Finite Group Databasebeta

← #36365331

Subgroups — #36365331

Order 22144.

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

normal subgroups only

This group has 464 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#26012299C212yesmaximal, complemented
2#26012295C212yesmaximal, complemented
3#26011881C211yesmaximal, characteristic, complemented
4#26011759C212yesmaximal, complemented
5#26011757C212yesmaximal, complemented
6#26011668C211yesmaximal, characteristic, complemented
7#26011660C212yesmaximal, complemented
8#26011610C212yesmaximal, complemented
9#454×C173C211yesmaximal, characteristic, complemented
10#19149725C2212yescomplemented
11#19149724C2211yescharacteristic
12#19149710—22no—
13#19149710C2212yescomplemented
14#19149707C2212yes—
15#19149706—22no—
16#19149704—22no—
17#19149704C2212yescomplemented
18#19149703—22no—
19#19149700—22no—
20#19149699—22no—
21#19149699C2212yes—
22C2×#14583813—22no—
23C2×#14583813C2212yescomplemented
24C2×#14583811C2212yescomplemented
25D8×D173—22no—
26#19149687C2212yescomplemented
27#19149686C2212yescomplemented
28#19149676C2211yescharacteristic, complemented
29#19149614C2212yescomplemented
30#19149607C2212yescomplemented
31#19149596C2211yescharacteristic, complemented
32#19149595C2212yescomplemented
33#19149590C2211yescharacteristic, complemented
342−1+4×C173C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C173C2212yescomplemented
36#124×C173C2211yescharacteristic, complemented
37#104×C173C2212yescomplemented
38#101×C173C2211yescharacteristic, complemented
39#14583827—22no—
40#14583826—22no—
41#14583826C2312yes—
42#14583825—22no—
43C2×D692—22no—
44C2×D692—41no—
45C2×D692C2312yes—
46C2×C22⋊D173—41no—
47D4×D173—22no—
48D4×D173—22no—
49D4×D173—42no—
50#14583816—22no—
51#14583816D412yescomplemented
52#14583815—22no—
53#14583814D412yescomplemented
54#14583813—22no—
55#14583813—22no—
56#14583813—22no—
57#14583813—22no—
58#14583813D412yescomplemented
59D1384—22no—
60#14583811—22no—
61#14583811—22no—
62#14583811—22no—
63#14583811D412yescomplemented
64#14583810—22no—
65#14583810C2312yes—
66#14583809—22no—
67C2×#13063436—22no—
68C2×#13063436C2312yes—
69C8×D173—22no—
70#41×C173—22no—
71#41×C173—41no—
72#41×C173C2312yes—
73C2×Q8×C173—22no—
74C2×Q8×C173C2311yescharacteristic
75C2×D4×C173C2311yescharacteristic
76SD16×C173—22no—
77D8×C173—22no—
78M16×C173C2312yes—
79C22⋊C4×C173—41no—
80C42×C173C2311yescharacteristic
81C22×D173—22no—
82C22×D173—41no—
83C22×D173—42no—
84Dic346C2×D412yes—
85D692—22no—
86D692—22no—
87D692—42no—
88D692—82no—
89D692C2×D412yes—
90C22⋊D173—22no—
91C22⋊D173—42no—
92C22⋊D173—42no—
93C2×Dic173—41no—
94C4×D173—22no—
95C4×D173—42no—
96#13063436—22no—
97#13063436C2×D412yes—
98#13063436C2×D412yes—
99C23×C173—21no—
100Q8×C173—22no—
101Q8×C173—41no—
102Q8×C173C2×D412yes—
103D4×C173—22no—
104D4×C173—22no—
105D4×C173—41no—
106D4×C173C2×D412yes—
107C2×C4×C173—22no—
108C2×C4×C173—21no—
109C2×C4×C173—21no—
110C2×C4×C173—42no—
111C2×C4×C173C2411yescharacteristic
112C8×C173—22no—
113C22×C173—22no—
114C22×C173—41no—
115C22×C173—81no—
116C22×C173H(5,𝔽2)11yescharacteristic
117C2×D173—22no—
118C2×D173—42no—
119C2×D173—82no—
120C2×D173—161no—
121C2×D173—162no—
122C4×C173—22no—
123C4×C173—41no—
124C4×C173—41no—
125C4×C173C22×D412yes—
126Dic173—22no—
127C2×C173—21no—
128C2×C173—42no—
129C2×C173—81no—
130C2×C173D4211yescharacteristic
131D173—42no—
132D173—82no—
133C173#328811yescharacteristic, complemented, Sylow
134#3288—1731nomaximal, Sylow
135#577—1732no—
136#576—1732no—
137#496—1731no—
138#472—1732no—
139#471—1732no—
140#455—1731no—
