Finite Group Databasebeta

← #36364038

Subgroups — #36364038

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×#19149706C211yesmaximal, characteristic, complemented
3C2×#19149702C211yesmaximal, characteristic, complemented
4C2×SD16×D173C211yesmaximal, characteristic, complemented
5#26012066C211yesmaximal, characteristic, complemented
6#26012059C211yesmaximal, characteristic, complemented
7#26011884C211yesmaximal, characteristic, complemented
8#26011880C211yesmaximal, characteristic, complemented
9#26011877C211yesmaximal, characteristic, complemented
10#26011876C211yesmaximal, characteristic, complemented
11#26011873C211yesmaximal, characteristic, complemented
12#26011865C211yesmaximal, characteristic, complemented
13#26011862C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#497×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583827C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic
20#19149711—24no—
21#19149706—24no—
22#19149702—24no—
23C2×#14583816C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic
25C2×#14583815C2211yescharacteristic, complemented
26C2×#14583814C2211yescharacteristic
27C2×#14583813C2211yescharacteristic
28C2×#14583813C2211yescharacteristic
29C2×#14583813C2211yescharacteristic, complemented
30C2×#14583813C2211yescharacteristic, complemented
31C2×#14583813C2211yescharacteristic, complemented
32SD16×D173—24no—
33C2×D1384C2211yescharacteristic
34C2×D1384C2211yescharacteristic, complemented
35C2×#14583811C2211yescharacteristic
36C2×#14583809C2211yescharacteristic
37C2×C8×D173C2211yescharacteristic
38#19149677C2211yescharacteristic, complemented
39#19149676C2211yescharacteristic
40#19149675C2211yescharacteristic
41#19149668C2211yescharacteristic
42#19149668C2211yescharacteristic, complemented
43#19149667C2211yescharacteristic, complemented
44#19149664—22no—
45#19149657—22no—
46#19149652—22no—
47#19149649—22no—
48#19149636C2211yescharacteristic
49#19149589C2211yescharacteristic, complemented
50C4×#13063436C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×Q16×C173C2211yescharacteristic, complemented
53C2×Q16×C173C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C4⋊Q8×C173C2211yescharacteristic, complemented
57#124×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—22no—
61#14583827—22no—
62#14583827—22no—
63#14583827—22no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—22no—
68C2×D692—21no—
69C2×D692—21no—
70C2×D692—22no—
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C2×C22⋊D173—22no—
76D4×D173—22no—
77D4×D173—22no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173C2311yescharacteristic
82#14583816—22no—
83#14583815—22no—
84#14583815—22no—
85#14583814—22no—
86#14583813—22no—
87#14583813—22no—
88#14583813—22no—
89#14583813—22no—
90#14583813—22no—
91D1384—22no—
92D1384—22no—
93#14583811—22no—
94#14583809D414yescomplemented
95C2×#13063436C2311yescharacteristic
96C2×#13063436C2311yescharacteristic
97C8×D173D414yescomplemented
98#14583805—22no—
99#14583805D412yes—
100#14583803—22no—
101#14583803—22no—
102#14583803—22no—
103#14583803D412yes—
104C4×Dic173C2311yescharacteristic
105C2×Q8×C173C2311yescharacteristic
106C2×Q8×C173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109Q16×C173—22no—
110Q16×C173—22no—
111SD16×C173—22no—
112SD16×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C4⋊C4×C173—22no—
116C22⋊C4×C173—22no—
117C42×C173C2311yescharacteristic
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173—42no—
122C22×D173—82no—
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—42no—
127D692—21no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—42no—
134D692—42no—
135D692—82no—
136C22⋊D173—42no—
137C22⋊D173—82no—
138C2×Dic173—21no—
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—
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157C2×C4×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—42no—
166C22×C173C22×D411yescharacteristic
167C2×D173—22no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—42no—
172C2×D173—42no—
173C2×D173—42no—
174C2×D173—84no—
