Finite Group Databasebeta

← #36363052

Subgroups — #36363052

Order 22144.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149708C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic
3#124×D173C211yesmaximal, characteristic, complemented
4C4×#14583825C211yesmaximal, characteristic, complemented
5#26011716C211yesmaximal, characteristic, complemented
6#26011714C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011710C211yesmaximal, characteristic, complemented
9#26011708C211yesmaximal, characteristic, complemented
10#26011706C211yesmaximal, characteristic, complemented
11#26011704C211yesmaximal, characteristic, complemented
12#26011513C211yesmaximal, characteristic, complemented
13#26011498C211yesmaximal, characteristic, complemented
14#26011494C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149708—24no—
20#19149703—24no—
21C2×Dic692C2211yescharacteristic, complemented
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×#14583809C2211yescharacteristic
29#19149668C2211yescharacteristic
30#19149667C2211yescharacteristic
31#19149666C2211yescharacteristic
32#19149665—21no—
33#19149665—21no—
34C4×Dic346C2211yescharacteristic, complemented
35#19149644C2211yescharacteristic, complemented
36D4×Dic173C2211yescharacteristic, complemented
37C4×C22⋊D173—21no—
38#19149639—21no—
39#19149637C2211yescharacteristic
40#19149635—21no—
41C22⋊C4×D173—21no—
42C22⋊C4×D173—21no—
43C2×C4×Dic173—21no—
44C42×D173C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149608C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic
52#19149600C2211yescharacteristic
53#19149599C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic
61C23×D173—21no—
62#14583825—22no—
63#14583825D414yes—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67Q8×D173—22no—
68Q8×D173—22no—
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72D4×D173—22no—
73D4×D173—22no—
74C22×Dic173—21no—
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173D411yescharacteristic
78C2×C4×D173D411yescharacteristic
79C2×C4×D173C2311yescharacteristic
80Dic692—22no—
81#14583816—22no—
82#14583815—22no—
83#14583814—22no—
84#14583814—22no—
85#14583811—22no—
86#14583809—22no—
87#14583809—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C4⋊Dic173C2311yescharacteristic
91#14583805—21no—
92#14583805—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101C4×Dic173—21no—
102C4×Dic173—21no—
103C4×Dic173—21no—
104C4×Dic173D411yescharacteristic
105C4×Dic173D411yescharacteristic
106C4×Dic173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110C22×C4×C173—21no—
111SD16×C173—22no—
112D8×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173—21no—
121C22×D173—42no—
122C22×D173—42no—
123C22×D173—42no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—42no—
129Dic346C2×D412yes—
130D692—21no—
131D692—21no—
132C22⋊D173—24no—
133C22⋊D173—42no—
134C2×Dic173—21no—
135C2×Dic173—24no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—42no—
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C4×D173—41no—
143C4×D173—41no—
144C4×D173—42no—
145C4×D173C2×D412yes—
146C4×D173#4112yes—
147#13063436—21no—
148#13063436—21no—
149C23×C173—21no—
150C23×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173C2×D412yes—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—42no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2×D411yescharacteristic
162C2×C4×C173C2411yescharacteristic
163C8×C173—21no—
164C8×C173—21no—
165C22×C173—22no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173C22×D411yescharacteristic
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—84no—
178C2×D173#12112yes—
179C4×C173—21no—
180C4×C173—21no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173C22×D411yescharacteristic
184C4×C173C2×#4111yescharacteristic
185Dic173—22no—
186Dic173—21no—
187Dic173—21no—
188Dic173—41no—
189Dic173—41no—
190Dic173#12112yes—
191C2×C173—42no—
192C2×C173—81no—
193C2×C173C2×#12111yescharacteristic
194C2×C173#57511yescharacteristic
195C2×C173#57511yescharacteristic
196D173—22no—
197D173—81no—
198C173#304011yescharacteristic, complemented, Sylow
199#3040—1731nomaximal, Sylow
200C2×#137—1731no—
201C2×AGL(1,ℤ/8)—1731no—
202C2×#124—1731no—
203C4×#41—1731no—
204#464—1731no—
205#464—1731no—
206#464—1731no—
207#464—1731no—
208#463—1731no—
209#463—1731no—
210#463—1731no—
211#463D17311yescharacteristic, complemented
212#423—1731no—
