Finite Group Databasebeta

← #36363700

Subgroups — #36363700

Order 22144.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012020C211yesmaximal, characteristic, complemented
2C4×D4×D173C211yesmaximal, characteristic, complemented
3#26011820C211yesmaximal, characteristic, complemented
4#26011803C211yesmaximal, characteristic, complemented
5#26011792C211yesmaximal, characteristic, complemented
6#26011791C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011640C211yesmaximal, characteristic, complemented
9#26011623C211yesmaximal, characteristic
10#26011603C211yesmaximal, characteristic, complemented
11#26011600C211yesmaximal, characteristic, complemented
12#26011594C211yesmaximal, characteristic, complemented
13#26011592C211yesmaximal, characteristic, complemented
14#26011547C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C22×C4×D173—21no—
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20#19149672—21no—
21#19149671C2211yescharacteristic, complemented
22#19149661—21no—
23#19149655C2211yescharacteristic
24#19149650C2211yescharacteristic
25Q8×Dic173C2211yescharacteristic, complemented
26C4×D692C2211yescharacteristic
27D4×Dic173C2211yescharacteristic
28D4×Dic173C2211yescharacteristic, complemented
29#19149641C2211yescharacteristic
30C4×C22⋊D173—21no—
31#19149639—21no—
32#19149639—21no—
33#19149635—21no—
34C4⋊C4×D173C2211yescharacteristic
35C22⋊C4×D173—21no—
36C42×D173C2211yescharacteristic
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic, complemented
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54#124×C173C2211yescharacteristic, complemented
55C4×D4×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63D4×D173—22no—
64D4×D173—22no—
65C22×Dic173—21no—
66C22×Dic173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—44no—
71C2×C4×D173D411yescharacteristic
72C2×C4×D173D411yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583813—22no—
75#14583811—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92C4×Dic173—21no—
93C4×Dic173D411yescharacteristic
94C4×Dic173D411yescharacteristic
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173—21no—
100SD16×C173—22no—
101D8×C173—22no—
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C42×C173C2311yescharacteristic
109C22×D173—24no—
110C22×D173—21no—
111C22×D173—42no—
112C22×D173—42no—
113C22×D173C2×D411yescharacteristic
114D692—21no—
115D692#4112yes—
116C22⋊D173—24no—
117C22⋊D173—42no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—42no—
122C2×Dic173—42no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—22no—
127C4×D173—21no—
128C4×D173—21no—
129C4×D173—41no—
130C4×D173—42no—
131C4×D173—42no—
132C4×D173—42no—
133#13063436—21no—
134#13063436—21no—
135C23×C173—21no—
136C23×C173—21no—
137Q8×C173—21no—
138Q8×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173#4112yes—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—42no—
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C8×C173—21no—
150C8×C173—21no—
151C22×C173—22no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173C22×D411yescharacteristic
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—22no—
161C2×D173—48no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173AGL(1,ℤ/8)12yes—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173C2×#4111yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—42no—
178C2×C173—81no—
179C2×C173#52511yescharacteristic
180C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C173#57511yescharacteristic
182D173—22no—
183D173—42no—
184C173#311411yescharacteristic, complemented, Sylow
185#3114—1731nomaximal, Sylow
186#529—1731no—
187C2×C4×D4—1731no—
188#483—1731no—
189#480—1731no—
190#478—1731no—
191#478—1731no—
192#463—1731no—
193#463D17311yescharacteristic, complemented
194#450—1731no—
195#447—1731no—
196#442—1731no—
197#442—1731no—
198#440—1731no—
