Finite Group Databasebeta

← #36362796

Subgroups — #36362796

Order 22144.

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

normal subgroups only

This group has 738 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
1C23×Dic346C211yesmaximal, characteristic, complemented
2C2×#19149709C212yesmaximal, complemented
3#26011646C214yesmaximal, complemented
4#26011629C214yesmaximal, complemented
5#26011499C212yesmaximal, complemented
6#26011455C211yesmaximal, characteristic, complemented
7#525×C173C211yesmaximal, characteristic
8C22×Dic346—22no—
9C22×Dic346—21no—
10C22×Dic346—28no—
11C22×Dic346C2214yes—
12C22×Dic346C2212yes—
13C23×Dic173—21no—
14#19149709—28no—
15C2×#14583816C2214yescomplemented
16C2×#14583814C2214yescomplemented
17C2×#14583810C2212yes—
18#19149602C2218yescomplemented
19C22⋊#13063436C2214yes—
20C2×#41×C173C2211yescharacteristic
21C23×C4×C173C2211yescharacteristic
22#123×C173—21no—
23C22⋊Q8×C173C2212yes—
24#121×C173C2212yes—
25#120×C173—21no—
26C4×D4×C173—22no—
27#117×C173C2211yescharacteristic
28C2×Dic346—28no—
29C2×Dic346—216no—
30C2×Dic346—216no—
31C2×Dic346—24no—
32C2×Dic346—24no—
33C2×Dic346—22no—
34C2×Dic346D418yescomplemented
35C2×Dic346C2314yes—
36C22×Dic173—28no—
37C22×Dic173—22no—
38C22×Dic173—24no—
39#14583816—28no—
40#14583814—28no—
41#14583810—24no—
42C2×#13063436C2314yes—
43C24×C173D411yescharacteristic
44#41×C173—24no—
45C2×Q8×C173C2311yescharacteristic
46C2×D4×C173—21no—
47C2×D4×C173—22no—
48C2×D4×C173C2311yescharacteristic
49C22×C4×C173—24no—
50C22×C4×C173—21no—
51C22×C4×C173D412yes—
52C22×C4×C173D411yescharacteristic
53C22×C4×C173C2312yes—
54C22×C4×C173C2311yescharacteristic
55C4⋊C4×C173—22no—
56C4⋊C4×C173C2312yes—
57C22⋊C4×C173—21no—
58C22⋊C4×C173—22no—
59C22⋊C4×C173—22no—
60C42×C173—21no—
61Dic346—216no—
62Dic346—28no—
63Dic346—24no—
64Dic346C2×D418yes—
65C2×Dic173—216no—
66C2×Dic173—28no—
67C2×Dic173—24no—
68#13063436—24no—
69C23×C173—22no—
70C23×C173—24no—
71C23×C173—21no—
72C23×C173C2×D411yescharacteristic
73C23×C173C2×D412yes—
74Q8×C173—22no—
75D4×C173—22no—
76D4×C173—42no—
77D4×C173—44no—
78C2×C4×C173—22no—
79C2×C4×C173—24no—
80C2×C4×C173—24no—
81C2×C4×C173—22no—
82C2×C4×C173—21no—
83C2×C4×C173—42no—
84C2×C4×C173C2×D414yes—
85C2×C4×C173C2×D412yes—
86C2×C4×C173C2×D411yescharacteristic
87C2×C4×C173C2411yescharacteristic
88C22×C173—28no—
89C22×C173—22no—
90C22×C173—22no—
91C22×C173—22no—
92C22×C173—41no—
93C22×C173—42no—
94C22×C173#12012yes—
95C22×C173#13714yescomplemented
96C22×C173C22×D411yescharacteristic
97C4×C173—22no—
98C4×C173—41no—
99C4×C173—42no—
100C4×C173#12012yes—
101C4×C173C22×D412yes—
102Dic173—28no—
103C2×C173—22no—
104C2×C173—24no—
105C2×C173—81no—
106C2×C173C2×#12011yescharacteristic
107C2×C173C2×#13712yes—
108C173#300411yescharacteristic, complemented, Sylow
