Finite Group Databasebeta

← #36364024

Subgroups — #36364024

Order 22144.

572 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C2×#19149708C211yesmaximal, characteristic, complemented
3C2×#19149706C211yesmaximal, characteristic, complemented
4C2×#19149703C211yesmaximal, characteristic, complemented
5#26012040C211yesmaximal, characteristic, complemented
6#124×D173C211yesmaximal, characteristic, complemented
7#26011884C211yesmaximal, characteristic, complemented
8#26011882C211yesmaximal, characteristic, complemented
9#26011879C211yesmaximal, characteristic, complemented
10#26011877C211yesmaximal, characteristic, complemented
11#26011875C211yesmaximal, characteristic, complemented
12#26011873C211yesmaximal, characteristic, complemented
13#26011873C211yesmaximal, characteristic, complemented
14C4×#14583809C211yesmaximal, characteristic, complemented
15#495×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic
20#19149711—24no—
21#19149708—24no—
22#19149706—24no—
23#19149703—24no—
24C2×Dic692C2211yescharacteristic, complemented
25C2×#14583816C2211yescharacteristic, complemented
26C2×#14583816C2211yescharacteristic, complemented
27C2×#14583815C2211yescharacteristic
28C2×#14583815C2211yescharacteristic, complemented
29C2×#14583814C2211yescharacteristic
30C2×#14583814C2211yescharacteristic, complemented
31C2×#14583813C2211yescharacteristic
32C2×#14583813C2211yescharacteristic, complemented
33C2×D1384C2211yescharacteristic
34C2×#14583811C2211yescharacteristic
35C2×#14583811C2211yescharacteristic
36C2×#14583809C2211yescharacteristic
37C2×#14583809C2211yescharacteristic
38#19149677C2211yescharacteristic, complemented
39#19149675C2211yescharacteristic, complemented
40#19149668C2211yescharacteristic
41#19149667C2211yescharacteristic
42#19149667C2211yescharacteristic, complemented
43#19149666C2211yescharacteristic
44#19149665—22no—
45#19149652—22no—
46#19149639—22no—
47C22⋊C4×D173—22no—
48C42×D173C2211yescharacteristic
49#19149589C2211yescharacteristic, complemented
50#19149589C2211yescharacteristic, complemented
51C4×#13063436C2211yescharacteristic, complemented
52C2×Q16×C173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57#124×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—22no—
61#14583827—22no—
62#14583825—22no—
63#14583825—22no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67Q8×D173—22no—
68Q8×D173—22no—
69C2×D692—21no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173—22no—
74C2×C22⋊D173—21no—
75D4×D173—22no—
76D4×D173—22no—
77C22×Dic173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173D412yes—
81C2×C4×D173C2311yescharacteristic
82Dic692—22no—
83#14583816—22no—
84#14583816—22no—
85#14583815—22no—
86#14583815—22no—
87#14583814—22no—
88#14583814—22no—
89#14583813—22no—
90#14583813—22no—
91D1384—22no—
92#14583811—22no—
93#14583811—22no—
94#14583809D414yescomplemented
95#14583809D414yescomplemented
96C2×#13063436C2311yescharacteristic
97C2×#13063436C2311yescharacteristic
98#14583805—22no—
99C22⋊Dic173—22no—
100#14583803—22no—
101#14583803—22no—
102#14583803—22no—
103C4×Dic173D412yes—
104C4×Dic173C2311yescharacteristic
105C2×Q8×C173C2311yescharacteristic
106C2×Q8×C173C2311yescharacteristic
107C2×D4×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109Q16×C173—22no—
110SD16×C173—22no—
111SD16×C173—22no—
112D8×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C22⋊C4×C173—22no—
116C22⋊C4×C173—22no—
117C42×C173C2311yescharacteristic
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—44no—
121C22×D173—42no—
122C22×D173C2×D411yescharacteristic
123Dic346—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—42no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—42no—
133C22⋊D173—44no—
134C22⋊D173—42no—
135C22⋊D173—42no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—42no—
139C2×Dic173C2×D412yes—
140C2×Dic173C2×D411yescharacteristic
141C4×D173—42no—
142C4×D173—42no—
143C4×D173—42no—
144C4×D173C2×D412yes—
145C4×D173C2×D412yes—
146#13063436C2×D412yes—
147#13063436C2×D412yes—
148C23×C173—21no—
149C23×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173C2×D412yes—
161C2×C4×C173C2411yescharacteristic
162C8×C173C2×D412yes—
163C8×C173C2×D412yes—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—42no—
167C22×C173—42no—
168C22×C173C22×D411yescharacteristic
169C2×D173—22no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—42no—
173C2×D173—42no—
174C2×D173—84no—
175C2×D173#12712yes—
176C4×C173—22no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173C22×D411yescharacteristic
180C4×C173C22×D411yescharacteristic
181Dic173—22no—
182Dic173—41no—
183Dic173—41no—
184Dic173#12712yes—
185C2×C173—81no—
186C2×C173—81no—
187C2×C173C2×#12711yescharacteristic
188C2×C173#57512yes—
189D173—22no—
190D173—81no—
191D173—81no—
192C173#315311yescharacteristic, complemented, Sylow
