Finite Group Databasebeta

← #36364612

Subgroups — #36364612

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#26012092C211yesmaximal, characteristic, complemented
2#26012054C211yesmaximal, characteristic, complemented
3#26011879C211yesmaximal, characteristic, complemented
4#26011866C211yesmaximal, characteristic, complemented
5#26011854C211yesmaximal, characteristic, complemented
6#26011844C211yesmaximal, characteristic, complemented
7#26011707C211yesmaximal, characteristic, complemented
8#26011706C211yesmaximal, characteristic, complemented
9#26011683C211yesmaximal, characteristic, complemented
10#26011637C211yesmaximal, characteristic, complemented
11#26011633C211yesmaximal, characteristic, complemented
12#26011626C211yesmaximal, characteristic, complemented
13#26011623C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#459×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic, complemented
19C2×#14583814C2211yescharacteristic, complemented
20C2×D1384C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23C2×#14583811C2211yescharacteristic, complemented
24#19149675C2211yescharacteristic
25#19149675C2211yescharacteristic, complemented
26#19149670C2211yescharacteristic, complemented
27#19149667C2211yescharacteristic
28#19149666C2211yescharacteristic
29#19149662—21no—
30#19149661—21no—
31#19149654C2211yescharacteristic
32#19149653—21no—
33#19149652—21no—
34#19149651—21no—
35#19149651—21no—
36#19149650C2211yescharacteristic
37#19149647—21no—
38#19149647—21no—
39C4×D692C2211yescharacteristic, complemented
40#19149636C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×D8×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic, complemented
54#127×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583825—41no—
61#14583825—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
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—
73D4×D173—41no—
74D4×D173—41no—
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173C2311yescharacteristic
78#14583815—41no—
79#14583814—41no—
80D1384—41no—
81#14583811—41no—
82#14583811—41no—
83#14583811—41no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C4⋊Dic173—21no—
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805D411yescharacteristic
90#14583805D411yescharacteristic
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803D411yescharacteristic
99#14583803D411yescharacteristic
100C4×Dic173C2311yescharacteristic
101C2×D4×C173—21no—
102C2×D4×C173—21no—
103C2×D4×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107D8×C173—41no—
108D8×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—42no—
117C22×D173—81no—
118C22×D173—81no—
119C22×D173C2×D411yescharacteristic
120Dic346—21no—
121Dic346—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—81no—
127C22⋊D173—42no—
128C22⋊D173—42no—
129C22⋊D173—81no—
130C22⋊D173—81no—
131C22⋊D173—81no—
132C22⋊D173—81no—
133C22⋊D173—81no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—81no—
143#13063436—21no—
144#13063436—21no—
145C23×C173—21no—
146C23×C173—21no—
147C23×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—81no—
155D4×C173—81no—
156C2×C4×C173—21no—
157C2×C4×C173—81no—
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C8×C173—21no—
162C8×C173—21no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173C22×D411yescharacteristic
173C2×D173—21no—
174C2×D173—21no—
175C2×D173—21no—
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—21no—
185C4×C173—41no—
186C4×C173H(5,𝔽2)11yescharacteristic
187C4×C173H(5,𝔽2)11yescharacteristic
188Dic173—21no—
189Dic173—21no—
190Dic173—21no—
191Dic173—41no—
192Dic173—41no—
193C2×C173—81no—
194C2×C173—81no—
195C2×C173—81no—
196C2×C173#53511yescharacteristic
197C2×C173#57611yescharacteristic
198C2×C173#57611yescharacteristic
199D173—41no—
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#459D17311yescharacteristic, complemented
214#449—1731no—
