Finite Group Databasebeta

← #36364034

Subgroups — #36364034

Order 22144.

564 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×#19149703C211yesmaximal, characteristic, complemented
3C2×#19149700C211yesmaximal, characteristic, complemented
4C2×SD16×D173C211yesmaximal, characteristic, complemented
5#26012061C211yesmaximal, characteristic, complemented
6#26012059C211yesmaximal, characteristic, complemented
7#26011886C211yesmaximal, characteristic, complemented
8#26011884C211yesmaximal, characteristic, complemented
9#26011881C211yesmaximal, characteristic, complemented
10#26011879C211yesmaximal, characteristic, complemented
11#26011872C211yesmaximal, characteristic, complemented
12#26011864C211yesmaximal, characteristic, complemented
13#26011862C211yesmaximal, characteristic, complemented
14#26011445C211yesmaximal, characteristic, complemented
15#495×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic
20#19149711—24no—
21#19149703—24no—
22#19149700—24no—
23C2×#14583816C2211yescharacteristic, complemented
24C2×#14583816C2211yescharacteristic, complemented
25C2×#14583815C2211yescharacteristic
26C2×#14583815C2211yescharacteristic
27C2×#14583815C2211yescharacteristic, complemented
28C2×#14583815C2211yescharacteristic, complemented
29C2×#14583814C2211yescharacteristic
30C2×#14583814C2211yescharacteristic
31C2×#14583813C2211yescharacteristic
32C2×#14583813C2211yescharacteristic, complemented
33SD16×D173—24no—
34C2×#14583811C2211yescharacteristic
35C2×#14583811C2211yescharacteristic, complemented
36C2×#14583809C2211yescharacteristic
37C2×C8×D173C2211yescharacteristic
38#19149679C2211yescharacteristic, complemented
39#19149677C2211yescharacteristic, complemented
40#19149676C2211yescharacteristic
41#19149675C2211yescharacteristic
42#19149668C2211yescharacteristic
43#19149666C2211yescharacteristic, complemented
44#19149661—22no—
45#19149656—22no—
46#19149652—22no—
47#19149649—22no—
48#19149636C2211yescharacteristic
49#19149590C2211yescharacteristic, complemented
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, 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×D692—22no—
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—22no—
76D4×D173—22no—
77D4×D173—22no—
78C22×Dic173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173C2311yescharacteristic
82#14583816—22no—
83#14583816—22no—
84#14583815—22no—
85#14583815—22no—
86#14583815—22no—
87#14583815—22no—
88#14583814—22no—
89#14583814—22no—
90#14583813—22no—
91#14583813—22no—
92#14583811—22no—
93#14583811—22no—
94#14583809D414yescomplemented
95C2×#13063436C2311yescharacteristic
96C2×#13063436C2311yescharacteristic
97C8×D173D414yescomplemented
98C4⋊Dic173—22no—
99#14583805—22no—
100#14583805D412yes—
101C22⋊Dic173—22no—
102#14583803—22no—
103#14583803D412yes—
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—42no—
121C22×D173—82no—
122C22×D173C2×D411yescharacteristic
123Dic346—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—42no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—42no—
133D692—82no—
134C22⋊D173—42no—
135C22⋊D173—42no—
136C22⋊D173—82no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173C2×D412yes—
141C2×Dic173C2×D411yescharacteristic
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#57712yes—
189D173—22no—
190D173—81no—
191D173—81no—
192C173#315411yescharacteristic, complemented, Sylow
193#3154—1731nomaximal, Sylow
194C2×#137—1731no—
195C2×AGL(1,ℤ/8)—1731no—
196C2×#135—1731no—
197C22×SD16—1731no—
198#536—1731no—
199#535—1731no—
200#497—1731no—
201#497—1731no—
202#496—1731no—
203#496—1731no—
204#495—1731no—
205#495D17311yescharacteristic, complemented
206#492—1731no—
207#492—1731no—
208#405—1731no—
209C2×#41—1731no—
210C2×#41—1731no—
211C22×Q8—1731no—
212C22×D4—1731no—
213#137—3464no—
214AGL(1,ℤ/8)—3464no—
215#135—3464no—
216C2×Q16—1731no—
217C2×Q16—1731no—
218C2×Q16C2×D17311yescharacteristic, complemented
219C2×SD16—3464no—
220C2×SD16—1731no—
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×SD16—1731no—
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16C2×D17311yescharacteristic
229C2×SD16C2×D17311yescharacteristic, complemented
230C2×D8—1731no—
231C2×D8—1731no—
232C2×D8C2×D17311yescharacteristic
233C2×M16—1731no—
234C22×C8—1731no—
235C4⋊Q8—1731no—
236C4⋊Q8—1731no—
237#127—1731no—
238#127—1731no—
239#124—1731no—
240#124—1731no—
241#124C2×D17311yescharacteristic
242#124C2×D17311yescharacteristic, complemented
243#123—3462no—
244C22⋊Q8—3462no—
245#121—3462no—
246#120—3462no—
247#117—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—3462no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4C22×D17311yescharacteristic
274C2×D4C22×D17311yescharacteristic
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—1731no—
279Q16—22no—
280Q16—3462no—
281Q16—3462no—
282SD16—22no—
283SD16—22no—
284SD16—3462no—
285SD16—3462no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290SD16—3462no—
291SD16—3462no—
292D8—22no—
293D8—3462no—
294D8—3462no—
295M16—1734no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—1734no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—3462no—
302C4⋊C4—3462no—
303C4⋊C4—1732no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—3462no—
307C22⋊C4—3462no—
308C22⋊C4—1732no—
309C42—1731no—
310C42C22×D17311yescharacteristic
311C23—21no—
312C23—21no—
313C23—3461no—
314C23—3461no—
315C23—13842no—
316C23—1731no—
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—13842no—
336D4—13842no—
337D4—6922no—
338D4—6922no—
339D4—6922no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—1732no—
345C2×C4—1731no—
346C2×C4—1732no—
347C2×C4—1732no—
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#2601229812yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC22×C173C22×D4
hypercenter#495D173
hypocenterC173#3154
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×#19149703C2semidirect
3. C2×#19149700C2semidirect
4. C2×SD16×D173C2semidirect
5. #26012061C2semidirect
6. #26012059C2semidirect
7. #26011886C2semidirect
8. #26011884C2semidirect
9. #26011881C2semidirect
10. #26011879C2semidirect
11. #26011872C2semidirect
12. #26011864C2semidirect
13. #26011862C2semidirect
14. #26011445C2semidirect
15. #495×C173C2semidirect
23. C2×#14583816C22semidirect
24. C2×#14583816C22semidirect
27. C2×#14583815C22semidirect
28. C2×#14583815C22semidirect
32. C2×#14583813C22semidirect
35. C2×#14583811C22semidirect
38. #19149679C22semidirect
39. #19149677C22semidirect
43. #19149666C22semidirect
49. #19149590C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
52. C2×Q16×C173C22semidirect
54. C2×SD16×C173C22semidirect
57. #124×C173C22semidirect
58. C4×C8×C173C22semidirect
94. #14583809D4semidirect
97. C8×D173D4semidirect
192. C173#3154semidirect
205. #495D173semidirect
218. C2×Q16C2×D173semidirect
229. C2×SD16C2×D173semidirect
242. #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×#191497032
C2×#191497002
C2×SD16×D1732
#260120612
#260120592
#260118862
#260118842
#260118812
#260118792
#260118722
#260118642
#260118622
#260114452
#495×C1732
#3154173