Finite Group Databasebeta

← #36363914

Subgroups — #36363914

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#26012059C211yesmaximal, characteristic
3#26011856C211yesmaximal, characteristic, complemented
4#26011854C211yesmaximal, characteristic, complemented
5#26011852C211yesmaximal, characteristic, complemented
6#26011847C211yesmaximal, characteristic, complemented
7#26011845C211yesmaximal, characteristic, complemented
8#26011839C211yesmaximal, characteristic, complemented
9#26011836C211yesmaximal, characteristic, complemented
10#26011494C211yesmaximal, characteristic, complemented
11#26011491C211yesmaximal, characteristic, complemented
12#26011486C211yesmaximal, characteristic, complemented
13#102×D173C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#489×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149678C2211yescharacteristic, complemented
21#19149677—21no—
22#19149676C2211yescharacteristic
23#19149675C2211yescharacteristic
24#19149671C2211yescharacteristic, complemented
25#19149670C2211yescharacteristic, complemented
26#19149669—21no—
27#19149668C2211yescharacteristic
28#19149657—21no—
29#19149655—21no—
30#19149652—21no—
31#19149652—21no—
32#19149649—21no—
33#19149649—21no—
34#19149637C2211yescharacteristic
35#19149636C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic
37#19149603C2211yescharacteristic, complemented
38#19149603C2211yescharacteristic, complemented
39#19149601C2211yescharacteristic
40#19149601C2211yescharacteristic
41#19149601C2211yescharacteristic, complemented
42#19149601C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49#19149589C2211yescharacteristic, complemented
50C4×#13063436C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C4⋊Q8×C173C2211yescharacteristic, complemented
53#124×C173C2211yescharacteristic
54#103×C173C2211yescharacteristic, complemented
55#103×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—22no—
61#14583827—22no—
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—
71D4×D173—22no—
72D4×D173—22no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76#14583809—22no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C8×D173—22no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805D411yescharacteristic
89#14583805D411yescharacteristic
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173—21no—
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—42no—
111C22×D173—81no—
112C22×D173—81no—
113C22×D173C2×D411yescharacteristic
114Dic346—81no—
115D692—21no—
116D692—21no—
117D692—21no—
118D692—21no—
119D692—42no—
120D692—81no—
121D692—81no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—42no—
131C4×D173—81no—
132C4×D173#4112yes—
133C4×D173#4112yes—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—81no—
140D4×C173—21no—
141D4×C173—21no—
142C2×C4×C173—21no—
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×C173C22×D411yescharacteristic
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
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×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173C2×#4111yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173#12412yes—
176C2×C173—81no—
177C2×C173C2×#12411yescharacteristic
178C2×C173#57611yescharacteristic
179C2×C173#57711yescharacteristic
180D173—22no—
181D173—81no—
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#489D17311yescharacteristic, complemented
191#488—1731no—
192#488—1731no—
193#486—1731no—
194#486—1731no—
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#127—1731no—
209#125—3461no—
210#125—1731no—
211#125—1731no—
212#124—1731no—
213#124C2×D17311yescharacteristic
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#103C2×D17311yescharacteristic, complemented
226#103C2×D17311yescharacteristic, complemented
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#102—1731no—
236#102—1731no—
237#102C2×D17311yescharacteristic
238#102C2×D17311yescharacteristic
239#97—1731no—
240#97C2×D17311yescharacteristic, complemented
241C4×C8—1731no—
242C4×C8—1731no—
243C24—3461no—
244#41—3462no—
245#41—3462no—
246C2×Q8—21no—
247C2×Q8—3461no—
248C2×Q8C22×D17311yescharacteristic
249C2×D4—3461no—
250C2×D4—3462no—
251C2×D4—3462no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—1731no—
258C2×D4—1731no—
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—21no—
283C22⋊C4—21no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—1731no—
288C22⋊C4—1731no—
289C42—3461no—
290C42—1731no—
291C42C22×D17311yescharacteristic
292C23—21no—
293C23—3461no—
294C23—3461no—
295C23—13841no—
296C23—13841no—
297C23—1731no—
298C23—6922no—
299Q8—21no—
300Q8—21no—
301Q8—81no—
302Q8—13841no—
303D4—21no—
304D4—21no—
305D4—3461no—
306D4—3461no—
307D4—3461no—
308D4—3461no—
309D4—13841no—
310D4—13841no—
311D4—13841no—
312D4—13841no—
313D4—6922no—
314D4—6922no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
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—3461no—
338C22—3461no—
339C22—13844no—
340C22—1732no—
341C22—6921no—
342C22—6921no—
343C22—6921no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—3461no—
354C4—3461no—
355C4—1732no—
356C4—6921no—
357C4—6921no—
358C4#41×D17311yescharacteristic
359C4#41×D17311yescharacteristic
360C2—81no—
361C2—3462no—
362C2—13841no—
363C2—13841no—
364C2#124×D17311yescharacteristic
365C2#2601229511yescharacteristic
366C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#489×C173C2
socleC22×C173C22×D4
prime radical#26012059C2
hypercenter#489D173
hypocenterC173#3136
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #26012118C2semidirect
3. #26011856C2semidirect
4. #26011854C2semidirect
5. #26011852C2semidirect
6. #26011847C2semidirect
7. #26011845C2semidirect
8. #26011839C2semidirect
9. #26011836C2semidirect
10. #26011494C2semidirect
11. #26011491C2semidirect
12. #26011486C2semidirect
13. #102×D173C2semidirect
14. #26011448C2semidirect
15. #489×C173C2semidirect
20. #19149678C22semidirect
24. #19149671C22semidirect
25. #19149670C22semidirect
37. #19149603C22semidirect
38. #19149603C22semidirect
41. #19149601C22semidirect
42. #19149601C22semidirect
47. #19149599C22semidirect
48. #19149599C22semidirect
49. #19149589C22semidirect
50. C4×#13063436C22semidirect
51. C8×Dic173C22semidirect
52. C4⋊Q8×C173C22semidirect
54. #103×C173C22semidirect
55. #103×C173C22semidirect
58. #97×C173C22semidirect
183. C173#3136semidirect
190. #489D173semidirect
206. C4⋊Q8C2×D173semidirect
225. #103C2×D173semidirect
226. #103C2×D173semidirect
240. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121182
#260120592
#260118562
#260118542
#260118522
#260118472
#260118452
#260118392
#260118362
#260114942
#260114912
#260114862
#102×D1732
#260114482
#489×C1732
#3136173