Finite Group Databasebeta

← C2×#26011718 #36363121

Subgroups — C2×#26011718 #36363121

Order 22144.

564 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14583814C211yesmaximal, characteristic, complemented
2C23×#13063436C211yesmaximal, characteristic, complemented
3C2×#19149658C211yesmaximal, characteristic, complemented
4#26011718C218yesmaximal, direct product factor
5C2×#19149610C211yesmaximal, characteristic, complemented
6C2×#19149602C211yesmaximal, characteristic, complemented
7C2×#19149600C211yesmaximal, characteristic, complemented
8C2×#121×C173C211yesmaximal, characteristic
9C22×Dic346C2211yescharacteristic
10C2×#14583814—24no—
11C2×#14583814C2214yescomplemented
12C22×#13063436—24no—
13C22×#13063436C2211yescharacteristic
14C22×#13063436C2211yescharacteristic
15C22×#13063436C2214yescomplemented
16#19149658—22no—
17#19149658C2214yescomplemented
18C2×C4⋊Dic173C2211yescharacteristic
19C2×#14583805—21no—
20C2×C22⋊Dic173—21no—
21#19149610C2214yescomplemented
22#19149602C2214yescomplemented
23#19149600C2214yescomplemented
24C22×D4×C173—21no—
25C22×D4×C173C2211yescharacteristic
26C23×C4×C173C2211yescharacteristic
27#121×C173—22no—
28#121×C173C2214yes—
29C2×C4⋊C4×C173C2211yescharacteristic
30C2×C22⋊C4×C173—21no—
31C2×Dic346—21no—
32C2×Dic346—22no—
33C2×Dic346—22no—
34C2×Dic346C2312yes—
35C22×Dic173—21no—
36C22×Dic173—21no—
37#14583814—28no—
38C2×#13063436—28no—
39C2×#13063436—22no—
40C2×#13063436D414yescomplemented
41C2×#13063436C2312yes—
42C2×#13063436C2312yes—
43C4⋊Dic173—21no—
44C4⋊Dic173C2312yes—
45#14583805—24no—
46C22⋊Dic173—24no—
47C24×C173—21no—
48C24×C173D411yescharacteristic
49C2×D4×C173—24no—
50C2×D4×C173—22no—
51C2×D4×C173—21no—
52C2×D4×C173—22no—
53C2×D4×C173—44no—
54C2×D4×C173C2312yes—
55C22×C4×C173—22no—
56C22×C4×C173—22no—
57C22×C4×C173—21no—
58C22×C4×C173D411yescharacteristic
59C22×C4×C173D412yes—
60C22×C4×C173C2312yes—
61C22×C4×C173C2311yescharacteristic
62C4⋊C4×C173—21no—
63C4⋊C4×C173C2312yes—
64C22⋊C4×C173—24no—
65Dic346—22no—
66Dic346—22no—
67Dic346—42no—
68C2×Dic173—22no—
69C2×Dic173—22no—
70C2×Dic173—42no—
71C2×Dic173—42no—
72#13063436—22no—
73#13063436C2×D414yescomplemented
74C23×C173—22no—
75C23×C173—22no—
76C23×C173—22no—
77C23×C173—42no—
78C23×C173—42no—
79C23×C173—42no—
80C23×C173SD1614yescomplemented
81C23×C173C2×D412yes—
82C23×C173C2×D411yescharacteristic
83D4×C173—22no—
84D4×C173—22no—
85D4×C173—48no—
86D4×C173—42no—
87C2×C4×C173—22no—
88C2×C4×C173—24no—
89C2×C4×C173—22no—
90C2×C4×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173—42no—
93C2×C4×C173C2×D412yes—
94C2×C4×C173C2×D412yes—
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173#4112yes—
97C2×C4×C173C2411yescharacteristic
98C22×C173—22no—
99C22×C173—22no—
100C22×C173—28no—
101C22×C173—42no—
102C22×C173—42no—
103C22×C173—42no—
104C22×C173—84no—
105C22×C173#12112yes—
106C22×C173C2×SD1612yes—
107C22×C173C2×SD1614yescomplemented
108C22×C173#13512yes—
109C22×C173C22×D411yescharacteristic
110C4×C173—22no—
111C4×C173—42no—
112C4×C173#12112yes—
113C4×C173C22×D411yescharacteristic
114C4×C173C2×#4111yescharacteristic
115Dic173—42no—
116Dic173—42no—
117C2×C173—24no—
118C2×C173—82no—
119C2×C173#46514yescomplemented
120C2×C173C2×#12111yescharacteristic
121C2×C173C22×SD1611yescharacteristic
122C2×C173C2×#13511yescharacteristic
123C173C2×#46511yescharacteristic, complemented, Sylow
