Finite Group Databasebeta

← C2×#26011721 #36363124

Subgroups — C2×#26011721 #36363124

Order 22144.

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

normal subgroups only

This group has 564 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
1C22×#14583815C211yesmaximal, characteristic, complemented
2C22×C8×D173C211yesmaximal, characteristic, complemented
3C2×#19149659C211yesmaximal, characteristic, complemented
4C2×#19149654C211yesmaximal, characteristic
5#26011721C218yesmaximal, direct product factor
6C2×#19149602C211yesmaximal, characteristic, complemented
7C2×#19149599C211yesmaximal, characteristic, complemented
8C2×#106×C173C211yesmaximal, characteristic, complemented
9C22×Dic346C2211yescharacteristic
10C22×D692—21no—
11C22×D692C2211yescharacteristic
12C22×C4×D173C2211yescharacteristic
13C2×#14583815—24no—
14C2×#14583815C2214yescomplemented
15C22×#13063436C2211yescharacteristic
16C2×C8×D173—24no—
17C2×C8×D173C2214yescomplemented
18#19149659—22no—
19#19149659C2214yescomplemented
20#19149654—22no—
21#19149654C2214yes—
22C2×C4⋊Dic173—21no—
23C2×#14583803—21no—
24C2×#14583803—21no—
25#19149602C2214yescomplemented
26#19149599C2214yescomplemented
27C22×C8×C173C2211yescharacteristic
28C2×C4⋊C4×C173C2211yescharacteristic
29C2×C4⋊C4×C173C2211yescharacteristic
30#106×C173C2214yescomplemented
31C23×D173—21no—
32C23×D173D411yescharacteristic
33C2×Dic346—21no—
34C2×Dic346—22no—
35C2×Dic346—22no—
36C2×Dic346C2312yes—
37C2×D692—24no—
38C2×D692—22no—
39C2×D692—21no—
40C2×D692—22no—
41C2×D692—44no—
42C2×D692C2312yes—
43C22×Dic173—21no—
44C22×Dic173D411yescharacteristic
45C2×C4×D173—22no—
46C2×C4×D173—22no—
47C2×C4×D173D412yes—
48C2×C4×D173C2312yes—
49#14583815—28no—
50C2×#13063436—22no—
51C2×#13063436C2312yes—
52C8×D173—28no—
53C4⋊Dic173—24no—
54#14583803—24no—
55#14583803—24no—
56C22×C4×C173—21no—
57C22×C4×C173—21no—
58C22×C4×C173C2311yescharacteristic
59C2×C8×C173D414yescomplemented
60C2×C8×C173C2312yes—
61C4⋊C4×C173—21no—
62C4⋊C4×C173—21no—
63C4⋊C4×C173C2312yes—
64C4⋊C4×C173C2312yes—
65C22×D173—22no—
66C22×D173—22no—
67C22×D173—22no—
68C22×D173—42no—
69C22×D173—42no—
70C22×D173—42no—
71C22×D173SD1614yescomplemented
72C22×D173C2×D412yes—
73Dic346—22no—
74Dic346—22no—
75Dic346—42no—
76D692—22no—
77D692—22no—
78D692—48no—
79D692—42no—
80C2×Dic173—22no—
81C2×Dic173—22no—
82C2×Dic173—42no—
83C2×Dic173C2×D412yes—
84C4×D173—24no—
85C4×D173—22no—
86C4×D173—22no—
87#13063436—22no—
88C23×C173C2×D411yescharacteristic
89C2×C4×C173—22no—
90C2×C4×C173—22no—
91C2×C4×C173—42no—
92C2×C4×C173—42no—
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173C2×D412yes—
95C2×C4×C173#4112yes—
96C2×C4×C173C2411yescharacteristic
97C8×C173C2×D414yescomplemented
98C22×C173#12112yes—
99C22×C173C2×SD1612yes—
100C22×C173#13512yes—
101C22×C173C22×D411yescharacteristic
102C2×D173—22no—
103C2×D173—22no—
104C2×D173—28no—
105C2×D173—42no—
106C2×D173—42no—
107C2×D173—42no—
108C2×D173—84no—
109C2×D173C2×SD1614yescomplemented
110C4×C173—42no—
111C4×C173—42no—
112C4×C173#12112yes—
113C4×C173C22×D411yescharacteristic
114C4×C173C2×#4111yescharacteristic
115Dic173—22no—
116Dic173—42no—
117C2×C173#46514yescomplemented
118C2×C173C2×#12111yescharacteristic
119C2×C173C22×SD1611yescharacteristic
120C2×C173C2×#13511yescharacteristic
121D173—24no—
122D173—82no—
123C173C2×#46511yescharacteristic, complemented, Sylow
124C2×#465—1731nomaximal, Sylow
125C22×SD16—1731no—
126C23×C8—1731no—
