Finite Group Databasebeta

← #36364611

Subgroups — #36364611

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012094C211yesmaximal, characteristic, complemented
2#26012055C211yesmaximal, characteristic, complemented
3#26011877C211yesmaximal, characteristic, complemented
4#26011866C211yesmaximal, characteristic, complemented
5#26011854C211yesmaximal, characteristic, complemented
6#26011847C211yesmaximal, characteristic, complemented
7#26011709C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011685C211yesmaximal, characteristic, complemented
10#26011684C211yesmaximal, characteristic, complemented
11#26011636C211yesmaximal, characteristic, complemented
12#26011627C211yesmaximal, characteristic, complemented
13#26011624C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C2×#14583813C2211yescharacteristic, complemented
18C2×D1384C2211yescharacteristic
19C2×D1384C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C2×#14583811C2211yescharacteristic, complemented
23#19149675C2211yescharacteristic
24#19149675C2211yescharacteristic
25#19149675C2211yescharacteristic, complemented
26#19149670C2211yescharacteristic, complemented
27#19149668C2211yescharacteristic
28#19149668C2211yescharacteristic
29#19149663—21no—
30#19149661—21no—
31#19149653—21no—
32#19149653—21no—
33#19149653C2211yescharacteristic
34#19149651—21no—
35#19149650—21no—
36#19149647—21no—
37#19149647—21no—
38D4×Dic173C2211yescharacteristic, complemented
39#19149641C2211yescharacteristic, complemented
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173—21no—
60C2×D692—21no—
61C2×D692—41no—
62C2×D692—41no—
63C2×D692C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C22×Dic173—21no—
74C2×C4×D173—21no—
75#14583813—41no—
76D1384—41no—
77D1384—41no—
78#14583811—41no—
79#14583811—41no—
80#14583811—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173—21no—
84C4⋊Dic173D411yescharacteristic
85C4⋊Dic173D411yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173D411yescharacteristic
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173C2311yescharacteristic
108SD16×C173—41no—
109D8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—42no—
118C22×D173—81no—
119C22×D173—81no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—42no—
127C22⋊D173—81no—
128C22⋊D173—81no—
129C22⋊D173—81no—
130C22⋊D173—81no—
131C22⋊D173—81no—
132C22⋊D173—81no—
133C22⋊D173—81no—
134C2×Dic173—21no—
135C2×Dic173—81no—
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—81no—
141#13063436—21no—
142#13063436—21no—
143C23×C173—21no—
144C23×C173—21no—
145C23×C173C2×D411yescharacteristic
146Q8×C173—21no—
147Q8×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—42no—
153D4×C173—81no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—42no—
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C8×C173—21no—
162C8×C173—21no—
163C22×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173C22×D411yescharacteristic
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—84no—
183C4×C173—21no—
184C4×C173—41no—
185C4×C173—41no—
186C4×C173H(5,𝔽2)11yescharacteristic
187C4×C173H(5,𝔽2)11yescharacteristic
188Dic173—21no—
189Dic173—21no—
190Dic173—21no—
191Dic173—21no—
192Dic173—41no—
193C2×C173—41no—
194C2×C173—81no—
195C2×C173—81no—
196C2×C173#53511yescharacteristic
197C2×C173#57611yescharacteristic
198C2×C173#57611yescharacteristic
199D173—81no—
200D173—81no—
201D173—81no—
202C173#322011yescharacteristic, complemented, Sylow
203#3220—1731nomaximal, Sylow
204#540—1731no—
205#535—1731no—
206#496—1731no—
207#493—1731no—
208#490—1731no—
209#488—1731no—
210#463—1731no—
211#463—1731no—
212#459—1731no—
213#459—1731no—
214#449—1731no—
215#449D17311yescharacteristic, complemented
216#447—1731no—
217#447—1731no—
218#433—1731no—
219C2×#41C2×D17311yescharacteristic
220C22×D4—1731no—
221C2×SD16—1731no—
222C2×SD16C2×D17311yescharacteristic, complemented
223C2×D8—1731no—
224C2×D8—1731no—
225C2×D8—1731no—
226C2×D8—1731no—
227C2×D8—1731no—
