Finite Group Databasebeta

← #36364650

Subgroups — #36364650

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#26012059C211yesmaximal, characteristic, complemented
3#26011881C211yesmaximal, characteristic, complemented
4#26011850C211yesmaximal, characteristic, complemented
5#26011838C211yesmaximal, characteristic, complemented
6#26011715C211yesmaximal, characteristic, complemented
7#26011708C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011684C211yesmaximal, characteristic, complemented
10#26011652C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011635C211yesmaximal, characteristic, complemented
13#26011623C211yesmaximal, characteristic, complemented
14#26011550C211yesmaximal, characteristic, complemented
15#495×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583813C2211yescharacteristic, complemented
20C2×#14583811C2211yescharacteristic
21C2×#14583811C2211yescharacteristic, complemented
22#19149679C2211yescharacteristic, complemented
23#19149676C2211yescharacteristic
24#19149675C2211yescharacteristic
25#19149672—21no—
26#19149668C2211yescharacteristic
27#19149665—21no—
28#19149663—21no—
29#19149656—21no—
30#19149655C2211yescharacteristic
31#19149652—21no—
32#19149652—21no—
33#19149650C2211yescharacteristic
34#19149649—21no—
35#19149649—21no—
36Q8×Dic173C2211yescharacteristic, complemented
37D4×Dic173C2211yescharacteristic, complemented
38#19149636C2211yescharacteristic
39#19149608C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic, complemented
52C2×Q16×C173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57#124×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—41no—
61#14583827—41no—
62C2×Dic346—21no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72D4×D173—41no—
73D4×D173—41no—
74C22×Dic173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583813—41no—
78#14583813—41no—
79#14583811—41no—
80#14583811—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173—21no—
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805D411yescharacteristic
89#14583805D411yescharacteristic
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×Q8×C173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103Q16×C173—41no—
104SD16×C173—41no—
105SD16×C173—41no—
106D8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
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—81no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—42no—
126D692—81no—
127D692—81no—
128C22⋊D173—42no—
129C22⋊D173—81no—
130C22⋊D173—81no—
131C22⋊D173—81no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—81no—
135C2×Dic173C2×D411yescharacteristic
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—42no—
141#13063436—21no—
142#13063436—21no—
143C23×C173—21no—
144C23×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148Q8×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153C2×C4×C173—21no—
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×C173C22×D411yescharacteristic
167C2×D173—21no—
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—84no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173H(5,𝔽2)11yescharacteristic
182C4×C173H(5,𝔽2)11yescharacteristic
183Dic173—21no—
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—81no—
189C2×C173—81no—
190C2×C173#53511yescharacteristic
191C2×C173#57611yescharacteristic
192C2×C173#57711yescharacteristic
193D173—41no—
194D173—81no—
195D173—81no—
196C173#322411yescharacteristic, complemented, Sylow
197#3224—1731nomaximal, Sylow
198#542—1731no—
199#535—1731no—
200#496—1731no—
201#495D17311yescharacteristic, complemented
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×Q16C2×D17311yescharacteristic, complemented
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16C2×D17311yescharacteristic
219C2×SD16C2×D17311yescharacteristic, complemented
220C2×D8—1731no—
221C2×D8—1731no—
222C2×D8C2×D17311yescharacteristic
223C4⋊Q8—1731no—
224#127—1731no—
225#127—1731no—
226#126—3461no—
227#124—3461no—
228#124—1731no—
229#124C2×D17311yescharacteristic
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#97—1731no—
255C4×C8C2×D17311yescharacteristic, complemented
256C24—3461no—
257#41—6921no—
258#41—6921no—
259C2×Q8—3461no—
260C2×Q8C22×D17311yescharacteristic
261C2×Q8C22×D17311yescharacteristic
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—1731no—
270C2×D4—1731no—
271C2×D4—6921no—
272C2×D4—6921no—
273C2×D4C22×D17311yescharacteristic
274C2×D4C22×D17311yescharacteristic
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4—1731no—
278Q16—41no—
279SD16—41no—
280SD16—41no—
281SD16—6921no—
282SD16—6921no—
283D8—41no—
284D8—6921no—
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—21no—
300C22⋊C4—21no—
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—3461no—
314C23—3461no—
315C23—13841no—
316C23—13841no—
317C23—1731no—
318C23—6922no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—13841no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—3461no—
329D4—3461no—
330D4—3461no—
331D4—3461no—
332D4—13841no—
333D4—13841no—
334D4—13841no—
335D4—13841no—
336D4—13841no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
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—3461no—
364C22—3461no—
365C22—3461no—
366C22—13844no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—3461no—
379C4—3461no—
380C4—3461no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—81no—
386C2—81no—
387C2—13841no—
388C2—13841no—
389C2—6921no—
390C2#2601205411yescharacteristic
391C2#2601229311yescharacteristic
392C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC22×C173C22×D4
hypercenter#495D173
hypocenterC173#3224
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #26012110C2semidirect
2. #26012059C2semidirect
3. #26011881C2semidirect
4. #26011850C2semidirect
5. #26011838C2semidirect
6. #26011715C2semidirect
7. #26011708C2semidirect
8. #26011689C2semidirect
9. #26011684C2semidirect
10. #26011652C2semidirect
11. #26011640C2semidirect
12. #26011635C2semidirect
13. #26011623C2semidirect
14. #26011550C2semidirect
15. #495×C173C2semidirect
19. C2×#14583813C22semidirect
21. C2×#14583811C22semidirect
22. #19149679C22semidirect
36. Q8×Dic173C22semidirect
37. D4×Dic173C22semidirect
39. #19149608C22semidirect
40. #19149608C22semidirect
41. #19149603C22semidirect
42. #19149603C22semidirect
45. #19149601C22semidirect
46. #19149601C22semidirect
51. #19149590C22semidirect
52. C2×Q16×C173C22semidirect
54. C2×SD16×C173C22semidirect
57. #124×C173C22semidirect
58. C4×C8×C173C22semidirect
196. C173#3224semidirect
201. #495D173semidirect
215. C2×Q16C2×D173semidirect
219. C2×SD16C2×D173semidirect
230. #124C2×D173semidirect
255. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121102
#260120592
#260118812
#260118502
#260118382
#260117152
#260117082
#260116892
#260116842
#260116522
#260116402
#260116352
#260116232
#260115502
#495×C1732
#3224173