Finite Group Databasebeta

← #36365180

Subgroups — #36365180

Order 22144.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149707C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3#26012159C211yesmaximal, characteristic, complemented
4#26012125C211yesmaximal, characteristic, complemented
5#26011866C211yesmaximal, characteristic, complemented
6#26011745C211yesmaximal, characteristic, complemented
7#26011690C211yesmaximal, characteristic, complemented
8#26011685C211yesmaximal, characteristic, complemented
9#26011637C211yesmaximal, characteristic, complemented
10#26011634C211yesmaximal, characteristic, complemented
11#26011626C211yesmaximal, characteristic, complemented
12#26011624C211yesmaximal, characteristic, complemented
13#26011578C211yesmaximal, characteristic, complemented
14#26011567C211yesmaximal, characteristic, complemented
15#469×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C2×#14583825C2211yescharacteristic, complemented
18C22×D692C2211yescharacteristic
19C2×D4×D173—21no—
20C2×D4×D173C2211yescharacteristic
21#19149707—24no—
22#19149703—24no—
23C2×#14583815C2211yescharacteristic
24C2×#14583815C2211yescharacteristic, complemented
25C2×#14583814C2211yescharacteristic, complemented
26C2×D1384C2211yescharacteristic
27C2×D1384C2211yescharacteristic, complemented
28C2×#14583811C2211yescharacteristic
29C2×#14583811C2211yescharacteristic, complemented
30C2×#14583811C2211yescharacteristic, complemented
31C2×#14583809C2211yescharacteristic, complemented
32#19149675—21no—
33#19149675C2211yescharacteristic
34#19149675C2211yescharacteristic, complemented
35#19149654C2211yescharacteristic
36#19149654C2211yescharacteristic, complemented
37#19149652—21no—
38#19149652—21no—
39#19149651—21no—
40#19149649—21no—
41#19149647—21no—
42#19149644C2211yescharacteristic, complemented
43#19149641C2211yescharacteristic
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149608C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic, complemented
51C22⋊#13063436C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53C8×Dic173C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic, complemented
55C2×M16×C173C2211yescharacteristic, complemented
56#121×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic, complemented
58#106×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63#14583826—22no—
64#14583826—41no—
65#14583825—22no—
66#14583825—41no—
67C2×Dic346C2311yescharacteristic
68C2×D692—22no—
69C2×D692—21no—
70C2×D692—21no—
71C2×D692—21no—
72C2×D692—41no—
73C2×D692C2311yescharacteristic
74C2×D692C2311yescharacteristic
75C2×D692C2311yescharacteristic
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80C2×C22⋊D173—21no—
81C2×C22⋊D173—21no—
82C2×C22⋊D173—21no—
83C2×C22⋊D173D411yescharacteristic
84D4×D173—22no—
85D4×D173—41no—
86D4×D173—44no—
87C22×Dic173—21no—
88C2×C4×D173—21no—
89C2×C4×D173—21no—
90C2×C4×D173C2311yescharacteristic
91#14583815—22no—
92#14583815D414yescomplemented
93#14583814—22no—
94D1384—22no—
95D1384—22no—
96#14583811—22no—
97#14583811—41no—
98#14583811—41no—
99#14583809—22no—
100C2×#13063436C2311yescharacteristic
101C2×#13063436C2311yescharacteristic
102#14583805—21no—
103#14583805D411yescharacteristic
104C22⋊Dic173D411yescharacteristic
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803D411yescharacteristic
109C4×Dic173—21no—
110C4×Dic173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173—21no—
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173C2311yescharacteristic
116D8×C173—22no—
117M16×C173—22no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22⋊C4×C173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—21no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—42no—
