Finite Group Databasebeta

← C2×#26011734 #36363148

Subgroups — C2×#26011734 #36363148

Order 22144.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14583809C211yesmaximal, characteristic, complemented
2C2×#19149655C211yesmaximal, characteristic, complemented
3C2×#19149650C211yesmaximal, characteristic
4#26011734C218yesmaximal, direct product factor
5C2×#19149613C211yesmaximal, characteristic, complemented
6C2×#19149603C211yesmaximal, characteristic, complemented
7C2×#19149600C211yesmaximal, characteristic, complemented
8C22×SD16×C173C211yesmaximal, characteristic, complemented
9C22×C22⋊D173—21no—
10C22×C4×D173C2211yescharacteristic
11C2×#14583809—24no—
12C2×#14583809C2214yescomplemented
13C22×#13063436C2211yescharacteristic
14#19149655—22no—
15#19149655C2214yescomplemented
16#19149650—22no—
17#19149650C2214yes—
18C2×C4⋊Dic173C2211yescharacteristic
19C2×C4⋊Dic173C2211yescharacteristic
20C2×#14583805—21no—
21C2×C22⋊Dic173—21no—
22C2×#14583803—21no—
23#19149613C2214yescomplemented
24#19149603C2214yescomplemented
25#19149600C2214yescomplemented
26C22×Q8×C173C2211yescharacteristic
27C22×D4×C173C2211yescharacteristic
28C2×SD16×C173—24no—
29C2×SD16×C173C2214yescomplemented
30C22×C8×C173C2211yescharacteristic
31C23×D173D411yescharacteristic
32C2×C22⋊D173—24no—
33C2×C22⋊D173—44no—
34C22×Dic173—21no—
35C22×Dic173—21no—
36C22×Dic173D411yescharacteristic
37C2×C4×D173—22no—
38C2×C4×D173—22no—
39C2×C4×D173D412yes—
40C2×C4×D173C2312yes—
41#14583809—28no—
42C2×#13063436—22no—
43C2×#13063436C2312yes—
44C4⋊Dic173—21no—
45C4⋊Dic173—21no—
46C4⋊Dic173C2312yes—
47C4⋊Dic173C2312yes—
48#14583805—24no—
49C22⋊Dic173—24no—
50#14583803—24no—
51C24×C173—21no—
52C2×Q8×C173—21no—
53C2×Q8×C173—22no—
54C2×Q8×C173—22no—
55C2×Q8×C173C2312yes—
56C2×D4×C173—22no—
57C2×D4×C173—21no—
58C2×D4×C173—22no—
59C2×D4×C173C2312yes—
60C22×C4×C173—21no—
61C22×C4×C173C2311yescharacteristic
62SD16×C173—28no—
63C2×C8×C173D414yescomplemented
64C2×C8×C173C2312yes—
65C22×D173—22no—
66C22×D173—22no—
67C22×D173—22no—
68C22×D173C2×D412yes—
69C22⋊D173—48no—
70C2×Dic173—22no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173—42no—
74C2×Dic173—42no—
75C2×Dic173C2×D412yes—
76C4×D173—24no—
77C4×D173—22no—
78C4×D173—22no—
79#13063436—22no—
80C23×C173—22no—
81C23×C173—42no—
82C23×C173—42no—
83C23×C173—42no—
84C23×C173C2×D411yescharacteristic
85Q8×C173—22no—
86Q8×C173—22no—
87Q8×C173—42no—
88D4×C173—22no—
89D4×C173—22no—
90D4×C173—42no—
91C2×C4×C173—22no—
92C2×C4×C173—42no—
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173C2×D412yes—
95C2×C4×C173#4112yes—
96C2×C4×C173C2411yescharacteristic
97C8×C173C2×D414yescomplemented
98C22×C173—42no—
99C22×C173—42no—
100C22×C173—42no—
101C22×C173—84no—
102C22×C173#12112yes—
103C22×C173AGL(1,ℤ/8)12yes—
104C22×C173#13712yes—
105C22×C173C22×D411yescharacteristic
106C2×D173—22no—
107C2×D173—22no—
108C2×D173—22no—
109C2×D173—44no—
110C4×C173—42no—
111C4×C173#12112yes—
112C4×C173C22×D411yescharacteristic
113C4×C173C2×#4111yescharacteristic
114Dic173—22no—
115Dic173—42no—
116Dic173—42no—
117C2×C173—82no—
118C2×C173#46814yescomplemented
119C2×C173C2×#12111yescharacteristic
120C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C173C2×#13711yescharacteristic
122D173—42no—
123C173C2×#46811yescharacteristic, complemented, Sylow
124C2×#468—1731nomaximal, Sylow
