Finite Group Databasebeta

← #36365361

Subgroups — #36365361

Order 22144.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012296C211yesmaximal, characteristic, complemented
2#26012295C212yesmaximal, complemented
3#26012292C211yesmaximal, characteristic, complemented
4#26011866C211yesmaximal, characteristic, complemented
5#26011762C211yesmaximal, characteristic, complemented
6#26011760C212yesmaximal, complemented
7#26011661C212yesmaximal, complemented
8#26011657C212yesmaximal, complemented
9#26011607C211yesmaximal, characteristic, complemented
10#26011606C211yesmaximal, characteristic, complemented
11#472×C173C211yesmaximal, characteristic, complemented
12#19149725C2212yescomplemented
13#19149724C2212yescomplemented
14#19149707—22no—
15#19149707C2212yescomplemented
16#19149707C2212yescomplemented
17#19149706—22no—
18#19149705C2212yescomplemented
19#19149703—22no—
20#19149703—22no—
21#19149701—21no—
22#19149701C2211yescharacteristic
23#19149700—22no—
24#19149699C2211yescharacteristic
25#19149698—21no—
26C2×D1384—21no—
27C2×D1384C2211yescharacteristic, complemented
28C2×#14583811C2212yescomplemented
29D8×D173—21no—
30D8×D173—22no—
31#19149687C2212yescomplemented
32#19149685C2211yescharacteristic, complemented
33#19149675C2212yescomplemented
34#19149614C2211yescharacteristic, complemented
35#19149606C2211yescharacteristic, complemented
36#19149605C2212yescomplemented
37#19149605C2211yescharacteristic, complemented
38#19149595C2212yescomplemented
39C8×Dic173C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C173C2212yescomplemented
41C2×D8×C173C2211yescharacteristic, complemented
42#131×C173C2211yescharacteristic, complemented
43#108×C173C2211yescharacteristic, complemented
44#100×C173C2212yescomplemented
45#14583827—22no—
46#14583826—22no—
47#14583826C2311yescharacteristic
48#14583826C2312yes—
49#14583825—22no—
50#14583825—22no—
51C2×D692—22no—
52C2×D692C2312yes—
53C2×C22⋊D173—42no—
54C2×C22⋊D173—42no—
55D4×D173—22no—
56D4×D173—22no—
57D4×D173—22no—
58D4×D173—42no—
59Dic692D411yescharacteristic, complemented
60#14583816D411yescharacteristic, complemented
61#14583815—22no—
62#14583815—22no—
63#14583815D412yescomplemented
64#14583814—22no—
65#14583814D411yescharacteristic, complemented
66#14583813D411yescharacteristic, complemented
67D1384—22no—
68D1384—22no—
69D1384—21no—
70D1384—21no—
71D1384D411yescharacteristic, complemented
72#14583811—22no—
73#14583811—22no—
74#14583811—22no—
75#14583811D411yescharacteristic, complemented
76#14583810C2311yescharacteristic
77#14583809—22no—
78#14583809—21no—
79C2×#13063436C2311yescharacteristic
80C8×D173—22no—
81C8×D173—21no—
82C4×Dic173C2311yescharacteristic
83#41×C173C2311yescharacteristic
84C2×D4×C173C2312yes—
85SD16×C173—22no—
86D8×C173—22no—
87D8×C173—21no—
88D8×C173—21no—
89M16×C173—21no—
90M16×C173C2311yescharacteristic
91M16×C173C2312yes—
92C2×C8×C173—21no—
93C2×C8×C173C2311yescharacteristic
94C22×D173—22no—
95C22×D173—42no—
96C22×D173—42no—
97Dic346C2×D412yes—
98Dic346C2×D411yescharacteristic
99D692—22no—
100D692—22no—
101D692—42no—
102D692C2×D412yes—
103D692C2×D411yescharacteristic
104C22⋊D173—22no—
105C22⋊D173—21no—
106C22⋊D173—42no—
107C22⋊D173—42no—
108C22⋊D173—42no—
109C22⋊D173—84no—
110C2×Dic173—21no—
111C2×Dic173—42no—
112C4×D173—21no—
113C4×D173—22no—
114C4×D173—42no—
115#13063436—21no—
116#13063436C2×D411yescharacteristic
117#13063436C2×D411yescharacteristic
118C23×C173—22no—
119Q8×C173C2×D411yescharacteristic
120D4×C173—21no—
121D4×C173—22no—
122D4×C173—22no—
123D4×C173C2×D411yescharacteristic
124C2×C4×C173—21no—
125C2×C4×C173C2411yescharacteristic
126C8×C173—22no—
127C8×C173—21no—
128C8×C173C2×D411yescharacteristic
129C8×C173C2×D411yescharacteristic
130C22×C173—21no—
131C22×C173—42no—
132C22×C173—82no—
133C22×C173H(5,𝔽2)11yescharacteristic
134C2×D173—21no—
135C2×D173—22no—
136C2×D173—42no—
137C2×D173—82no—
138C2×D173—162no—
139C2×D173—162no—
140C4×C173—21no—
141C4×C173C22×D411yescharacteristic
142C4×C173C22×D411yescharacteristic
143Dic173—22no—
144Dic173—22no—
145Dic173—21no—
146C2×C173—21no—
147C2×C173—41no—
148C2×C173—82no—
149C2×C173D4211yescharacteristic
150D173—41no—
151D173—42no—
152D173—82no—
153C173#329211yescharacteristic, complemented, Sylow
154#3292—1731nomaximal, Sylow
155#576—1732no—
156#576—1731no—
157#576—1731no—
