Finite Group Databasebeta

← #36364661

Subgroups — #36364661

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012114C211yesmaximal, characteristic, complemented
2#26012058C211yesmaximal, characteristic, complemented
3#26011877C211yesmaximal, characteristic, complemented
4#26011872C211yesmaximal, characteristic, complemented
5#26011849C211yesmaximal, characteristic, complemented
6#26011836C211yesmaximal, characteristic, complemented
7#26011711C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011689C211yesmaximal, characteristic, complemented
10#26011685C211yesmaximal, characteristic, complemented
11#26011651C211yesmaximal, characteristic, complemented
12#26011638C211yesmaximal, characteristic, complemented
13#26011625C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583816C2211yescharacteristic, complemented
18C2×#14583815C2211yescharacteristic, complemented
19C2×#14583813C2211yescharacteristic, complemented
20C2×D1384C2211yescharacteristic, complemented
21C2×#14583811C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23#19149678C2211yescharacteristic, complemented
24#19149675C2211yescharacteristic
25#19149675C2211yescharacteristic
26#19149674—21no—
27#19149668C2211yescharacteristic, complemented
28#19149666C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149665—21no—
31#19149661—21no—
32#19149659—21no—
33#19149653C2211yescharacteristic
34#19149650—21no—
35#19149650—21no—
36#19149648—21no—
37#19149648—21no—
38Q8×Dic173C2211yescharacteristic, complemented
39#19149641C2211yescharacteristic, complemented
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73#14583816—41no—
74#14583815—41no—
75#14583813—41no—
76D1384—41no—
77#14583811—41no—
78#14583811—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173—21no—
82C4⋊Dic173D411yescharacteristic
83C4⋊Dic173D411yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99C24×C173—21no—
100C2×Q8×C173C2311yescharacteristic
101C2×D4×C173—21no—
102C2×D4×C173—41no—
103C2×D4×C173—41no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107SD16×C173—41no—
108SD16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—81no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692—21no—
123C22⋊D173—42no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C22⋊D173—81no—
127C22⋊D173—81no—
128C22⋊D173—81no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C4×D173—42no—
136C4×D173—81no—
137#13063436—21no—
138#13063436—21no—
139C23×C173—21no—
140C23×C173—21no—
141C23×C173—42no—
142C23×C173—81no—
143C23×C173—81no—
144C23×C173C2×D411yescharacteristic
145Q8×C173—21no—
146Q8×C173—21no—
147D4×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—42no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C8×C173—21no—
159C8×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—84no—
170C22×C173C22×D411yescharacteristic
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173H(5,𝔽2)11yescharacteristic
181C4×C173H(5,𝔽2)11yescharacteristic
182Dic173—21no—
183Dic173—21no—
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—41no—
189C2×C173—81no—
190C2×C173—81no—
191C2×C173#53511yescharacteristic
192C2×C173#57611yescharacteristic
193C2×C173#57711yescharacteristic
194D173—81no—
195D173—81no—
196C173#322411yescharacteristic, complemented, Sylow
197#3224—1731nomaximal, Sylow
198#542—1731no—
199#535—1731no—
200#496—1731no—
201#495—1731no—
202#489—1731no—
203#486—1731no—
204#464—1731no—
205#463—1731no—
206#460—1731no—
207#459—1731no—
208#452—1731no—
209#450D17311yescharacteristic, complemented
210#449—1731no—
211#447—1731no—
212#433—1731no—
213C2×#41—1731no—
214C22×D4C2×D17311yescharacteristic
215C2×Q16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16C2×D17311yescharacteristic
219C2×SD16C2×D17311yescharacteristic
220C2×D8—1731no—
221C2×D8—1731no—
222C2×D8—1731no—
223C4⋊Q8—1731no—
224#127—1731no—
225#127—1731no—
226#126—3461no—
227#124—3461no—
228#124—1731no—
229#124—1731no—
230#124—1731no—
231#123—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8C2×D17311yescharacteristic
234#121—3461no—
235#121—3461no—
236#121—1731no—
237#120—3461no—
238#120—3461no—
239C4×Q8—1731no—
240C4×D4—1731no—
241#117—1731no—
242C4⋊C8—1731no—
243C4⋊C8—1731no—
244#103—1731no—
245#103—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102C2×D17311yescharacteristic
251#102C2×D17311yescharacteristic
252C22⋊C8—1731no—
253C22⋊C8C2×D17311yescharacteristic
254#97—1731no—
255C4×C8—1731no—
256C24—21no—
257#41—6921no—
258#41—6921no—
259C2×Q8—3461no—
260C2×Q8—1731no—
261C2×Q8C22×D17311yescharacteristic
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4C22×D17311yescharacteristic
274C2×D4C22×D17311yescharacteristic
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4C22×D17311yescharacteristic
278Q16—6921no—
279SD16—41no—
280SD16—41no—
281SD16—6921no—
282SD16—6921no—
283D8—6921no—
284D8—6921no—
285D8—6921no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8C22×D17311yescharacteristic
289C2×C8C22×D17311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4—1731no—
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—1731no—
307C22⋊C4—1731no—
308C42—3461no—
309C42—1731no—
310C42—1731no—
311C23—21no—
312C23—21no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—3461no—
317C23—3461no—
318C23D4×D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—3461no—
322Q8—3461no—
323Q8—13841no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—3461no—
330D4—3461no—
331D4—3461no—
332D4—3461no—
333D4—13841no—
334D4—13841no—
335D4—13841no—
336D4—13841no—
337D4—13841no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—13841no—
346C2×C4—1731no—
347C2×C4—1731no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—6922no—
351C2×C4D4×D17311yescharacteristic
352C2×C4C23×D17311yescharacteristic
353C8—21no—
354C8—21no—
355C8—3461no—
356C8—3461no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—3461no—
378C4—3461no—
379C4—3461no—
380C4—3461no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—13841no—
389C2—13841no—
390C2#2601205611yescharacteristic
391C2#2601229511yescharacteristic
392C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
hypercenter#450D173
hypocenterC173#3224
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012114C2semidirect
2. #26012058C2semidirect
3. #26011877C2semidirect
4. #26011872C2semidirect
5. #26011849C2semidirect
6. #26011836C2semidirect
7. #26011711C2semidirect
8. #26011705C2semidirect
9. #26011689C2semidirect
10. #26011685C2semidirect
11. #26011651C2semidirect
12. #26011638C2semidirect
13. #26011625C2semidirect
14. #26011549C2semidirect
15. #450×C173C2semidirect
17. C2×#14583816C22semidirect
18. C2×#14583815C22semidirect
19. C2×#14583813C22semidirect
20. C2×D1384C22semidirect
23. #19149678C22semidirect
27. #19149668C22semidirect
38. Q8×Dic173C22semidirect
39. #19149641C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
45. #19149601C22semidirect
48. #19149599C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
196. C173#3224semidirect
209. #450D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260120582
#260118772
#260118722
#260118492
#260118362
#260117112
#260117052
#260116892
#260116852
#260116512
#260116382
#260116252
#260115492
#450×C1732
#3224173