125C22×SD16D17311yescharacteristic, complemented
126C22×M16—1731no—
127C2×C22⋊Q8—1731no—
128C2×#121—1731no—
129#468—1738no—
130C2×#107—1731no—
131C2×#103—1731no—
132C2×#102—1731no—
133C22×Q8C2×D17311yescharacteristic
134C22×D4—3461no—
135C22×D4C2×D17311yescharacteristic
136C23×C4—1731no—
137C2×SD16—24no—
138C2×SD16C2×D17314yescomplemented
139C2×M16—3464no—
140C2×M16—1734no—
141C22×C8—1731no—
142C22×C8C2×D17311yescharacteristic
143C22⋊Q8—3462no—
144C22⋊Q8—1734no—
145#121—3462no—
146#121—1734no—
147C2×C4⋊C4—3461no—
148C2×C4⋊C4—1731no—
149C2×C4⋊C4—1731no—
150C2×C22⋊C4—3461no—
151C2×C22⋊C4—3461no—
152#107—1734no—
153#103—1734no—
154#102—1734no—
155C24—21no—
156C24—1731no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D17312yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—3464no—
165C2×D4—6924no—
166C2×D4C22×D17312yes—
167C22×C4—21no—
168C22×C4—3462no—
169C22×C4—3462no—
170C22×C4—3461no—
171C22×C4—3461no—
172C22×C4—1731no—
173C22×C4—1732no—
174C22×C4—1732no—
175C22×C4C22×D17311yescharacteristic
176SD16—28no—
177M16—3468no—
178C2×C8—3462no—
179C2×C8—1732no—
180C2×C8C22⋊D17314yescomplemented
181C2×C8C22×D17312yes—
182C4⋊C4—3464no—
183C4⋊C4—3461no—
184C4⋊C4—3461no—
185C4⋊C4—1732no—
186C4⋊C4—1732no—
187C22⋊C4—3464no—
188C22⋊C4—3464no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—3462no—
194C23—3462no—
195C23—3462no—
196C23—1732no—
197C23D4×D17311yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—6928no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—3462no—
208C2×C4—3464no—
209C2×C4—3462no—
210C2×C4—3462no—
211C2×C4—3462no—
212C2×C4—3462no—
213C2×C4—1732no—
214C2×C4—6922no—
215C2×C4—6922no—
216C2×C4D4×D17311yescharacteristic
217C2×C4C2×C22⋊D17312yes—
218C2×C4#1458382512yes—
219C2×C4C23×D17311yescharacteristic
220C8—3462no—
221C8C2×C22⋊D17314yescomplemented
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—3462no—
227C22—3462no—
228C22—3462no—
229C22—6924no—
230C22#1914965012yes—
231C22#1914970612yes—
232C22#1914970812yes—
233C22C2×D4×D17311yescharacteristic
234C4—42no—
235C4—3462no—
236C4—6922no—
237C4—6922no—
238C4#1914965012yes—
239C4C22×C22⋊D17311yescharacteristic
240C4C2×#1458382511yescharacteristic
241C2—82no—
242C2—6922no—
243C2#2601173414yesdirect product factor
244C2C2×#1914965011yescharacteristic
245C2C2×#1914970611yescharacteristic
246C2C2×#1914970811yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC22×SD16×C173C2
socleC23×C173C2×D4
prime radicalC2×#19149650C2
hypercenterC22×SD16D173
hypocenterC173C2×#468
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14583809C2semidirect
2. C2×#19149655C2semidirect
4. #26011734C2direct (N × H)
5. C2×#19149613C2semidirect
6. C2×#19149603C2semidirect
7. C2×#19149600C2semidirect
8. C22×SD16×C173C2semidirect
12. C2×#14583809C22semidirect
15. #19149655C22semidirect
23. #19149613C22semidirect
24. #19149603C22semidirect
25. #19149600C22semidirect
29. C2×SD16×C173C22semidirect
63. C2×C8×C173D4semidirect
97. C8×C173C2×D4semidirect
118. C2×C173#468semidirect
123. C173C2×#468semidirect
125. C22×SD16D173semidirect
138. C2×SD16C2×D173semidirect
180. C2×C8C22⋊D173semidirect
221. C8C2×C22⋊D173semidirect
243. C2#26011734direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838092
C2×#191496552
C2×#191496502
#260117342
C2×#191496132
C2×#191496032
C2×#191496002
C22×SD16×C1732
C2×#468173