158#493—1731no—
159#472—1731no—
160#472—1732no—
161#472D17311yescharacteristic, complemented
162#453—1732no—
163#453—1732no—
164#443—1731no—
165#443—1731no—
166H(5,𝔽2)—1732no—
167H(5,𝔽2)—1732no—
168AGL(1,ℤ/8)—3462no—
169AGL(1,ℤ/8)—3462no—
170AGL(1,ℤ/8)—3462no—
171AGL(1,ℤ/8)—3462no—
172AGL(1,ℤ/8)—1732no—
173AGL(1,ℤ/8)—1732no—
174AGL(1,ℤ/8)—1732no—
175AGL(1,ℤ/8)C2×D17312yescomplemented
176#135—3461no—
177#135—3461no—
178#135—3462no—
179#135—1731no—
180#135—1731no—
181C2×D8—3461no—
182C2×D8—3462no—
183C2×D8—3461no—
184C2×D8—1732no—
185C2×D8—1731no—
186C2×D8C2×D17311yescharacteristic, complemented
187#131—1732no—
188#131—1731no—
189#131C2×D17311yescharacteristic, complemented
190#127—1732no—
191#108—1731no—
192#108C2×D17311yescharacteristic, complemented
193#104—1731no—
194#104—1732no—
195#104—1731no—
196#100—1732no—
197#100C2×D17312yescomplemented
198C4×C8—1731no—
199#41—3462no—
200#41—3462no—
201#41—3462no—
202#41—3462no—
203#41—1731no—
204#41—1732no—
205#41C22×D17311yescharacteristic
206C2×D4—3462no—
207C2×D4—3462no—
208C2×D4—3462no—
209C2×D4—3462no—
210C2×D4—1732no—
211C2×D4—6922no—
212C2×D4—6922no—
213C2×D4—6922no—
214C2×D4C22×D17312yes—
215Q16—1731no—
216Q16—1731no—
217SD16—22no—
218SD16—3462no—
219SD16—3462no—
220SD16—3462no—
221SD16—1731no—
222SD16—1731no—
223SD16—1732no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3462no—
228D8—3462no—
229D8—3462no—
230D8—3462no—
231D8—3462no—
232D8—3461no—
233D8—3461no—
234D8—1731no—
235D8—1731no—
236M16—21no—
237M16—3462no—
238M16—3461no—
239M16—1731no—
240M16C22×D17312yes—
241M16C22×D17311yescharacteristic
242C2×C8—21no—
243C2×C8—3462no—
244C2×C8—3461no—
245C2×C8—1731no—
246C2×C8C22×D17311yescharacteristic
247C42—1731no—
248C23—22no—
249C23—3462no—
250C23—6922no—
251C23—6922no—
252Q8—1731no—
253Q8—1732no—
254Q8D4×D17311yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—3462no—
259D4—3461no—
260D4—3462no—
261D4—3462no—
262D4—13844no—
263D4—1731no—
264D4—1732no—
265D4—6922no—
266D4—6922no—
267D4—6922no—
268D4—6922no—
269D4D4×D17311yescharacteristic
270C2×C4—21no—
271C2×C4—3461no—
272C2×C4—3462no—
273C2×C4—3461no—
274C2×C4—6922no—
275C2×C4—6922no—
276C2×C4C23×D17311yescharacteristic
277C8—21no—
278C8—22no—
279C8—3461no—
280C8—1731no—
281C8—1731no—
282C8D4×D17311yescharacteristic
283C8D4×D17311yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—3462no—
288C22—3461no—
289C22—13842no—
290C22—6922no—
291C22—27682no—
292C22—27682no—
293C22#1914972511yescharacteristic
294C4—21no—
295C4—3462no—
296C4—3462no—
297C4—3461no—
298C4C2×D4×D17311yescharacteristic
299C4C2×D4×D17311yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—13842no—
304C2—6922no—
305C2—6921no—
306C2#2601212511yescharacteristic
subgroupG/H
centerC2#26012125
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#472×C173C2
socleC2×C173D42
hypercenter#472D173
hypocenterC173#3292
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #26012296C2semidirect
2. #26012295C2semidirect
3. #26012292C2semidirect
4. #26011866C2semidirect
5. #26011762C2semidirect
6. #26011760C2semidirect
7. #26011661C2semidirect
8. #26011657C2semidirect
9. #26011607C2semidirect
10. #26011606C2semidirect
11. #472×C173C2semidirect
12. #19149725C22semidirect
13. #19149724C22semidirect
15. #19149707C22semidirect
16. #19149707C22semidirect
18. #19149705C22semidirect
27. C2×D1384C22semidirect
28. C2×#14583811C22semidirect
31. #19149687C22semidirect
32. #19149685C22semidirect
33. #19149675C22semidirect
34. #19149614C22semidirect
35. #19149606C22semidirect
36. #19149605C22semidirect
37. #19149605C22semidirect
38. #19149595C22semidirect
39. C8×Dic173C22semidirect
40. AGL(1,ℤ/8)×C173C22semidirect
41. C2×D8×C173C22semidirect
42. #131×C173C22semidirect
43. #108×C173C22semidirect
44. #100×C173C22semidirect
59. Dic692D4semidirect
60. #14583816D4semidirect
63. #14583815D4semidirect
65. #14583814D4semidirect
66. #14583813D4semidirect
71. D1384D4semidirect
75. #14583811D4semidirect
153. C173#3292semidirect
161. #472D173semidirect
175. AGL(1,ℤ/8)C2×D173semidirect
186. C2×D8C2×D173semidirect
189. #131C2×D173semidirect
192. #108C2×D173semidirect
197. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122962
#260122952
#260122922
#260118662
#260117622
#260117602
#260116612
#260116572
#260116072
#260116062
#472×C1732
#3292173