Finite Group Databasebeta

← #36364548

Subgroups — #36364548

Order 22144.

444 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012038C211yesmaximal, characteristic, complemented
2#124×D173C211yesmaximal, characteristic, complemented
3#26011879C211yesmaximal, characteristic, complemented
4#26011877C211yesmaximal, characteristic, complemented
5#26011852C211yesmaximal, characteristic, complemented
6#26011849C211yesmaximal, characteristic, complemented
7#26011707C211yesmaximal, characteristic, complemented
8#26011706C211yesmaximal, characteristic, complemented
9#26011703C211yesmaximal, characteristic, complemented
10#26011631C211yesmaximal, characteristic, complemented
11#26011628C211yesmaximal, characteristic, complemented
12#26011626C211yesmaximal, characteristic, complemented
13#26011623C211yesmaximal, characteristic, complemented
14#26011547C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×Q8×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19C2×#14583815C2211yescharacteristic, complemented
20C2×#14583814C2211yescharacteristic, complemented
21C2×#14583813C2211yescharacteristic, complemented
22C2×D1384C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24C2×#14583811C2211yescharacteristic
25#19149678C2211yescharacteristic, complemented
26#19149675C2211yescharacteristic, complemented
27#19149668C2211yescharacteristic
28#19149667C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149665—21no—
31#19149665—21no—
32#19149665—21no—
33#19149654C2211yescharacteristic
34#19149652—21no—
35#19149650C2211yescharacteristic
36C4×D692C2211yescharacteristic, complemented
37C4×C22⋊D173—21no—
38C22⋊C4×D173—21no—
39C22⋊C4×D173—21no—
40C42×D173C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic, complemented
53C2×D8×C173C2211yescharacteristic
54#124×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60#14583826—44no—
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64Q8×D173—41no—
65Q8×D173—41no—
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73D4×D173—41no—
74D4×D173—41no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173D411yescharacteristic
78C2×C4×D173D411yescharacteristic
79C2×C4×D173C2311yescharacteristic
80#14583815—41no—
81#14583814—41no—
82#14583813—41no—
83D1384—41no—
84#14583811—41no—
85#14583811—41no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99C4×Dic173D411yescharacteristic
100C4×Dic173D411yescharacteristic
101C4×Dic173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173—21no—
106SD16×C173—41no—
107D8×C173—41no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C22⋊C4×C173—21no—
114C42×C173C2311yescharacteristic
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—42no—
118C22×D173—42no—
119C22×D173—42no—
120C22×D173C2×D411yescharacteristic
121Dic346—21no—
122Dic346—21no—
123Dic346—42no—
124Dic346—42no—
125D692—21no—
126D692—21no—
127D692—21no—
128D692—21no—
129D692—42no—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C4×D173—22no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—41no—
143C4×D173—42no—
144C4×D173—42no—
145#13063436—21no—
146#13063436—21no—
147C23×C173—21no—
148C23×C173—21no—
149Q8×C173—21no—
150Q8×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—42no—
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C8×C173—21no—
162C8×C173—21no—
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—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C2×D173—84no—
179C2×D173AGL(1,ℤ/8)12yes—
180C4×C173—21no—
181C4×C173—21no—
182C4×C173—41no—
183C4×C173—41no—
184C4×C173H(5,𝔽2)11yescharacteristic
185C4×C173H(5,𝔽2)11yescharacteristic
186Dic173—21no—
187Dic173—21no—
188Dic173—21no—
189Dic173—41no—
190Dic173—41no—
191C2×C173—81no—
192C2×C173—81no—
193C2×C173#53511yescharacteristic
194C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
195C2×C173#57511yescharacteristic
196D173—22no—
197D173—81no—
198D173—81no—
199C173#321311yescharacteristic, complemented, Sylow
200#3213—1731nomaximal, Sylow
201#532—1731no—
202C2×#124—1731no—
203#496—1731no—
204#496—1731no—
205#489—1731no—
206#489—1731no—
207#463—1731no—
208#463—1731no—
209#463—1731no—
210#463D17311yescharacteristic, complemented
211#448—1731no—
212#448—1731no—
213#447—1731no—
214#447—1731no—
215#432—1731no—
216C2×#41—3461no—
217C22×Q8—1731no—
218C22×D4—1731no—
219C2×SD16—1731no—
220C2×SD16—1731no—
