Finite Group Databasebeta

← #36363912

Subgroups — #36363912

Order 22144.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012118C211yesmaximal, characteristic, complemented
2#26012054C211yesmaximal, characteristic
3#26011855C211yesmaximal, characteristic, complemented
4#26011854C211yesmaximal, characteristic, complemented
5#26011853C211yesmaximal, characteristic, complemented
6#26011849C211yesmaximal, characteristic, complemented
7#26011846C211yesmaximal, characteristic, complemented
8#26011844C211yesmaximal, characteristic, complemented
9#26011836C211yesmaximal, characteristic, complemented
10#26011494C211yesmaximal, characteristic, complemented
11#26011489C211yesmaximal, characteristic, complemented
12#26011484C211yesmaximal, characteristic, complemented
13#102×D173C211yesmaximal, characteristic, complemented
14#26011445C211yesmaximal, characteristic, complemented
15#486×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149678C2211yescharacteristic, complemented
21#19149677—21no—
22#19149675C2211yescharacteristic
23#19149671C2211yescharacteristic, complemented
24#19149670C2211yescharacteristic, complemented
25#19149669—21no—
26#19149667C2211yescharacteristic
27#19149666C2211yescharacteristic
28#19149657—21no—
29#19149655—21no—
30#19149651—21no—
31#19149651—21no—
32#19149647—21no—
33#19149647—21no—
34#19149637C2211yescharacteristic
35#19149636C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic
37#19149604C2211yescharacteristic, complemented
38#19149604C2211yescharacteristic, complemented
39#19149602C2211yescharacteristic, complemented
40#19149602C2211yescharacteristic, complemented
41#19149601C2211yescharacteristic
42#19149601C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic
45#19149600C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149590C2211yescharacteristic, complemented
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C4⋊Q8×C173C2211yescharacteristic, complemented
53#127×C173C2211yescharacteristic
54#102×C173C2211yescharacteristic
55#102×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583825—22no—
61#14583825—22no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69D4×D173—22no—
70D4×D173—22no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583809—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—22no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87#14583805D411yescharacteristic
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173—21no—
98C2×D4×C173—21no—
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—42no—
110C22×D173—81no—
111C22×D173—81no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—81no—
116D692—21no—
117D692—21no—
118C22⋊D173—42no—
119C22⋊D173—42no—
120C22⋊D173—81no—
121C22⋊D173—81no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—42no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—81no—
130C4×D173#4112yes—
131C4×D173#4112yes—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135C23×C173—21no—
136Q8×C173—81no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—81no—
142D4×C173—81no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C8×C173—21no—
149C8×C173—21no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173C22×D411yescharacteristic
157C2×D173—21no—
158C2×D173—21no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—84no—
163C2×D173#12412yes—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173C2×#4111yescharacteristic
169C4×C173C2×#4111yescharacteristic
170Dic173—21no—
171Dic173—21no—
172Dic173—41no—
173Dic173—41no—
174Dic173—41no—
175Dic173#12412yes—
176C2×C173—81no—
177C2×C173—81no—
178C2×C173C2×#12411yescharacteristic
179C2×C173#57611yescharacteristic
180C2×C173#57711yescharacteristic
181D173—22no—
182D173—81no—
183C173#313611yescharacteristic, complemented, Sylow
184#3136—1731nomaximal, Sylow
185#544—1731no—
186#535—1731no—
187#490—1731no—
188#490—1731no—
189#489—1731no—
190#489—1731no—
191#488—1731no—
192#488—1731no—
193#486—1731no—
194#486D17311yescharacteristic, complemented
195#418—1731no—