131C22×D173—81no—
132C22×D173—81no—
133C22×D173C2×D411yescharacteristic
134Dic346—21no—
135Dic346C2×D412yescomplemented
136D692—21no—
137D692—21no—
138D692—21no—
139D692—21no—
140D692—21no—
141D692—42no—
142D692—42no—
143D692—81no—
144D692C2×D412yes—
145C22⋊D173—22no—
146C22⋊D173—42no—
147C22⋊D173—42no—
148C22⋊D173—42no—
149C22⋊D173—42no—
150C22⋊D173—42no—
151C22⋊D173—81no—
152C22⋊D173—81no—
153C2×Dic173—21no—
154C2×Dic173—42no—
155C2×Dic173C2×D411yescharacteristic
156C2×Dic173C2×D411yescharacteristic
157C2×Dic173C2×D411yescharacteristic
158C4×D173—21no—
159C4×D173—21no—
160C4×D173—42no—
161C4×D173—42no—
162#13063436—21no—
163#13063436—21no—
164C23×C173—21no—
165C23×C173—21no—
166C23×C173C2×D411yescharacteristic
167D4×C173—21no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173—21no—
171D4×C173—42no—
172D4×C173—81no—
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—21no—
176C2×C4×C173—21no—
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2411yescharacteristic
179C8×C173—21no—
180C8×C173C2×D412yescomplemented
181C22×C173—21no—
182C22×C173—21no—
183C22×C173—21no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173C22×D411yescharacteristic
191C2×D173—21no—
192C2×D173—22no—
193C2×D173—21no—
194C2×D173—21no—
195C2×D173—41no—
196C2×D173—41no—
197C2×D173—41no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—84no—
204C2×D173—84no—
205C4×C173—21no—
206C4×C173—41no—
207C4×C173—41no—
208C4×C173C22×D411yescharacteristic
209C4×C173H(5,𝔽2)11yescharacteristic
210Dic173—21no—
211Dic173—22no—
212Dic173—21no—
213Dic173AGL(1,ℤ/8)12yescomplemented
214C2×C173—41no—
215C2×C173—81no—
216C2×C173—81no—
217C2×C173D4211yescharacteristic
218C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C173#57611yescharacteristic
220D173—41no—
221D173—42no—
222D173—81no—
223D173—81no—
224C173#327411yescharacteristic, complemented, Sylow
225#3274—1731nomaximal, Sylow
226C2×AGL(1,ℤ/8)—1731no—
227C2×AGL(1,ℤ/8)—1731no—
228#550—1731no—
229D42—1731no—
230#493—1731no—
231#469—1731no—
232#469D17311yescharacteristic, complemented
233#460—1731no—
234#459—1731no—
235#449—1731no—
236#449—1731no—
237#447—1731no—
238#447—1731no—
239C4×SD16—1731no—
240#436—1731no—
241C2×#41—1731no—
242C2×#41—1731no—
243C22×D4—3461no—
244C22×D4—1731no—
245C22×D4—1731no—
246AGL(1,ℤ/8)—3464no—
247AGL(1,ℤ/8)—3464no—
248C2×SD16—1731no—
249C2×SD16—1731no—
250C2×SD16—1731no—
251C2×D8—1731no—
252C2×D8—1731no—
253C2×D8—1731no—
254C2×D8—1731no—
255C2×D8—1731no—
256C2×D8C2×D17311yescharacteristic, complemented
257C2×M16—1731no—
258C2×M16C2×D17311yescharacteristic, complemented
259#127—3461no—
260#127—1731no—
261#127—1731no—
262#121—3461no—
263#121—3461no—
264#121—3461no—
265#121—1731no—
266#121—1731no—
267#121C2×D17311yescharacteristic
268#121C2×D17311yescharacteristic, complemented
269#120—3461no—
270#120—3461no—
271C4×Q8—1731no—
272C4×D4—3461no—
273C4×D4—1731no—
274C4×D4—1731no—
275#106C2×D17311yescharacteristic, complemented
276C4⋊C8—1731no—
277#103—1731no—
278#102—1731no—
279#102—1731no—
280#102—1731no—
281#102C2×D17311yescharacteristic, complemented
282#102C2×D17311yescharacteristic, complemented
283C22⋊C8—1731no—
284C22⋊C8—1731no—
285C4×C8—1731no—
286C24—3461no—
287C24—3461no—
288#41—3462no—
289#41—3462no—
290#41—6921no—
291#41—6921no—
292C2×Q8—1731no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3462no—
296C2×D4—3461no—
297C2×D4—3461no—
298C2×D4—3462no—
299C2×D4—3461no—
300C2×D4—3461no—
301C2×D4—3461no—
302C2×D4—3461no—
303C2×D4—3461no—
304C2×D4—3461no—
305C2×D4—3461no—
306C2×D4—3461no—
307C2×D4—1731no—
308C2×D4—1731no—
309C2×D4—1731no—
310C2×D4—1731no—
311C2×D4—6921no—
312C2×D4—6921no—
313C2×D4—6924no—
314C2×D4C22×D17311yescharacteristic
315C2×D4C22×D17311yescharacteristic
