Finite Group Databasebeta

← #36365382

Subgroups — #36365382

Order 22144.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012199C211yesmaximal, characteristic, complemented
2#26012125C211yesmaximal, characteristic
3#26011836C211yesmaximal, characteristic, complemented
4#26011811C211yesmaximal, characteristic, complemented
5#26011807C211yesmaximal, characteristic, complemented
6#26011780C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011641C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011576C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic, complemented
13C2×#19149601C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic, complemented
21C2×C8×D173C2211yescharacteristic, complemented
22#19149678C2211yescharacteristic, complemented
23#19149675C2211yescharacteristic
24#19149669—21no—
25#19149659—21no—
26#19149656—21no—
27#19149655C2211yescharacteristic, complemented
28#19149654C2211yescharacteristic
29#19149652—21no—
30#19149649—21no—
31#19149647—21no—
32Q8×Dic173C2211yescharacteristic, complemented
33#19149641C2211yescharacteristic
34C4×C22⋊D173C2211yescharacteristic
35#19149639—21no—
36C2×C4⋊Dic173C2211yescharacteristic, complemented
37C4⋊C4×D173C2211yescharacteristic, complemented
38#19149610C2211yescharacteristic, complemented
39#19149610C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149601C2211yescharacteristic
42#19149601C2211yescharacteristic
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173—21no—
61C2×Dic346—21no—
62C2×D692—22no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—41no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173D411yescharacteristic
72D4×D173—22no—
73D4×D173—41no—
74D4×D173—44no—
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79#14583813—22no—
80#14583813—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—41no—
84C4⋊Dic173—21no—
85C4⋊Dic173—21no—
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91#14583805D411yescharacteristic
92C22⋊Dic173D411yescharacteristic
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103SD16×C173—41no—
104C2×C8×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173—42no—
117C22×D173—42no—
118C22×D173—81no—
119C22×D173—81no—
120C22×D173C2×D411yescharacteristic
121Dic346—81no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—42no—
126D692—42no—
127D692#4112yes—
128C22⋊D173—42no—
129C22⋊D173—42no—
130C22⋊D173—81no—
131C22⋊D173SD1614yescomplemented
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—
141C4×D173—81no—
142#13063436—21no—
143#13063436—21no—
144C23×C173—21no—
145C23×C173C2×D411yescharacteristic
146Q8×C173—21no—
147Q8×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—42no—
151C2×C4×C173—21no—
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—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173C2×SD1612yes—
166C22×C173C22×D411yescharacteristic
167C2×D173—21no—
168C2×D173—21no—
169C2×D173—22no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—84no—
176C2×D173—84no—
177C2×D173C2×SD1612yes—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173C2×#4111yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173C2×SD1612yes—
189C2×C173—22no—
190C2×C173—81no—
191C2×C173#54611yescharacteristic
192C2×C173C22×SD1611yescharacteristic
193C2×C173#57611yescharacteristic
194D173—22no—
195D173—42no—
196D173—81no—
197C173#329511yescharacteristic, complemented, Sylow
198#3295—1731nomaximal, Sylow
199#554—1731no—
200D42—1731no—
201#486—1731no—
202#481—1731no—
203#481—1731no—
204#476—1731no—
205#465—1731no—
206#465D17311yescharacteristic, complemented
207#460—1731no—
208#450—1731no—
209#450—1731no—
210C4×SD16—1731no—
211C8×D4—1731no—
212C2×#102—1731no—
213C2×#102—1731no—
214C22×D4—3461no—
215C22×D4—1731no—
216C22×D4—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16C2×D17311yescharacteristic, complemented
220C22×C8—1731no—
221C22×C8C2×D17311yescharacteristic, complemented
222C4⋊Q8—1731no—
223#127—1731no—
224#125—3461no—
225C22⋊Q8—3461no—
226C22⋊Q8—3461no—
227C22⋊Q8—1731no—
228C22⋊Q8C2×D17311yescharacteristic, complemented
229#121—3461no—
230#121—1731no—
231#121C2×D17311yescharacteristic
