Finite Group Databasebeta

← #36365357

Subgroups — #36365357

Order 22144.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#26012303C212yesmaximal, complemented
2#26012298C212yesmaximal, complemented
3#26011864C211yesmaximal, characteristic, complemented
4#26011755C212yesmaximal, complemented
5#26011754C212yesmaximal, complemented
6#26011675C212yesmaximal, complemented
7#26011660C212yesmaximal, complemented
8#26011609C211yesmaximal, characteristic, complemented
9#443×C173C211yesmaximal, characteristic, complemented
10#19149726C2212yes—
11#19149725C2212yescomplemented
12#19149712—22no—
13#19149712C2212yescomplemented
14#19149711—22no—
15#19149709C2212yescomplemented
16#19149707—22no—
17#19149707C2212yes—
18#19149704C2212yes—
19#19149703—22no—
20#19149701—22no—
21#19149701C2211yescharacteristic
22#19149700—22no—
23C2×#14583815—22no—
24C2×#14583815C2212yescomplemented
25C2×#14583815C2212yescomplemented
26SD16×D173—22no—
27#19149685C2212yescomplemented
28#19149679C2211yescharacteristic, complemented
29#19149676C2211yescharacteristic
30#19149615C2211yescharacteristic, complemented
31#19149607C2212yescomplemented
32#19149597C2212yescomplemented
33#19149595C2212yescomplemented
34#135×C173C2211yescharacteristic
35#131×C173C2212yescomplemented
36#108×C173C2211yescharacteristic, complemented
37#104×C173C2212yescomplemented
38C4×C8×C173C2211yescharacteristic, complemented
39#14583827—22no—
40#14583826—22no—
41#14583826—22no—
42#14583826C2312yes—
43#14583825—22no—
44#14583825—42no—
45C2×Dic346—22no—
46C2×Dic346C2312yes—
47Q8×D173—22no—
48C2×D692—22no—
49C2×D692—41no—
50C2×D692C2312yes—
51D4×D173—22no—
52D4×D173—42no—
53Dic692—22no—
54Dic692D411yescharacteristic, complemented
55#14583816—22no—
56#14583815—22no—
57#14583815—22no—
58#14583815—22no—
59#14583815—22no—
60#14583815—22no—
61#14583815—22no—
62#14583815D412yescomplemented
63#14583814—22no—
64#14583813—22no—
65D1384—22no—
66D1384D411yescharacteristic, complemented
67#14583811—22no—
68#14583810C2312yes—
69#14583809—22no—
70C8×D173—22no—
71C4⋊Dic173—41no—
72#41×C173C2312yes—
73Q16×C173D411yescharacteristic, complemented
74SD16×C173D412yescomplemented
75D8×C173D411yescharacteristic, complemented
76M16×C173—22no—
77M16×C173C2312yes—
78C2×C8×C173—22no—
79C2×C8×C173C2311yescharacteristic
80C2×C8×C173C2311yescharacteristic
81C42×C173C2311yescharacteristic
82C22×D173—22no—
83C22×D173—42no—
84Dic346—22no—
85Dic346—22no—
86Dic346—42no—
87Dic346C2×D412yes—
88D692—22no—
89D692—22no—
90D692—42no—
91D692—82no—
92D692C2×D412yes—
93C22⋊D173—22no—
94C22⋊D173—42no—
95C22⋊D173—42no—
96C2×Dic173—22no—
97C2×Dic173—42no—
98C4×D173—22no—
99C4×D173—42no—
100C4×D173—42no—
101#13063436—22no—
102Q8×C173C2×D412yes—
103D4×C173—22no—
104D4×C173C2×D412yes—
105C2×C4×C173—21no—
106C2×C4×C173—22no—
107C2×C4×C173C2411yescharacteristic
108C8×C173—22no—
109C8×C173C2×D411yescharacteristic
110C8×C173C2×D411yescharacteristic
111C8×C173C2×D411yescharacteristic
112C8×C173C2×D411yescharacteristic
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×C173C22×D411yescharacteristic
122C4×C173C22×D411yescharacteristic
123Dic173—22no—
124Dic173—42no—
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—1732no—
138#453—1732no—
139#443—1731no—
140#443D17311yescharacteristic, complemented
1412−1+4—1732no—
142H(5,𝔽2)—1732no—
143#137—3462no—
144#137—3462no—
145#137—1732no—
146#137—1732no—
147AGL(1,ℤ/8)—3462no—
148AGL(1,ℤ/8)—3462no—
149AGL(1,ℤ/8)—1732no—
150AGL(1,ℤ/8)—1732no—
151#135—3462no—
152#135—3462no—
153#135—1731no—