109#3004—1731nomaximal, Sylow
110C23×Q8—1731no—
111C2×#137—1732no—
112#525D17311yescharacteristic
113#451—1734no—
114#448—1734no—
115#419—1732no—
116#407—1731no—
117C2×#41C2×D17311yescharacteristic
118C22×Q8—3468no—
119C22×Q8—3462no—
120C22×Q8—3461no—
121C22×Q8—1734no—
122C22×Q8—1732no—
123C23×C4—3461no—
124C23×C4C2×D17311yescharacteristic
125#137—3468no—
126C2×Q16—1734no—
127C2×SD16—1734no—
128C2×M16—1732no—
129#123—21no—
130C22⋊Q8C2×D17312yes—
131#121C2×D17312yes—
132#120—21no—
133C4×D4—22no—
134#117C2×D17311yescharacteristic
135#103—1738no—
136C22⋊C8—1734no—
137C24C22⋊D17311yescharacteristic
138#41—24no—
139C2×Q8—3468no—
140C2×Q8—34616no—
141C2×Q8—34616no—
142C2×Q8—3464no—
143C2×Q8—3464no—
144C2×Q8—3462no—
145C2×Q8—1738no—
146C2×Q8—1734no—
147C2×Q8C22×D17311yescharacteristic
148C2×D4—22no—
149C2×D4—21no—
150C2×D4C22×D17311yescharacteristic
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—3468no—
154C22×C4—3464no—
155C22×C4—3462no—
156C22×C4C22⋊D17312yes—
157C22×C4C22⋊D17311yescharacteristic
158C22×C4C22×D17312yes—
159C22×C4C22×D17311yescharacteristic
160Q16—3468no—
161SD16—3468no—
162M16—3464no—
163C2×C8—1734no—
164C4⋊C4—22no—
165C4⋊C4C22×D17312yes—
166C22⋊C4—21no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—21no—
170C23—22no—
171C23—24no—
172C23—21no—
173C23C2×C22⋊D17311yescharacteristic
174C23C2×C22⋊D17312yes—
175Q8—22no—
176Q8—34616no—
177Q8—3468no—
178Q8—3464no—
179Q8—1738no—
180D4—22no—
181D4—42no—
182D4—44no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—34616no—
190C2×C4—3468no—
191C2×C4—3464no—
192C2×C4D4×D17314yes—
193C2×C4C2×C22⋊D17312yes—
194C2×C4C2×C22⋊D17311yescharacteristic
195C2×C4C23×D17311yescharacteristic
196C8—3464no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—41no—
202C22—42no—
203C22#1914964712yes—
204C22#1914970914yescomplemented
205C22C22×C22⋊D17311yescharacteristic
206C4—22no—
207C4—41no—
208C4—42no—
209C4—3468no—
210C4#1914964712yes—
211C4C2×D4×D17312yes—
212C2—22no—
213C2—24no—
214C2—81no—
215C2C2×#1914964711yescharacteristic
216C2C2×#1914970912yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#525×C173C2
socleC22×C173C22×D4
hypercenter#525D173
hypocenterC173#3004
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×Dic346C2semidirect
2. C2×#19149709C2semidirect
3. #26011646C2semidirect
4. #26011629C2semidirect
5. #26011499C2semidirect
6. #26011455C2semidirect
15. C2×#14583816C22semidirect
16. C2×#14583814C22semidirect
18. #19149602C22semidirect
34. C2×Dic346D4semidirect
95. C22×C173#137semidirect
108. C173#3004semidirect
204. C22#19149709semidirect

Maximal subgroups

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

subgroupindex
C23×Dic3462
C2×#191497092
#260116462
#260116292
#260114992
#260114552
#525×C1732
#3004173