Finite Group Databasebeta

← #36365353

Subgroups — #36365353

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#26012300C212yesmaximal, complemented
2#26012299C212yesmaximal, complemented
3#26011865C211yesmaximal, characteristic, complemented
4#26011757C212yesmaximal, complemented
5#26011756C212yesmaximal, complemented
6#26011673C211yesmaximal, characteristic, complemented
7#26011660C212yesmaximal, complemented
8#26011605C212yesmaximal, complemented
9#456×C173C211yesmaximal, characteristic, complemented
10#19149727C2211yescharacteristic
11#19149725C2212yescomplemented
12#19149711—22no—
13#19149710—22no—
14#19149710C2212yescomplemented
15#19149707C2212yes—
16#19149706—22no—
17#19149704—22no—
18#19149704C2212yes—
19#19149702—22no—
20#19149699—22no—
21#19149699C2212yes—
22C2×#14583813—22no—
23C2×#14583813C2212yescomplemented
24C2×#14583813C2212yescomplemented
25SD16×D173—22no—
26#19149687C2212yescomplemented
27#19149686C2212yescomplemented
28#19149676C2211yescharacteristic
29#19149614C2212yescomplemented
30#19149607C2212yescomplemented
31#19149598C2211yescharacteristic, complemented
32#19149595C2212yescomplemented
33C4×#13063436C2211yescharacteristic, complemented
342−1+4×C173C2211yescharacteristic
35#137×C173C2212yescomplemented
36C4⋊Q8×C173C2211yescharacteristic, complemented
37#104×C173C2212yescomplemented
38#101×C173C2211yescharacteristic, complemented
39#14583827—22no—
40#14583827—22no—
41#14583826—22no—
42#14583826—41no—
43#14583826C2312yes—
44Q8×D173—22no—
45C2×D692—22no—
46C2×D692—41no—
47C2×D692C2312yes—
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#14583813—22no—
59#14583813—22no—
60#14583813D412yescomplemented
61D1384—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×Q8×C173C2311yescharacteristic
76Q16×C173—22no—
77SD16×C173—22no—
78M16×C173C2312yes—
79C4⋊C4×C173—41no—
80C42×C173C2311yescharacteristic
81C22×D173—22no—
82C22×D173—42no—
83Dic346—41no—
84Dic346C2×D412yes—
85D692—22no—
86D692—22no—
87D692—42no—
88D692—41no—
89D692—82no—
90D692C2×D412yes—
91C22⋊D173—22no—
92C22⋊D173—42no—
93C4×D173—22no—
94C4×D173—42no—
95C4×D173—42no—
96#13063436—22no—
97#13063436C2×D412yes—
98#13063436C2×D412yes—
99Q8×C173—22no—
100Q8×C173—22no—
101Q8×C173—41no—
102Q8×C173C2×D412yes—
103D4×C173—22no—
104D4×C173—41no—
105D4×C173C2×D412yes—
106C2×C4×C173—21no—
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×C173H(5,𝔽2)11yescharacteristic
115C2×D173—22no—
116C2×D173—42no—
117C2×D173—82no—
118C2×D173—162no—
119C4×C173—22no—
120C4×C173—22no—
121C4×C173—41no—
122C4×C173—41no—
123C4×C173C22×D412yes—
124Dic173—22no—
125C2×C173—21no—
126C2×C173—42no—
127C2×C173D4211yescharacteristic
128D173—42no—
129D173—82no—
130C173#329111yescharacteristic, complemented, Sylow
131#3291—1731nomaximal, Sylow
132#577—1732no—
133#577—1732no—
134#492—1731no—
135#471—1732no—
136#471—1732no—
137#456—1731no—
138#456D17311yescharacteristic, complemented
139#453—1732no—
140#443—1732no—
1412−1+4—1731no—
1422−1+4C2×D17311yescharacteristic
143H(5,𝔽2)—1732no—
144#137—3462no—
145#137—3462no—
146#137—1732no—
147#137C2×D17312yescomplemented