175C2×D173#12712yes—
176C4×C173—22no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173C22×D411yescharacteristic
181C4×C173C22×D411yescharacteristic
182Dic173—22no—
183Dic173—41no—
184Dic173#12712yes—
185C2×C173—81no—
186C2×C173C2×#12711yescharacteristic
187C2×C173#57712yes—
188D173—22no—
189D173—81no—
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#497D17311yescharacteristic, complemented
202#496—1731no—
203#496—1731no—
204#495—1731no—
205#495—1731no—
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×Q16C2×D17311yescharacteristic, complemented
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
230C2×D8—1731no—
231C2×D8—1731no—
232C2×D8—1731no—
233C2×M16—1731no—
234C22×C8—1731no—
235C4⋊Q8—1731no—
236C4⋊Q8C2×D17311yescharacteristic, complemented
237#127—1731no—
238#127—1731no—
239#124—1731no—
240#124—1731no—
241#124—1731no—
242#124C2×D17311yescharacteristic
243#123—3462no—
244C22⋊Q8—3462no—
245#121—3462no—
246#120—3462no—
247#117—1731no—
248#97—1731no—
249#97C2×D17311yescharacteristic, complemented
250C4×C8—1731no—
251C4×C8—1731no—
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×Q8C22×D17311yescharacteristic
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×D4—1731no—
274C2×D4C22×D17311yescharacteristic
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—1731no—
279Q16—22no—
280Q16—22no—
281Q16—3462no—
282SD16—22no—
283SD16—22no—
284SD16—3462no—
285SD16—3462no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290SD16—3462no—
291SD16—3462no—
292D8—3462no—
293D8—3462no—
294D8—3462no—
295M16—1734no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—1734no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—3462no—
303C4⋊C4—1732no—
304C22⋊C4—22no—
305C22⋊C4—3462no—
306C22⋊C4—3462no—
307C22⋊C4—3462no—
308C22⋊C4—1732no—
309C42—1731no—
310C42C22×D17311yescharacteristic
311C23—21no—
312C23—3461no—
313C23—3461no—
314C23—3461no—
315C23—13842no—
316C23—1731no—
317C23—6922no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—3461no—
325Q8—3461no—
326Q8—6922no—
327D4—21no—
328D4—21no—
329D4—3461no—
330D4—3461no—
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—21no—
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—42no—
359C22—3462no—
360C22—13844no—
361C22—1732no—
362C22—6921no—
363C22—6921no—
364C22—6921no—
365C22—6922no—
366C22—6922no—
367C22—6922no—
368C22C2×D4×D17311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3462no—
374C4—1732no—
375C4—6921no—
376C4C2×D4×D17311yescharacteristic
377C4C2×D4×D17311yescharacteristic
378C2—81no—
379C2—3462no—
380C2—13841no—
381C2—13841no—
382C2—13841no—
383C2#127×D17311yescharacteristic
384C2#2601230012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#497×C173C2
socleC22×C173C22×D4
hypercenter#497D173
hypocenterC173#3154
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×#19149706C2semidirect
3. C2×#19149702C2semidirect
4. C2×SD16×D173C2semidirect
5. #26012066C2semidirect
6. #26012059C2semidirect
7. #26011884C2semidirect
8. #26011880C2semidirect
9. #26011877C2semidirect
10. #26011876C2semidirect
11. #26011873C2semidirect
12. #26011865C2semidirect
13. #26011862C2semidirect
14. #26011448C2semidirect
15. #497×C173C2semidirect
23. C2×#14583816C22semidirect
25. C2×#14583815C22semidirect
29. C2×#14583813C22semidirect
30. C2×#14583813C22semidirect
31. C2×#14583813C22semidirect
34. C2×D1384C22semidirect
38. #19149677C22semidirect
42. #19149668C22semidirect
43. #19149667C22semidirect
49. #19149589C22semidirect
50. C4×#13063436C22semidirect
51. C8×Dic173C22semidirect
52. C2×Q16×C173C22semidirect
53. C2×Q16×C173C22semidirect
56. C4⋊Q8×C173C22semidirect
58. #97×C173C22semidirect
94. #14583809D4semidirect
97. C8×D173D4semidirect
192. C173#3154semidirect
201. #497D173semidirect
217. C2×Q16C2×D173semidirect
218. C2×Q16C2×D173semidirect
236. C4⋊Q8C2×D173semidirect
249. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×#191497062
C2×#191497022
C2×SD16×D1732
#260120662
#260120592
#260118842
#260118802
#260118772
#260118762
#260118732
#260118652
#260118622
#260114482
#497×C1732
#3154173