Finite Group Databasebeta

← #36364652

Subgroups — #36364652

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#26012110C211yesmaximal, characteristic, complemented
2#26012054C211yesmaximal, characteristic, complemented
3#26011873C211yesmaximal, characteristic, complemented
4#26011851C211yesmaximal, characteristic, complemented
5#26011837C211yesmaximal, characteristic, complemented
6#26011711C211yesmaximal, characteristic, complemented
7#26011704C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011684C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011623C211yesmaximal, characteristic, complemented
14#26011553C211yesmaximal, characteristic, complemented
15#496×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583814C2211yescharacteristic, complemented
20C2×#14583813C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22#19149679C2211yescharacteristic, complemented
23#19149675C2211yescharacteristic
24#19149672—21no—
25#19149667C2211yescharacteristic
26#19149666C2211yescharacteristic
27#19149665—21no—
28#19149663—21no—
29#19149656—21no—
30#19149655C2211yescharacteristic
31#19149651—21no—
32#19149651—21no—
33#19149650C2211yescharacteristic
34#19149647—21no—
35#19149647—21no—
36Q8×Dic173C2211yescharacteristic, complemented
37D4×Dic173C2211yescharacteristic, complemented
38#19149636C2211yescharacteristic
39#19149608C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149604C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic, complemented
56#127×C173C2211yescharacteristic
57#124×C173C2211yescharacteristic, complemented
58#97×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583825—41no—
61#14583825—41no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70D4×D173—41no—
71D4×D173—41no—
72C22×Dic173—21no—
73C22×Dic173—21no—
74C2×C4×D173C2311yescharacteristic
75#14583816—41no—
76#14583814—41no—
77#14583813—41no—
78#14583811—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805D411yescharacteristic
87#14583805D411yescharacteristic
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105SD16×C173—41no—
106SD16×C173—41no—
107D8×C173—41no—
108D8×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—81no—
117C22×D173—81no—
118C22×D173C2×D411yescharacteristic
119Dic346—21no—
120Dic346—21no—
121Dic346—81no—
122D692—21no—
123D692—21no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C22⋊D173—81no—
127C22⋊D173—81no—
128C22⋊D173—81no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—42no—
133C2×Dic173—81no—
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136C2×Dic173C2×D411yescharacteristic
137C4×D173—21no—
138C4×D173—21no—
139#13063436—21no—
140#13063436—21no—
141C23×C173—21no—
142C23×C173—21no—
143C23×C173—21no—
144Q8×C173—21no—
145Q8×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—81no—
153D4×C173—81no—
154C2×C4×C173—21no—
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C8×C173—21no—
159C8×C173—21no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173C22×D411yescharacteristic
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—84no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173H(5,𝔽2)11yescharacteristic
181C4×C173H(5,𝔽2)11yescharacteristic
182Dic173—21no—
183Dic173—21no—
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—81no—
189C2×C173—81no—
190C2×C173—81no—
191C2×C173#53511yescharacteristic
192C2×C173#57611yescharacteristic
193C2×C173#57711yescharacteristic
194D173—41no—
195D173—81no—
196C173#322411yescharacteristic, complemented, Sylow
197#3224—1731nomaximal, Sylow
198#542—1731no—
199#535—1731no—
200#496D17311yescharacteristic, complemented
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#450—1731no—
210#449—1731no—
211#447—1731no—
212#433—1731no—
213C2×#41—1731no—
214C22×D4—1731no—
215C2×Q16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16C2×D17311yescharacteristic
219C2×SD16C2×D17311yescharacteristic, complemented
220C2×D8—1731no—
221C2×D8C2×D17311yescharacteristic
222C2×D8C2×D17311yescharacteristic, complemented
223C4⋊Q8—1731no—
224#127—1731no—
225#127C2×D17311yescharacteristic
226#126—3461no—
227#124—3461no—
228#124—1731no—
229#124—1731no—
230#124C2×D17311yescharacteristic, complemented
231#123—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8—1731no—
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#102—1731no—
251#102—1731no—
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97C2×D17311yescharacteristic, complemented
255C4×C8—1731no—
256C24—3461no—
257#41—6921no—
258#41—6921no—
259C2×Q8—3461no—
260C2×Q8—1731no—
261C2×Q8C22×D17311yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—1731no—
270C2×D4—6921no—
271C2×D4—6921no—
272C2×D4C22×D17311yescharacteristic
273C2×D4C22×D17311yescharacteristic
274C2×D4C22×D17311yescharacteristic
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4—1731no—
278Q16—6921no—
279SD16—41no—
280SD16—41no—
281SD16—6921no—
282SD16—6921no—
283D8—41no—
284D8—41no—
285D8—6921no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8C22×D17311yescharacteristic
289C2×C8C22×D17311yescharacteristic
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4—1731no—
296C4⋊C4—1731no—
297C22⋊C4—21no—
298C22⋊C4—21no—
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—
310C42C22×D17311yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—3461no—
315C23—13841no—
316C23—13841no—
317C23—1731no—
318C23—6922no—
319Q8—21no—
320Q8—21no—
321Q8—3461no—
322Q8—3461no—
323Q8—13841no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—81no—
331D4—81no—
332D4—3461no—
333D4—3461no—
334D4—13841no—
335D4—13841no—
336D4—13841no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—13841no—
346C2×C4—1731no—
347C2×C4—1731no—
348C2×C4—1731no—
349C2×C4—6922no—
350C2×C4D4×D17311yescharacteristic
351C2×C4D4×D17311yescharacteristic
352C2×C4C23×D17311yescharacteristic
353C8—21no—
354C8—21no—
355C8—3461no—
356C8—3461no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—13844no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—3461no—
378C4—3461no—
379C4—3461no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—81no—
386C2—81no—
387C2—81no—
388C2—13841no—
389C2—6921no—
390C2#2601205411yescharacteristic
391C2#2601229511yescharacteristic
392C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3224
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26012110C2semidirect
2. #26012054C2semidirect
3. #26011873C2semidirect
4. #26011851C2semidirect
5. #26011837C2semidirect
6. #26011711C2semidirect
7. #26011704C2semidirect
8. #26011689C2semidirect
9. #26011684C2semidirect
10. #26011650C2semidirect
11. #26011640C2semidirect
12. #26011633C2semidirect
13. #26011623C2semidirect
14. #26011553C2semidirect
15. #496×C173C2semidirect
18. C2×#14583816C22semidirect
19. C2×#14583814C22semidirect
22. #19149679C22semidirect
36. Q8×Dic173C22semidirect
37. D4×Dic173C22semidirect
39. #19149608C22semidirect
40. #19149608C22semidirect
41. #19149604C22semidirect
42. #19149604C22semidirect
47. #19149600C22semidirect
48. #19149600C22semidirect
51. C4×#13063436C22semidirect
53. C2×SD16×C173C22semidirect
55. C2×D8×C173C22semidirect
57. #124×C173C22semidirect
58. #97×C173C22semidirect
196. C173#3224semidirect
200. #496D173semidirect
219. C2×SD16C2×D173semidirect
222. C2×D8C2×D173semidirect
230. #124C2×D173semidirect
254. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121102
#260120542
#260118732
#260118512
#260118372
#260117112
#260117042
#260116892
#260116842
#260116502
#260116402
#260116332
#260116232
#260115532
#496×C1732
#3224173