Finite Group Databasebeta

← #36353526

Subgroups — #36353526

Order 22144.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#26012327C211yesmaximal, characteristic
2#26011507C212yesmaximal, complemented
3#26011488C212yesmaximal, complemented
4#26011465C211yesmaximal, characteristic, complemented
5#408×C173C211yesmaximal, characteristic, complemented
6#19149727C412yescomplemented
7#19149726—22no—
8#19149725—22no—
9#19149724—22no—
10#19149724C412yescomplemented
11C2×#14583826—22no—
12C2×#14583826C2212yes—
13#41×D173—24no—
14#19149638C2212yescomplemented
15#19149605—24no—
16#19149604—22no—
17#19149601—22no—
18#19149594—22no—
19C2×#41×C173C2211yescharacteristic
20C2×M16×C173C2211yescharacteristic, complemented
21C22×C8×C173C2211yescharacteristic, complemented
22C22⋊C8×C173—22no—
23#14583827—24no—
24#14583826—24no—
25#14583826—24no—
26#14583826—24no—
27#14583826—22no—
28#14583826—22no—
29#14583826C2×C414yes—
30#14583826D414yes—
31#14583825—28no—
32#14583825—24no—
33C2×Dic346—22no—
34C2×Dic346D412yes—
35Q8×D173—24no—
36C2×D692—22no—
37C2×D692D412yes—
38C2×C22⋊D173—24no—
39C2×C22⋊D173—22no—
40D4×D173—28no—
41D4×D173—24no—
42C2×C4×D173—24no—
43C2×C4×D173—22no—
44C4⋊Dic173—22no—
45C22⋊Dic173—22no—
46C4×Dic173—22no—
47#41×C173—22no—
48#41×C173—22no—
49C2×Q8×C173C2×C411yescharacteristic
50C2×D4×C173C2×C411yescharacteristic
51C2×D4×C173D412yes—
52C22×C4×C173D412yes—
53C22×C4×C173C2311yescharacteristic
54M16×C173—22no—
55C2×C8×C173—22no—
56C2×C8×C173—21no—
57C2×C8×C173—21no—
58C22×D173—24no—
59C22×D173—22no—
60Dic346—24no—
61Dic346—22no—
62Dic346C22⋊C414yes—
63D692—24no—
64D692—22no—
65D692C22⋊C414yes—
66C22⋊D173—28no—
67C22⋊D173—24no—
68C22⋊D173—24no—
69C2×Dic173—24no—
70C2×Dic173—22no—
71C2×Dic173—42no—
72C4×D173—28no—
73C4×D173—24no—
74C4×D173—24no—
75C23×C173C22⋊C412yes—
76C23×C173C2×D411yescharacteristic
77Q8×C173—22no—
78D4×C173—24no—
79D4×C173—22no—
80C2×C4×C173—22no—
81C2×C4×C173—22no—
82C2×C4×C173C22⋊C412yes—
83C2×C4×C173C22×C411yescharacteristic
84C2×C4×C173C2×D412yes—
85C2×C4×C173C2×D412yes—
86C2×C4×C173C2×D411yescharacteristic
87C8×C173—41no—
88C8×C173—41no—
89C22×C173—22no—
90C22×C173—41no—
91C22×C173—42no—
92C22×C173C2×C22⋊C411yescharacteristic
93C22×C173#12012yes—
94C2×D173—24no—
95C2×D173—82no—
96C2×D173—84no—
97C4×C173—22no—
98C4×C173C2×C22⋊C412yes—
99C4×C173#12011yescharacteristic
100C4×C173#12011yescharacteristic
101Dic173—24no—
102Dic173—82no—
103C2×C173—22no—
104C2×C173—21no—
105C2×C173—42no—
106C2×C173#37911yescharacteristic
107D173—44no—
108C173#179411yescharacteristic, complemented, Sylow
109#1794—1731nomaximal, Sylow
110#585—1731no—
111#421—1732no—
112#416—1732no—
113#410—1731no—
114#408D17311yescharacteristic, complemented
1152−1+4—3462no—
1162−1+4—1732no—
117H(5,𝔽2)—3462no—
118H(5,𝔽2)—3462no—
119H(5,𝔽2)—1732no—
120C2×#41—3464no—
121C2×#41—3462no—
122C2×#41—1732no—
123C2×#41C2×D17311yescharacteristic
124C2×M16C2×D17311yescharacteristic, complemented
125C22×C8C2×D17311yescharacteristic, complemented
126#117—1732no—
127#104—3464no—
128#103—3462no—
129#102—3462no—
130#99—3462no—
131C22⋊C8—22no—
132#41—22no—
133#41—22no—
134#41—3464no—
135#41—3464no—
136#41—3468no—
137#41—3464no—
138#41—3464no—
139#41—3462no—
140#41—3462no—
141#41—3464no—
142#41—1734no—
143#41—1734no—
144C2×Q8—3462no—
145C2×Q8—3464no—
146C2×Q8—1732no—
147C2×Q8C4×D17311yescharacteristic
148C2×D4—3462no—
149C2×D4—3468no—
150C2×D4—3464no—
151C2×D4—3462no—
152C2×D4—3464no—
153C2×D4—1732no—
154C2×D4C4×D17311yescharacteristic
155C2×D4D69212yes—
156C22×C4—3464no—
157C22×C4—3462no—
158C22×C4C22⋊D17312yes—
159C22×C4C22×D17311yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—3462no—
165C22⋊C4—3462no—
166C42—3462no—
167C23—3464no—
168C23—3462no—
169C23#1458380312yes—
170C23C2×D69211yescharacteristic
171Q8—22no—
172Q8—3464no—
173Q8—3462no—
174Q8—1734no—
175D4—24no—
176D4—22no—
177D4—3464no—
178D4—3468no—
179D4—3464no—
180D4—3464no—
181D4—3462no—
182D4—1734no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—3468no—
186C2×C4—3464no—
187C2×C4—3462no—
188C2×C4—3464no—
189C2×C4—3464no—
190C2×C4—6922no—
191C2×C4#1458380312yes—
192C2×C4C2×C4×D17311yescharacteristic
193C2×C4D4×D17312yes—
194C2×C4D4×D17312yes—
195C2×C4C2×C22⋊D17311yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—3464no—
202C22—13842no—
203C22—13844no—
204C22C2×#1458380311yescharacteristic
205C22#1914964912yes—
206C4—22no—
207C4—3464no—
208C4—13842no—
209C4C22⋊C4×D17312yes—
210C4#1914964711yescharacteristic
211C4#1914964711yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—6924no—
216C2#2601134611yescharacteristic
subgroupG/H
centerC4#19149647
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#408×C173C2
socleC2×C173#379
prime radical#26012327C2
cosocleC22×C4×C173C23
hypercenter#408D173
hypocenterC173#1794
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #26011507C2semidirect
3. #26011488C2semidirect
4. #26011465C2semidirect
5. #408×C173C2semidirect
6. #19149727C4semidirect
10. #19149724C4semidirect
14. #19149638C22semidirect
20. C2×M16×C173C22semidirect
21. C22×C8×C173C22semidirect
108. C173#1794semidirect
114. #408D173semidirect
124. C2×M16C2×D173semidirect
125. C22×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260115072
#260114882
#260114652
#408×C1732
#1794173