193#3153—1731nomaximal, Sylow
194C2×#137—1731no—
195C2×#137—1731no—
196C2×AGL(1,ℤ/8)—1731no—
197C2×AGL(1,ℤ/8)—1731no—
198#532—1731no—
199C2×#124—1731no—
200#497—1731no—
201#497—1731no—
202#496—1731no—
203#496—1731no—
204#495—1731no—
205#495—1731no—
206#495—1731no—
207#495D17311yescharacteristic, complemented
208C4×M16—1731no—
209C2×#41—1731no—
210C2×#41—1731no—
211C22×Q8—1731no—
212C22×D4—1731no—
213#137—3464no—
214#137—3464no—
215AGL(1,ℤ/8)—3464no—
216AGL(1,ℤ/8)—3464no—
217C2×Q16—1731no—
218C2×Q16—1731no—
219C2×Q16—1731no—
220C2×Q16C2×D17311yescharacteristic, complemented
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×SD16—1731no—
226C2×SD16—1731no—
227C2×SD16C2×D17311yescharacteristic
228C2×SD16C2×D17311yescharacteristic, complemented
229C2×D8—1731no—
230C2×D8—1731no—
231C2×D8—1731no—
232C2×D8C2×D17311yescharacteristic
233C2×M16—1731no—
234C2×M16—1731no—
235C4⋊Q8—1731no—
236#127—1731no—
237#124—3462no—
238#124—1731no—
239#124—1731no—
240#124—1731no—
241#124—1731no—
242#124C2×D17311yescharacteristic
243#124C2×D17311yescharacteristic, complemented
244#121—3462no—
245C4×D4—3462no—
246C2×C22⋊C4—3462no—
247C2×C42—1731no—
248#97—1731no—
249#97—1731no—
250C4×C8—1731no—
251C4×C8C2×D17311yescharacteristic, complemented
252C24—3461no—
253#41—3462no—
254#41—3462no—
255#41—3462no—
256#41—3462no—
257C2×Q8—3461no—
258C2×Q8—3462no—
259C2×Q8—3462no—
260C2×Q8—1731no—
261C2×Q8—1731no—
262C2×Q8C22×D17311yescharacteristic
263C2×Q8C22×D17311yescharacteristic
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3462no—
267C2×D4—3462no—
268C2×D4—3462no—
269C2×D4—3461no—
270C2×D4—1731no—
271C2×D4—1731no—
272C2×D4C22×D17311yescharacteristic
273C2×D4C22×D17311yescharacteristic
274C22×C4—3461no—
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4—1731no—
278C22×C4—1732no—
279Q16—22no—
280Q16—3462no—
281Q16—3462no—
282Q16—3462no—
283SD16—22no—
284SD16—22no—
285SD16—3462no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290SD16—3462no—
291D8—22no—
292D8—3462no—
293D8—3462no—
294D8—3462no—
295M16—1734no—
296M16—1734no—
297C2×C8—1731no—
298C2×C8—1731no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—3462no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—3462no—
305C22⋊C4—3462no—
306C22⋊C4—3462no—
307C22⋊C4—3462no—
308C42—1732no—
309C42—1731no—
310C42C22×D17311yescharacteristic
311C23—21no—
312C23—21no—
313C23—3461no—
314C23—3461no—
315C23—1731no—
316C23—6924no—
317C23—6922no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—3461no—
323Q8—3461no—
324Q8—3461no—
325Q8—3461no—
326Q8—6922no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—3461no—
332D4—3461no—
333D4—3461no—
334D4—3461no—
335D4—6922no—
336D4—6924no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—1732no—
344C2×C4—1731no—
345C2×C4—1732no—
346C2×C4—1732no—
347C2×C4—6922no—
348C2×C4—6922no—
349C2×C4—6922no—
350C2×C4—6922no—
351C2×C4D4×D17312yes—
352C2×C4C23×D17311yescharacteristic
353C8—1732no—
354C8—1732no—
355C8D4×D17312yes—
356C8D4×D17312yes—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—3462no—
362C22—13844no—
363C22—1732no—
364C22—6921no—
365C22—6921no—
366C22—6922no—
367C22—6922no—
368C22C2×D4×D17311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—3462no—
373C4—1732no—
374C4—6921no—
375C4—6921no—
376C4C2×D4×D17311yescharacteristic
377C4C2×D4×D17311yescharacteristic
378C2—81no—
379C2—81no—
380C2—3462no—
381C2—13841no—
382C2—13841no—
383C2#127×D17311yescharacteristic
384C2#2601228812yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC22×C173C22×D4
hypercenter#495D173
hypocenterC173#3153
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×#19149708C2semidirect
3. C2×#19149706C2semidirect
4. C2×#19149703C2semidirect
5. #26012040C2semidirect
6. #124×D173C2semidirect
7. #26011884C2semidirect
8. #26011882C2semidirect
9. #26011879C2semidirect
10. #26011877C2semidirect
11. #26011875C2semidirect
12. #26011873C2semidirect
13. #26011873C2semidirect
14. C4×#14583809C2semidirect
15. #495×C173C2semidirect
24. C2×Dic692C22semidirect
25. C2×#14583816C22semidirect
26. C2×#14583816C22semidirect
28. C2×#14583815C22semidirect
30. C2×#14583814C22semidirect
32. C2×#14583813C22semidirect
38. #19149677C22semidirect
39. #19149675C22semidirect
42. #19149667C22semidirect
49. #19149589C22semidirect
50. #19149589C22semidirect
51. C4×#13063436C22semidirect
52. C2×Q16×C173C22semidirect
54. C2×SD16×C173C22semidirect
57. #124×C173C22semidirect
58. C4×C8×C173C22semidirect
94. #14583809D4semidirect
95. #14583809D4semidirect
192. C173#3153semidirect
207. #495D173semidirect
220. C2×Q16C2×D173semidirect
228. C2×SD16C2×D173semidirect
243. #124C2×D173semidirect
251. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×#191497082
C2×#191497062
C2×#191497032
#260120402
#124×D1732
#260118842
#260118822
#260118792
#260118772
#260118752
#260118732
C4×#145838092
#495×C1732
#3153173