127C2×C22⋊Q8—1731no—
128C2×#121—1731no—
129#465—1738no—
130C2×#106D17311yescharacteristic, complemented
131C2×#103—1731no—
132C2×#102—1731no—
133C22×Q8—1731no—
134C22×D4—3461no—
135C22×D4—1731no—
136C23×C4—1731no—
137C2×SD16—3464no—
138C2×SD16—1734no—
139C22×C8—3464no—
140C22×C8—1731no—
141C22×C8—1734no—
142C22×C8C2×D17311yescharacteristic
143C22⋊Q8—3462no—
144C22⋊Q8—1734no—
145#121—3462no—
146#121—1734no—
147C2×C4⋊C4—3461no—
148C2×C4⋊C4C2×D17311yescharacteristic
149C2×C4⋊C4C2×D17311yescharacteristic
150C2×C22⋊C4—3461no—
151C2×C22⋊C4—3461no—
152#106C2×D17314yescomplemented
153#103—1734no—
154#102—1734no—
155C24—3461no—
156C24—1731no—
157C2×Q8—3461no—
158C2×Q8—3462no—
159C2×Q8—3462no—
160C2×Q8—1732no—
161C2×D4—3464no—
162C2×D4—3461no—
163C2×D4—3462no—
164C2×D4—3462no—
165C2×D4—1732no—
166C2×D4—6924no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—3462no—
170C22×C4—3462no—
171C22×C4—3461no—
172C22×C4—1731no—
173C22×C4—1732no—
174C22×C4—1732no—
175C22×C4C22×D17311yescharacteristic
176SD16—3468no—
177C2×C8—3462no—
178C2×C8—3468no—
179C2×C8—1732no—
180C2×C8D69214yescomplemented
181C2×C8C22×D17312yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—3464no—
185C4⋊C4C22×D17312yes—
186C4⋊C4C22×D17312yes—
187C22⋊C4—3464no—
188C22⋊C4—3464no—
189C23—3462no—
190C23—3462no—
191C23—3462no—
192C23—1734no—
193C23—1732no—
194C23—6922no—
195C23—6922no—
196C23—6922no—
197C23D4×D17311yescharacteristic
198Q8—3462no—
199Q8—3462no—
200Q8—6922no—
201D4—3462no—
202D4—3462no—
203D4—6928no—
204D4—6922no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—3462no—
210C2×C4—3464no—
211C2×C4—3462no—
212C2×C4—3462no—
213C2×C4—3462no—
214C2×C4—1732no—
215C2×C4—6922no—
216C2×C4D4×D17311yescharacteristic
217C2×C4C2×D69212yes—
218C2×C4#1458382712yes—
219C2×C4C23×D17311yescharacteristic
220C8—3462no—
221C8C2×D69214yescomplemented
222C22—3468no—
223C22—3462no—
224C22—3462no—
225C22—13844no—
226C22—1734no—
227C22—6922no—
228C22—6922no—
229C22—6922no—
230C22#1914965412yes—
231C22SD16×D17312yes—
232C22#1914970012yes—
233C22C2×D4×D17311yescharacteristic
234C4—42no—
235C4—42no—
236C4—3462no—
237C4—6922no—
238C4#1914965412yes—
239C4C22×D69211yescharacteristic
240C4C2×#1458382711yescharacteristic
241C2—3464no—
242C2—13842no—
243C2#2601172114yesdirect product factor
244C2C2×#1914965411yescharacteristic
245C2C2×SD16×D17311yescharacteristic
246C2C2×#1914970011yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#106×C173C2
socleC23×C173C2×D4
prime radicalC2×#19149654C2
hypercenterC2×#106D173
hypocenterC173C2×#465
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. C22×#14583815C2semidirect
2. C22×C8×D173C2semidirect
3. C2×#19149659C2semidirect
5. #26011721C2direct (N × H)
6. C2×#19149602C2semidirect
7. C2×#19149599C2semidirect
8. C2×#106×C173C2semidirect
14. C2×#14583815C22semidirect
17. C2×C8×D173C22semidirect
19. #19149659C22semidirect
25. #19149602C22semidirect
26. #19149599C22semidirect
30. #106×C173C22semidirect
59. C2×C8×C173D4semidirect
71. C22×D173SD16semidirect
97. C8×C173C2×D4semidirect
109. C2×D173C2×SD16semidirect
117. C2×C173#465semidirect
123. C173C2×#465semidirect
130. C2×#106D173semidirect
152. #106C2×D173semidirect
180. C2×C8D692semidirect
221. C8C2×D692semidirect
243. C2#26011721direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838152
C22×C8×D1732
C2×#191496592
C2×#191496542
#260117212
C2×#191496022
C2×#191495992
C2×#106×C1732
C2×#465173