Finite Group Databasebeta

← #36362927

Subgroups — #36362927

Order 22144.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#26012289C212yesmaximal, complemented
2#26012065C211yesmaximal, characteristic, complemented
3#26011674C214yesmaximal, complemented
4#26011664C214yesmaximal, complemented
5#26011508C212yesmaximal, complemented
6C2×#19149598C211yesmaximal, characteristic, complemented
7C2×2−1+4×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9#19149710—22no—
10#19149709C2214yescomplemented
11#19149705—22no—
12#19149704C2214yescomplemented
13#19149699—24no—
14C2×#14583810C2212yes—
15#19149677C2212yescomplemented
16#19149668C2212yescomplemented
17#19149662—22no—
18#19149655—22no—
19#19149638C2211yescharacteristic
20#19149606C2218yescomplemented
21#19149598C2214yescomplemented
222−1+4×C173—24no—
232−1+4×C173—22no—
242−1+4×C173C2214yes—
25C2×#41×C173—22no—
26C2×#41×C173—22no—
27C2×#41×C173C2212yes—
28C22×Q8×C173—22no—
29C22×Q8×C173C2211yescharacteristic
30#14583826—22no—
31#14583826—22no—
32C2×Dic346C2311yescharacteristic
33C2×D692C2311yescharacteristic
34C2×C22⋊D173—21no—
35C2×C4×D173—21no—
36#14583816—24no—
37#14583814—24no—
38#14583813—24no—
39#14583811—24no—
40#14583810—22no—
41#14583810C2314yes—
42C2×#13063436—22no—
43C4⋊Dic173—21no—
44#14583805—24no—
45C22⋊Dic173—21no—
46#14583803—24no—
47C4×Dic173C2312yes—
48#41×C173—28no—
49#41×C173—28no—
50#41×C173—28no—
51#41×C173—28no—
52#41×C173—22no—
53#41×C173—22no—
54#41×C173D414yes—
55#41×C173C2314yes—
56C2×Q8×C173—22no—
57C2×Q8×C173—28no—
58C2×Q8×C173—28no—
59C2×Q8×C173—22no—
60C2×Q8×C173—22no—
61C2×Q8×C173D412yes—
62C2×Q8×C173D412yes—
63C2×Q8×C173C2312yes—
64C2×D4×C173—22no—
65C2×D4×C173—22no—
66C2×D4×C173D412yes—
67C22×C4×C173—24no—
68C22×C4×C173—22no—
69C22×C4×C173D412yes—
70C22×C4×C173C2311yescharacteristic
71C22×D173—21no—
72Dic346—22no—
73D692—22no—
74C22⋊D173—41no—
75C22⋊D173—41no—
76C2×Dic173—21no—
77C2×Dic173—22no—
78C4×D173—42no—
79#13063436—24no—
80C23×C173—22no—
81C23×C173C2×D411yescharacteristic
82Q8×C173—28no—
83Q8×C173—22no—
84Q8×C173—24no—
85Q8×C173—24no—
86Q8×C173C2×D414yes—
87D4×C173—28no—
88D4×C173—24no—
89D4×C173—24no—
90D4×C173—22no—
91D4×C173C2×D414yes—
92C2×C4×C173—216no—
93C2×C4×C173—22no—
94C2×C4×C173—22no—
95C2×C4×C173—22no—
96C2×C4×C173—24no—
97C2×C4×C173—24no—
98C2×C4×C173C2×D414yes—
99C2×C4×C173C2×D411yescharacteristic
100C2×C4×C173C2×D412yes—
101C2×C4×C173C2×D412yes—
102C2×C4×C173C2411yescharacteristic
103C22×C173—24no—
104C22×C173—41no—
105C22×C173—44no—
106C22×C173#12011yescharacteristic
107C22×C173#12011yescharacteristic
108C22×C173C22×D411yescharacteristic
109C2×D173—42no—
110C2×D173—41no—
111C4×C173—24no—
112C4×C173—24no—
113C4×C173#12012yes—
114C4×C173C22×D412yes—
115Dic173—42no—
116Dic173—41no—
117C2×C173—21no—
118C2×C173—21no—
119C2×C173—21no—
120C2×C173—44no—
121C2×C173C2×#12011yescharacteristic
122D173—81no—
123C173#302011yescharacteristic, complemented, Sylow