124C2×#465—1731nomaximal, Sylow
125C22×SD16—1731no—
126C23×C8—1731no—
127C2×C22⋊Q8—1731no—
128C2×#121D17311yescharacteristic
129#465—1738no—
130C2×#106—1731no—
131C2×#103—1731no—
132C2×#102—1731no—
133C22×Q8—1731no—
134C22×D4—21no—
135C22×D4C2×D17311yescharacteristic
136C23×C4C2×D17311yescharacteristic
137C2×SD16—3464no—
138C2×SD16—1734no—
139C22×C8—3464no—
140C22×C8—1731no—
141C22×C8—1734no—
142C22×C8—1731no—
143C22⋊Q8—3462no—
144C22⋊Q8—1734no—
145#121—22no—
146#121C2×D17314yes—
147C2×C4⋊C4—3461no—
148C2×C4⋊C4—1731no—
149C2×C4⋊C4C2×D17311yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—3461no—
152#106—1734no—
153#103—1734no—
154#102—1734no—
155C24—21no—
156C24C22⋊D17311yescharacteristic
157C2×Q8—3461no—
158C2×Q8—3462no—
159C2×Q8—3462no—
160C2×Q8—1732no—
161C2×D4—24no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—44no—
166C2×D4C22×D17312yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—3461no—
171C22×C4—3461no—
172C22×C4C22⋊D17312yes—
173C22×C4C22⋊D17311yescharacteristic
174C22×C4C22×D17312yes—
175C22×C4C22×D17311yescharacteristic
176SD16—3468no—
177C2×C8—3462no—
178C2×C8—3468no—
179C2×C8—1732no—
180C2×C8—1734no—
181C2×C8—1732no—
182C4⋊C4—21no—
183C4⋊C4—3464no—
184C4⋊C4—3461no—
185C4⋊C4—1732no—
186C4⋊C4C22×D17312yes—
187C22⋊C4—24no—
188C22⋊C4—3464no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23#1458381414yescomplemented
196C23C2×C22⋊D17312yes—
197C23C2×C22⋊D17311yescharacteristic
198Q8—3462no—
199Q8—3462no—
200Q8—6922no—
201D4—22no—
202D4—22no—
203D4—48no—
204D4—42no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—3462no—
212C2×C4—3462no—
213C2×C4—6922no—
214C2×C4—6922no—
215C2×C4D4×D17312yes—
216C2×C4C2×C22⋊D17312yes—
217C2×C4C2×C22⋊D17311yescharacteristic
218C2×C4#1458382512yes—
219C2×C4C23×D17311yescharacteristic
220C8—3462no—
221C8—1734no—
222C22—28no—
223C22—22no—
224C22—22no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—84no—
229C22#1914965112yes—
230C22C2×#1458381412yes—
231C22C2×#1458381414yescomplemented
232C22#1914969912yes—
233C22C22×C22⋊D17311yescharacteristic
234C4—22no—
235C4—42no—
236C4—6922no—
237C4—6922no—
238C4#1914965112yes—
239C4C2×D4×D17311yescharacteristic
240C4C2×#1458382511yescharacteristic
241C2—24no—
242C2—82no—
243C2#2601171814yesdirect product factor
244C2C2×#1914965111yescharacteristic
245C2C22×#1458381411yescharacteristic
246C2C2×#1914969911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radicalC2×#121×C173C2
socleC23×C173C2×D4
hypercenterC2×#121D173
hypocenterC173C2×#465
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14583814C2semidirect
2. C23×#13063436C2semidirect
3. C2×#19149658C2semidirect
4. #26011718C2direct (N × H)
5. C2×#19149610C2semidirect
6. C2×#19149602C2semidirect
7. C2×#19149600C2semidirect
11. C2×#14583814C22semidirect
15. C22×#13063436C22semidirect
17. #19149658C22semidirect
21. #19149610C22semidirect
22. #19149602C22semidirect
23. #19149600C22semidirect
40. C2×#13063436D4semidirect
73. #13063436C2×D4semidirect
80. C23×C173SD16semidirect
107. C22×C173C2×SD16semidirect
119. C2×C173#465semidirect
123. C173C2×#465semidirect
195. C23#14583814semidirect
231. C22C2×#14583814semidirect
243. C2#26011718direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838142
C23×#130634362
C2×#191496582
#260117182
C2×#191496102
C2×#191496022
C2×#191496002
C2×#121×C1732
C2×#465173