215#449—1731no—
216#447—1731no—
217#447—1731no—
218#433—1731no—
219C2×#41—1731no—
220C22×D4—1731no—
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×D8—1731no—
224C2×D8—1731no—
225C2×D8—1731no—
226C2×D8—1731no—
227C2×D8C2×D17311yescharacteristic
228C2×D8C2×D17311yescharacteristic, complemented
229#127—1731no—
230#127—1731no—
231#127C2×D17311yescharacteristic
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#121—1731no—
243#120—3461no—
244#120—3461no—
245C4×D4—1731no—
246C4×D4C2×D17311yescharacteristic, complemented
247#117—1731no—
248C4⋊C8—1731no—
249C4⋊C8C2×D17311yescharacteristic, complemented
250#103—1731no—
251#103—1731no—
252#102—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102C2×D17311yescharacteristic
257#102C2×D17311yescharacteristic, complemented
258C22⋊C8—1731no—
259C22⋊C8—1731no—
260#97—1731no—
261C4×C8—1731no—
262C24—3461no—
263#41—6921no—
264#41—6921no—
265C2×Q8—1731no—
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—6921no—
279C2×D4—6921no—
280C2×D4C22×D17311yescharacteristic
281C2×D4C22×D17311yescharacteristic
282C2×D4C22×D17311yescharacteristic
283C22×C4—21no—
284C22×C4—3461no—
285C22×C4—3461no—
286C22×C4—1731no—
287SD16—6921no—
288SD16—6921no—
289D8—41no—
290D8—41no—
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—
316C42C22×D17311yescharacteristic
317C23—21no—
318C23—21no—
319C23—21no—
320C23—3461no—
321C23—3461no—
322C23—13841no—
323C23—13841no—
324C23—1731no—
325C23—6922no—
326Q8—3461no—
327Q8—3461no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—81no—
335D4—81no—
336D4—3461no—
337D4—3461no—
338D4—3461no—
339D4—3461no—
340D4—13841no—
341D4—13841no—
342D4—13841no—
343D4—13841no—
344D4—13841no—
345D4—13841no—
346D4—6922no—
347D4—6922no—
348C2×C4—21no—
349C2×C4—81no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—13841no—
355C2×C4—1731no—
356C2×C4—1731no—
357C2×C4—1731no—
358C2×C4—6922no—
359C2×C4D4×D17311yescharacteristic
360C2×C4D4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—21no—
364C8—3461no—
365C8—3461no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—3461no—
376C22—3461no—
377C22—3461no—
378C22—13844no—
379C22—6921no—
380C22—6921no—
381C22—6921no—
382C22—6921no—
383C22—6921no—
384C22—6921no—
385C22C2×D4×D17311yescharacteristic
386C4—21no—
387C4—21no—
388C4—41no—
389C4—3461no—
390C4—3461no—
391C4—3461no—
392C4—6921no—
393C4—6921no—
394C4#1914972411yescharacteristic
395C4#1914972511yescharacteristic
396C2—81no—
397C2—81no—
398C2—81no—
399C2—13841no—
400C2—13841no—
401C2—6921no—
402C2#2601205711yescharacteristic
403C2#2601229211yescharacteristic
404C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3220
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #26012092C2semidirect
2. #26012054C2semidirect
3. #26011879C2semidirect
4. #26011866C2semidirect
5. #26011854C2semidirect
6. #26011844C2semidirect
7. #26011707C2semidirect
8. #26011706C2semidirect
9. #26011683C2semidirect
10. #26011637C2semidirect
11. #26011633C2semidirect
12. #26011626C2semidirect
13. #26011623C2semidirect
14. #26011551C2semidirect
15. #459×C173C2semidirect
18. C2×#14583815C22semidirect
19. C2×#14583814C22semidirect
23. C2×#14583811C22semidirect
25. #19149675C22semidirect
26. #19149670C22semidirect
39. C4×D692C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149604C22semidirect
43. #19149602C22semidirect
46. #19149600C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
53. C2×D8×C173C22semidirect
55. C4×D4×C173C22semidirect
56. C4⋊C8×C173C22semidirect
58. #102×C173C22semidirect
202. C173#3220semidirect
213. #459D173semidirect
228. C2×D8C2×D173semidirect
246. C4×D4C2×D173semidirect
249. C4⋊C8C2×D173semidirect
257. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120922
#260120542
#260118792
#260118662
#260118542
#260118442
#260117072
#260117062
#260116832
#260116372
#260116332
#260116262
#260116232
#260115512
#459×C1732
#3220173