Finite Group Databasebeta

← #416×D173 #36362055

Subgroups — #416×D173 #36362055

Order 22144.

792 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C211yesmaximal, characteristic
2C22×C8×D173C211yesmaximal, characteristic, complemented
3#117×D173C211yesmaximal, characteristic, complemented
4#26011488C211yesmaximal, characteristic, complemented
5#26011487C211yesmaximal, characteristic, complemented
6#26011486C212yesmaximal, complemented
7#26011485C211yesmaximal, characteristic, complemented
8#26011484C212yesmaximal, complemented
9#103×D173C212yesmaximal, complemented
10#102×D173C212yesmaximal, complemented
11#416×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C2×#14583826—21no—
14C2×#14583826C2211yescharacteristic
15C2×#14583825—21no—
16C2×#14583825C2211yescharacteristic
17#41×D173—24no—
18#41×D173C418yescomplemented
19C2×Q8×D173C2211yescharacteristic
20C2×D4×D173—21no—
21C2×D4×D173C2211yescharacteristic
22C22×C4×D173—21no—
23C22×C4×D173C2211yescharacteristic
24C22×#13063436C2211yescharacteristic, complemented
25C2×C8×D173—24no—
26C2×C8×D173C2212yes—
27C2×C8×D173C2212yescomplemented
28#19149638C2211yescharacteristic, complemented
29#19149637C2212yescomplemented
30#19149636—21no—
31#19149635—21no—
32C4⋊C4×D173C2212yes—
33C22⋊C4×D173—21no—
34C42×D173—21no—
35#19149604C2212yescomplemented
36#19149603C2212yescomplemented
37#19149602C2212yescomplemented
38#19149601C2212yescomplemented
39#19149600C2212yescomplemented
40#19149599C2212yescomplemented
41C2×#41×C173C2211yescharacteristic
42C22×C8×C173C2211yescharacteristic, complemented
43#117×C173C2211yescharacteristic, complemented
44#103×C173C2212yescomplemented
45#102×C173C2212yescomplemented
46C23×D173—21no—
47C23×D173D411yescharacteristic
48#14583827—22no—
49#14583827C2×C414yescomplemented
50#14583826—28no—
51#14583826—22no—
52#14583826C2×C414yescomplemented
53#14583825—28no—
54#14583825—22no—
55#14583825C2×C414yescomplemented
56C2×Dic346—21no—
57C2×Dic346C2311yescharacteristic
58Q8×D173—22no—
59Q8×D173C2×C414yes—
60C2×D692—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65D4×D173—28no—
66D4×D173—22no—
67D4×D173C2×C414yes—
68C22×Dic173—21no—
69C22×Dic173D411yescharacteristic
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—28no—
73C2×C4×D173—22no—
74C2×C4×D173—21no—
75C2×C4×D173C2×C414yes—
76C2×C4×D173D411yescharacteristic
77C2×C4×D173D411yescharacteristic
78C2×C4×D173D414yes—
79C2×C4×D173C2311yescharacteristic
80C2×C4×D173C2311yescharacteristic
81C2×#13063436—22no—
82C2×#13063436C2312yes—
83C8×D173—24no—
84C8×D173—24no—
85C4⋊Dic173C2312yes—
86#14583805—22no—
87C22⋊Dic173—21no—
88#14583803—22no—
89C4×Dic173—22no—
90C4×Dic173—21no—
91#41×C173—22no—
92#41×C173C2×C414yescomplemented
93C2×Q8×C173C2311yescharacteristic
94C2×D4×C173—21no—
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173—21no—
97C22×C4×C173C2311yescharacteristic
98C2×C8×C173—22no—
99C2×C8×C173C2312yes—
100C4⋊C4×C173C2312yes—
101C22⋊C4×C173—21no—
102C42×C173—21no—
103C22×D173—24no—
104C22×D173—21no—
105C22×D173—24no—
106C22×D173—44no—
107C22×D173C22⋊C414yes—
108C22×D173C2×D411yescharacteristic
109C22×D173C2×D411yescharacteristic