124#3020—1731nomaximal, Sylow
125C2×2−1+4D17311yescharacteristic, complemented
126#575—1732no—
127#536—1731no—
128#456—1734no—
129#454—1734no—
130#421—1732no—
131C2×#101—1731no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D17314yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1731no—
138C2×#41C2×D17312yes—
139C22×Q8—22no—
140C22×Q8C2×D17311yescharacteristic
141#137—3462no—
142#137—1734no—
143AGL(1,ℤ/8)—3462no—
144AGL(1,ℤ/8)—1734no—
145#135—3464no—
146C2×M16—1732no—
147C4⋊Q8—1732no—
148#124—1732no—
149#123—3462no—
150C22⋊Q8—3462no—
151#117—1731no—
152#104—1738no—
153#101—1734no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—3462no—
161#41—3462no—
162#41C22⋊D17314yes—
163#41C22×D17314yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1731no—
170C2×Q8C22⋊D17312yes—
171C2×Q8C22⋊D17312yes—
172C2×Q8C22×D17312yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—3461no—
176C2×D4—1731no—
177C2×D4C22⋊D17312yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—3461no—
181C22×C4C22⋊D17312yes—
182C22×C4C22×D17311yescharacteristic
183Q16—3464no—
184SD16—3464no—
185SD16—3464no—
186D8—3464no—
187M16—3462no—
188M16—1734no—
189C2×C8—3462no—
190C4⋊C4—3461no—
191C4⋊C4—3464no—
192C22⋊C4—3461no—
193C22⋊C4—3464no—
194C42—1732no—
195C23—22no—
196C23—3461no—
197C23C2×C22⋊D17311yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—3462no—
203Q8C2×C22⋊D17314yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—3462no—
209D4—6921no—
210D4—6921no—
211D4C2×C22⋊D17314yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—3462no—
219C2×C4—3461no—
220C2×C4—6922no—
221C2×C4C2×C22⋊D17314yes—
222C2×C4C2×C22⋊D17311yescharacteristic
223C2×C4C2×C22⋊D17312yes—
224C2×C4C2×C22⋊D17312yes—
225C2×C4C23×D17311yescharacteristic
226C8—3464no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—6922no—
231C22—6921no—
232C22#1914964811yescharacteristic
233C22#1914964811yescharacteristic
234C22C22×C22⋊D17311yescharacteristic
235C4—24no—
236C4—24no—
237C4—6922no—
238C4—6921no—
239C4#1914964812yes—
240C4C22×C22⋊D17312yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—13841no—
246C2C2×#1914964811yescharacteristic
subgroupG/H
centerC2C2×#19149648
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×2−1+4×C173C2
socleC2×C173C2×#120
hypercenterC2×2−1+4D173
hypocenterC173#3020
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #26012289C2semidirect
2. #26012065C2semidirect
3. #26011674C2semidirect
4. #26011664C2semidirect
5. #26011508C2semidirect
6. C2×#19149598C2semidirect
7. C2×2−1+4×C173C2semidirect
10. #19149709C22semidirect
12. #19149704C22semidirect
15. #19149677C22semidirect
16. #19149668C22semidirect
20. #19149606C22semidirect
21. #19149598C22semidirect
123. C173#3020semidirect
125. C2×2−1+4D173semidirect

Maximal subgroups

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

subgroupindex
#260122892
#260120652
#260116742
#260116642
#260115082
C2×#191495982
C2×2−1+4×C1732
#3020173