141#454D17311yescharacteristic, complemented
142#453—1732no—
143#444—1732no—
1442−1+4C2×D17311yescharacteristic, complemented
145H(5,𝔽2)—1732no—
146H(5,𝔽2)—1731no—
147#137—3462no—
148#137—1732no—
149AGL(1,ℤ/8)—3462no—
150AGL(1,ℤ/8)—3462no—
151AGL(1,ℤ/8)—3462no—
152AGL(1,ℤ/8)—1732no—
153AGL(1,ℤ/8)—1732no—
154AGL(1,ℤ/8)C2×D17312yescomplemented
155#135—3462no—
156#135—3462no—
157#135—1732no—
158C2×SD16—3462no—
159C2×SD16—1732no—
160C2×D8—3462no—
161C2×D8—1732no—
162#131—1732no—
163#131—1732no—
164#127—1731no—
165#124C2×D17311yescharacteristic, complemented
166#108—1732no—
167#104—1732no—
168#104C2×D17312yescomplemented
169#101C2×D17311yescharacteristic, complemented
170#100—1731no—
171#100—1732no—
172#97—1731no—
173#41—22no—
174#41—41no—
175#41—3462no—
176#41—3462no—
177#41—3462no—
178#41—1732no—
179#41C22×D17312yes—
180C2×Q8—22no—
181C2×Q8C22×D17311yescharacteristic
182C2×D4—3462no—
183C2×D4—3462no—
184C2×D4—3462no—
185C2×D4—1732no—
186C2×D4—6921no—
187C2×D4—6922no—
188C2×D4—6921no—
189C2×D4C22×D17311yescharacteristic
190Q16—3462no—
191Q16—1732no—
192SD16—22no—
193SD16—3462no—
194SD16—3462no—
195SD16—3462no—
196SD16—3462no—
197SD16—3462no—
198SD16—1732no—
199SD16—1732no—
200D8—22no—
201D8—3462no—
202D8—3462no—
203D8—3462no—
204D8—3462no—
205D8—1732no—
206M16—3462no—
207M16—3462no—
208M16—1732no—
209M16C22×D17312yes—
210C2×C8—3462no—
211C2×C8—3462no—
212C2×C8—1732no—
213C22⋊C4—41no—
214C42C22×D17311yescharacteristic
215C23—21no—
216C23—3462no—
217C23—6922no—
218C23—6921no—
219Q8—22no—
220Q8—41no—
221Q8—1732no—
222Q8D4×D17312yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—3462no—
227D4—3462no—
228D4—3462no—
229D4—13842no—
230D4—1732no—
231D4—6922no—
232D4—6922no—
233D4—6922no—
234D4D4×D17312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—3462no—
240C2×C4—6922no—
241C2×C4—6921no—
242C2×C4C23×D17311yescharacteristic
243C8—22no—
244C8—3462no—
245C8—1732no—
246C8—1732no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—3462no—
251C22—13842no—
252C22—6922no—
253C22—27682no—
254C22—27681no—
255C22#1914972411yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—3462no—
260C4C2×D4×D17312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—13842no—
265C2—6922no—
266C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#454×C173C2
socleC2×C173D42
hypercenter#454D173
hypocenterC173#3288
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26012299C2semidirect
2. #26012295C2semidirect
3. #26011881C2semidirect
4. #26011759C2semidirect
5. #26011757C2semidirect
6. #26011668C2semidirect
7. #26011660C2semidirect
8. #26011610C2semidirect
9. #454×C173C2semidirect
10. #19149725C22semidirect
13. #19149710C22semidirect
17. #19149704C22semidirect
23. C2×#14583813C22semidirect
24. C2×#14583811C22semidirect
26. #19149687C22semidirect
27. #19149686C22semidirect
28. #19149676C22semidirect
29. #19149614C22semidirect
30. #19149607C22semidirect
31. #19149596C22semidirect
32. #19149595C22semidirect
33. #19149590C22semidirect
34. 2−1+4×C173C22semidirect
35. AGL(1,ℤ/8)×C173C22semidirect
36. #124×C173C22semidirect
37. #104×C173C22semidirect
38. #101×C173C22semidirect
51. #14583816D4semidirect
53. #14583814D4semidirect
58. #14583813D4semidirect
63. #14583811D4semidirect
133. C173#3288semidirect
141. #454D173semidirect
144. 2−1+4C2×D173semidirect
154. AGL(1,ℤ/8)C2×D173semidirect
165. #124C2×D173semidirect
168. #104C2×D173semidirect
169. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122992
#260122952
#260118812
#260117592
#260117572
#260116682
#260116602
#260116102
#454×C1732
#3288173