110Dic346—21no—
111Dic346—24no—
112Dic346C22×C412yes—
113D692—24no—
114D692—21no—
115D692C22×C412yes—
116C22⋊D173—24no—
117C22⋊D173—24no—
118C22⋊D173—24no—
119C2×Dic173—21no—
120C2×Dic173—24no—
121C2×Dic173—22no—
122C2×Dic173C22⋊C414yes—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—24no—
126C4×D173—24no—
127C4×D173—24no—
128C4×D173—44no—
129C4×D173C22⋊C414yes—
130C4×D173C22⋊C414yes—
131C4×D173C22×C412yes—
132C4×D173C22×C412yes—
133C4×D173C2×D412yes—
134C4×D173C2×D412yes—
135#13063436—22no—
136C23×C173—21no—
137C23×C173C2×D411yescharacteristic
138Q8×C173—21no—
139Q8×C173C22×C412yes—
140D4×C173—24no—
141D4×C173—21no—
142D4×C173C22×C412yes—
143C2×C4×C173—21no—
144C2×C4×C173—24no—
145C2×C4×C173—22no—
146C2×C4×C173C22×C412yes—
147C2×C4×C173C2×D412yes—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—22no—
151C22×C173—22no—
152C22×C173—22no—
153C22×C173—42no—
154C22×C173C2×C22⋊C412yes—
155C22×C173C22×D411yescharacteristic
156C2×D173—22no—
157C2×D173—28no—
158C2×D173—22no—
159C2×D173—42no—
160C2×D173—48no—
161C2×D173C2×C22⋊C412yes—
162C2×D173C2×C22⋊C412yes—
163C2×D173#13514yes—
164C4×C173—22no—
165C4×C173—42no—
166C4×C173C2×C22⋊C412yes—
167C4×C173C23×C411yescharacteristic
168C4×C173C22×D411yescharacteristic
169Dic173—22no—
170Dic173—42no—
171Dic173C2×C22⋊C412yes—
172Dic173C2×C22⋊C412yes—
173C2×C173—22no—
174C2×C173—42no—
175C2×C173C22×C22⋊C411yescharacteristic
176C2×C173C2×#13512yes—
177D173—22no—
178D173—42no—
179D173#41614yesdirect product factor
180C173C2×#41611yescharacteristic, complemented, Sylow
181C2×#416—1731nomaximal, Sylow
182C22×#41—1731no—
183C23×C8—1731no—
184C2×#117—1731no—
185#416—1731no—
186#416—1731no—
187#416—1731no—
188#416—1732no—
189#416—1732no—
190#416D17311yescharacteristic, direct product factor
191C2×#103—1732no—
192C2×#102—1732no—
193C2×#41—3461no—
194C2×#41—3461no—
195C2×#41—3464no—
196C2×#41—1738no—
197C2×#41—1731no—
198C2×#41—1731no—
199C2×#41—1731no—
200C2×#41C2×D17311yescharacteristic
201C22×Q8—1731no—
202C22×D4—3461no—
203C22×D4—1731no—
204C23×C4—3461no—
205C23×C4—1731no—
206C22×C8—3464no—
207C22×C8—1731no—
208C22×C8—1732no—
209C22×C8—1732no—
210C22×C8C2×D17311yescharacteristic, complemented
211#117—3461no—
212#117—3461no—
213#117—1731no—
214#117—1732no—
215#117C2×D17311yescharacteristic, complemented
216C2×C4⋊C4—1732no—
217C2×C22⋊C4—3461no—
218C2×C42—3461no—
219#103—1732no—
220#103—1732no—
221#103—1732no—
222#103C2×D17312yescomplemented
223#102—1732no—
224#102—1732no—
225#102—1732no—
226#102C2×D17312yescomplemented
227C24—3461no—
228C24—1731no—
229#41—22no—
230#41—3468no—
231#41—3468no—
232#41—3462no—
233#41—3462no—
234#41—3462no—
235#41—1734no—
236#41—1734no—
237#41—1734no—
238#41C4×D17314yescomplemented
239C2×Q8—3461no—
240C2×Q8—3462no—
241C2×Q8—1734no—
242C2×Q8—1731no—
243C2×Q8C22×D17311yescharacteristic
244C2×D4—21no—
245C2×D4—3461no—
246C2×D4—3468no—
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—3462no—
250C2×D4—3461no—
251C2×D4—1734no—
252C2×D4—1731no—
253C2×D4C22×D17311yescharacteristic
254C22×C4—21no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—3468no—
258C22×C4—3462no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4—1734no—