213#419—1731no—
214#418—1731no—
215C2×#41—1731no—
216C22×Q8—1731no—
217C22×D4—1731no—
218#137—3464no—
219AGL(1,ℤ/8)—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×SD16—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×SD16C2×D17311yescharacteristic, complemented
226C2×D8—1731no—
227C2×D8C2×D17311yescharacteristic
228C2×M16—1731no—
229C2×M16—1731no—
230#124—3461no—
231#124—3461no—
232#124—1731no—
233#124—1731no—
234#124—1731no—
235#124C2×D17311yescharacteristic
236C4×Q8—1731no—
237C4×Q8—1731no—
238C4×D4—3461no—
239C4×D4—3461no—
240C4×D4—1731no—
241C4×D4C2×D17311yescharacteristic, complemented
242#117—3461no—
243#117—1731no—
244C2×C22⋊C4—3461no—
245C2×C22⋊C4—3461no—
246C2×C42—3461no—
247C2×C42—1731no—
248C4⋊C8—1731no—
249C4⋊C8—1731no—
250C4⋊C8—1731no—
251C4⋊C8C2×D17311yescharacteristic, complemented
252#103—1731no—
253#103—1731no—
254#103—1731no—
255#103C2×D17311yescharacteristic, complemented
256#102—1731no—
257#102—1731no—
258#102—1731no—
259#102C2×D17311yescharacteristic
260C24—3461no—
261#41—3462no—
262#41—1734no—
263C2×Q8—3461no—
264C2×Q8—3462no—
265C2×Q8—3462no—
266C2×Q8—1731no—
267C2×Q8—1731no—
268C2×Q8C22×D17311yescharacteristic
269C2×D4—3461no—
270C2×D4—3462no—
271C2×D4—3462no—
272C2×D4—3461no—
273C2×D4—1731no—
274C2×D4C22×D17311yescharacteristic
275C2×D4C22×D17311yescharacteristic
276C22×C4—21no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4—1731no—
281C22×C4—1731no—
282C22×C4—1731no—
283Q16—3462no—
284Q16—3462no—
285SD16—22no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289D8—22no—
290D8—3462no—
291M16—3462no—
292M16—3462no—
293C2×C8—1731no—
294C2×C8—1731no—
295C2×C8C22×D17311yescharacteristic
296C2×C8C22×D17311yescharacteristic
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—1731no—
300C4⋊C4C22×D17311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C42—3461no—
313C42—3461no—
314C42—3461no—
315C42—1731no—
316C42—1731no—
317C42—1731no—
318C42C22×D17311yescharacteristic
319C23—21no—
320C23—21no—
321C23—3461no—
322C23—1731no—
323C23—6922no—
324C23—6922no—
325C23—6922no—
326Q8—21no—
327Q8—21no—
328Q8—3461no—
329Q8—3461no—
330Q8—3461no—
331Q8—1732no—
332Q8—6922no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—3464no—
337D4—3461no—
338D4—3461no—
339D4—6922no—
340D4D4×D17312yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—3461no—
345C2×C4—3464no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—1732no—
352C2×C4—1732no—
353C2×C4—6921no—
354C2×C4—6921no—
355C2×C4—6922no—
356C2×C4—6922no—
357C2×C4D4×D17311yescharacteristic
358C2×C4D4×D17311yescharacteristic
359C2×C4C23×D17311yescharacteristic
360C8—21no—
361C8—21no—
362C8—3461no—
363C8—3461no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—3461no—
371C22—3461no—
372C22—13844no—
373C22—1732no—
374C22—6921no—
375C22—6921no—
376C22—6921no—
377C22C2×D4×D17311yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—3462no—
383C4—3461no—
384C4—3461no—
385C4—1732no—
386C4—6921no—
387C4—6921no—
388C4C2×D4×D17311yescharacteristic
389C4#41×D17311yescharacteristic
390C2—42no—
391C2—81no—
392C2—3462no—
393C2—13841no—
394C2#121×D17311yescharacteristic
395C2#2601228711yescharacteristic
396C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149703C2
hypercenter#463D173
hypocenterC173#3040
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
3. #124×D173C2semidirect
4. C4×#14583825C2semidirect
5. #26011716C2semidirect
6. #26011714C2semidirect
7. #26011712C2semidirect
8. #26011710C2semidirect
9. #26011708C2semidirect
10. #26011706C2semidirect
11. #26011704C2semidirect
12. #26011513C2semidirect
13. #26011498C2semidirect
14. #26011494C2semidirect
15. #463×C173C2semidirect
21. C2×Dic692C22semidirect
22. C2×#14583816C22semidirect
25. C2×#14583814C22semidirect
34. C4×Dic346C22semidirect
35. #19149644C22semidirect
36. D4×Dic173C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149603C22semidirect
50. #19149602C22semidirect
54. C2×SD16×C173C22semidirect
57. C4×D4×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
198. C173#3040semidirect
211. #463D173semidirect
225. C2×SD16C2×D173semidirect
241. C4×D4C2×D173semidirect
251. C4⋊C8C2×D173semidirect
255. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C2×#191497032
#124×D1732
C4×#145838252
#260117162
#260117142
#260117122
#260117102
#260117082
#260117062
#260117042
#260115132
#260114982
#260114942
#463×C1732
#3040173