232#120—3461no—
233#120—3461no—
234C4×Q8—1731no—
235C4×D4—3461no—
236C4×D4—1731no—
237C4×D4—1731no—
238C2×C4⋊C4—1731no—
239C2×C4⋊C4—1731no—
240#106—1731no—
241#106—1731no—
242#106C2×D17311yescharacteristic, complemented
243C4⋊C8—1731no—
244#103C2×D17311yescharacteristic, complemented
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252#102—1731no—
253#102C2×D17311yescharacteristic, complemented
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C4×C8—1731no—
257C24—3461no—
258C24—3461no—
259C2×Q8—3461no—
260C2×Q8C22×D17311yescharacteristic
261C2×D4—21no—
262C2×D4—3462no—
263C2×D4—3461no—
264C2×D4—3462no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—1731no—
270C2×D4—1731no—
271C2×D4—1731no—
272C2×D4—6921no—
273C2×D4—6921no—
274C2×D4—6924no—
275C2×D4C22×D17311yescharacteristic
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—1731no—
280C22×C4C22×D17311yescharacteristic
281SD16—41no—
282SD16—3462no—
283SD16—6921no—
284C2×C8—41no—
285C2×C8—1731no—
286C2×C8—1731no—
287C2×C8—6921no—
288C2×C8C22×D17311yescharacteristic
289C2×C8C22×D17311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—1731no—
297C4⋊C4—1731no—
298C4⋊C4—1731no—
299C4⋊C4C22×D17311yescharacteristic
300C4⋊C4C22×D17311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—1731no—
307C22⋊C4—1731no—
308C42—3461no—
309C42—1731no—
310C23—21no—
311C23—3461no—
312C23—3461no—
313C23—13841no—
314C23—13841no—
315C23—1731no—
316C23—6922no—
317C23—6922no—
318C23—6922no—
319C23—6922no—
320C23D4×D17311yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—13841no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—3461no—
328D4—3461no—
329D4—3461no—
330D4—13841no—
331D4—1734no—
332D4—1732no—
333D4—6922no—
334D4—6922no—
335D4—6922no—
336D4—6922no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—13841no—
347C2×C4—13841no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—6922no—
352C2×C4D4×D17311yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3461no—
357C8—3461no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3461no—
364C22—3461no—
365C22—3462no—
366C22—13844no—
367C22—13844no—
368C22—1732no—
369C22—6921no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22SD16×D17312yes—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3461no—
381C4—3461no—
382C4—1732no—
383C4—6921no—
384C4—6921no—
385C4#41×D17311yescharacteristic
386C4#1914972411yescharacteristic
387C2—22no—
388C2—81no—
389C2—3462no—
390C2—13841no—
391C2—6922no—
392C2#2601213011yescharacteristic
393C2C2×SD16×D17311yescharacteristic
394C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
prime radical#26012125C2
hypercenter#465D173
hypocenterC173#3295
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #26012199C2semidirect
3. #26011836C2semidirect
4. #26011811C2semidirect
5. #26011807C2semidirect
6. #26011780C2semidirect
7. #26011723C2semidirect
8. #26011689C2semidirect
9. #26011641C2semidirect
10. #26011640C2semidirect
11. #26011576C2semidirect
12. C8×C22⋊D173C2semidirect
13. C2×#19149601C2semidirect
14. #102×D173C2semidirect
15. #465×C173C2semidirect
20. C2×#14583813C22semidirect
21. C2×C8×D173C22semidirect
22. #19149678C22semidirect
27. #19149655C22semidirect
32. Q8×Dic173C22semidirect
36. C2×C4⋊Dic173C22semidirect
37. C4⋊C4×D173C22semidirect
38. #19149610C22semidirect
39. #19149610C22semidirect
40. #19149608C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
45. #19149600C22semidirect
46. #19149600C22semidirect
48. #19149599C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
52. C2×SD16×C173C22semidirect
53. C22×C8×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
56. #106×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
131. C22⋊D173SD16semidirect
197. C173#3295semidirect
206. #465D173semidirect
219. C2×SD16C2×D173semidirect
221. C22×C8C2×D173semidirect
228. C22⋊Q8C2×D173semidirect
242. #106C2×D173semidirect
244. #103C2×D173semidirect
253. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121992
#260121252
#260118362
#260118112
#260118072
#260117802
#260117232
#260116892
#260116412
#260116402
#260115762
C8×C22⋊D1732
C2×#191496012
#102×D1732
#465×C1732
#3295173