207C2×SD16—1732no—
208C2×SD16C2×D17312yescomplemented
209C22×C8—1732no—
210#121—3461no—
211#121—3461no—
212#121—1731no—
213#121—1731no—
214#121—1731no—
215#121C2×D17311yescharacteristic
216C2×C4⋊C4—1731no—
217C2×C22⋊C4—3461no—
218#103—1732no—
219#103—1732no—
220#103—1732no—
221#103C2×D17312yescomplemented
222C22⋊C8—1731no—
223C22⋊C8—1732no—
224C22⋊C8C2×D17311yescharacteristic, complemented
225C24—3461no—
226C24—1731no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—3468no—
230C2×Q8—3464no—
231C2×Q8—3462no—
232C2×Q8—3462no—
233C2×Q8—3468no—
234C2×Q8—3468no—
235C2×Q8—3464no—
236C2×Q8—3464no—
237C2×Q8—3464no—
238C2×Q8—3461no—
239C2×Q8—1738no—
240C2×Q8—1732no—
241C2×Q8C22×D17312yes—
242C2×D4—21no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—3464no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4—6924no—
250C2×D4C22×D17311yescharacteristic
251C22×C4—21no—
252C22×C4—3468no—
253C22×C4—3462no—
254C22×C4—3462no—
255C22×C4—3464no—
256C22×C4—3461no—
257C22×C4—3461no—
258C22×C4—1731no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—24no—
265SD16—3464no—
266SD16—3464no—
267SD16—3464no—
268C2×C8—3464no—
269C2×C8—1732no—
270C2×C8C22×D17312yes—
271C4⋊C4—3461no—
272C4⋊C4—1731no—
273C4⋊C4C22×D17311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C23—21no—
279C23—3462no—
280C23—3462no—
281C23—3461no—
282C23—1734no—
283C23—1731no—
284C23—1731no—
285C23—6922no—
286C23—6922no—
287C23—6922no—
288C23D4×D17311yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—3468no—
292Q8—3464no—
293Q8—3468no—
294Q8—3462no—
295Q8—1734no—
296Q8D4×D17314yescomplemented
297D4—22no—
298D4—42no—
299D4—3462no—
300D4—6922no—
301D4—6922no—
302D4—6922no—
303D4—6922no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—3468no—
309C2×C4—3468no—
310C2×C4—3462no—
311C2×C4—3462no—
312C2×C4—3464no—
313C2×C4—3462no—
314C2×C4—3462no—
315C2×C4—3462no—
316C2×C4—3461no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—1734no—
320C2×C4—6922no—
321C2×C4D4×D17311yescharacteristic
322C2×C4C23×D17311yescharacteristic
323C8—22no—
324C8—3462no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—3468no—
331C22—3461no—
332C22—3461no—
333C22—13844no—
334C22—1732no—
335C22—1732no—
336C22—1732no—
337C22—1732no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22SD16×D17312yescomplemented
342C22C2×D4×D17311yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—3464no—
347C4—3461no—
348C4—1732no—
349C4—6921no—
350C4C2×D4×D17312yes—
351C2—22no—
352C2—81no—
353C2—3462no—
354C2—13841no—
355C2—1734no—
356C2#120×D17311yescharacteristic
357C2C2×SD16×D17311yescharacteristic
358C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
prime radical#121×D173C2
hypercenter#448D173
hypocenterC173C2×#448
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D173C2semidirect
2. C2×SD16×D173C2semidirect
4. #26011632C2semidirect
5. #26011631C2semidirect
6. #26011630C2semidirect
7. #26011629C2semidirect
8. #26011628C2semidirect
9. #103×D173C2semidirect
10. C22⋊C8×D173C2semidirect
11. #448×C173C2semidirect
13. C22×Dic346C22semidirect
18. C2×Q8×D173C22semidirect
23. C2×#14583815C22semidirect
24. C2×#14583814C22semidirect
25. C2×#14583813C22semidirect
35. #19149604C22semidirect
36. #19149603C22semidirect
37. #19149602C22semidirect
38. C22⋊#13063436C22semidirect
39. #19149591C22semidirect
40. C22×Q8×C173C22semidirect
41. C2×SD16×C173C22semidirect
43. #103×C173C22semidirect
44. C22⋊C8×C173C22semidirect
58. Q8×D173D4semidirect
104. C22×D173SD16semidirect
111. Dic346C2×D4semidirect
135. Q8×C173C2×D4semidirect
150. C22×C173C2×SD16semidirect
161. C2×D173C2×SD16semidirect
178. D173#448direct (N × H)
179. C173C2×#448semidirect
189. #448D173direct (N × H)
199. C22×Q8C2×D173semidirect
208. C2×SD16C2×D173semidirect
221. #103C2×D173semidirect
224. C22⋊C8C2×D173semidirect
296. Q8D4×D173semidirect
341. C22SD16×D173semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1732
C2×SD16×D1732
#121×D1732
#260116322
#260116312
#260116302
#260116292
#260116282
#103×D1732
C22⋊C8×D1732
#448×C1732
C2×#448173