Finite Group Databasebeta

← #485×D173 #36363451

Subgroups — #485×D173 #36363451

Order 22144.

416 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D173C211yesmaximal, characteristic
2#117×D173C211yesmaximal, characteristic
3#26011835C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011833C211yesmaximal, characteristic, complemented
6#26011832C211yesmaximal, characteristic, complemented
7#26011831C211yesmaximal, characteristic, complemented
8#26011830C211yesmaximal, characteristic, complemented
9#26011829C211yesmaximal, characteristic, complemented
10#107×D173C211yesmaximal, characteristic, complemented
11#106×D173C211yesmaximal, characteristic, complemented
12#103×D173C211yesmaximal, characteristic, complemented
13#103×D173C211yesmaximal, characteristic, complemented
14C22⋊C8×D173C211yesmaximal, characteristic
15#485×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18C2×C8×D173C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149659C2211yescharacteristic
21#19149658C2211yescharacteristic
22#19149657—21no—
23#19149656—21no—
24#19149655C2211yescharacteristic
25#19149638C2211yescharacteristic
26#19149637C2211yescharacteristic
27#19149637C2211yescharacteristic
28#19149636—21no—
29#19149635—21no—
30C4⋊C4×D173—21no—
31C4⋊C4×D173C2211yescharacteristic
32C4⋊C4×D173C2211yescharacteristic
33C4⋊C4×D173C2211yescharacteristic
34C22⋊C4×D173—21no—
35C22⋊C4×D173—21no—
36C42×D173—21no—
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149603C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173D411yescharacteristic
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62Q8×D173—22no—
63Q8×D173—22no—
64C22×Dic173D411yescharacteristic
65C2×C4×D173—22no—
66C2×C4×D173—22no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173D411yescharacteristic
72C2×C4×D173D411yescharacteristic
73C2×C4×D173C2311yescharacteristic
74C2×C4×D173C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—22no—
78C8×D173—22no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C2×Q8×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173—21no—
108C22×D173—22no—
109C22×D173—22no—
110C22×D173—22no—
111C22×D173C2×D411yescharacteristic
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—42no—
116C2×Dic173—22no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—22no—
124C4×D173—21no—
125C4×D173—21no—
126C4×D173—22no—
127C4×D173—42no—
128C4×D173—42no—
129C4×D173—42no—
130C4×D173—42no—
131C4×D173#4112yes—
132C4×D173#4112yes—
133#13063436—21no—
134#13063436—21no—
135C23×C173C2×D411yescharacteristic
136Q8×C173—21no—
137Q8×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—44no—
156C2×D173#12312yes—
157C2×D173#13512yes—
158C2×D173#13712yes—
159C4×C173—21no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173—41no—
164C4×C173C2×#4111yescharacteristic
165C4×C173C2×#4111yescharacteristic
166Dic173—21no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173#12312yes—
172C2×C173—41no—
173C2×C173C2×#12311yescharacteristic
174C2×C173C2×#13511yescharacteristic
175C2×C173C2×#13711yescharacteristic
176D173—41no—
177D173#48514yesdirect product factor
178C173C2×#48511yescharacteristic, complemented, Sylow
179C2×#485—1731nomaximal, Sylow
180C2×C22⋊Q8—1731no—
181C2×#117—1731no—
182#485—1731no—
183#485—1731no—
184#485—1731no—
185#485—1731no—
186#485—1731no—
187#485—1731no—
188#485—1731no—
189#485D17311yescharacteristic, direct product factor
190C2×#107—1731no—
191C2×#106—1731no—
192C2×#103—1731no—
193C2×#103—1731no—
194C2×C22⋊C8—1731no—
195C22×Q8—1731no—
196C23×C4—1731no—
197C22×C8—1731no—
198C22×C8—1731no—
199C22⋊Q8—3461no—
200C22⋊Q8—3461no—
201C22⋊Q8—1731no—
202C22⋊Q8—1731no—
203C22⋊Q8—1731no—
204C22⋊Q8C2×D17311yescharacteristic
205#117—3461no—
206#117—3461no—
207#117—1731no—
208#117—1731no—
209#117—1731no—
210#117C2×D17311yescharacteristic
211C2×C4⋊C4—3461no—
212C2×C4⋊C4—1731no—
213C2×C4⋊C4—1731no—
214C2×C4⋊C4—1731no—
215C2×C22⋊C4—3461no—
216C2×C22⋊C4—3461no—
217C2×C42—3461no—
218#107—1731no—
219#107—1731no—
220#107—1731no—
221#107C2×D17311yescharacteristic, complemented
222#106—1731no—
223#106—1731no—
224#106—1731no—
225#106C2×D17311yescharacteristic, complemented
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103—1731no—
232#103C2×D17311yescharacteristic, complemented
233#103C2×D17311yescharacteristic, complemented
234C22⋊C8—1731no—
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237C22⋊C8C2×D17311yescharacteristic
238C24—1731no—
239C2×Q8—3461no—
240C2×Q8—3462no—
241C2×Q8—3462no—
242C2×Q8—1731no—
243C2×Q8C22×D17311yescharacteristic
244C22×C4—3462no—
245C22×C4—3462no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4—1731no—
253C22×C4—1731no—
254C22×C4—1731no—
255C22×C4C22×D17311yescharacteristic
256C2×C8—3462no—
257C2×C8—3462no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4C22×D17311yescharacteristic
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C42—21no—
284C42—3461no—
285C42—3461no—
286C42—3461no—
287C23—3462no—
288C23—3462no—
289C23—3462no—
290C23—1731no—
291C23—1731no—
292C23D4×D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3461no—
296Q8—3461no—
297Q8—6922no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—3462no—
305C2×C4—3462no—
306C2×C4—3462no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—1731no—
314C2×C4—1731no—
315C2×C4—1732no—
316C2×C4—1732no—
317C2×C4—6922no—
318C2×C4—6922no—
319C2×C4—6922no—
320C2×C4—6922no—
321C2×C4D4×D17311yescharacteristic
322C2×C4C23×D17311yescharacteristic
323C8—21no—
324C8—21no—
325C8—3461no—
326C8—3461no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—3461no—
331C22—3461no—
332C22—3461no—
333C22—1732no—
334C22—1732no—
335C22—1732no—
336C22—6924no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3461no—
344C4—1732no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4#41×D17311yescharacteristic
350C4#41×D17311yescharacteristic
351C2—41no—
352C2—1734no—
353C2—6921no—
354C2#123×D17311yescharacteristic
355C2#135×D17311yescharacteristic
356C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#485×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#485D173
hypocenterC173C2×#485
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #26011835C2semidirect
4. #26011834C2semidirect
5. #26011833C2semidirect
6. #26011832C2semidirect
7. #26011831C2semidirect
8. #26011830C2semidirect
9. #26011829C2semidirect
10. #107×D173C2semidirect
11. #106×D173C2semidirect
12. #103×D173C2semidirect
13. #103×D173C2semidirect
15. #485×C173C2semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149610C22semidirect
42. #19149610C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149603C22semidirect
46. #19149603C22semidirect
47. #19149602C22semidirect
48. #19149602C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. #103×C173C22semidirect
57. #103×C173C22semidirect
177. D173#485direct (N × H)
178. C173C2×#485semidirect
189. #485D173direct (N × H)
221. #107C2×D173semidirect
225. #106C2×D173semidirect
232. #103C2×D173semidirect
233. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1732
#117×D1732
#260118352
#260118342
#260118332
#260118322
#260118312
#260118302
#260118292
#107×D1732
#106×D1732
#103×D1732
C22⋊C8×D1732
#485×C1732
C2×#485173