221C2×SD16—1731no—
222C2×SD16C2×D17311yescharacteristic, complemented
223C2×D8—1731no—
224C2×D8—1731no—
225C2×D8—1731no—
226C2×D8C2×D17311yescharacteristic
227C4⋊Q8—1731no—
228#127—1731no—
229#124—3461no—
230#124—3461no—
231#124—3461no—
232#124—1731no—
233#124—1731no—
234#124—1731no—
235#124C2×D17311yescharacteristic
236#121—3461no—
237#121—1731no—
238#121—1731no—
239C4×D4—3461no—
240C4×D4—1731no—
241C4×D4C2×D17311yescharacteristic, complemented
242C2×C22⋊C4—3461no—
243C2×C22⋊C4—3461no—
244C2×C42—1731no—
245C4⋊C8—1731no—
246C4⋊C8C2×D17311yescharacteristic, complemented
247#103—1731no—
248#103—1731no—
249#103—1731no—
250#103C2×D17311yescharacteristic, complemented
251#102—1731no—
252#102—1731no—
253#102—1731no—
254#102C2×D17311yescharacteristic
255C22⋊C8—1731no—
256C22⋊C8—1731no—
257#97—1731no—
258#97—1731no—
259C24—3461no—
260#41—6924no—
261C2×Q8—3461no—
262C2×Q8—3461no—
263C2×Q8—1731no—
264C2×Q8—6921no—
265C2×Q8—6921no—
266C2×Q8C22×D17311yescharacteristic
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—6921no—
275C2×D4—6921no—
276C2×D4C22×D17311yescharacteristic
277C2×D4C22×D17311yescharacteristic
278C22×C4—21no—
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—1731no—
282C22×C4—1731no—
283C22×C4—1731no—
284SD16—41no—
285SD16—6921no—
286SD16—6921no—
287SD16—6921no—
288D8—41no—
289D8—6921no—
290D8—6921no—
291D8—6921no—
292C2×C8—1731no—
293C2×C8—1731no—
294C2×C8C22×D17311yescharacteristic
295C2×C8C22×D17311yescharacteristic
296C4⋊C4—3461no—
297C4⋊C4—1731no—
298C4⋊C4C22×D17311yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C42—1731no—
312C42—1731no—
313C42—1731no—
314C42C22×D17311yescharacteristic
315C23—21no—
316C23—21no—
317C23—3461no—
318C23—3461no—
319C23—1731no—
320C23—6922no—
321C23—6922no—
322C23—6922no—
323Q8—21no—
324Q8—21no—
325Q8—3461no—
326Q8—3461no—
327Q8—6922no—
328Q8—6922no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—3461no—
334D4—3461no—
335D4—3461no—
336D4—3461no—
337D4—6922no—
338D4—6922no—
339D4—6922no—
340D4—6922no—
341D4—6922no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—3462no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—1731no—
351C2×C4—1731no—
352C2×C4—1731no—
353C2×C4—6921no—
354C2×C4—6922no—
355C2×C4—6922no—
356C2×C4D4×D17311yescharacteristic
357C2×C4D4×D17311yescharacteristic
358C2×C4C23×D17311yescharacteristic
359C8—21no—
360C8—21no—
361C8—3461no—
362C8—3461no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—3461no—
370C22—3461no—
371C22—13844no—
372C22—1732no—
373C22—6921no—
374C22—6921no—
375C22—6921no—
376C22—6921no—
377C22—6921no—
378C22—6921no—
379C22C2×D4×D17311yescharacteristic
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—3461no—
385C4—3461no—
386C4—3461no—
387C4—6921no—
388C4—6921no—
389C4#1914972411yescharacteristic
390C4#1914972511yescharacteristic
391C2—81no—
392C2—81no—
393C2—3462no—
394C2—13841no—
395C2—13841no—
396C2#2601205711yescharacteristic
397C2AGL(1,ℤ/8)×D17311yescharacteristic
398C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
hypercenter#463D173
hypocenterC173#3213
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. #26012038C2semidirect
2. #124×D173C2semidirect
3. #26011879C2semidirect
4. #26011877C2semidirect
5. #26011852C2semidirect
6. #26011849C2semidirect
7. #26011707C2semidirect
8. #26011706C2semidirect
9. #26011703C2semidirect
10. #26011631C2semidirect
11. #26011628C2semidirect
12. #26011626C2semidirect
13. #26011623C2semidirect
14. #26011547C2semidirect
15. #463×C173C2semidirect
19. C2×#14583815C22semidirect
20. C2×#14583814C22semidirect
21. C2×#14583813C22semidirect
25. #19149678C22semidirect
26. #19149675C22semidirect
36. C4×D692C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149604C22semidirect
43. #19149603C22semidirect
44. #19149602C22semidirect
50. #19149589C22semidirect
51. #19149589C22semidirect
52. C2×SD16×C173C22semidirect
55. C4×D4×C173C22semidirect
56. C4⋊C8×C173C22semidirect
57. #103×C173C22semidirect
199. C173#3213semidirect
210. #463D173semidirect
222. C2×SD16C2×D173semidirect
241. C4×D4C2×D173semidirect
246. C4⋊C8C2×D173semidirect
250. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120382
#124×D1732
#260118792
#260118772
#260118522
#260118492
#260117072
#260117062
#260117032
#260116312
#260116282
#260116262
#260116232
#260115472
#463×C1732
#3213173