Finite Group Databasebeta

← #36365358

Subgroups — #36365358

Order 22144.

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

normal subgroups only

This group has 460 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#26012302C212yesmaximal, complemented
2#26012301C212yesmaximal, complemented
3#26011863C211yesmaximal, characteristic, complemented
4#26011753C212yesmaximal, complemented
5#26011752C212yesmaximal, complemented
6#26011675C212yesmaximal, complemented
7#26011658C211yesmaximal, characteristic, complemented
8#26011605C212yesmaximal, complemented
9#453×C173C211yesmaximal, characteristic, complemented
10#19149726C2212yes—
11#19149724C2211yescharacteristic, complemented
12#19149712C2212yescomplemented
13#19149709—22no—
14#19149709C2212yescomplemented
15#19149708—22no—
16#19149705—22no—
17#19149705C2212yes—
18#19149703—22no—
19#19149699—22no—
20#19149699C2212yes—
21#19149698—22no—
22C2×#14583814—22no—
23C2×#14583814C2212yescomplemented
24C2×#14583814C2212yescomplemented
25SD16×D173—22no—
26#19149687C2212yescomplemented
27#19149686C2212yescomplemented
28#19149679C2211yescharacteristic, complemented
29#19149614C2212yescomplemented
30#19149607C2212yescomplemented
31#19149597C2212yescomplemented
32#19149596C2211yescharacteristic, complemented
33C4×#13063436C2211yescharacteristic, complemented
34H(5,𝔽2)×C173C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C173C2212yes—
36#127×C173C2211yescharacteristic
37#104×C173C2212yescomplemented
38#100×C173C2211yescharacteristic, complemented
39#14583826—22no—
40#14583826C2312yes—
41#14583825—22no—
42#14583825—22no—
43#14583825—42no—
44C2×Dic346—22no—
45C2×Dic346C2312yes—
46Q8×D173—22no—
47C2×C22⋊D173—41no—
48D4×D173—22no—
49Dic692—22no—
50#14583816—22no—
51#14583816D412yescomplemented
52#14583815—22no—
53#14583814—22no—
54#14583814—22no—
55#14583814—22no—
56#14583814—22no—
57#14583814—22no—
58#14583814—22no—
59#14583814D412yescomplemented
60#14583813D412yescomplemented
61#14583811—22no—
62#14583811D412yescomplemented
63#14583810—22no—
64#14583810C2312yes—
65#14583809—22no—
66C2×#13063436—22no—
67C2×#13063436C2312yes—
68C8×D173—22no—
69C4⋊Dic173—41no—
70#41×C173—22no—
71#41×C173C2312yes—
72C2×D4×C173—22no—
73C2×D4×C173—41no—
74C2×D4×C173—41no—
75C2×D4×C173C2311yescharacteristic
76C2×D4×C173C2311yescharacteristic
77SD16×C173—22no—
78D8×C173—22no—
79M16×C173C2312yes—
80C42×C173C2311yescharacteristic
81C22×D173—41no—
82Dic346—22no—
83Dic346—22no—
84Dic346—42no—
85Dic346C2×D412yes—
86D692C2×D412yes—
87C22⋊D173—22no—
88C22⋊D173—42no—
89C22⋊D173—42no—
90C2×Dic173—22no—
91C2×Dic173—41no—
92C2×Dic173—42no—
93C4×D173—22no—
94C4×D173—42no—
95#13063436—22no—
96#13063436C2×D412yes—
97#13063436C2×D412yes—
98C23×C173—21no—
99C23×C173—21no—
100C23×C173—41no—
101Q8×C173C2×D412yes—
102D4×C173—22no—
103D4×C173—22no—
104D4×C173—22no—
105D4×C173—42no—
106D4×C173—82no—
107D4×C173C2×D412yes—
108C2×C4×C173—21no—
109C2×C4×C173—22no—
110C2×C4×C173—41no—
111C2×C4×C173C2411yescharacteristic
112C8×C173—22no—
113C22×C173—22no—
114C22×C173—41no—
115C22×C173—41no—
116C22×C173—81no—
117C22×C173—81no—
118C22×C173—161no—
119C22×C173H(5,𝔽2)11yescharacteristic
120C2×D173—22no—
121C2×D173—161no—
122C4×C173—22no—
123C4×C173—22no—
124C4×C173C22×D412yes—
125Dic173—22no—
126Dic173—42no—
127C2×C173—21no—
128C2×C173—42no—
129C2×C173—81no—
130C2×C173—81no—
131C2×C173D4211yescharacteristic
132D173—42no—
133C173#329111yescharacteristic, complemented, Sylow
