Finite Group Databasebeta

← #36364029

Subgroups — #36364029

Order 22144.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#19149712C212yesmaximal, complemented
2C2×#19149707C212yesmaximal, complemented
3#26012041C211yesmaximal, characteristic, complemented
4C2×#19149667C211yesmaximal, characteristic, complemented
5#26011885C212yesmaximal, complemented
6#26011880C212yesmaximal, complemented
7#26011875C214yesmaximal, complemented
8C4×M16×C173C211yesmaximal, characteristic, complemented
9C2×#14583826C2212yes—
10C22×Dic346C2211yescharacteristic
11C22×D692C2211yescharacteristic
12#19149712—28no—
13#19149707—28no—
14C2×Dic692C2214yescomplemented
15C2×#14583815C2214yes—
16C2×#14583815C2214yescomplemented
17C2×D1384C2214yes—
18#19149679C2211yescharacteristic, complemented
19#19149676C2211yescharacteristic, complemented
20#19149667—22no—
21#19149667C2212yes—
22#19149667C2212yes—
23#19149667C2212yescomplemented
24#19149653—22no—
25C4×D692—22no—
26C2×#14583803—22no—
27C2×M16×C173C2212yes—
28C2×C42×C173C2211yescharacteristic
29#97×C173C2212yescomplemented
30C4×C8×C173C2212yescomplemented
31C23×D173—21no—
32#14583826—24no—
33#14583826—24no—
34C2×Dic346—21no—
35C2×Dic346—24no—
36C2×Dic346C2312yes—
37C2×Dic346C2312yes—
38C2×D692—24no—
39C2×D692—22no—
40C2×D692—21no—
41C2×D692C2312yes—
42C2×D692C2312yes—
43C2×C22⋊D173—22no—
44C22×Dic173—21no—
45C2×C4×D173—22no—
46Dic692—28no—
47#14583815—28no—
48#14583815—28no—
49D1384—28no—
50C4⋊Dic173—22no—
51#14583803—24no—
52#14583803—24no—
53#14583803—24no—
54C22×C4×C173D412yes—
55C22×C4×C173C2311yescharacteristic
56M16×C173D418yescomplemented
57C2×C8×C173C2314yes—
58C42×C173D412yes—
59C42×C173C2311yescharacteristic
60C42×C173C2311yescharacteristic
61C22×D173—22no—
62C22×D173—22no—
63C22×D173—44no—
64C22×D173—42no—
65Dic346—22no—
66Dic346—24no—
67Dic346—22no—
68Dic346—42no—
69D692—22no—
70D692—24no—
71D692—22no—
72D692—44no—
73D692—42no—
74C22⋊D173—44no—
75C2×Dic173—22no—
76C2×Dic173—22no—
77C2×Dic173—42no—
78C4×D173—44no—
79C23×C173C2×D411yescharacteristic
80C2×C4×C173—42no—
81C2×C4×C173C2×D412yes—
82C2×C4×C173C2×D412yes—
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D414yes—
85C2×C4×C173C2411yescharacteristic
86C8×C173C2×D418yes—
87C22×C173—22no—
88C22×C173#12712yes—
89C22×C173C22×D411yescharacteristic
90C2×D173—42no—
91C2×D173—42no—
92C2×D173—44no—
93C2×D173—44no—
94C2×D173—84no—
95C4×C173—22no—
96C4×C173—22no—
97C4×C173#12712yes—
98C4×C173C22×D412yes—
99Dic173—42no—
100Dic173—42no—
101C2×C173—22no—
102C2×C173C2×#12711yescharacteristic
103C2×C173#57512yes—
104D173—82no—
105D173—82no—
106C173#315311yescharacteristic, complemented, Sylow
107#3153—1731nomaximal, Sylow
108C2×#137—1732no—
109C2×AGL(1,ℤ/8)—1732no—
110#532—1731no—
111C2×#124—1731no—
112#497—1732no—