154#135C2×D17311yescharacteristic
155C2×SD16—3462no—
156C2×SD16—3462no—
157C2×SD16—1732no—
158C2×SD16—1732no—
159#131—1732no—
160#131C2×D17312yescomplemented
161C4⋊Q8—1731no—
162#127—1731no—
163#108—1731no—
164#108C2×D17311yescharacteristic, complemented
165#104—1732no—
166#104C2×D17312yescomplemented
167#101—1732no—
168#100—1732no—
169C4×C8C2×D17311yescharacteristic, complemented
170#41—3462no—
171#41—3462no—
172#41—3462no—
173#41—3462no—
174#41—1732no—
175#41—6922no—
176#41C22×D17312yes—
177C2×Q8—3462no—
178C2×Q8—3462no—
179C2×Q8—1732no—
180C2×D4—3462no—
181C2×D4—3462no—
182C2×D4—1732no—
183C2×D4—6921no—
184C2×D4—6922no—
185Q16—3462no—
186Q16—3462no—
187Q16—1731no—
188Q16D69211yescharacteristic, complemented
189SD16—3462no—
190SD16—3462no—
191SD16—3462no—
192SD16—3462no—
193SD16—3462no—
194SD16—3462no—
195SD16—3462no—
196SD16—3462no—
197SD16—1732no—
198SD16D69212yescomplemented
199D8—3462no—
200D8—3462no—
201D8—1731no—
202D8D69211yescharacteristic, complemented
203M16—22no—
204M16—3462no—
205M16—1732no—
206M16C22×D17312yes—
207C2×C8—22no—
208C2×C8—3462no—
209C2×C8C22×D17311yescharacteristic
210C2×C8C22×D17311yescharacteristic
211C4⋊C4—6921no—
212C42C22×D17311yescharacteristic
213C23—3462no—
214C23—6922no—
215Q8—3462no—
216Q8—3462no—
217Q8—1732no—
218Q8—6922no—
219Q8C2×D69212yes—
220D4—22no—
221D4—3462no—
222D4—3462no—
223D4—3462no—
224D4—13842no—
225D4—1732no—
226D4—6922no—
227D4—6922no—
228D4—6922no—
229D4C2×D69212yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—3462no—
233C2×C4—3462no—
234C2×C4—6922no—
235C2×C4—6922no—
236C2×C4—6922no—
237C2×C4C23×D17311yescharacteristic
238C8—22no—
239C8—3462no—
240C8D4×D17311yescharacteristic
241C8D4×D17311yescharacteristic
242C8C2×D69211yescharacteristic
243C8C2×D69211yescharacteristic
244C22—22no—
245C22—3462no—
246C22—13842no—
247C22—6922no—
248C22—27682no—
249C22#1914972511yescharacteristic
250C4—22no—
251C4—22no—
252C4—3462no—
253C4—6922no—
254C4C2×D4×D17311yescharacteristic
255C4C22×D69211yescharacteristic
256C2—21no—
257C2—42no—
258C2—13842no—
259C2—6922no—
260C2D4×D69211yescharacteristic
subgroupG/H
centerC2D4×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#443×C173C2
socleC2×C173D42
hypercenter#443D173
hypocenterC173#3291
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #26012303C2semidirect
2. #26012298C2semidirect
3. #26011864C2semidirect
4. #26011755C2semidirect
5. #26011754C2semidirect
6. #26011675C2semidirect
7. #26011660C2semidirect
8. #26011609C2semidirect
9. #443×C173C2semidirect
11. #19149725C22semidirect
13. #19149712C22semidirect
15. #19149709C22semidirect
24. C2×#14583815C22semidirect
25. C2×#14583815C22semidirect
27. #19149685C22semidirect
28. #19149679C22semidirect
30. #19149615C22semidirect
31. #19149607C22semidirect
32. #19149597C22semidirect
33. #19149595C22semidirect
35. #131×C173C22semidirect
36. #108×C173C22semidirect
37. #104×C173C22semidirect
38. C4×C8×C173C22semidirect
54. Dic692D4semidirect
62. #14583815D4semidirect
66. D1384D4semidirect
73. Q16×C173D4semidirect
74. SD16×C173D4semidirect
75. D8×C173D4semidirect
130. C173#3291semidirect
140. #443D173semidirect
160. #131C2×D173semidirect
164. #108C2×D173semidirect
166. #104C2×D173semidirect
169. C4×C8C2×D173semidirect
188. Q16D692semidirect
198. SD16D692semidirect
202. D8D692semidirect

Maximal subgroups

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

subgroupindex
#260123032
#260122982
#260118642
#260117552
#260117542
#260116752
#260116602
#260116092
#443×C1732
#3291173