228C2×D8C2×D17311yescharacteristic, complemented
229#127—1731no—
230#127—1731no—
231#127—1731no—
232#125—1731no—
233#124—1731no—
234#124—1731no—
235#123—3461no—
236#123—3461no—
237#121—3461no—
238#121—3461no—
239#121—3461no—
240#121—3461no—
241#121—1731no—
242#121C2×D17311yescharacteristic
243#120—3461no—
244#120—3461no—
245C4×D4—1731no—
246C4×D4—1731no—
247#117—1731no—
248C4⋊C8—1731no—
249C4⋊C8—1731no—
250#103—1731no—
251#103C2×D17311yescharacteristic, complemented
252#102—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102C2×D17311yescharacteristic, complemented
258C22⋊C8—1731no—
259C22⋊C8C2×D17311yescharacteristic
260#97—1731no—
261C4×C8—1731no—
262C24—3461no—
263#41—41no—
264#41—41no—
265C2×Q8C22×D17311yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—1731no—
277C2×D4—1731no—
278C2×D4—1731no—
279C2×D4—6921no—
280C2×D4—6921no—
281C2×D4C22×D17311yescharacteristic
282C2×D4C22×D17311yescharacteristic
283C22×C4—21no—
284C22×C4—3461no—
285C22×C4—3461no—
286C22×C4C22×D17311yescharacteristic
287SD16—41no—
288SD16—6921no—
289D8—41no—
290D8—6921no—
291D8—6921no—
292D8—6921no—
293D8—6921no—
294D8—6921no—
295C2×C8—1731no—
296C2×C8—1731no—
297C2×C8C22×D17311yescharacteristic
298C2×C8C22×D17311yescharacteristic
299C4⋊C4—3461no—
300C4⋊C4—3461no—
301C4⋊C4—1731no—
302C4⋊C4—1731no—
303C4⋊C4—1731no—
304C4⋊C4C22×D17311yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—1731no—
314C22⋊C4—1731no—
315C42—1731no—
316C42—1731no—
317C23—21no—
318C23—21no—
319C23—3461no—
320C23—3461no—
321C23—3461no—
322C23—13841no—
323C23—13841no—
324C23—6922no—
325C23D4×D17311yescharacteristic
326Q8—21no—
327Q8—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—81no—
334D4—3461no—
335D4—3461no—
336D4—3461no—
337D4—3461no—
338D4—3461no—
339D4—3461no—
340D4—13841no—
341D4—13841no—
342D4—13841no—
343D4—13841no—
344D4—13841no—
345D4—13841no—
346D4—13841no—
347D4—6922no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—3461no—
354C2×C4—13841no—
355C2×C4—13841no—
356C2×C4—1731no—
357C2×C4—1731no—
358C2×C4—1731no—
359C2×C4—1731no—
360C2×C4D4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—21no—
364C8—3461no—
365C8—3461no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—13844no—
376C22—6921no—
377C22—6921no—
378C22—6921no—
379C22—6921no—
380C22—6921no—
381C22—6921no—
382C22—6921no—
383C22—6921no—
384C22—6921no—
385C22C2×D4×D17311yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—3461no—
390C4—3461no—
391C4—3461no—
392C4—3461no—
393C4—6921no—
394C4#1914972411yescharacteristic
395C4#1914972411yescharacteristic
396C2—41no—
397C2—81no—
398C2—81no—
399C2—13841no—
400C2—13841no—
401C2—13841no—
402C2#2601205611yescharacteristic
403C2#2601229311yescharacteristic
404C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3220
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012094C2semidirect
2. #26012055C2semidirect
3. #26011877C2semidirect
4. #26011866C2semidirect
5. #26011854C2semidirect
6. #26011847C2semidirect
7. #26011709C2semidirect
8. #26011708C2semidirect
9. #26011685C2semidirect
10. #26011684C2semidirect
11. #26011636C2semidirect
12. #26011627C2semidirect
13. #26011624C2semidirect
14. #26011549C2semidirect
15. #449×C173C2semidirect
17. C2×#14583813C22semidirect
22. C2×#14583811C22semidirect
25. #19149675C22semidirect
26. #19149670C22semidirect
38. D4×Dic173C22semidirect
39. #19149641C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149603C22semidirect
46. #19149600C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
53. C2×SD16×C173C22semidirect
54. C2×D8×C173C22semidirect
56. #103×C173C22semidirect
57. #102×C173C22semidirect
202. C173#3220semidirect
215. #449D173semidirect
222. C2×SD16C2×D173semidirect
228. C2×D8C2×D173semidirect
251. #103C2×D173semidirect
257. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120942
#260120552
#260118772
#260118662
#260118542
#260118472
#260117092
#260117082
#260116852
#260116842
#260116362
#260116272
#260116242
#260115492
#449×C1732
#3220173