148AGL(1,ℤ/8)—3462no—
149AGL(1,ℤ/8)—3462no—
150AGL(1,ℤ/8)—1732no—
151AGL(1,ℤ/8)—1732no—
152#135—3462no—
153#135—3462no—
154#135—1732no—
155C2×SD16—3462no—
156C2×SD16—3462no—
157C2×SD16—1732no—
158C2×SD16—1732no—
159#131—1732no—
160#131—1732no—
161C4⋊Q8C2×D17311yescharacteristic, complemented
162#127—1731no—
163#108—1732no—
164#104—1732no—
165#104C2×D17312yescomplemented
166#101—1731no—
167#101C2×D17311yescharacteristic, complemented
168#100—1732no—
169C4×C8—1731no—
170#41—22no—
171#41—41no—
172#41—3462no—
173#41—3462no—
174#41—3462no—
175#41—1732no—
176#41—6921no—
177#41C22×D17312yes—
178C2×Q8—22no—
179C2×Q8—3462no—
180C2×Q8C22×D17311yescharacteristic
181C2×Q8C22×D17311yescharacteristic
182C2×D4—3462no—
183C2×D4—3462no—
184C2×D4—1732no—
185C2×D4—6921no—
186C2×D4—6922no—
187Q16—22no—
188Q16—3462no—
189Q16—1732no—
190SD16—22no—
191SD16—3462no—
192SD16—3462no—
193SD16—3462no—
194SD16—3462no—
195SD16—3462no—
196SD16—3462no—
197SD16—3462no—
198SD16—1732no—
199SD16—1732no—
200D8—3462no—
201D8—3462no—
202D8—1732no—
203M16—3462no—
204M16—3462no—
205M16—1732no—
206M16C22×D17312yes—
207C2×C8—3462no—
208C2×C8—3462no—
209C2×C8—1732no—
210C4⋊C4—41no—
211C42C22×D17311yescharacteristic
212C23—3462no—
213C23—6922no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—1732no—
218Q8—6921no—
219Q8D4×D17312yes—
220D4—22no—
221D4—41no—
222D4—3462no—
223D4—3462no—
224D4—3462no—
225D4—13842no—
226D4—1732no—
227D4—6922no—
228D4—6921no—
229D4—6922no—
230D4D4×D17312yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3462no—
237C2×C4—6922no—
238C2×C4—6922no—
239C2×C4C23×D17311yescharacteristic
240C8—22no—
241C8—3462no—
242C8—1732no—
243C8—1732no—
244C22—22no—
245C22—3462no—
246C22—13842no—
247C22—6922no—
248C22—27682no—
249C22#1914972411yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—3462no—
255C4C2×D4×D17312yes—
256C2—21no—
257C2—42no—
258C2—13842no—
259C2—6922no—
260C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#456×C173C2
socleC2×C173D42
hypercenter#456D173
hypocenterC173#3291
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26012300C2semidirect
2. #26012299C2semidirect
3. #26011865C2semidirect
4. #26011757C2semidirect
5. #26011756C2semidirect
6. #26011673C2semidirect
7. #26011660C2semidirect
8. #26011605C2semidirect
9. #456×C173C2semidirect
11. #19149725C22semidirect
14. #19149710C22semidirect
23. C2×#14583813C22semidirect
24. C2×#14583813C22semidirect
26. #19149687C22semidirect
27. #19149686C22semidirect
29. #19149614C22semidirect
30. #19149607C22semidirect
31. #19149598C22semidirect
32. #19149595C22semidirect
33. C4×#13063436C22semidirect
35. #137×C173C22semidirect
36. C4⋊Q8×C173C22semidirect
37. #104×C173C22semidirect
38. #101×C173C22semidirect
51. #14583816D4semidirect
53. #14583814D4semidirect
60. #14583813D4semidirect
63. #14583811D4semidirect
130. C173#3291semidirect
138. #456D173semidirect
147. #137C2×D173semidirect
161. C4⋊Q8C2×D173semidirect
165. #104C2×D173semidirect
167. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123002
#260122992
#260118652
#260117572
#260117562
#260116732
#260116602
#260116052
#456×C1732
#3291173