113#496—1732no—
114#495—1734no—
115C4×M16D17311yescharacteristic, complemented
116C2×#41—1732no—
117C22×Q8—1731no—
118C22×D4—1731no—
119#137—3468no—
120AGL(1,ℤ/8)—3468no—
121C2×Q16—1734no—
122C2×SD16—1734no—
123C2×SD16—1734no—
124C2×D8—1734no—
125C2×M16C2×D17312yes—
126C4⋊Q8—1731no—
127#127—1731no—
128#124—3462no—
129#124—1732no—
130#124—1732no—
131#124—1732no—
132#121—3462no—
133C4×D4—3462no—
134C2×C22⋊C4—3462no—
135C2×C42C2×D17311yescharacteristic
136#97C2×D17312yescomplemented
137C4×C8C2×D17312yescomplemented
138C24—3461no—
139#41—3464no—
140#41—3464no—
141C2×Q8—3461no—
142C2×Q8—3464no—
143C2×Q8—1732no—
144C2×Q8—1732no—
145C2×D4—3462no—
146C2×D4—3464no—
147C2×D4—3462no—
148C2×D4—3461no—
149C2×D4—1732no—
150C2×D4—1732no—
151C22×C4—3461no—
152C22×C4—3462no—
153C22×C4D69212yes—
154C22×C4C22×D17311yescharacteristic
155Q16—3468no—
156SD16—3468no—
157SD16—3468no—
158D8—3468no—
159M16D69218yescomplemented
160C2×C8C22×D17314yes—
161C4⋊C4—3462no—
162C22⋊C4—3464no—
163C22⋊C4—3464no—
164C22⋊C4—3464no—
165C42D69212yes—
166C42C22×D17311yescharacteristic
167C42C22×D17311yescharacteristic
168C23—3462no—
169C23—3462no—
170C23—6924no—
171C23—6922no—
172C23C2×D69211yescharacteristic
173Q8—3462no—
174Q8—3464no—
175Q8—3462no—
176Q8—6922no—
177D4—3462no—
178D4—3464no—
179D4—3462no—
180D4—6924no—
181D4—6924no—
182D4—6922no—
183C2×C4—42no—
184C2×C4—3462no—
185C2×C4—3462no—
186C2×C4—6922no—
187C2×C4—6924no—
188C2×C4C2×D69212yes—
189C2×C4C2×D69212yes—
190C2×C4C2×D69211yescharacteristic
191C2×C4C2×D69214yes—
192C2×C4C23×D17311yescharacteristic
193C8C2×D69218yes—
194C22—22no—
195C22—13844no—
196C22—6922no—
197C22—6922no—
198C22—6924no—
199C22—6924no—
200C22#1914967612yes—
201C22C22×D69211yescharacteristic
202C4—22no—
203C4—22no—
204C4—6922no—
205C4—6922no—
206C4#1914967612yes—
207C4C22×D69212yes—
208C2—22no—
209C2—13842no—
210C2—13842no—
211C2C2×#1914967611yescharacteristic
212C2#2601229112yes—
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×M16×C173C2
socleC22×C173C22×D4
hypercenterC4×M16D173
hypocenterC173#3153
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C2×#19149707C2semidirect
3. #26012041C2semidirect
4. C2×#19149667C2semidirect
5. #26011885C2semidirect
6. #26011880C2semidirect
7. #26011875C2semidirect
8. C4×M16×C173C2semidirect
14. C2×Dic692C22semidirect
16. C2×#14583815C22semidirect
18. #19149679C22semidirect
19. #19149676C22semidirect
23. #19149667C22semidirect
29. #97×C173C22semidirect
30. C4×C8×C173C22semidirect
56. M16×C173D4semidirect
106. C173#3153semidirect
115. C4×M16D173semidirect
136. #97C2×D173semidirect
137. C4×C8C2×D173semidirect
159. M16D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×#191497072
#260120412
C2×#191496672
#260118852
#260118802
#260118752
C4×M16×C1732
#3153173