199#440—1731no—
200#432—1731no—
201C22×D4—1731no—
202C23×C4—3461no—
203C2×SD16—1731no—
204C2×SD16C2×D17311yescharacteristic
205C2×D8—1731no—
206C2×D8C2×D17311yescharacteristic
207#126—3461no—
208#125—1731no—
209#124C2×D17311yescharacteristic, complemented
210#123—3461no—
211C22⋊Q8—1731no—
212#121—1731no—
213C4×Q8—1731no—
214C4×D4—3461no—
215C4×D4—3461no—
216C4×D4—3461no—
217C4×D4—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4—1731no—
221C4×D4C2×D17311yescharacteristic
222#117—3461no—
223C2×C4⋊C4—1731no—
224C2×C22⋊C4—3461no—
225C2×C42—1731no—
226#107—1731no—
227#107—1731no—
228#106—1731no—
229#106—1731no—
230C4⋊C8—1731no—
231C4⋊C8C2×D17311yescharacteristic, complemented
232#103—1731no—
233#103C2×D17311yescharacteristic, complemented
234#102—1731no—
235#102—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102C2×D17311yescharacteristic, complemented
240C22⋊C8—1731no—
241C22⋊C8—1731no—
242#97—1731no—
243#97—1731no—
244C24—3461no—
245C2×Q8C22×D17311yescharacteristic
246C2×D4—3462no—
247C2×D4—3462no—
248C2×D4—3461no—
249C2×D4—3461no—
250C2×D4—1731no—
251C2×D4C22×D17311yescharacteristic
252C2×D4C22×D17311yescharacteristic
253C22×C4—21no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—6924no—
263SD16—22no—
264SD16—3462no—
265D8—22no—
266D8—3462no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8C22×D17311yescharacteristic
270C2×C8C22×D17311yescharacteristic
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4—1731no—
279C4⋊C4C22×D17311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C42—3461no—
290C42—1731no—
291C42—1731no—
292C42—1731no—
293C42C22×D17311yescharacteristic
294C23—21no—
295C23—21no—
296C23—3464no—
297C23—3461no—
298C23—1731no—
299C23—6922no—
300C23—6922no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—3464no—
307D4—3461no—
308D4—1732no—
309D4—6922no—
310D4#1458382512yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3462no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—6921no—
324C2×C4—6922no—
325C2×C4—6922no—
326C2×C4—6922no—
327C2×C4—6922no—
328C2×C4—6922no—
329C2×C4D4×D17311yescharacteristic
330C2×C4D4×D17311yescharacteristic
331C2×C4C23×D17311yescharacteristic
332C8—21no—
333C8—21no—
334C8—3461no—
335C8—3461no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—3461no—
343C22—3461no—
344C22—3462no—
345C22—1732no—
346C22—6928no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—3461no—
355C4—3461no—
356C4—3461no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4C2×#1458382511yescharacteristic
362C2—42no—
363C2—81no—
364C2—3462no—
365C2—6922no—
366C2#2601200111yescharacteristic
367C2AGL(1,ℤ/8)×D17311yescharacteristic
368C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
prime radical#26011623C2
hypercenter#463D173
hypocenterC173#3114
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #26012020C2semidirect
2. C4×D4×D173C2semidirect
3. #26011820C2semidirect
4. #26011803C2semidirect
5. #26011792C2semidirect
6. #26011791C2semidirect
7. #26011703C2semidirect
8. #26011640C2semidirect
10. #26011603C2semidirect
11. #26011600C2semidirect
12. #26011594C2semidirect
13. #26011592C2semidirect
14. #26011547C2semidirect
15. #463×C173C2semidirect
21. #19149671C22semidirect
25. Q8×Dic173C22semidirect
28. D4×Dic173C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149603C22semidirect
47. #19149600C22semidirect
50. #19149589C22semidirect
51. #19149589C22semidirect
54. #124×C173C22semidirect
56. C4⋊C8×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
184. C173#3114semidirect
193. #463D173semidirect
209. #124C2×D173semidirect
231. C4⋊C8C2×D173semidirect
233. #103C2×D173semidirect
239. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120202
C4×D4×D1732
#260118202
#260118032
#260117922
#260117912
#260117032
#260116402
#260116232
#260116032
#260116002
#260115942
#260115922
#260115472
#463×C1732
#3114173