196#417—1731no—
197#416—1731no—
198C2×#102—1731no—
199#405—1731no—
200C2×#41—1731no—
201C22×D4—1731no—
202C2×M16—1731no—
203C22×C8—1731no—
204C4⋊Q8—3461no—
205C4⋊Q8—1731no—
206C4⋊Q8C2×D17311yescharacteristic, complemented
207#127—1731no—
208#127C2×D17311yescharacteristic
209#125—3461no—
210#125—1731no—
211#125—1731no—
212#124—1731no—
213#124—1731no—
214C22⋊Q8—3461no—
215C22⋊Q8—3461no—
216#121—3461no—
217#121—3461no—
218#120—3461no—
219#120—3461no—
220#117—1731no—
221#117—1731no—
222C2×C4⋊C4—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102C2×D17311yescharacteristic
236#102C2×D17311yescharacteristic
237#102C2×D17311yescharacteristic, complemented
238#102C2×D17311yescharacteristic, complemented
239#97—1731no—
240#97—1731no—
241C4×C8—1731no—
242C4×C8C2×D17311yescharacteristic, complemented
243C24—3461no—
244#41—3462no—
245#41—3462no—
246C2×Q8—21no—
247C2×Q8—3461no—
248C2×Q8—1731no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—3461no—
252C2×D4—3462no—
253C2×D4—3462no—
254C2×D4—3461no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—1731no—
258C2×D4C22×D17311yescharacteristic
259C2×D4C22×D17311yescharacteristic
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263M16—3462no—
264C2×C8—3462no—
265C2×C8—1731no—
266C2×C8—1731no—
267C2×C8C22×D17311yescharacteristic
268C2×C8C22×D17311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—3461no—
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⋊C4—1731no—
280C4⋊C4C22×D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—1731no—
288C22⋊C4—1731no—
289C42—3461no—
290C42—1731no—
291C42C22×D17311yescharacteristic
292C23—21no—
293C23—21no—
294C23—3461no—
295C23—13841no—
296C23—13841no—
297C23—1731no—
298C23—6922no—
299Q8—81no—
300Q8—3461no—
301Q8—3461no—
302Q8—13841no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—81no—
309D4—3461no—
310D4—3461no—
311D4—13841no—
312D4—13841no—
313D4—6922no—
314D4—6922no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—13841no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—1731no—
324C2×C4—1732no—
325C2×C4—1732no—
326C2×C4—6922no—
327C2×C4D4×D17311yescharacteristic
328C2×C4D4×D17311yescharacteristic
329C2×C4C23×D17311yescharacteristic
330C8—21no—
331C8—21no—
332C8—3461no—
333C8—3461no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—3461no—
341C22—3461no—
342C22—13844no—
343C22—1732no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—3461no—
353C4—3461no—
354C4—1732no—
355C4—6921no—
356C4—6921no—
357C4—6921no—
358C4#41×D17311yescharacteristic
359C4#41×D17311yescharacteristic
360C2—81no—
361C2—81no—
362C2—3462no—
363C2—13841no—
364C2#124×D17311yescharacteristic
365C2#2601229211yescharacteristic
366C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#486×C173C2
socleC22×C173C22×D4
prime radical#26012054C2
hypercenter#486D173
hypocenterC173#3136
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #26012118C2semidirect
3. #26011855C2semidirect
4. #26011854C2semidirect
5. #26011853C2semidirect
6. #26011849C2semidirect
7. #26011846C2semidirect
8. #26011844C2semidirect
9. #26011836C2semidirect
10. #26011494C2semidirect
11. #26011489C2semidirect
12. #26011484C2semidirect
13. #102×D173C2semidirect
14. #26011445C2semidirect
15. #486×C173C2semidirect
20. #19149678C22semidirect
23. #19149671C22semidirect
24. #19149670C22semidirect
37. #19149604C22semidirect
38. #19149604C22semidirect
39. #19149602C22semidirect
40. #19149602C22semidirect
45. #19149600C22semidirect
46. #19149600C22semidirect
49. #19149590C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
52. C4⋊Q8×C173C22semidirect
56. #102×C173C22semidirect
57. #102×C173C22semidirect
58. C4×C8×C173C22semidirect
183. C173#3136semidirect
194. #486D173semidirect
206. C4⋊Q8C2×D173semidirect
237. #102C2×D173semidirect
238. #102C2×D173semidirect
242. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121182
#260120542
#260118552
#260118542
#260118532
#260118492
#260118462
#260118442
#260118362
#260114942
#260114892
#260114842
#102×D1732
#260114452
#486×C1732
#3136173