134#3291—1731nomaximal, Sylow
135#577—1732no—
136#577—1732no—
137#492—1731no—
138#471—1732no—
139#471—1732no—
140#456—1732no—
141#453—1731no—
142#453D17311yescharacteristic, complemented
143#443—1732no—
1442−1+4—1732no—
145H(5,𝔽2)—1731no—
146H(5,𝔽2)C2×D17311yescharacteristic, complemented
147#137—3462no—
148#137—3462no—
149#137—1732no—
150#137—1732no—
151AGL(1,ℤ/8)—3462no—
152AGL(1,ℤ/8)—3462no—
153AGL(1,ℤ/8)—1732no—
154AGL(1,ℤ/8)C2×D17312yes—
155#135—3462no—
156#135—3462no—
157#135—1732no—
158C2×SD16—3462no—
159C2×SD16—3462no—
160C2×SD16—1732no—
161C2×SD16—1732no—
162#131—1732no—
163#131—1732no—
164C4⋊Q8—1731no—
165#127C2×D17311yescharacteristic
166#108—1732no—
167#104—1732no—
168#104C2×D17312yescomplemented
169#101—1732no—
170#100—1731no—
171#100C2×D17311yescharacteristic, complemented
172C4×C8—1731no—
173#41—22no—
174#41—3462no—
175#41—3462no—
176#41—3462no—
177#41—1732no—
178#41—6922no—
179#41C22×D17312yes—
180C2×Q8—3462no—
181C2×Q8—3462no—
182C2×Q8—1732no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3462no—
187C2×D4—6921no—
188C2×D4C22×D17311yescharacteristic
189C2×D4C22×D17311yescharacteristic
190Q16—3462no—
191Q16—3462no—
192Q16—1732no—
193SD16—22no—
194SD16—3462no—
195SD16—3462no—
196SD16—3462no—
197SD16—3462no—
198SD16—3462no—
199SD16—3462no—
200SD16—3462no—
201SD16—1732no—
202SD16—1732no—
203D8—22no—
204D8—3462no—
205D8—1732no—
206M16—3462no—
207M16—3462no—
208M16—1732no—
209M16C22×D17312yes—
210C2×C8—3462no—
211C2×C8—3462no—
212C2×C8—1732no—
213C4⋊C4—6921no—
214C42C22×D17311yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—6921no—
219Q8—3462no—
220Q8—3462no—
221Q8—1732no—
222Q8—6922no—
223Q8D4×D17312yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—3462no—
230D4—1732no—
231D4—6922no—
232D4—6922no—
233D4D4×D17312yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—3462no—
238C2×C4—3462no—
239C2×C4—6922no—
240C2×C4—6921no—
241C2×C4—6922no—
242C2×C4C23×D17311yescharacteristic
243C8—22no—
244C8—3462no—
245C8—1732no—
246C8—1732no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—3462no—
254C22—27681no—
255C22#1914972411yescharacteristic
256C4—22no—
257C4—22no—
258C4—3462no—
259C4—6922no—
260C4C2×D4×D17312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—6922no—
266C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#453×C173C2
socleC2×C173D42
hypercenter#453D173
hypocenterC173#3291
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26012302C2semidirect
2. #26012301C2semidirect
3. #26011863C2semidirect
4. #26011753C2semidirect
5. #26011752C2semidirect
6. #26011675C2semidirect
7. #26011658C2semidirect
8. #26011605C2semidirect
9. #453×C173C2semidirect
11. #19149724C22semidirect
12. #19149712C22semidirect
14. #19149709C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583814C22semidirect
26. #19149687C22semidirect
27. #19149686C22semidirect
28. #19149679C22semidirect
29. #19149614C22semidirect
30. #19149607C22semidirect
31. #19149597C22semidirect
32. #19149596C22semidirect
33. C4×#13063436C22semidirect
34. H(5,𝔽2)×C173C22semidirect
37. #104×C173C22semidirect
38. #100×C173C22semidirect
51. #14583816D4semidirect
59. #14583814D4semidirect
60. #14583813D4semidirect
62. #14583811D4semidirect
133. C173#3291semidirect
142. #453D173semidirect
146. H(5,𝔽2)C2×D173semidirect
168. #104C2×D173semidirect
171. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123022
#260123012
#260118632
#260117532
#260117522
#260116752
#260116582
#260116052
#453×C1732
#3291173