316C22×C4—3461no—
317C22×C4—3461no—
318C22×C4—3461no—
319C22×C4—1731no—
320C22×C4C22×D17311yescharacteristic
321SD16—3462no—
322SD16—3462no—
323SD16—1734no—
324D8—22no—
325D8—3462no—
326D8—3462no—
327D8—3462no—
328D8—6921no—
329D8—6921no—
330M16—22no—
331M16—3462no—
332C2×C8—1731no—
333C2×C8—1731no—
334C2×C8C22×D17311yescharacteristic
335C2×C8C22×D17311yescharacteristic
336C4⋊C4—3461no—
337C4⋊C4—1731no—
338C4⋊C4C22×D17311yescharacteristic
339C4⋊C4C22×D17311yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—3461no—
343C22⋊C4—3461no—
344C22⋊C4—3461no—
345C22⋊C4—1731no—
346C22⋊C4—1731no—
347C42—3461no—
348C42—1731no—
349C23—21no—
350C23—21no—
351C23—3461no—
352C23—3461no—
353C23—3461no—
354C23—13841no—
355C23—13841no—
356C23—1731no—
357C23—6922no—
358C23—6922no—
359C23—6922no—
360C23—6922no—
361C23D4×D17311yescharacteristic
362Q8—3461no—
363Q8—1732no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—81no—
370D4—3462no—
371D4—3461no—
372D4—3461no—
373D4—3461no—
374D4—3461no—
375D4—3461no—
376D4—13841no—
377D4—13841no—
378D4—13841no—
379D4—1732no—
380D4—6922no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4—6922no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—1731no—
395C2×C4—1731no—
396C2×C4—1731no—
397C2×C4—6922no—
398C2×C4—6922no—
399C2×C4—6922no—
400C2×C4D4×D17311yescharacteristic
401C2×C4C23×D17311yescharacteristic
402C8—21no—
403C8—3461no—
404C8—3461no—
405C8D4×D17312yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—3461no—
416C22—3461no—
417C22—3461no—
418C22—3462no—
419C22—13844no—
420C22—13844no—
421C22—6921no—
422C22—6921no—
423C22—6921no—
424C22—6921no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22C2×D4×D17311yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—3461no—
434C4—3461no—
435C4—3462no—
436C4—1732no—
437C4C2×D4×D17311yescharacteristic
438C4#1914972511yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—13841no—
443C2—13841no—
444C2—6921no—
445C2—6922no—
446C2#2601212511yescharacteristic
447C2AGL(1,ℤ/8)×D17311yescharacteristic
448C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#19149707C2semidirect
2. C2×#19149703C2semidirect
3. #26012159C2semidirect
4. #26012125C2semidirect
5. #26011866C2semidirect
6. #26011745C2semidirect
7. #26011690C2semidirect
8. #26011685C2semidirect
9. #26011637C2semidirect
10. #26011634C2semidirect
11. #26011626C2semidirect
12. #26011624C2semidirect
13. #26011578C2semidirect
14. #26011567C2semidirect
15. #469×C173C2semidirect
16. C2×#14583826C22semidirect
17. C2×#14583825C22semidirect
24. C2×#14583815C22semidirect
25. C2×#14583814C22semidirect
27. C2×D1384C22semidirect
29. C2×#14583811C22semidirect
30. C2×#14583811C22semidirect
31. C2×#14583809C22semidirect
34. #19149675C22semidirect
36. #19149654C22semidirect
42. #19149644C22semidirect
46. #19149608C22semidirect
47. #19149602C22semidirect
49. #19149599C22semidirect
50. #19149599C22semidirect
51. C22⋊#13063436C22semidirect
52. #19149591C22semidirect
53. C8×Dic173C22semidirect
54. C2×D8×C173C22semidirect
55. C2×M16×C173C22semidirect
57. #121×C173C22semidirect
58. #106×C173C22semidirect
59. #102×C173C22semidirect
60. #102×C173C22semidirect
92. #14583815D4semidirect
135. Dic346C2×D4semidirect
180. C8×C173C2×D4semidirect
213. Dic173AGL(1,ℤ/8)semidirect
224. C173#3274semidirect
232. #469D173semidirect
256. C2×D8C2×D173semidirect
258. C2×M16C2×D173semidirect
268. #121C2×D173semidirect
275. #106C2×D173semidirect
281. #102C2×D173semidirect
282. #102C2×D173semidirect
405. C8D4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
C2×#191497032
#260121592
#260121252
#260118662
#260117452
#260116902
#260116852
#260116372
#260116342
#260116262
#260116242
#260115782
#260115672
#469×C1732
#3274173