263C22×C4—1734no—
264C22×C4—1731no—
265C22×C4—1731no—
266C22×C4—1731no—
267C22×C4—1731no—
268C22×C4C22×D17311yescharacteristic
269C2×C8—22no—
270C2×C8—3464no—
271C2×C8—3464no—
272C2×C8—3462no—
273C2×C8—1732no—
274C2×C8C22×D17312yes—
275C4⋊C4—3462no—
276C4⋊C4—1732no—
277C4⋊C4C22×D17312yes—
278C22⋊C4—21no—
279C22⋊C4—3462no—
280C22⋊C4—3461no—
281C42—21no—
282C42—3462no—
283C42—3461no—
284C23—21no—
285C23—3464no—
286C23—3464no—
287C23—3461no—
288C23—1734no—
289C23—1731no—
290C23—1731no—
291C23—6924no—
292C23D4×D17311yescharacteristic
293Q8—21no—
294Q8—3464no—
295Q8—3461no—
296Q8—1732no—
297Q8C2×C4×D17312yes—
298D4—24no—
299D4—21no—
300D4—3464no—
301D4—3464no—
302D4—3464no—
303D4—3464no—
304D4—3461no—
305D4—1732no—
306D4C2×C4×D17312yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—3464no—
311C2×C4—3464no—
312C2×C4—3461no—
313C2×C4—3464no—
314C2×C4—3464no—
315C2×C4—3462no—
316C2×C4—1734no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—1734no—
320C2×C4—1734no—
321C2×C4—1732no—
322C2×C4—1732no—
323C2×C4—1732no—
324C2×C4—1732no—
325C2×C4—6924no—
326C2×C4C2×C4×D17312yes—
327C2×C4D4×D17312yes—
328C2×C4D4×D17311yescharacteristic
329C2×C4C23×D17311yescharacteristic
330C8—22no—
331C8—3462no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—3468no—
336C22—3462no—
337C22—3462no—
338C22—1732no—
339C22—1732no—
340C22—1734no—
341C22—6928no—
342C22—6922no—
343C22C22⋊C4×D17312yes—
344C22C2×D4×D17311yescharacteristic
345C4—22no—
346C4—42no—
347C4—3462no—
348C4—1732no—
349C4—1732no—
350C4—6922no—
351C4C22⋊C4×D17312yes—
352C4C22×C4×D17311yescharacteristic
353C4C2×D4×D17311yescharacteristic
354C2—22no—
355C2—42no—
356C2—3462no—
357C2—1734no—
358C2—6922no—
359C2C2×C22⋊C4×D17311yescharacteristic
360C2#135×D17312yes—
subgroupG/H
centerC2×C4D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
prime radicalC2×#41×D173C2
cosocleC2×C4×C173C24
hypercenter#416D173
hypocenterC173C2×#416
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×C8×D173C2semidirect
3. #117×D173C2semidirect
4. #26011488C2semidirect
5. #26011487C2semidirect
6. #26011486C2semidirect
7. #26011485C2semidirect
8. #26011484C2semidirect
9. #103×D173C2semidirect
10. #102×D173C2semidirect
11. #416×C173C2semidirect
18. #41×D173C4semidirect
24. C22×#13063436C22semidirect
27. C2×C8×D173C22semidirect
28. #19149638C22semidirect
29. #19149637C22semidirect
35. #19149604C22semidirect
36. #19149603C22semidirect
37. #19149602C22semidirect
38. #19149601C22semidirect
39. #19149600C22semidirect
40. #19149599C22semidirect
42. C22×C8×C173C22semidirect
43. #117×C173C22semidirect
44. #103×C173C22semidirect
45. #102×C173C22semidirect
49. #14583827C2×C4semidirect
52. #14583826C2×C4semidirect
55. #14583825C2×C4semidirect
92. #41×C173C2×C4semidirect
179. D173#416direct (N × H)
180. C173C2×#416semidirect
190. #416D173direct (N × H)
210. C22×C8C2×D173semidirect
215. #117C2×D173semidirect
222. #103C2×D173semidirect
226. #102C2×D173semidirect
238. #41C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1732
C22×C8×D1732
#117×D1732
#260114882
#260114872
#260114862
#260114852
#260114842
#103×D1732
#102×D1732
#416×C1732
C2×#416173